diff --git a/lmfdb/groups/abstract/main.py b/lmfdb/groups/abstract/main.py
index df7050d750..a6c23164e7 100644
--- a/lmfdb/groups/abstract/main.py
+++ b/lmfdb/groups/abstract/main.py
@@ -1186,6 +1186,16 @@ def normal_autdiagram(label):
     title = f"Diagram of normal subgroups up to automorphism for group {label}"
     return _subgroup_diagram(label, title, only=("normal", "aut"), style="normal_autdiagram")
 
+@abstract_page.route("/maximal_diagram/<label>")
+def maximal_diagram(label):
+    title = f"Diagram of maximal subgroups up to conjugation for group {label}"
+    return _subgroup_diagram(label, title, only=("maximal", ""), style="maximal_diagram")
+
+@abstract_page.route("/maximal_autdiagram/<label>")
+def maximal_autdiagram(label):
+    title = f"Diagram of maximal subgroups up to automorphism for group {label}"
+    return _subgroup_diagram(label, title, only=("maximal", "aut"), style="maximal_autdiagram")
+
 def show_type(ab, nil, solv, smith, nilcls, dlen, clen):
     # arguments - ["abelian", "nilpotent", "solvable", "smith_abelian_invariants", "nilpotency_class", "derived_length", "composition_length"]
     if ab:
@@ -2023,11 +2033,11 @@ def diagram_js(gp, layers, display_opts, aut=False, normal=False):
     return [ll, layers[1]], order_lookup
 
 def diagram_js_string(gp, only=None):
-    glist = [[], [], [], []]
-    order_lookup = [[], [], [], []]
+    glist = [[], [], [], [], [], []]
+    order_lookup = [[], [], [], [], [], []]
     display_opts = defaultdict(int)
     limit = (100 if only is None else 0)
-    for i, pair in enumerate([("subgroup", ""), ("subgroup", "aut"), ("normal", ""), ("normal", "aut")]):
+    for i, pair in enumerate([("subgroup", ""), ("subgroup", "aut"), ("normal", ""), ("normal", "aut"), ("maximal", ""), ("maximal", "aut")]):
         sub_all, sub_aut = pair
         if (only is None or only == pair) and gp.diagram_count(sub_all, sub_aut, limit=limit):
             glist[i], order_lookup[i] = diagram_js(gp, gp.subgroup_lattice(sub_all, sub_aut), display_opts, aut=bool(sub_aut), normal=(sub_all == "normal"))
diff --git a/lmfdb/groups/abstract/templates/abstract-show-group.html b/lmfdb/groups/abstract/templates/abstract-show-group.html
index 9543ec7291..2bcee2da1a 100644
--- a/lmfdb/groups/abstract/templates/abstract-show-group.html
+++ b/lmfdb/groups/abstract/templates/abstract-show-group.html
@@ -403,6 +403,7 @@ <h3>{{KNOWL('group.subgroup.diagram.lmfdb','Subgroup diagram')}} and profile</h3
     <div class="sub_divider">
       <button class='sub_subgroup sub_active' onclick="select_subgroup_mode('subgroup'); return false">all subgroups</button>
       <button class='sub_normal sub_inactive' onclick="select_subgroup_mode('normal'); return false">normal subgroups</button>
+      <button class='sub_maximal sub_inactive' onclick="select_subgroup_mode('maximal'); return false">maximal subgroups</button>
     </div>
     <div class="sub_divider">
       <button class='sub_ sub_active' onclick="select_subgroup_mode(''); return false">up to conjugacy</button>
@@ -412,7 +413,7 @@ <h3>{{KNOWL('group.subgroup.diagram.lmfdb','Subgroup diagram')}} and profile</h3
 
 {# Just header information for the diagrams #}
 
-{% for sub_all in ["subgroup", "normal"] %}
+{% for sub_all in ["subgroup", "normal", "maximal"] %}
   {% for sub_aut in ["", "aut"] %}
     {% set sub_desc, sub_summary, sub_cnt = gp.get_diagram_info(sub_all, sub_aut, limit=100) %}
     <div class="{{sub_all}}_{{sub_aut}}diagram" style="display:none">
@@ -450,7 +451,7 @@ <h4>{{sub_desc}}</h4>
   {{ info.dojs|safe }}
 </script>
 
