Finite Group Databasebeta

← #9294

Subgroups — #9294

Order 256.

245 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 69 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#3208C211yesmaximal, characteristic
2#2961C211yesmaximal, characteristic, complemented
3#1961C211yesmaximal, characteristic
4#1545C211yesmaximal, characteristic
5#1518C211yesmaximal, characteristic, complemented
6#1286C211yesmaximal, characteristic, complemented
7#1275C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#127C2211yescharacteristic
10#492C412yescomplemented
11#486C412yescomplemented
12#460—22no—
13#450—22no—
14#446—22no—
15#446—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#424—21no—
19#423C2211yescharacteristic
20C2×#102—21no—
21C2×#102—21no—
22#405—21no—
23C2×#97C2211yescharacteristic
24#382C2211yescharacteristic
25#331—22no—
26#328—22no—
27C22×D4—21no—
28C22×D4—21no—
29C22×D4—21no—
30C2×SD16—24no—
31C2×M16—21no—
32C22×C8D411yescharacteristic
33C22×C8D411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35C4⋊Q8C2×C411yescharacteristic
36#127—21no—
37#127—21no—
38#127—21no—
39#127C2×C412yes—
40#125—21no—
41C22⋊Q8—21no—
42C22⋊Q8D412yes—
43C4×Q8—21no—
44C4×Q8D412yes—
45#117—21no—
46C2×C4⋊C4—21no—
47C2×C42C2311yescharacteristic
48C4⋊C8—24no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8D412yes—
53#102—24no—
54#102—42no—
55#102—42no—
56C22⋊C8—21no—
57C22⋊C8—21no—
58#97—22no—
59#97—21no—
60#97—21no—
61C4×C8—22no—
62C4×C8—21no—
63C4×C8C2×C412yes—
64#95—21no—
65#95—21no—
66C24—41no—
67C2×Q8—21no—
68C2×Q8C22⋊C412yes—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—82no—
80C2×D4—82no—
81C22×C4—41no—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4#4111yescharacteristic
85SD16—84no—
86M16—41no—
87M16—41no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—21no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—41no—
96C2×C8SD1612yescomplemented
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4C22⋊C412yes—
105C22⋊C4—22no—
106C42—22no—
107C42C22×C411yescharacteristic
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42#4111yescharacteristic
111C23—42no—
112C23—82no—
113C23—82no—
114C23—81no—
115C23—81no—
116C23#12111yescharacteristic
117Q8—42no—
118Q8—81no—
119D4—44no—
120D4—42no—
121D4—42no—
122D4—82no—
123D4—82no—
124D4—82no—
125D4—81no—
126D4—81no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4C2×C22⋊C411yescharacteristic
135C2×C4C4×D411yescharacteristic
136C2×C4C4×D411yescharacteristic
137C2×C4#12011yescharacteristic
138C2×C4#12111yescharacteristic
139C2×C4#12311yescharacteristic
140C2×C4C2×SD1611yescharacteristic
141C2×C4#13511yescharacteristic
142C2×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C4#13711yescharacteristic
144C8—22no—
145C8—42no—
146C8—41no—
147C8—42no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22—82no—
152C22—82no—
153C22—82no—
154C22—162no—
155C22—162no—
156C22#38611yescharacteristic
157C22#46011yescharacteristic
158C22#46411yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4—42no—
165C4—161no—
166C4C4×SD1611yescharacteristic
167C4#44011yescharacteristic
168C4#44811yescharacteristic
169C4#44911yescharacteristic
170C2—21no—
171C2—21no—
172C2—162no—
173C2#188011yescharacteristic
174C2#190311yescharacteristic
175C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
LCSG ▷ 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
2. #2961C2semidirect
5. #1518C2semidirect
6. #1286C2semidirect
7. #1275C2semidirect
10. #492C4semidirect
11. #486C4semidirect
96. C2×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#32082
#29612
#19612
#15452
#15182
#12862
#12752