Finite Group Databasebeta

← #93184

Subgroups — #93184

Order 512.

435 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8552C211yesmaximal, characteristic
2#8544C212yesmaximal
3C22×#124C2211yescharacteristic
4#1819—22no—
5#1819C414yescomplemented
6#1784C412yescomplemented
7#1781—22no—
8#1462—22no—
9C23×Q8C2×C411yescharacteristic
10C23×D4D411yescharacteristic
11C2×#124—44no—
12C2×#124—42no—
13C22×C22⋊C4C2×C412yes—
14C22×C22⋊C4D412yes—
15C22×C42Q811yescharacteristic
16C2×#101—42no—
17C2×#99—44no—
18#407—22no—
19#379—24no—
20#342—44no—
21#342—44no—
22#323—44no—
23C25C22⋊C412yes—
24C22×Q8—22no—
25C22×Q8—42no—
26C22×Q8—82no—
27C22×D4—24no—
28C22×D4—41no—
29C22×D4—82no—
30C23×C4C4211yescharacteristic
31C23×C4C22⋊C411yescharacteristic
32C23×C4C4⋊C411yescharacteristic
33C23×C4C4⋊C412yes—
34C2×M16—42no—
35#124—416no—
36C2×C22⋊C4—24no—
37C2×C22⋊C4—24no—
38C2×C22⋊C4—24no—
39C2×C22⋊C4—42no—
40C2×C22⋊C4—42no—
41C2×C22⋊C4—42no—
42C2×C22⋊C4—44no—
43C2×C22⋊C4—44no—
44C2×C22⋊C4—44no—
45C2×C42—42no—
46C2×C42—41no—
47#101—84no—
48#99—88no—
49C22⋊C8—44no—
50C24—22no—
51C24—22no—
52C24—82no—
53C24—82no—
54C24—82no—
55C24#9511yescharacteristic
56C24#9914yes—
57C2×Q8—44no—
58C2×Q8—44no—
59C2×Q8—84no—
60C2×Q8—82no—
61C2×Q8—84no—
62C2×D4—48no—
63C2×D4—44no—
64C2×D4—82no—
65C2×D4—84no—
66C2×D4—82no—
67C22×C4—22no—
68C22×C4—21no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—42no—
76C22×C4—81no—
77C22×C4—81no—
78C22×C4—82no—
79C22×C4—84no—
80C22×C4#9912yes—
81M16—84no—
82C2×C8—82no—
83C22⋊C4—48no—
84C22⋊C4—48no—
85C22⋊C4—88no—
86C22⋊C4—88no—
87C22⋊C4—88no—
88C42—44no—
89C23—24no—
90C23—41no—
91C23—42no—
92C23—44no—
93C23—44no—
94C23—44no—
95C23—44no—
96C23—44no—
97C23—82no—
98C23—82no—
99C23—82no—
100C23—82no—
101C23—168no—
102C23#34212yes—
103C23#34211yescharacteristic
104Q8—88no—
105D4—88no—
106C2×C4—24no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—82no—
118C2×C4—82no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4—164no—
122C2×C4—164no—
123C2×C4—164no—
124C8—162no—
125C22—21no—
126C22—22no—
127C22—22no—
128C22—22no—
129C22—22no—
130C22—44no—
131C22—44no—
132C22—44no—
133C22—44no—
134C22—164no—
135C22—164no—
136C22—164no—
137C22#130411yescharacteristic
138C4—42no—
139C4—42no—
140C4—81no—
141C4—82no—
142C4—324no—
143C2—41no—
144C2—42no—
145C2—84no—
146C2#731011yescharacteristic
147C2#731312yes—
subgroupG/H
center, socleC22#1304
commutatorC23×C4C42
Frattini, cosocleC22×#124C22
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C22×#124 ▷ C24 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #1819C4semidirect
6. #1784C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#85522
#85442