-{% for sub_all in ["subgroup", "normal"] %}
+{% for sub_all in ["subgroup", "normal", "maximal"] %}
   {% for sub_aut in ["", "aut"] %}
     {% set sub_profile, sub_desc, sub_summary = gp.get_profile(sub_all, sub_aut) %}
     <div class="{{sub_all}}_{{sub_aut}}profile" style="display: none">
diff --git a/lmfdb/groups/abstract/test_abstract_groups.py b/lmfdb/groups/abstract/test_abstract_groups.py
index 4d6faea241..7236fda104 100644
--- a/lmfdb/groups/abstract/test_abstract_groups.py
+++ b/lmfdb/groups/abstract/test_abstract_groups.py
@@ -1,5 +1,42 @@
+import re
+from ast import literal_eval
+
 from lmfdb.tests import LmfdbTest
 
+# The graphs that diagram_js_string builds for a group page, in order
+DIAGRAM_MODES = [("subgroup", ""), ("subgroup", "aut"), ("normal", ""), ("normal", "aut"),
+                 ("maximal", ""), ("maximal", "aut")]
+
+
+def diagram_graphs(page, label):
+    r"""
+    The graphs passed to make_sdiagram on a group page or a fullpage diagram: one
+    entry for each mode of DIAGRAM_MODES, empty when that diagram is not drawn.
+    Returns None when the page contains no subgroup diagram at all.
+    """
+    m = re.search(r'make_sdiagram\("subdiagram", "%s", (.*)\);' % re.escape(label), page)
+    if m is None:
+        return None
+    graphs, order_lookup, layers = literal_eval("(" + m.group(1) + ")")
+    assert len(graphs) == len(DIAGRAM_MODES)
+    return graphs
+
+
+def graph_content(graph):
+    r"""
+    The short labels of the nodes and the edges of one graph from diagram_graphs
+    """
+    nodes, edges = graph
+    return {node[1] for node in nodes}, {tuple(edge) for edge in edges}
+
+
+def profile_content(page, cls):
+    r"""
+    The contents of one of the subgroup profile divs on a group page
+    """
+    return re.search(r'<div class="%s"[^>]*>(.*?)</div>' % cls, page, re.DOTALL).group(1)
+
+
 class AbGpsTest(LmfdbTest):
     # All tests should pass
 
