Finite Group Databasebeta

← #97069

Subgroups — #97069

Order 512.

1600 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C22×#384C211yesmaximal, characteristic
2#8299C216yesmaximal
3C23×C4⋊C4C413yes—
4C23×C4⋊C4C2213yes—
5C23×C42C411yescharacteristic
6C2×#384—212no—
7#1779C2214yes—
8C2×#324—212no—
9C24×C4C2×C413yes—
10C24×C4C2×C413yes—
11C24×C4C2311yescharacteristic
12C22×C4⋊C4—212no—
13C22×C4⋊C4—26no—
14C22×C4⋊C4—212no—
15C22×C4⋊C4—412no—
16C22×C4⋊C4—412no—
17C22×C4⋊C4D4112yes—
18C22×C42—26no—
19C22×C42—26no—
20C22×C42C814yescomplemented
21C2×C22⋊C8—212no—
22#384—232no—
23#324—424no—
24C25C22×C411yescharacteristic
25C23×C4—26no—
26C23×C4—26no—
27C23×C4—212no—
28C23×C4—212no—
29C23×C4—26no—
30C23×C4—46no—
31C23×C4—46no—
32C23×C4—412no—
33C23×C4C22⋊C416yes—
34C23×C4C22⋊C416yes—
35C23×C4C2×C816yes—
36C23×C4M1616yes—
37C23×C4C2×D416yes—
38C22×C8—44no—
39C2×C4⋊C4—224no—
40C2×C4⋊C4—224no—
41C2×C4⋊C4—224no—
42C2×C4⋊C4—412no—
43C2×C4⋊C4—448no—
44C2×C4⋊C4—448no—
45C2×C4⋊C4—412no—
46C2×C4⋊C4—412no—
47C2×C42—28no—
48C2×C42—224no—
49C2×C42—224no—
50C22⋊C8—424no—
51C24—26no—
52C24—26no—
53C24C2×C22⋊C413yes—
54C24C22×C811yescharacteristic
55C24C2×M1613yes—
56C22×C4—224no—
57C22×C4—248no—
58C22×C4—224no—
59C22×C4—26no—
60C22×C4—212no—
61C22×C4—26no—
62C22×C4—212no—
63C22×C4—26no—
64C22×C4—26no—
65C22×C4—224no—
66C22×C4—224no—
67C22×C4—212no—
68C22×C4—26no—
69C22×C4—212no—
70C22×C4—412no—
71C22×C4—412no—
72C22×C4—412no—
73C22×C4—412no—
74C22×C4—412no—
75C22×C4—88no—
76C22×C4—824no—
77C22×C4C22⋊C8112yes—
78C22×C4#99112yes—
79C22×C4#101112yes—
80C22×C4#12014yes—
81C2×C8—812no—
82C4⋊C4—448no—
83C4⋊C4—448no—
84C42—232no—
85C23—28no—
86C23—224no—
87C23—26no—
88C23—26no—
89C23—212no—
90C23—26no—
91C23—26no—
92C23#37911yescharacteristic
93C23C2×C22⋊C813yes—
94C23#40713yes—
95C23C2×#9916yes—
96C23C2×#10116yes—
97C2×C4—248no—
98C2×C4—248no—
99C2×C4—212no—
100C2×C4—212no—
101C2×C4—212no—
102C2×C4—212no—
103C2×C4—212no—
104C2×C4—212no—
105C2×C4—212no—
106C2×C4—46no—
107C2×C4—412no—
108C2×C4—812no—
109C2×C4—824no—
110C2×C4—812no—
111C2×C4#324124yescomplemented
112C8—164no—
113C22—28no—
114C22—224no—
115C22—26no—
116C22—26no—
117C22—212no—
118C22—26no—
119C22—26no—
120C22C2×#32416yes—
121C22#146316yes—
122C22#177911yescharacteristic
123C22#178113yes—
124C22#178413yes—
125C4—224no—
126C4—44no—
127C4—86no—
128C2—26no—
129C2—26no—
130C2#829913yes—
131C2#854611yescharacteristic
132C2#855113yes—
subgroupG/H
center, socleC23#379
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
20. C22×C42C8semidirect
111. C2×C4#324semidirect

Maximal subgroups

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

subgroupindex
C22×#3842
#82992