Finite Group Databasebeta

← #97561

Subgroups — #97561

Order 512.

1169 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1169 conjugacy classes of subgroups — too many to draw. Only its 98 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#35463C211yesmaximal, characteristic
2#8304C216yesmaximal, complemented
3C2×#534C413yes—
4C22×#123C2213yes—
5#2539—26no—
6#2501—26no—
7#2273C411yescharacteristic
8#1779C2214yes—
9C2×#324—26no—
10C2×#323—26no—
11C23×D4C2×C413yes—
12C24×C4C2311yescharacteristic
13#534—412no—
14C2×#123—26no—
15C2×#123—212no—
16C2×#123—412no—
17C2×#121—26no—
18C2×#120—26no—
19C22×C4⋊C4D416yes—
20C22×C22⋊C4C2×C413yes—
21C22×C22⋊C4D416yes—
22C2×C22⋊C8—212no—
23#399—212no—
24#392—24no—
25#390—212no—
26#384—24no—
27C4×C22⋊C4—26no—
28#324—412no—
29#323—412no—
30C25C22⋊C413yes—
31C25C22×C411yescharacteristic
32C22×D4—26no—
33C22×D4—26no—
34C22×D4—46no—
35C22×D4—412no—
36C23×C4—46no—
37C23×C4C22⋊C413yes—
38C23×C4C22⋊C416yes—
39C23×C4C2×D416yes—
40C22×C8—44no—
41#123—448no—
42#123—424no—
43#121—412no—
44#121—412no—
45#120—424no—
46C2×C4⋊C4—26no—
47C2×C4⋊C4—212no—
48C2×C4⋊C4—212no—
49C2×C4⋊C4—46no—
50C2×C22⋊C4—26no—
51C2×C22⋊C4—212no—
52C2×C22⋊C4—212no—
53C2×C22⋊C4—212no—
54C2×C22⋊C4—212no—
55C2×C22⋊C4—46no—
56C2×C42—24no—
57C22⋊C8—412no—
58C22⋊C8—412no—
59#95—24no—
60C24—26no—
61C24—23no—
62C24—26no—
63C24—44no—
64C24—412no—
65C24—46no—
66C24C2×C22⋊C413yes—
67C2×D4—412no—
68C2×D4—412no—
69C2×D4—412no—
70C2×D4—412no—
71C2×D4—812no—
72C2×D4—824no—
73C22×C4—26no—
74C22×C4—26no—
75C22×C4—26no—
76C22×C4—26no—
77C22×C4—23no—
78C22×C4—26no—
79C22×C4—412no—
80C22×C4—412no—
81C22×C4—46no—
82C22×C4—88no—
83C22×C4#99112yes—
84C22×C4#12014yes—
85C2×C8—812no—
86C4⋊C4—424no—
87C4⋊C4—412no—
88C22⋊C4—224no—
89C22⋊C4—424no—
90C22⋊C4—424no—
91C42—84no—
92C23—28no—
93C23—26no—
94C23—212no—
95C23—412no—
96C23—43no—
97C23—46no—
98C23—412no—
99C23—86no—
100C23—86no—
101C23—86no—
102C23—86no—
103C23—1612no—
104C23#37911yescharacteristic
105C23C2×#9916yes—
106D4—824no—
107D4—1612no—
108C2×C4—212no—
109C2×C4—212no—
110C2×C4—412no—
111C2×C4—412no—
112C2×C4—46no—
113C2×C4—43no—
114C2×C4—86no—
115C2×C4—86no—
116C2×C4—86no—
117C2×C4—86no—
118C8—164no—
119C22—26no—
120C22—26no—
121C22—26no—
122C22—412no—
123C22—88no—
124C22—812no—
125C22—86no—
126C22—83no—
127C22—1612no—
128C22#178113yes—
129C22#179114yes—
130C4—412no—
131C4—44no—
132C4—83no—
133C2—46no—
134C2—163no—
135C2#853916yes—
136C2#854611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC25C22×C4
Frattini, cosocleC24×C4C23
prime radical#35463C2
LCSG ▷ C25 ▷ C23 ▷ 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
2. #8304C2semidirect

Maximal subgroups

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

subgroupindex
#354632
#83042