@@ -90,3 +127,124 @@ def test_underlying_data(self):
     def test_subgroups(self):
         self.check_args("/Groups/Abstract/sub/78125.1385.15625.A","Group of order 31250000")
         self.check_args("/Groups/Abstract/sub/16384.mv.8._.BQX",'The ambient group is <a title="Abelian group [group.abelian]"')
+
+    def test_maximal_subgroups(self):
+        r"""
+        Check the maximal subgroup mode of the subgroup diagram/profile
+        """
+        # S4 has maximal subgroups A4 (2.a1.a1), D4 (3.a1.a1) and S3 (4.a1.a1),
+        # none of which are related by an automorphism; 1.a1.a1 is S4 itself,
+        # which sits at the top of the maximal diagram
+        page = self.tc.get("/Groups/Abstract/24.12").get_data(as_text=True)
+        assert "maximal subgroups</button>" in page
+        profile = profile_content(page, "maximal_profile")
+        assert "Classes of maximal subgroups up to conjugation" in profile
+        assert "A_4" in profile and "D_4" in profile and "S_3" in profile
+        autprofile = profile_content(page, "maximal_autprofile")
+        assert "Classes of maximal subgroups up to automorphism" in autprofile
+        assert "A_4" in autprofile and "D_4" in autprofile and "S_3" in autprofile
+        graphs = diagram_graphs(page, "24.12")
+        for mode in [("maximal", ""), ("maximal", "aut")]:
+            nodes, edges = graph_content(graphs[DIAGRAM_MODES.index(mode)])
+            assert nodes == {"1.a1.a1", "2.a1.a1", "3.a1.a1", "4.a1.a1"}
+            assert edges == {("2.a1.a1", "1.a1.a1"), ("3.a1.a1", "1.a1.a1"), ("4.a1.a1", "1.a1.a1")}
+
+    def test_maximal_subgroups_diagram(self):
+        r"""
+        Check that the maximal subgroup diagram is a star with the whole group on
+        top, even when the full subgroup diagram is too large to be displayed
+        """
+        from lmfdb.groups.abstract.web_groups import WebAbstractGroup
+        # 32.45 has 118 classes of subgroups, too many to draw, but only 15 classes
+        # of maximal subgroups, which fuse into 2 classes up to automorphism
+        G = WebAbstractGroup("32.45")
+        assert G.diagram_count("subgroup", "", limit=100) == 0
+        assert G.diagram_count("maximal", "", limit=100) == 16
+        assert G.diagram_count("maximal", "aut", limit=100) == 3
+        nodes, edges = G.subgroup_lattice("maximal", "")
+        top = [H for H in nodes if H.quotient_order == 1]
+        maxima = [H for H in nodes if H.maximal]
+        assert len(top) == 1
+        assert len(maxima) == 15
+        assert len(nodes) == len(maxima) + 1
+        assert {tuple(edge) for edge in edges} == {(H.short_label, top[0].short_label) for H in maxima}
+        # Up to automorphism there is exactly one node for each autjugacy class
+        aut_nodes, aut_edges = G.subgroup_lattice("maximal", "aut")
+        assert len(aut_nodes) == G.diagram_count("maximal", "aut")
+        assert len({H.aut_label for H in aut_nodes}) == len(aut_nodes)
+        assert {H.aut_label for H in aut_nodes} == {H.aut_label for H in nodes}
+        aut_top = [H for H in aut_nodes if H.quotient_order == 1]
+        assert len(aut_top) == 1
+        assert len(aut_edges) == len(aut_nodes) - 1
+        assert {target for source, target in aut_edges} == {aut_top[0].short_label}
+        # Only the two maximal diagrams and the normal diagram up to automorphism
+        # are small enough to be drawn on the group page
+        page = self.tc.get("/Groups/Abstract/32.45").get_data(as_text=True)
+        graphs = diagram_graphs(page, "32.45")
+        assert [DIAGRAM_MODES[i] for i, graph in enumerate(graphs) if graph] == [
+            ("normal", "aut"), ("maximal", ""), ("maximal", "aut")]
+        assert graph_content(graphs[DIAGRAM_MODES.index(("maximal", ""))]) == (
+            {H.short_label for H in nodes},
+            {(H.short_label, top[0].short_label) for H in maxima})
+        assert graph_content(graphs[DIAGRAM_MODES.index(("maximal", "aut"))]) == (
+            {H.short_label for H in aut_nodes},
+            {tuple(edge) for edge in aut_edges})
+        profile = profile_content(page, "maximal_profile")
+        assert "x 14" in profile  # 14 conjugate copies of C_2^2 x C_4
+
+    def test_maximal_subgroups_fullpage(self):
+        r"""
+        Check that the fullpage maximal subgroup diagrams contain the right graph
+        """
+        from lmfdb.groups.abstract.web_groups import WebAbstractGroup
+        for label, title, mode in [
+                ("24.12", "Diagram of maximal subgroups up to conjugation", ("maximal", "")),
+                ("24.12", "Diagram of maximal subgroups up to automorphism", ("maximal", "aut")),
+                ("32.45", "Diagram of maximal subgroups up to conjugation", ("maximal", "")),
+                ("32.45", "Diagram of maximal subgroups up to automorphism", ("maximal", "aut"))]:
+            sub_all, sub_aut = mode
+            G = WebAbstractGroup(label)
+            shown = [H for H in G.subgroups.values() if H.maximal or H.quotient_order == 1]
+            response = self.tc.get(f"/Groups/Abstract/{sub_all}_{sub_aut}diagram/{label}")
+            assert response.status_code == 200
+            page = response.get_data(as_text=True)
+            assert title in page
+            assert f'show_info("{sub_all}_{sub_aut}diagram")' in page
+            graphs = diagram_graphs(page, label)
+            assert graphs is not None, f"no diagram on the fullpage {sub_all} {sub_aut} diagram for {label}"
+            # The fullpage version contains only the requested diagram
+            assert [DIAGRAM_MODES[i] for i, graph in enumerate(graphs) if graph] == [mode]
+            drawn, edges = graph_content(graphs[DIAGRAM_MODES.index(mode)])
+            whole_group, = [H.short_label for H in shown if H.quotient_order == 1]
+            assert edges == {(short_label, whole_group) for short_label in drawn - {whole_group}}
+            if sub_aut:
+                # one representative from each autjugacy class, so 3 nodes for 32.45
+                assert drawn <= {H.short_label for H in shown}
+                assert len(drawn) == G.diagram_count(sub_all, sub_aut)
+                assert ({H.aut_label for H in shown if H.short_label in drawn}
+                        == {H.aut_label for H in shown})
+            else:
+                assert drawn == {H.short_label for H in shown}
+
+    def test_maximal_subgroups_without_inclusions(self):
+        r"""
+        Check that maximal subgroups are shown for a group whose subgroup
+        inclusions were never computed
+        """
+        from lmfdb.groups.abstract.web_groups import WebAbstractGroup
+        G = WebAbstractGroup("1024.dip")
+        assert G.maximal_subgroups_known
+        assert not G.subgroup_inclusions_known
+        assert G.maximal_profile
+        # No diagram can be drawn without the inclusions, but the profile is still shown
+        assert G.diagram_count("maximal", "", limit=100) == 0
+        page = self.tc.get("/Groups/Abstract/1024.dip").get_data(as_text=True)
+        assert diagram_graphs(page, "1024.dip") is None
+        profile = profile_content(page, "maximal_profile")
+        assert "not computed" not in profile
+        # 255 classes of maximal subgroups, all of order 512
+        assert "Order 512:" in profile
+        assert r"$C_2^6:D_4$</a> x 248" in profile
+        assert r"$D_4\times C_2^6$</a> x 3" in profile
+        assert r"$C_2^8.C_2$</a> x 3" in profile
+        assert r"$C_2^9$</a>" in profile
diff --git a/lmfdb/groups/abstract/web_groups.py b/lmfdb/groups/abstract/web_groups.py
index 5ece3ccf86..a0e60318b1 100644
--- a/lmfdb/groups/abstract/web_groups.py
+++ b/lmfdb/groups/abstract/web_groups.py
@@ -1274,6 +1274,26 @@ def normal_autprofile(self):
                     seen.add(s.aut_label)
             return self._finalize_profile(by_order)
 
