Finite Group Databasebeta

← #11515

Subgroups — #11515

Order 256.

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

normal subgroups only

This group has 209 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#2317C211yesmaximal, characteristic
2#1464C212yesmaximal
3#1324C214yesmaximal, complemented
4C2×#117C411yescharacteristic
5C2×#117C2211yescharacteristic
6C22×C4⋊C4C411yescharacteristic
7#432—22no—
8#389—22no—
9#385—22no—
10#384C412yes—
11#382C2212yes—
12#324C2214yes—
13#323—22no—
14C23×C4—21no—
15C23×C4C2×C411yescharacteristic
16#117—24no—
17#117—42no—
18C2×C4⋊C4—21no—
19C2×C4⋊C4—21no—
20C2×C4⋊C4—24no—
21C2×C4⋊C4—42no—
22C2×C4⋊C4C2×C412yes—
23C2×C4⋊C4C2×C411yescharacteristic
24C2×C4⋊C4C2311yescharacteristic
25C2×C22⋊C4—21no—
26C2×C22⋊C4D412yes—
27C2×C42C2×C412yes—
28C2×C42D412yes—
29C4⋊C8—42no—
30C22⋊C8—24no—
31#97—42no—
32#95—24no—
33C24C22⋊C411yescharacteristic
34C22×C4—22no—
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×C4C22×C411yescharacteristic
44C22×C4C2×D412yes—
45C2×C8—44no—
46C4⋊C4—24no—
47C4⋊C4—41no—
48C4⋊C4—41no—
49C4⋊C4—42no—
50C4⋊C4—42no—
51C4⋊C4—84no—
52C22⋊C4—24no—
53C22⋊C4—42no—
54C42—22no—
55C42—22no—
56C42—22no—
57C42M1614yes—
58C23—21no—
59C23—21no—
60C23—41no—
61C23—41no—
62C23#10112yes—
63C23C2×C22⋊C411yescharacteristic
64C2×C4—28no—
65C2×C4—21no—
66C2×C4—21no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—24no—
70C2×C4—22no—
71C2×C4—22no—
72C2×C4—42no—
73C2×C4—42no—
74C2×C4—42no—
75C2×C4—42no—
76C2×C4—44no—
77C2×C4—42no—
78C2×C4—84no—
79C2×C4#9912yes—
80C2×C4C2×M1612yes—
81C8—84no—
82C22—21no—
83C22—21no—
84C22—22no—
85C22—42no—
86C22—42no—
87C22—41no—
88C22—41no—
89C22#40711yescharacteristic
90C22C2×#9911yescharacteristic
91C22C2×#10111yescharacteristic
92C4—22no—
93C4—22no—
94C4—44no—
95C4—42no—
96C4—42no—
97C4—82no—
98C2—21no—
99C2—21no—
100C2—42no—
101C2#146311yescharacteristic
102C2#212911yescharacteristic
103C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×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

Maximal subgroups

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

subgroupindex
#23172
#14642
#13242