Finite Group Databasebeta

← #93183

Subgroups — #93183

Order 512.

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

normal subgroups only

This group has 413 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#8553C212yesmaximal
2#8543C211yesmaximal, characteristic
3C22×#123C2211yescharacteristic
4#1821C414yescomplemented
5#1819—22no—
6#1781C412yes—
7#1462—22no—
8#1462—22no—
9C23×D4C2×C411yescharacteristic
10C24×C4Q811yescharacteristic
11C2×#123—44no—
12C2×#123—42no—
13C22×C4⋊C4C2×C412yes—
14C22×C22⋊C4D411yescharacteristic
15C22×C22⋊C4D412yes—
16C2×#99—42no—
17#407—24no—
18#379—22no—
19#343—44no—
20#342—44no—
21#323—44no—
22#323—44no—
23C25C22⋊C411yescharacteristic
24C25C4⋊C411yescharacteristic
25C22×D4—22no—
26C22×D4—42no—
27C22×D4—82no—
28C23×C4—41no—
29C23×C4—42no—
30C23×C4—82no—
31C23×C4C4211yescharacteristic
32C23×C4C22⋊C412yes—
33C23×C4C4⋊C412yes—
34C2×M16—44no—
35#123—416no—
36C2×C4⋊C4—24no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4—42no—
39C2×C22⋊C4—24no—
40C2×C22⋊C4—24no—
41C2×C22⋊C4—24no—
42C2×C22⋊C4—41no—
43C2×C22⋊C4—42no—
44C2×C22⋊C4—44no—
45C2×C22⋊C4—42no—
46#99—84no—
47C22⋊C8—44no—
48C22⋊C8—44no—
49C24—22no—
50C24—22no—
51C24—21no—
52C24—42no—
53C24—82no—
54C24—81no—
55C24—82no—
56C24#9511yescharacteristic
57C24#9912yes—
58C2×D4—44no—
59C2×D4—44no—
60C2×D4—82no—
61C2×D4—84no—
62C2×D4—84no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—44no—
71C22×C4—81no—
72C22×C4—82no—
73C22×C4—82no—
74C22×C4—84no—
75C22×C4—82no—
76C22×C4—84no—
77C22×C4—82no—
78C22×C4—82no—
79C22×C4#9914yes—
80M16—88no—
81C2×C8—84no—
82C4⋊C4—48no—
83C22⋊C4—44no—
84C22⋊C4—48no—
85C22⋊C4—84no—
86C22⋊C4—88no—
87C23—22no—
88C23—24no—
89C23—22no—
90C23—41no—
91C23—42no—
92C23—41no—
93C23—42no—
94C23—44no—
95C23—42no—
96C23—44no—
97C23—41no—
98C23—82no—
99C23—82no—
100C23—82no—
101C23—82no—
102C23—82no—
103C23—88no—
104C23—164no—
105C23#34211yescharacteristic
106C23#34212yes—
107D4—88no—
108C2×C4—24no—
109C2×C4—44no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—82no—
114C2×C4—82no—
115C2×C4—88no—
116C2×C4—82no—
117C2×C4—82no—
118C2×C4—82no—
119C2×C4—82no—
120C2×C4—162no—
121C2×C4—164no—
122C8—164no—
123C22—21no—
124C22—22no—
125C22—22no—
126C22—22no—
127C22—22no—
128C22—44no—
129C22—42no—
130C22—42no—
131C22—44no—
132C22—82no—
133C22—84no—
134C22—82no—
135C22—84no—
136C22—82no—
137C22—162no—
138C22—164no—
139C22#130411yescharacteristic
140C4—44no—
141C4—81no—
142C4—82no—
143C4—322no—
144C2—41no—
145C2—42no—
146C2—82no—
147C2—82no—
148C2#731111yescharacteristic
149C2#731312yes—
subgroupG/H
center, socleC22#1304
commutatorC23×C4C42
Frattini, cosocleC22×#123C22
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G
LECSG ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#85532
#85432