Finite Group Databasebeta

← #11500

Subgroups — #11500

Order 256.

353 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#381C211yesmaximal, characteristic
2#1462C212yesmaximal
3#1317C214yesmaximal, complemented
4C24×C4C411yescharacteristic
5C22×C22⋊C4C411yescharacteristic
6C22×C22⋊C4C2211yescharacteristic
7#407—22no—
8#381—22no—
9#381C412yes—
10#381C2212yes—
11C2×#95—22no—
12#323—22no—
13#323C2214yes—
14C25C2×C411yescharacteristic
15C23×C4—21no—
16C23×C4—21no—
17C23×C4—24no—
18C23×C4—21no—
19C23×C4—42no—
20C23×C4—42no—
21C23×C4C2×C412yes—
22C23×C4D412yes—
23C2×M16—42no—
24C2×C22⋊C4—21no—
25C2×C22⋊C4—42no—
26C2×C22⋊C4—42no—
27C2×C22⋊C4C2×C412yes—
28C2×C22⋊C4C2×C411yescharacteristic
29C2×C22⋊C4D412yes—
30C2×C22⋊C4C2311yescharacteristic
31C22⋊C8—24no—
32C22⋊C8—42no—
33#95—24no—
34#95—24no—
35C24—22no—
36C24—22no—
37C24—22no—
38C24—44no—
39C24C22⋊C411yescharacteristic
40C24C22⋊C411yescharacteristic
41C24C22×C411yescharacteristic
42C22×C4—24no—
43C22×C4—24no—
44C22×C4—22no—
45C22×C4—22no—
46C22×C4—22no—
47C22×C4—22no—
48C22×C4—22no—
49C22×C4—416no—
50C22×C4—42no—
51C22×C4—42no—
52C22×C4—42no—
53C22×C4—42no—
54C22×C4—42no—
55C22×C4—42no—
56C22×C4—44no—
57C22×C4—42no—
58C22×C4—44no—
59C22×C4C22⋊C412yes—
60C22×C4M1614yescomplemented
61C22×C4C2×D412yes—
62M16—84no—
63C2×C8—44no—
64C22⋊C4—41no—
65C22⋊C4—42no—
66C22⋊C4—41no—
67C22⋊C4—42no—
68C22⋊C4—42no—
69C23—24no—
70C23—21no—
71C23—21no—
72C23—28no—
73C23—21no—
74C23—21no—
75C23—21no—
76C23—21no—
77C23—42no—
78C23—42no—
79C23—48no—
80C23—48no—
81C23—44no—
82C23—44no—
83C23#9912yes—
84C23#10012yes—
85C23C2×C22⋊C411yescharacteristic
86C23C2×M1612yes—
87C2×C4—24no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—48no—
91C2×C4—48no—
92C2×C4—44no—
93C2×C4—44no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—44no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—44no—
100C2×C4—42no—
101C2×C4—82no—
102C2×C4—84no—
103C8—84no—
104C22—24no—
105C22—24no—
106C22—22no—
107C22—22no—
108C22—22no—
109C22—22no—
110C22—22no—
111C22—42no—
112C22—42no—
113C22—42no—
114C22—416no—
115C22—42no—
116C22—42no—
117C22—42no—
118C22#35214yes—
119C22#40711yescharacteristic
120C22C2×#9911yescharacteristic
121C22C2×#10011yescharacteristic
122C4—42no—
123C4—42no—
124C4—44no—
125C4—82no—
126C2—21no—
127C2—24no—
128C2—21no—
129C2—42no—
130C2—42no—
131C2#146211yescharacteristic
132C2C2×#35212yes—
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. #1317C2semidirect
60. C22×C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#3812
#14622
#13172