+    @lazy_attribute
+    def maximal_profile(self):
+        if self.has_subgroups and self.maximal_subgroups_known:
+            by_order = defaultdict(Counter)
+            for s in self.subgroups.values():
+                if s.maximal:
+                    by_order[s.subgroup_order][s.subgroup, s.subgroup_hash, s.subgroup_tex] += s.conjugacy_class_count
+            return self._finalize_profile(by_order)
+
+    @lazy_attribute
+    def maximal_autprofile(self):
+        if self.has_subgroups and self.maximal_subgroups_known and (self.all_subgroups_known or self.complements_known or self.outer_equivalence):
+            seen = set()
+            by_order = defaultdict(Counter)
+            for s in self.subgroups.values():
+                if s.maximal and s.aut_label not in seen:
+                    by_order[s.subgroup_order][s.subgroup, s.subgroup_hash, s.subgroup_tex] += 1
+                    seen.add(s.aut_label)
+            return self._finalize_profile(by_order)
+
     def _display_profile(self, profile, aut):
         def display_profile_line(order, subs):
             l = []
@@ -1312,6 +1332,12 @@ def _normal_summary(self):
             return f"All normal subgroups of index up to {self.normal_index_bound} or order up to {self.normal_order_bound} are shown. <br>"
         return ""
 
