Finite Group Databasebeta

← #10294

Subgroups — #10294

Order 256.

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

normal subgroups only

This group has 320 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#3229C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic
3#2858C211yesmaximal, characteristic
4#1994C211yesmaximal, characteristic, complemented
5#1546C211yesmaximal, characteristic, complemented
6#1545C211yesmaximal, characteristic
7#1276C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#127C2211yescharacteristic
11C2×#124—22no—
12#498—21no—
13#492—21no—
14#487—21no—
15#486—21no—
16#483—22no—
17#475—22no—
18#460—22no—
19#450—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#390—22no—
24#390—22no—
25C43C2211yescharacteristic
26#339—22no—
27#332—22no—
28#331—22no—
29C22×Q8—21no—
30C22×D4—21no—
31C22×D4—21no—
32C22×D4D412yes—
33C2×SD16—24no—
34C2×M16—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#127—21no—
42#127—21no—
43#127—21no—
44#127—21no—
45#125—21no—
46#124—44no—
47#124—44no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D412yes—
55C2×C22⋊C4—28no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42D412yes—
59C2×C42C2311yescharacteristic
60#106—24no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64C4⋊C8D412yes—
65#103—24no—
66#102—24no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C4×C8—21no—
70C4×C8—21no—
71C24—22no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—42no—
77C2×Q8—42no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—44no—
81C2×D4—44no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—42no—
87C2×D4—42no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93SD16—84no—
94M16—42no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—41no—
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—21no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C22⋊C4—416no—
109C22⋊C4—41no—
110C42—22no—
111C42—42no—
112C42—42no—
113C42—41no—
114C42SD1614yescomplemented
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42#4111yescharacteristic
119C23—44no—
120C23—44no—
121C23—42no—
122C23—44no—
123C23#12011yescharacteristic
124Q8—42no—
125Q8—42no—
126Q8—82no—
127Q8—81no—
128Q8—81no—
129D4—44no—
130D4—82no—
131D4—82no—
132D4—82no—
133D4—82no—
134D4—81no—
135D4—81no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—24no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—44no—
145C2×C4#12011yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12711yescharacteristic
151C2×C4C2×SD1612yes—
152C2×C4AGL(1,ℤ/8)12yes—
153C8—42no—
154C8—42no—
155C8—81no—
156C22—21no—
157C22—21no—
158C22—88no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22#39211yescharacteristic
163C22#45012yes—
164C4—21no—
165C4—21no—
166C4—44no—
167C4—81no—
168C4—81no—
169C4—82no—
170C4#45412yes—
171C4#45512yes—
172C4#46012yes—
173C4#49211yescharacteristic
174C4#49611yescharacteristic
175C2—22no—
176C2—162no—
177C2#200311yescharacteristic
178C2#200411yescharacteristic
179C2#201411yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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
1. #3229C2semidirect
4. #1994C2semidirect
5. #1546C2semidirect
7. #1276C2semidirect
114. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#32292
#32082
#28582
#19942
#15462
#15452
#12762