Finite Group Databasebeta

← #11428

Subgroups — #11428

Order 256.

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

normal subgroups only

This group has 263 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#2310C211yesmaximal, characteristic
2#1458C212yesmaximal
3#1317C214yesmaximal
4C23×Q8C411yescharacteristic
5C2×#117C411yescharacteristic
6C2×#117C2211yescharacteristic
7#431—22no—
8#391—22no—
9#391—22no—
10#381C412yes—
11#381C2212yes—
12#324—22no—
13#323C2214yes—
14C22×Q8—21no—
15C22×Q8—21no—
16C22×Q8—22no—
17C22×Q8—42no—
18C22×Q8—42no—
19C22×Q8C814yescomplemented
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C412yes—
22#117—42no—
23#117—42no—
24C2×C4⋊C4—21no—
25C2×C4⋊C4D412yes—
26C2×C22⋊C4C2×C412yes—
27C2×C22⋊C4C2×C411yescharacteristic
28C2×C22⋊C4C2311yescharacteristic
29C2×C42—21no—
30C2×C42D412yes—
31C4⋊C8—42no—
32C22⋊C8—24no—
33C4×C8—42no—
34#95—24no—
35#95—24no—
36C24C22×C411yescharacteristic
37C2×Q8—24no—
38C2×Q8—24no—
39C2×Q8—48no—
40C2×Q8—42no—
41C2×Q8—42no—
42C2×Q8—48no—
43C2×Q8—42no—
44C2×Q8—42no—
45C22×C4—22no—
46C22×C4—22no—
47C22×C4—22no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—42no—
51C22×C4—42no—
52C22×C4C22⋊C411yescharacteristic
53C22×C4C22⋊C412yes—
54C22×C4C22⋊C411yescharacteristic
55C22×C4C2×C812yes—
56C22×C4C2×C814yes—
57C22×C4M1612yes—
58C22×C4C2×D412yes—
59C2×C8—44no—
60C4⋊C4—42no—
61C4⋊C4—42no—
62C22⋊C4—41no—
63C22⋊C4—41no—
64C22⋊C4—42no—
65C42—22no—
66C42—22no—
67C42—42no—
68C23—21no—
69C23—21no—
70C23—42no—
71C23C2×C22⋊C411yescharacteristic
72C23C22×C811yescharacteristic
73C23C2×M1611yescharacteristic
74Q8—48no—
75Q8—48no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—24no—
79C2×C4—24no—
80C2×C4—24no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—44no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—44no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—44no—
90C2×C4—42no—
91C2×C4C22⋊C812yes—
92C2×C4C22⋊C812yes—
93C2×C4#9912yes—
94C2×C4#10012yes—
95C8—84no—
96C22—22no—
97C22—21no—
98C22—21no—
99C22—41no—
100C22—41no—
101C22—42no—
102C22—42no—
103C22C2×C22⋊C811yescharacteristic
104C22C2×#9911yescharacteristic
105C22C2×#10011yescharacteristic
106C4—22no—
107C4—42no—
108C4—42no—
109C4—44no—
110C4—82no—
111C4#32314yes—
112C2—21no—
113C2—21no—
114C2—41no—
115C2—41no—
116C2C2×#32311yescharacteristic
117C2#212312yes—
subgroupG/H
center, socleC22C2×#99
commutatorC23C22×C8
Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C22×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. C22×Q8C8semidirect

Maximal subgroups

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

subgroupindex
#23102
#14582
#13172