Finite Group Databasebeta

← #9336

Subgroups — #9336

Order 256.

180 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 57 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#3236C211yesmaximal, characteristic
2#2987C211yesmaximal, characteristic
3#1554C211yesmaximal, characteristic
4#1534C211yesmaximal, characteristic
5#1470C211yesmaximal, characteristic
6#1308C211yesmaximal, characteristic
7#1287C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×#125C2211yescharacteristic
10#501—21no—
11#491—21no—
12#484C412yes—
13#483C412yes—
14#479—22no—
15#446—22no—
16#446—22no—
17#433—22no—
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#333—21no—
24#332—21no—
25#330—22no—
26#324—22no—
27C4⋊Q8D411yescharacteristic
28C4⋊Q8D411yescharacteristic
29#125—21no—
30#125—21no—
31#125—21no—
32#125—21no—
33#125—21no—
34C22⋊Q8—21no—
35C22⋊Q8C2×C412yes—
36C4×Q8—21no—
37C4×Q8D412yes—
38#117—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C42C2311yescharacteristic
44#107—22no—
45#106—22no—
46C4⋊C8—24no—
47C4⋊C8—21no—
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52#103—22no—
53#103—22no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57C22⋊C8C2×C412yes—
58C22⋊C8D412yes—
59#97—22no—
60#97—21no—
61#97—21no—
62#97—21no—
63#97D412yes—
64C4×C8—22no—
65C4×C8—21no—
66C2×Q8—21no—
67C2×Q8C22⋊C412yes—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4#4111yescharacteristic
73C2×C8—22no—
74C2×C8—22no—
75C2×C8—22no—
76C2×C8—41no—
77C2×C8—41no—
78C4⋊C4—22no—
79C4⋊C4—21no—
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—22no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4C22⋊C412yes—
92C22⋊C4—22no—
93C42—22no—
94C42C2×D411yescharacteristic
95C42C2×D411yescharacteristic
96C42C2×D411yescharacteristic
97C42#4111yescharacteristic
98C23C4×D411yescharacteristic
99Q8—42no—
100Q8—81no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—81no—
110C2×C4C2×C22⋊C411yescharacteristic
111C2×C4C4×D411yescharacteristic
112C2×C4#12011yescharacteristic
113C2×C4#12111yescharacteristic
114C2×C4#12111yescharacteristic
115C2×C4#12311yescharacteristic
116C8—42no—
117C8—42no—
118C8—42no—
119C8—81no—
120C8—81no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22#38611yescharacteristic
125C4—21no—
126C4—21no—
127C4—21no—
128C4—21no—
129C4—21no—
130C4—21no—
131C4—42no—
132C4—42no—
133C4—81no—
134C4—81no—
135C2—41no—
136C2#188111yescharacteristic
137C2#190511yescharacteristic
138C2#191511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#32362
#29872
#15542
#15342
#14702
#13082
#12872