Finite Group Databasebeta

← #9478

Subgroups — #9478

Order 256.

305 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 65 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#2484C211yesmaximal, characteristic
2#2274C211yesmaximal, characteristic
3#1826C212yesmaximal
4#1463C211yesmaximal, characteristic
5#1284C212yesmaximal
6C2×#117—21no—
7C2×#117C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9#407C2212yes—
10#400C412yescomplemented
11#398C412yescomplemented
12#387—22no—
13#385—21no—
14#381—21no—
15#381C2211yescharacteristic
16C4×C4⋊C4—24no—
17C4×C22⋊C4—21no—
18C4×C22⋊C4C2212yes—
19#339—24no—
20#324—22no—
21C23×C4D411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4C2311yescharacteristic
24C2×M16—22no—
25#117—28no—
26#117—22no—
27#117—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4D412yes—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C42—22no—
40C2×C42—21no—
41C2×C42—22no—
42C2×C42D414yes—
43C22⋊C8—22no—
44C22⋊C8—22no—
45#95—22no—
46#95—22no—
47#95—22no—
48#95—22no—
49#95—22no—
50#95—21no—
51#95C2×C412yes—
52C24C2×D411yescharacteristic
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—24no—
56C22×C4—24no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4C22⋊C412yes—
65C22×C4C22⋊C412yes—
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D412yes—
68C22×C4C2×D411yescharacteristic
69C22×C4#4112yes—
70M16—44no—
71C2×C8—42no—
72C4⋊C4—28no—
73C4⋊C4—24no—
74C4⋊C4—24no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—44no—
78C22⋊C4—28no—
79C22⋊C4—42no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C42—28no—
83C42—44no—
84C42—42no—
85C42—82no—
86C23—22no—
87C23—22no—
88C23—41no—
89C23C2×C22⋊C411yescharacteristic
90C23#12011yescharacteristic
91C23#12311yescharacteristic
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—24no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—24no—
98C2×C4—28no—
99C2×C4—22no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—44no—
104C2×C4—44no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—81no—
115C2×C4#10414yes—
116C2×C4C4×D412yes—
117C2×C4#12112yes—
118C8—82no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—44no—
127C22#38611yescharacteristic
128C22C2×#10412yes—
129C22#45812yes—
130C4—22no—
131C4—22no—
132C4—24no—
133C4—48no—
134C4—42no—
135C4—82no—
136C4—81no—
137C4—81no—
138C2—22no—
139C2—22no—
140C2—41no—
141C2#189011yescharacteristic
142C2#189611yescharacteristic
143C2#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. #400C4semidirect
11. #398C4semidirect

Maximal subgroups

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

subgroupindex
#24842
#22742
#18262
#14632
#12842