+    @cached_method
+    def _maximal_summary(self):
+        if not self.maximal_subgroups_known:
+            return "Maximal subgroups have not been computed. <br>"
+        return ""
+
     @lazy_attribute
     def subgroup_order_bound(self):
         if self.subgroup_index_bound == 0 or self.subgroup_index_bound is None:
@@ -1342,6 +1368,14 @@ def get_profile(self, sub_all, sub_aut):
                 profile = self.subgroup_profile
                 desc = "Classes of subgroups up to conjugation"
             summary = self._subgroup_summary(in_profile=True)
+        elif sub_all == "maximal":
+            if sub_aut:
+                profile = self.maximal_autprofile
+                desc = "Classes of maximal subgroups up to automorphism"
+            else:
+                profile = self.maximal_profile
+                desc = "Classes of maximal subgroups up to conjugation"
+            summary = self._maximal_summary()
         else:
             if sub_aut:
                 profile = self.normal_autprofile
@@ -1356,7 +1390,7 @@ def get_profile(self, sub_all, sub_aut):
 
     @cached_method
     def diagram_count(self, sub_all, sub_aut, limit=0):
-        # The number of subgroups shown in the diagram of this type; sub_all can be "subgroup" or "normal" and sub_aut can be "aut" or ""
+        # The number of subgroups shown in the diagram of this type; sub_all can be "subgroup", "normal" or "maximal" and sub_aut can be "aut" or ""
         # If limit is nonzero, then a count of 0 is returned (indicating that the diagram should not be shown) when there would be more nodes than the limit.
         if not self.subgroup_inclusions_known:
             return 0
@@ -1382,7 +1416,13 @@ def impose_limit(n):
                     return 0
                 return impose_limit(len(subs))
         else:
-            subs = [H for H in self.subgroups.values() if H.normal]
+            if sub_all == "maximal":
+                if not self.maximal_subgroups_known:
+                    return 0
+                # We include the whole group at the top of the diagram
+                subs = [H for H in self.subgroups.values() if H.maximal or H.quotient_order == 1]
+            else:
+                subs = [H for H in self.subgroups.values() if H.normal]
             if sub_aut:
                 if any(H.aut_label is None or H.diagramx is None for H in subs):
                     # We don't know subgroups up to automorphism or can't lay out the subgroups
@@ -1402,6 +1442,12 @@ def get_diagram_info(self, sub_all, sub_aut, limit=0):
             else:
                 desc = "Classes of subgroups up to conjugation"
             summary = self._subgroup_summary(in_profile=False)
+        elif sub_all == "maximal":
+            if sub_aut:
+                desc = "Classes of maximal subgroups up to automorphism"
+            else:
+                desc = "Classes of maximal subgroups up to conjugation"
+            summary = self._maximal_summary()
         else:
             if sub_aut:
                 desc = "Normal subgroups up to automorphism"
@@ -1425,10 +1471,10 @@ def get_diagram_info(self, sub_all, sub_aut, limit=0):
         return desc, summary, count
 
     def diagram_classes(self):
-        # Which combinations of subgroup/normal and conj/aut have a diagram
+        # Which combinations of subgroup/normal/maximal and conj/aut have a diagram
         # Note that it's possible that the only diagrams "shown" will be for cases where there's a link to a fullpage version.
         classes = []
-        for sub_all in ["subgroup", "normal"]:
+        for sub_all in ["subgroup", "normal", "maximal"]:
             for sub_aut in ["", "aut"]:
                 if self.diagram_count(sub_all, sub_aut) > 0:
                     classes.append(f"{sub_all}_{sub_aut}diagram")
@@ -1451,6 +1497,19 @@ def test(H):
 
             def contains(G):
                 return [h for h in G.contains if test(subs[h])]
+        elif sub_all == "maximal":
+            def test(H):
+                # We include the whole group so that the diagram is connected
+                if H.maximal or H.quotient_order == 1:
+                    by_aut[H.aut_label].add(H.short_label)
+                    return True
+
+            def contains(G):
+                # Every maximal subgroup is covered by the whole group at the top;
+                # no inclusions among maximal subgroups themselves
+                if G.quotient_order == 1:
+                    return [H.short_label for H in nodes if H is not G]
+                return []
         else:
             def test(H):
                 if H.normal:
@@ -1493,7 +1552,7 @@ def contains(G):
 
     def fullpage_links(self, getpositions=False):
         s = ""
-        for sub_all in ["subgroup", "normal"]:
+        for sub_all in ["subgroup", "normal", "maximal"]:
             for sub_aut in ["", "aut"]:
                 cls = f'{sub_all}_{sub_aut}diagram'
                 s += f'<div class="{cls}">\n'
diff --git a/lmfdb/static/groups.js b/lmfdb/static/groups.js
index a16d8a9310..fefdfca9d3 100644
--- a/lmfdb/static/groups.js
+++ b/lmfdb/static/groups.js
@@ -129,11 +129,13 @@ function mytogglevert(use_big_on_top) {
 
