Finite Group Databasebeta

← #10115

Subgroups — #10115

Order 256.

233 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 53 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#3255C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic
3#3139C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic
5#1526C211yesmaximal, characteristic
6#1485C211yesmaximal, characteristic
7#1299C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#494—21no—
12#491—22no—
13#487—21no—
14#487—21no—
15#487—21no—
16#475—21no—
17#475—21no—
18#470—22no—
19#462—21no—
20#462—21no—
21#462—22no—
22#451—22no—
23#431C2211yescharacteristic
24#423C2211yescharacteristic
25#419—22no—
26C4×M16C2211yescharacteristic
27C4×M16C2211yescharacteristic
28#343—22no—
29#330—22no—
30Q8⋊C8—22no—
31C22×Q8D412yes—
32C2×Q16—24no—
33C2×Q16—22no—
34C2×M16—21no—
35C2×M16—21no—
36C2×M16D412yes—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44C4⋊Q8D411yescharacteristic
45#125—21no—
46#125—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55#117—21no—
56#117—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C42C2311yescharacteristic
60#106—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64#103—24no—
65#103—24no—
66#103—24no—
67#103—22no—
68#103—22no—
69C22⋊C8—21no—
70#97—21no—
71#97D412yes—
72C4×C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75C2×Q8—22no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—22no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—44no—
82C22×C4—22no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4#4111yescharacteristic
86Q16—82no—
87Q16—84no—
88M16—22no—
89M16—42no—
90M16—41no—
91M16—42no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—41no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C42—41no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C23#12111yescharacteristic
121Q8—44no—
122Q8—42no—
123Q8—41no—
124Q8—41no—
125Q8—44no—
126Q8—41no—
127Q8—41no—
128Q8—42no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—82no—
139C2×C4#12011yescharacteristic
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12711yescharacteristic
145C8—42no—
146C8—42no—
147C8—42no—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22#39211yescharacteristic
153C4—21no—
154C4—21no—
155C4—21no—
156C4—82no—
157C4—81no—
158C4—81no—
159C4—81no—
160C4—81no—
161C4#45512yes—
162C4#45511yescharacteristic
163C4#45611yescharacteristic
164C4#47312yes—
165C2—41no—
166C2#200411yescharacteristic
167C2#200711yescharacteristic
168C2#202011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C2×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
#32552
#32092
#31392
#19972
#15262
#14852
#12992