Finite Group Databasebeta

← #11518

Subgroups — #11518

Order 256.

223 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 55 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#1464C212yesmaximal
3#1326C214yesmaximal
4C2×#117C411yescharacteristic
5C2×#117C2211yescharacteristic
6C22×C42C411yescharacteristic
7#432—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—24no—
17#117—42no—
18C2×C4⋊C4—21no—
19C2×C4⋊C4—22no—
20C2×C4⋊C4—22no—
21C2×C4⋊C4D412yes—
22C2×C22⋊C4C2×C412yes—
23C2×C22⋊C4C2×C411yescharacteristic
24C2×C22⋊C4C2311yescharacteristic
25C2×C42—21no—
26C2×C42—21no—
27C2×C42—21no—
28C2×C42—42no—
29C2×C42D412yes—
30C4⋊C8—42no—
31C22⋊C8—24no—
32#97—42no—
33#95—22no—
34#95—22no—
35C24C22×C411yescharacteristic
36C22×C4—22no—
37C22×C4—22no—
38C22×C4—22no—
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—44no—
42C22×C4—42no—
43C22×C4C22⋊C411yescharacteristic
44C22×C4C22⋊C412yes—
45C22×C4C22⋊C411yescharacteristic
46C22×C4C2×D412yes—
47C2×C8—44no—
48C4⋊C4—24no—
49C4⋊C4—42no—
50C4⋊C4—44no—
51C4⋊C4—44no—
52C22⋊C4—22no—
53C22⋊C4—41no—
54C22⋊C4—41no—
55C22⋊C4M1614yes—
56C42—24no—
57C42—22no—
58C42—22no—
59C42—42no—
60C42—42no—
61C23—21no—
62C23—21no—
63C23—42no—
64C23C2×C22⋊C411yescharacteristic
65C23C2×M1612yes—
66C2×C4—28no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—21no—
70C2×C4—21no—
71C2×C4—24no—
72C2×C4—24no—
73C2×C4—22no—
74C2×C4—22no—
75C2×C4—42no—
76C2×C4—42no—
77C2×C4—42no—
78C2×C4—44no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—42no—
82C2×C4—44no—
83C2×C4#9912yes—
84C2×C4#10112yes—
85C8—84no—
86C22—22no—
87C22—22no—
88C22—42no—
89C22—42no—
90C22—42no—
91C22#40711yescharacteristic
92C22C2×#9911yescharacteristic
93C22C2×#10111yescharacteristic
94C4—22no—
95C4—22no—
96C4—44no—
97C4—42no—
98C4—42no—
99C4—44no—
100C2—21no—
101C2—21no—
102C2—42no—
103C2#146311yescharacteristic
104C2#212012yes—
subgroupG/H
center, socleC22C2×#99
commutator, prime radicalC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#22962
#14642
#13262