Finite Group Databasebeta

← #11519

Subgroups — #11519

Order 256.

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

normal subgroups only

This group has 271 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#2315C211yesmaximal, characteristic
2#1462C212yesmaximal
3#1326C214yesmaximal, complemented
4C22×C22⋊C4C411yescharacteristic
5C22×C22⋊C4C411yescharacteristic
6C22×C22⋊C4C2211yescharacteristic
7#407—22no—
8#385—22no—
9#385C412yes—
10#385C2212yes—
11#380—22no—
12#323—22no—
13#323C2214yes—
14C25C2×C411yescharacteristic
15C23×C4—21no—
16C23×C4C2×C412yes—
17C23×C4D412yes—
18C2×M16—42no—
19C2×C4⋊C4—22no—
20C2×C4⋊C4—22no—
21C2×C22⋊C4—21no—
22C2×C22⋊C4—24no—
23C2×C22⋊C4—21no—
24C2×C22⋊C4—21no—
25C2×C22⋊C4—41no—
26C2×C22⋊C4—41no—
27C2×C22⋊C4—41no—
28C2×C22⋊C4—41no—
29C2×C22⋊C4C2×C412yes—
30C2×C22⋊C4C2×C411yescharacteristic
31C2×C22⋊C4D412yes—
32C2×C22⋊C4C2311yescharacteristic
33C22⋊C8—24no—
34C22⋊C8—42no—
35#95—22no—
36#95—22no—
37C24—21no—
38C24—21no—
39C24—22no—
40C24—22no—
41C24—82no—
42C24C22⋊C411yescharacteristic
43C24C22⋊C411yescharacteristic
44C24C22×C411yescharacteristic
45C22×C4—22no—
46C22×C4—22no—
47C22×C4—24no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—42no—
51C22×C4—82no—
52C22×C4—82no—
53C22×C4—82no—
54C22×C4C22⋊C412yes—
55C22×C4C2×D412yes—
56M16—84no—
57C2×C8—44no—
58C4⋊C4—82no—
59C4⋊C4—82no—
60C22⋊C4—24no—
61C22⋊C4—22no—
62C22⋊C4—41no—
63C22⋊C4—42no—
64C22⋊C4—41no—
65C22⋊C4—42no—
66C22⋊C4—41no—
67C22⋊C4—41no—
68C22⋊C4M1614yescomplemented
69C23—24no—
70C23—24no—
71C23—21no—
72C23—21no—
73C23—21no—
74C23—21no—
75C23—21no—
76C23—21no—
77C23—42no—
78C23—42no—
79C23—42no—
80C23—82no—
81C23—82no—
82C23—84no—
83C23—84no—
84C23#9911yescharacteristic
85C23#9911yescharacteristic
86C23#10112yes—
87C23C2×C22⋊C411yescharacteristic
88C23C2×M1612yes—
89C2×C4—24no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4—44no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—82no—
101C2×C4—84no—
102C2×C4—84no—
103C8—84no—
104C22—22no—
105C22—22no—
106C22—22no—
107C22—21no—
108C22—21no—
109C22—22no—
110C22—22no—
111C22—42no—
112C22—42no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—84no—
120C22—84no—
121C22#40711yescharacteristic
122C22C2×#9911yescharacteristic
123C22C2×#10111yescharacteristic
124C4—44no—
125C4—42no—
126C4—42no—
127C4—82no—
128C2—21no—
129C2—21no—
130C2—41no—
131C2—41no—
132C2—42no—
133C2—42no—
134C2#146311yescharacteristic
135C2#212112yes—
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1326C2semidirect
68. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#23152
#14622
#13262