Finite Group Databasebeta

← #9178

Subgroups — #9178

Order 256.

223 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 67 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#3078C211yesmaximal, characteristic
2#2356C211yesmaximal, characteristic
3#1518C211yesmaximal, characteristic
4#1503C211yesmaximal, characteristic
5#1479C211yesmaximal, characteristic
6#1303C211yesmaximal, characteristic
7#1286C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#479—22no—
11#477C412yescomplemented
12#475C412yescomplemented
13#431C2211yescharacteristic
14#431C2211yescharacteristic
15#428—21no—
16#423C2211yescharacteristic
17#419—21no—
18#391—21no—
19#387—22no—
20#383—21no—
21#382C2211yescharacteristic
22C4×C4⋊C4—21no—
23C4×C4⋊C4C2211yescharacteristic
24#376—21no—
25#328—21no—
26#328—21no—
27#328—22no—
28Q8⋊C8—22no—
29C22×Q8D411yescharacteristic
30C2×M16—21no—
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33#125—21no—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8C2×C412yes—
41C4×Q8D412yes—
42#117—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42—22no—
49C2×C42—21no—
50C2×C42C2311yescharacteristic
51#107—22no—
52#106—22no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8C2×C412yes—
57#103—22no—
58#103—22no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C4×C8—21no—
62C4×C8—21no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—21no—
67#95D411yescharacteristic
68#95D411yescharacteristic
69#95D412yes—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8C22⋊C412yes—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86M16—41no—
87M16—41no—
88C2×C8—22no—
89C2×C8—41no—
90C2×C8—41no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—42no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4C22⋊C412yes—
104C22⋊C4—41no—
105C42—22no—
106C42—22no—
107C42—44no—
108C42—41no—
109C42—41no—
110C42—41no—
111C42C22×C411yescharacteristic
112C42C2×D411yescharacteristic
113C42#4111yescharacteristic
114C42#4111yescharacteristic
115C23#12011yescharacteristic
116Q8—22no—
117Q8—22no—
118Q8—41no—
119Q8—41no—
120Q8—81no—
121Q8—81no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4C2×C22⋊C411yescharacteristic
141C2×C4C4×D411yescharacteristic
142C2×C4C4×D411yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12311yescharacteristic
146C2×C4#13511yescharacteristic
147C2×C4#13511yescharacteristic
148C2×C4#13711yescharacteristic
149C2×C4#13711yescharacteristic
150C8—42no—
151C8—81no—
152C8—81no—
153C22—21no—
154C22—21no—
155C22#38611yescharacteristic
156C22#45211yescharacteristic
157C22#45211yescharacteristic
158C4—21no—
159C4—21no—
160C4—21no—
161C4—21no—
162C4—42no—
163C4—42no—
164C4—42no—
165C4—81no—
166C4—82no—
167C4—81no—
168C4—81no—
169C4#41611yescharacteristic
170C4#41911yescharacteristic
171C4#48511yescharacteristic
172C4#48511yescharacteristic
173C2—21no—
174C2—21no—
175C2#187311yescharacteristic
176C2#190511yescharacteristic
177C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#120
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
11. #477C4semidirect
12. #475C4semidirect

Maximal subgroups

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

subgroupindex
#30782
#23562
#15182
#15032
#14792
#13032
#12862