Finite Group Databasebeta

← #9026

Subgroups — #9026

Order 256.

308 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 75 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×#398C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic
3#1823C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1294C212yesmaximal
6C22×C4⋊C4—21no—
7C22×C4⋊C4C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10#398—22no—
11#398C414yescomplemented
12#387—24no—
13#381—21no—
14#380C2212yes—
15C4×C22⋊C4—21no—
16C2×#95—21no—
17C2×#95C2211yescharacteristic
18#340—24no—
19#324—22no—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—24no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—44no—
33C2×C4⋊C4C2×C412yes—
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4D412yes—
36C2×C4⋊C4D414yes—
37C2×C22⋊C4—22no—
38C2×C42—22no—
39C22⋊C8—22no—
40C22⋊C8—22no—
41#95—22no—
42#95—24no—
43#95—24no—
44#95—21no—
45#95—22no—
46#95C2×C412yes—
47C24C2×D411yescharacteristic
48C22×C4—24no—
49C22×C4—22no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—24no—
58C22×C4—42no—
59C22×C4—42no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4C22⋊C412yes—
67C22×C4C22⋊C412yes—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D412yes—
70C22×C4C2×D411yescharacteristic
71C22×C4#4112yes—
72C2×C8—44no—
73C2×C8—42no—
74C4⋊C4—24no—
75C4⋊C4—24no—
76C4⋊C4—24no—
77C4⋊C4—44no—
78C4⋊C4—48no—
79C4⋊C4—42no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C42—82no—
83C23—22no—
84C23—22no—
85C23C2×C22⋊C411yescharacteristic
86C23#12011yescharacteristic
87C23#12311yescharacteristic
88C23#13512yes—
89C23AGL(1,ℤ/8)12yes—
90C2×C4—24no—
91C2×C4—24no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—42no—
99C2×C4—48no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—44no—
110C2×C4—44no—
111C2×C4—44no—
112C2×C4—81no—
113C2×C4—82no—
114C2×C4—82no—
115C2×C4C4×D412yes—
116C2×C4#12112yes—
117C8—82no—
118C22—24no—
119C22—24no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22#38611yescharacteristic
125C22#41611yescharacteristic
126C22#41811yescharacteristic
127C22#44912yes—
128C22#45612yes—
129C22#45812yes—
130C22#48212yes—
131C4—22no—
132C4—22no—
133C4—44no—
134C4—42no—
135C4—42no—
136C4—84no—
137C4—82no—
138C2—21no—
139C2—21no—
140C2—22no—
141C2#160914yes—
142C2#187511yescharacteristic
143C2#188611yescharacteristic
144C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #398C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
#23132
#18232
C2×#3242
#12942