Finite Group Databasebeta

← #10293

Subgroups — #10293

Order 256.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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#3209C211yesmaximal, characteristic
3#2868C211yesmaximal, characteristic
4#1994C211yesmaximal, characteristic
5#1549C211yesmaximal, characteristic
6#1548C211yesmaximal, characteristic
7#1276C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8—22no—
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#500—21no—
13#494—21no—
14#487—21no—
15#487—21no—
16#484—22no—
17#477—22no—
18#462—22no—
19#451—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#384—22no—
24#384—22no—
25C43C2211yescharacteristic
26#339—22no—
27#333—22no—
28#332—22no—
29C22×Q8—21no—
30C22×Q8D412yes—
31C2×Q16—24no—
32C2×M16—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—48no—
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#125—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49#117—21no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D412yes—
55C2×C42—21no—
56C2×C42—21no—
57C2×C42D412yes—
58C2×C42C2311yescharacteristic
59#107—24no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63C4⋊C8D412yes—
64#103—24no—
65#103—24no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C4×C8—21no—
69C4×C8—21no—
70C2×Q8—21no—
71C2×Q8—22no—
72C2×Q8—21no—
73C2×Q8—22no—
74C2×Q8—44no—
75C2×Q8—44no—
76C2×Q8—41no—
77C2×Q8—41no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84Q16—84no—
85M16—42no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—41no—
89C4⋊C4—21no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—22no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—416no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—41no—
105C42—24no—
106C42—24no—
107C42—22no—
108C42—41no—
109C42Q1614yescomplemented
110C42C2×D411yescharacteristic
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C42#4111yescharacteristic
114C23#12011yescharacteristic
115Q8—42no—
116Q8—44no—
117Q8—42no—
118Q8—82no—
119Q8—82no—
120Q8—82no—
121C2×C4—24no—
122C2×C4—24no—
123C2×C4—24no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4#12011yescharacteristic
133C2×C4#12011yescharacteristic
134C2×C4#12111yescharacteristic
135C2×C4#12111yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12711yescharacteristic
138C2×C4C2×Q1612yes—
139C2×C4#13712yes—
140C8—42no—
141C8—42no—
142C8—81no—
143C22—21no—
144C22—21no—
145C22#39211yescharacteristic
146C22#45112yes—
147C4—28no—
148C4—21no—
149C4—21no—
150C4—82no—
151C4—82no—
152C4—81no—
153C4—81no—
154C4#45312yes—
155C4#45612yes—
156C4#46212yes—
157C4#49411yescharacteristic
158C4#49711yescharacteristic
159C2—22no—
160C2#200211yescharacteristic
161C2#200511yescharacteristic
162C2#201611yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#120
LCSG ▷ C2×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
109. C42Q16semidirect

Maximal subgroups

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

subgroupindex
#32302
#32092
#28682
#19942
#15492
#15482
#12762