Finite Group Databasebeta

← #9286

Subgroups — #9286

Order 256.

187 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 59 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#3231C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic
3#1531C211yesmaximal, characteristic
4#1531C211yesmaximal, characteristic
5#1497C211yesmaximal, characteristic
6#1287C211yesmaximal, characteristic
7#1275C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10#500C411yescharacteristic
11#498C411yescharacteristic
12#487C411yescharacteristic
13#487C411yescharacteristic
14#485—22no—
15#479—22no—
16#446—22no—
17#446—21no—
18C8×Q8—21no—
19#433—22no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24C2×#97C2211yescharacteristic
25#330—22no—
26#330—22no—
27Q8⋊C8—21no—
28Q8⋊C8—21no—
29C22×C8—21no—
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34#125—21no—
35#125D412yes—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39C22⋊Q8—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8—21no—
44#117—21no—
45#117D412yes—
46C2×C42C2311yescharacteristic
47#107—22no—
48#106—22no—
49C4⋊C8—22no—
50C4⋊C8—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55#103—22no—
56#103—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61#97—22no—
62#97—21no—
63#97—21no—
64#97—21no—
65#97D412yes—
66C4×C8—22no—
67C4×C8C2×C411yescharacteristic
68C4×C8C2×C411yescharacteristic
69C4×C8D411yescharacteristic
70C4×C8D411yescharacteristic
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C22×C4C2×D411yescharacteristic
76C22×C4#4111yescharacteristic
77C22×C4#4111yescharacteristic
78C2×C8—22no—
79C2×C8—21no—
80C2×C8—21no—
81C2×C8—21no—
82C2×C8—22no—
83C2×C8—21no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4C22⋊C412yes—
98C4⋊C4C22⋊C412yes—
99C22⋊C4—22no—
100C22⋊C4—41no—
101C42—22no—
102C42—41no—
103C42C22×C411yescharacteristic
104C42C2×D411yescharacteristic
105C42C2×D411yescharacteristic
106C42C2×D411yescharacteristic
107C23#12311yescharacteristic
108Q8—42no—
109Q8—41no—
110Q8—41no—
111Q8—41no—
112Q8—41no—
113Q8—81no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4C2×C22⋊C411yescharacteristic
123C2×C4C4×D411yescharacteristic
124C2×C4C4×D411yescharacteristic
125C2×C4#12011yescharacteristic
126C2×C4#12111yescharacteristic
127C2×C4#12111yescharacteristic
128C8—41no—
129C8—42no—
130C8—41no—
131C8—41no—
132C8—41no—
133C8—42no—
134C8—81no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22#38611yescharacteristic
139C4—21no—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—21no—
144C4—42no—
145C4—42no—
146C4—81no—
147C4—81no—
148C4#44311yescharacteristic
149C4#44411yescharacteristic
150C2—41no—
151C2#187611yescharacteristic
152C2#190211yescharacteristic
153C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#123
LCSG ▷ C42 ▷ 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
#32312
#29852
#15312
#14972
#12872
#12752