Finite Group Databasebeta

← #9460

Subgroups — #9460

Order 256.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#2503C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic
3#1821C211yesmaximal, characteristic
4#1511C212yesmaximal
5#1284C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#407C2212yes—
9#400—22no—
10#391C412yescomplemented
11#387—24no—
12#382C412yescomplemented
13#381—21no—
14#381—21no—
15#381C2212yes—
16#380—21no—
17C4×C22⋊C4—21no—
18C4×C22⋊C4C2211yescharacteristic
19#343—22no—
20#328—24no—
21C22×Q8C2×C411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C2×M16—22no—
26C22⋊Q8—22no—
27C22⋊Q8—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C412yes—
34C2×C4⋊C4D412yes—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C42—22no—
39C2×C42C2×C412yes—
40C22⋊C8—22no—
41C22⋊C8—22no—
42#95—22no—
43#95—22no—
44#95—22no—
45#95—22no—
46#95—22no—
47#95—22no—
48#95—21no—
49#95—22no—
50#95D414yes—
51C24C2×D411yescharacteristic
52C2×Q8—22no—
53C2×Q8—42no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—24no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4C22⋊C412yes—
66C22×C4C22⋊C412yes—
67C22×C4C22×C411yescharacteristic
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D412yes—
70C22×C4#4112yes—
71M16—44no—
72C2×C8—42no—
73C4⋊C4—24no—
74C4⋊C4—24no—
75C4⋊C4—22no—
76C4⋊C4—42no—
77C4⋊C4—82no—
78C22⋊C4—44no—
79C22⋊C4—42no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C42—42no—
83C42—82no—
84C23—22no—
85C23—22no—
86C23—41no—
87C23C2×C22⋊C411yescharacteristic
88C23#12011yescharacteristic
89C23#12311yescharacteristic
90Q8—82no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—44no—
103C2×C4—42no—
104C2×C4—44no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—44no—
111C2×C4—81no—
112C2×C4—81no—
113C2×C4C4×D412yes—
114C2×C4#12112yes—
115C8—82no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—44no—
124C22#38611yescharacteristic
125C22#41012yes—
126C22#45512yes—
127C4—22no—
128C4—22no—
129C4—42no—
130C4—42no—
131C4—84no—
132C4—82no—
133C4—81no—
134C4—81no—
135C2—22no—
136C2—22no—
137C2—41no—
138C2#188511yescharacteristic
139C2#189911yescharacteristic
140C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #391C4semidirect
12. #382C4semidirect

Maximal subgroups

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

subgroupindex
#25032
#23132
#18212
#15112
#12842