Finite Group Databasebeta

← #9022

Subgroups — #9022

Order 256.

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

normal subgroups only

This group has 286 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×#400C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic
3#1824C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1294C212yesmaximal
6C22×C4⋊C4C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8C2×C22⋊C8C2212yes—
9#400—22no—
10#400C414yescomplemented
11#387—24no—
12#381—21no—
13#380C2212yes—
14C4×C22⋊C4—21no—
15C2×#95—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—21no—
31C2×C4⋊C4C2×C412yes—
32C2×C4⋊C4C2×C412yes—
33C2×C4⋊C4D412yes—
34C2×C4⋊C4D414yes—
35C2×C22⋊C4—22no—
36C2×C42—22no—
37C22⋊C8—22no—
38C22⋊C8—22no—
39#95—22no—
40#95—24no—
41#95—24no—
42#95—24no—
43#95—21no—
44#95—22no—
45#95C2×C412yes—
46C24C2×D411yescharacteristic
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—21no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—24no—
56C22×C4—42no—
57C22×C4—42no—
58C22×C4—42no—
59C22×C4—42no—
60C22×C4—44no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—42no—
64C22×C4C22⋊C412yes—
65C22×C4C22⋊C412yes—
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D412yes—
68C22×C4C2×D411yescharacteristic
69C22×C4#4112yes—
70C2×C8—44no—
71C2×C8—42no—
72C4⋊C4—24no—
73C4⋊C4—24no—
74C4⋊C4—24no—
75C4⋊C4—44no—
76C4⋊C4—42no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C42—82no—
80C23—22no—
81C23—22no—
82C23C2×C22⋊C411yescharacteristic
83C23#12011yescharacteristic
84C23#12311yescharacteristic
85C23#13512yes—
86C23#13712yes—
87C2×C4—24no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—44no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—44no—
105C2×C4—44no—
106C2×C4—44no—
107C2×C4—81no—
108C2×C4—84no—
109C2×C4—82no—
110C2×C4—82no—
111C2×C4C4×D412yes—
112C2×C4#12112yes—
113C8—82no—
114C22—24no—
115C22—24no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—21no—
120C22#38611yescharacteristic
121C22#41611yescharacteristic
122C22#41911yescharacteristic
123C22#45212yes—
124C22#45412yes—
125C22#45812yes—
126C22#48512yes—
127C4—22no—
128C4—22no—
129C4—42no—
130C4—42no—
131C4—82no—
132C4—84no—
133C4—82no—
134C2—21no—
135C2—21no—
136C2—22no—
137C2#161514yes—
138C2#187311yescharacteristic
139C2#188811yescharacteristic
140C2#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
10. #400C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4002
#23132
#18242
C2×#3242
#12942