Finite Group Databasebeta

← #11452

Subgroups — #11452

Order 256.

231 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 71 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#2296C211yesmaximal, characteristic
2#1458C212yesmaximal
3#1326C214yesmaximal
4C2×#117C411yescharacteristic
5C2×#117C2211yescharacteristic
6C22×C42C411yescharacteristic
7#431—22no—
8#389—22no—
9#385C412yes—
10#385C2212yes—
11C4×C4⋊C4—22no—
12#324—22no—
13#323C2214yes—
14C23×C4C2×C411yescharacteristic
15C23×C4C2×C412yes—
16#117—22no—
17#117—41no—
18#117—41no—
19#117C814yescomplemented
20C2×C4⋊C4—21no—
21C2×C4⋊C4—22no—
22C2×C4⋊C4—22no—
23C2×C4⋊C4D412yes—
24C2×C22⋊C4C2×C412yes—
25C2×C22⋊C4C2×C411yescharacteristic
26C2×C22⋊C4C2311yescharacteristic
27C2×C42—21no—
28C2×C42—21no—
29C2×C42—21no—
30C2×C42—41no—
31C2×C42—41no—
32C2×C42D412yes—
33C4⋊C8—42no—
34C22⋊C8—24no—
35C4×C8—42no—
36#95—22no—
37#95—22no—
38C24C22×C411yescharacteristic
39C22×C4—21no—
40C22×C4—21no—
41C22×C4—22no—
42C22×C4—22no—
43C22×C4—22no—
44C22×C4—44no—
45C22×C4—42no—
46C22×C4C22⋊C411yescharacteristic
47C22×C4C22⋊C412yes—
48C22×C4C22⋊C411yescharacteristic
49C22×C4C2×C812yes—
50C22×C4M1612yes—
51C22×C4C2×D412yes—
52C2×C8—44no—
53C4⋊C4—24no—
54C4⋊C4—42no—
55C4⋊C4—44no—
56C4⋊C4—44no—
57C22⋊C4—22no—
58C22⋊C4—41no—
59C22⋊C4—41no—
60C22⋊C4C2×C814yes—
61C42—24no—
62C42—22no—
63C42—22no—
64C42—42no—
65C42—41no—
66C42—41no—
67C23—21no—
68C23—21no—
69C23—42no—
70C23C2×C22⋊C411yescharacteristic
71C23C22×C811yescharacteristic
72C23C2×M1611yescharacteristic
73C2×C4—21no—
74C2×C4—21no—
75C2×C4—22no—
76C2×C4—22no—
77C2×C4—22no—
78C2×C4—22no—
79C2×C4—24no—
80C2×C4—24no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—42no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—44no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—44no—
91C2×C4C22⋊C812yes—
92C2×C4C22⋊C812yes—
93C2×C4#9911yescharacteristic
94C2×C4#9911yescharacteristic
95C2×C4#10112yes—
96C8—84no—
97C22—22no—
98C22—21no—
99C22—21no—
100C22—41no—
101C22—41no—
102C22—42no—
103C22—42no—
104C22C2×C22⋊C811yescharacteristic
105C22C2×#9911yescharacteristic
106C22C2×#10111yescharacteristic
107C4—22no—
108C4—44no—
109C4—42no—
110C4—42no—
111C4—44no—
112C4#32412yes—
113C4#32412yes—
114C2—21no—
115C2—21no—
116C2—41no—
117C2—41no—
118C2C2×#32411yescharacteristic
119C2#212012yes—
subgroupG/H
centerC2×C4#99
commutatorC23C22×C8
Frattini, cosocleC2×C22⋊C4C23
socleC22C2×#99
prime radicalC24C22×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #117C8semidirect

Maximal subgroups

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

subgroupindex
#22962
#14582
#13262