Finite Group Databasebeta

← #9288

Subgroups — #9288

Order 256.

205 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 69 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#3230C211yesmaximal, characteristic
2#2961C211yesmaximal, characteristic
3#1993C211yesmaximal, characteristic
4#1548C211yesmaximal, characteristic
5#1518C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic
7#1275C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#500C412yescomplemented
11#487C412yescomplemented
12#484—22no—
13#477—22no—
14#446—22no—
15#446—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#424—21no—
19#423C2211yescharacteristic
20#405—21no—
21C2×#97C2211yescharacteristic
22#382C2211yescharacteristic
23#340—21no—
24#340—21no—
25#333—22no—
26#328—22no—
27C22×Q8—21no—
28C2×M16—21no—
29C22×C8D411yescharacteristic
30C22×C8D411yescharacteristic
31C4⋊Q8—21no—
32C4⋊Q8—21no—
33C4⋊Q8—21no—
34C4⋊Q8C2×C412yes—
35C4⋊Q8C2×C411yescharacteristic
36C4⋊Q8C2×C411yescharacteristic
37#125—21no—
38C22⋊Q8—21no—
39C22⋊Q8D412yes—
40C4×Q8—21no—
41C4×Q8D412yes—
42#117—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C42C2311yescharacteristic
47#107—24no—
48C4⋊C8—24no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8D412yes—
53#103—24no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56#97—22no—
57#97—21no—
58#97—21no—
59C4×C8—22no—
60C4×C8—21no—
61C4×C8C2×C412yes—
62#95—21no—
63#95—21no—
64C2×Q8—21no—
65C2×Q8—22no—
66C2×Q8—41no—
67C2×Q8—82no—
68C2×Q8C22⋊C412yes—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4#4111yescharacteristic
74M16—41no—
75M16—41no—
76C2×C8—21no—
77C2×C8—21no—
78C2×C8—22no—
79C2×C8—22no—
80C2×C8—21no—
81C2×C8—21no—
82C2×C8—21no—
83C2×C8—41no—
84C2×C8Q1612yes—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4C22⋊C412yes—
97C22⋊C4—22no—
98C42—22no—
99C42C22×C411yescharacteristic
100C42C2×D411yescharacteristic
101C42C2×D411yescharacteristic
102C42#4111yescharacteristic
103C23#12111yescharacteristic
104Q8—42no—
105Q8—81no—
106Q8—82no—
107Q8—82no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—82no—
114C2×C4—81no—
115C2×C4—81no—
116C2×C4—81no—
117C2×C4C2×C22⋊C411yescharacteristic
118C2×C4C4×D411yescharacteristic
119C2×C4C4×D411yescharacteristic
120C2×C4#12011yescharacteristic
121C2×C4#12111yescharacteristic
122C2×C4#12311yescharacteristic
123C2×C4C2×Q1611yescharacteristic
124C2×C4#13511yescharacteristic
125C2×C4#13711yescharacteristic
126C2×C4#13711yescharacteristic
127C8—22no—
128C8—42no—
129C8—41no—
130C8—42no—
131C8—81no—
132C22—21no—
133C22—21no—
134C22#38611yescharacteristic
135C22#46211yescharacteristic
136C22#46411yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—42no—
142C4—42no—
143C4—82no—
144C4—161no—
145C4C4×Q1611yescharacteristic
146C4#44111yescharacteristic
147C4#45111yescharacteristic
148C4#45211yescharacteristic
149C2—21no—
150C2—21no—
151C2#187811yescharacteristic
152C2#190311yescharacteristic
153C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #500C4semidirect
11. #487C4semidirect

Maximal subgroups

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

subgroupindex
#32302
#29612
#19932
#15482
#15182
#12862
#12752