 function mytoggleheights(use_order_for_height) {
   var who_old = whoisshowing;
-  if (use_order_for_height && (whoisshowing < 4)) {
-    whoisshowing += 4;
+  // The first half of glist lays subgroups out by number of prime divisors, the second half by order
+  var numtypes = glist.length / 2;
+  if (use_order_for_height && (whoisshowing < numtypes)) {
+    whoisshowing += numtypes;
   }
-  if ((! use_order_for_height) && whoisshowing > 3) {
-    whoisshowing -= 4;
+  if ((! use_order_for_height) && whoisshowing >= numtypes) {
+    whoisshowing -= numtypes;
   }
   if(who_old != whoisshowing) {
     glist[whoisshowing].highlit = null;
@@ -158,19 +160,20 @@ function getpositions() {
   return mylist;
 }
 
-var styles=['subgroup_diagram', 'subgroup_profile', 'subgroup_autdiagram', 'subgroup_autprofile', 'normal_diagram', 'normal_profile', 'normal_autdiagram', 'normal_autprofile'];
-var mode_pairs = [['subgroup', 'normal'], ['', 'aut'], ['diagram', 'profile']];
+var styles=['subgroup_diagram', 'subgroup_profile', 'subgroup_autdiagram', 'subgroup_autprofile', 'normal_diagram', 'normal_profile', 'normal_autdiagram', 'normal_autprofile', 'maximal_diagram', 'maximal_profile', 'maximal_autdiagram', 'maximal_autprofile'];
+var mode_groups = [['subgroup', 'normal', 'maximal'], ['', 'aut'], ['diagram', 'profile']];
 function select_subgroup_mode(mode) {
-  var cls, thismode, opposite_mode, piece;
+  var cls, thismode, others, piece;
   cls = "";
-  for (var i = 0; i < mode_pairs.length; i++) {
-    for (var j = 0; j < 2; j++) {
-      thismode = mode_pairs[i][j];
+  others = [];
+  for (var i = 0; i < mode_groups.length; i++) {
+    for (var j = 0; j < mode_groups[i].length; j++) {
+      thismode = mode_groups[i][j];
       if (thismode == mode) {
-        opposite_mode = mode_pairs[i][1-j];
         if ($("button.sub_" + mode).hasClass("sub_active")) {
           return; // already active
         }
+        others = mode_groups[i].filter(function(m) { return m != mode; });
         piece = mode;
         break;
       }
@@ -185,8 +188,10 @@ function select_subgroup_mode(mode) {
   }
   $("button.sub_" + mode).removeClass("sub_inactive");
   $("button.sub_" + mode).addClass("sub_active");
-  $("button.sub_" + opposite_mode).removeClass("sub_active");
-  $("button.sub_" + opposite_mode).addClass("sub_inactive");
+  for (var i = 0; i < others.length; i++) {
+    $("button.sub_" + others[i]).removeClass("sub_active");
+    $("button.sub_" + others[i]).addClass("sub_inactive");
+  }
   show_info(cls);
 }
 
@@ -197,15 +202,19 @@ function show_info(style) {
   }
   $('div.'+style).show();
   if (style.endsWith("diagram")) {
+    // Order within glist: subgroup, subgroup_aut, normal, normal_aut, maximal, maximal_aut,
+    // first laid out by number of prime divisors, then the same six laid out by order
     whoisshowing = 0;
     if (style.endsWith("autdiagram")) {
       whoisshowing += 1;
     }
-    if (heightstyle=='order') {
-      whoisshowing += 4;
-    }
     if (style.startsWith("normal")) {
       whoisshowing += 2;
+    } else if (style.startsWith("maximal")) {
+      whoisshowing += 4;
+    }
+    if (heightstyle=='order') {
+      whoisshowing += glist.length / 2;
     }
     sdiagram.newgraph(glist[whoisshowing]);
     sdiagram.setSize();