Finite Group Databasebeta

← #11449

Subgroups — #11449

Order 256.

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

normal subgroups only

This group has 217 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#2317C211yesmaximal, characteristic
2#1458C212yesmaximal
3#1324C214yesmaximal, complemented
4C2×#117C411yescharacteristic
5C2×#117C2211yescharacteristic
6C22×C4⋊C4C411yescharacteristic
7#431—22no—
8#389—22no—
9#385—22no—
10#384C412yes—
11#382C2212yes—
12#324C2214yes—
13#323—22no—
14C23×C4—21no—
15C23×C4C2×C411yescharacteristic
16#117—22no—
17#117—42no—
18#117C814yescomplemented
19C2×C4⋊C4—21no—
20C2×C4⋊C4—21no—
21C2×C4⋊C4—24no—
22C2×C4⋊C4—42no—
23C2×C4⋊C4C2×C412yes—
24C2×C4⋊C4C2×C411yescharacteristic
25C2×C4⋊C4C2311yescharacteristic
26C2×C22⋊C4—21no—
27C2×C22⋊C4D412yes—
28C2×C42C2×C412yes—
29C2×C42D412yes—
30C4⋊C8—42no—
31C22⋊C8—24no—
32C4×C8—42no—
33#95—24no—
34C24C22⋊C411yescharacteristic
35C22×C4—22no—
36C22×C4—22no—
37C22×C4—22no—
38C22×C4—22no—
39C22×C4—42no—
40C22×C4—82no—
41C22×C4C22⋊C412yes—
42C22×C4C22⋊C411yescharacteristic
43C22×C4C2×C812yes—
44C22×C4M1612yes—
45C22×C4C22×C411yescharacteristic
46C22×C4C2×D412yes—
47C2×C8—44no—
48C4⋊C4—24no—
49C4⋊C4—41no—
50C4⋊C4—41no—
51C4⋊C4—42no—
52C4⋊C4—42no—
53C4⋊C4—84no—
54C22⋊C4—24no—
55C22⋊C4—42no—
56C42—22no—
57C42—22no—
58C42—22no—
59C42C2×C814yes—
60C23—21no—
61C23—21no—
62C23—41no—
63C23—41no—
64C23C22⋊C812yes—
65C23C2×C22⋊C411yescharacteristic
66C2×C4—21no—
67C2×C4—21no—
68C2×C4—24no—
69C2×C4—24no—
70C2×C4—24no—
71C2×C4—22no—
72C2×C4—22no—
73C2×C4—42no—
74C2×C4—42no—
75C2×C4—42no—
76C2×C4—42no—
77C2×C4—44no—
78C2×C4—42no—
79C2×C4—84no—
80C2×C4C22⋊C812yes—
81C2×C4#9912yes—
82C2×C4#10112yes—
83C2×C4C22×C811yescharacteristic
84C2×C4C2×M1611yescharacteristic
85C8—84no—
86C22—21no—
87C22—21no—
88C22—22no—
89C22—42no—
90C22—42no—
91C22—41no—
92C22—41no—
93C22C2×C22⋊C811yescharacteristic
94C22C2×#9911yescharacteristic
95C22C2×#10111yescharacteristic
96C4—21no—
97C4—21no—
98C4—44no—
99C4—42no—
100C4—42no—
101C4—82no—
102C4#32414yes—
103C2—21no—
104C2—21no—
105C2—42no—
106C2C2×#32411yescharacteristic
107C2#212911yescharacteristic
108C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC2×C4C22×C8
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1324C2semidirect
18. #117C8semidirect

Maximal subgroups

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

subgroupindex
#23172
#14582
#13242