Finite Group Databasebeta

← #11512

Subgroups — #11512

Order 256.

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

normal subgroups only

This group has 289 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×#384C211yesmaximal, characteristic
2#1463C212yesmaximal
3#1325C214yesmaximal, complemented
4C22×C4⋊C4—22no—
5C22×C4⋊C4C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7C22×C42C411yescharacteristic
8#407—22no—
9#384—22no—
10#384C412yes—
11#384C2212yes—
12#324—22no—
13#324C2214yes—
14C23×C4—21no—
15C23×C4—21no—
16C23×C4C2×C411yescharacteristic
17C23×C4D412yes—
18C2×M16—42no—
19C2×C4⋊C4—24no—
20C2×C4⋊C4—21no—
21C2×C4⋊C4—24no—
22C2×C4⋊C4—41no—
23C2×C4⋊C4—41no—
24C2×C4⋊C4—41no—
25C2×C4⋊C4—41no—
26C2×C4⋊C4—48no—
27C2×C4⋊C4C2×C412yes—
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4D412yes—
30C2×C4⋊C4C2311yescharacteristic
31C2×C42—21no—
32C2×C42—24no—
33C2×C42C2×C412yes—
34C22⋊C8—24no—
35C22⋊C8—42no—
36C24C22⋊C411yescharacteristic
37C22×C4—24no—
38C22×C4—24no—
39C22×C4—24no—
40C22×C4—21no—
41C22×C4—21no—
42C22×C4—22no—
43C22×C4—22no—
44C22×C4—22no—
45C22×C4—22no—
46C22×C4—22no—
47C22×C4—22no—
48C22×C4—44no—
49C22×C4—44no—
50C22×C4—41no—
51C22×C4—41no—
52C22×C4C22⋊C411yescharacteristic
53C22×C4C22⋊C412yes—
54C22×C4C22×C411yescharacteristic
55C22×C4C2×D412yes—
56M16—84no—
57C2×C8—44no—
58C4⋊C4—41no—
59C4⋊C4—42no—
60C4⋊C4—41no—
61C4⋊C4—42no—
62C4⋊C4—48no—
63C4⋊C4—48no—
64C4⋊C4—42no—
65C42—24no—
66C42—22no—
67C42M1614yescomplemented
68C23—22no—
69C23—22no—
70C23—21no—
71C23—21no—
72C23#9911yescharacteristic
73C23#9911yescharacteristic
74C23C2×C22⋊C411yescharacteristic
75C2×C4—24no—
76C2×C4—216no—
77C2×C4—24no—
78C2×C4—24no—
79C2×C4—24no—
80C2×C4—21no—
81C2×C4—24no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—44no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—44no—
91C2×C4—42no—
92C2×C4—82no—
93C2×C4—82no—
94C2×C4—82no—
95C2×C4#10112yes—
96C2×C4C2×M1612yes—
97C8—84no—
98C22—24no—
99C22—24no—
100C22—21no—
101C22—21no—
102C22—21no—
103C22—21no—
104C22—21no—
105C22—21no—
106C22#35312yes—
107C22#35612yes—
108C22#40711yescharacteristic
109C22C2×#9911yescharacteristic
110C22C2×#10111yescharacteristic
111C4—28no—
112C4—22no—
113C4—22no—
114C4—42no—
115C4—42no—
116C4—82no—
117C2—21no—
118C2—22no—
119C2—22no—
120C2—21no—
121C2#146311yescharacteristic
122C2C2×#35311yescharacteristic
123C2C2×#35611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1325C2semidirect
67. C42M16semidirect

Maximal subgroups

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

subgroupindex
C2×#3842
#14632
#13252