Finite Group Databasebeta

← #9467

Subgroups — #9467

Order 256.

369 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 77 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#2972C211yesmaximal, characteristic
2#2409C211yesmaximal, characteristic
3#1799C211yesmaximal, characteristic
4#1787C212yesmaximal, complemented
5#1289C212yesmaximal, complemented
6C23×Q8C2211yescharacteristic
7C2×C22⋊Q8—21no—
8C2×C4×D4C2211yescharacteristic
9C8×D4—24no—
10#431—21no—
11C2×#103—24no—
12C2×#99—21no—
13C2×C22⋊C8—21no—
14C2×C22⋊C8C2212yes—
15#395C412yescomplemented
16#393C412yescomplemented
17#391—21no—
18#386—21no—
19#385—21no—
20#380C2212yes—
21#380C2211yescharacteristic
22#342—21no—
23C22×Q8—21no—
24C22×Q8—22no—
25C22×Q8—22no—
26C22×Q8—44no—
27C22×Q8C2×C412yes—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33C22×C8D414yes—
34C22⋊Q8—44no—
35C4×D4—22no—
36C4×D4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—22no—
48#103—48no—
49#99—42no—
50C22⋊C8—22no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53C4×C8—22no—
54#95—22no—
55#95—22no—
56#95—22no—
57#95C2×C412yes—
58C24C22⋊C411yescharacteristic
59C24C2×D411yescharacteristic
60C2×Q8—24no—
61C2×Q8—28no—
62C2×Q8—24no—
63C2×Q8—44no—
64C2×Q8—44no—
65C2×Q8—42no—
66C2×Q8—48no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4SD1612yes—
72C2×D4Q1612yes—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—42no—
78C22×C4—44no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22×C411yescharacteristic
86C22×C4C2×D412yes—
87C22×C4C2×D411yescharacteristic
88C22×C4#4112yes—
89C2×C8—24no—
90C2×C8—44no—
91C2×C8—82no—
92C4⋊C4—22no—
93C4⋊C4—42no—
94C4⋊C4—44no—
95C4⋊C4—44no—
96C22⋊C4—22no—
97C22⋊C4—22no—
98C22⋊C4—44no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C42—21no—
103C42—21no—
104C23—21no—
105C23—21no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23#10312yescomplemented
111C23C2×C22⋊C411yescharacteristic
112C23#12112yes—
113C23C2×SD1611yescharacteristic
114C23C2×Q1611yescharacteristic
115C23#13512yes—
116Q8—28no—
117Q8—44no—
118Q8—44no—
119Q8—44no—
120D4—24no—
121D4—24no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—24no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—48no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#10312yescomplemented
145C2×C4C4×D412yes—
146C2×C4#12011yescharacteristic
147C2×C4#12311yescharacteristic
148C8—44no—
149C22—21no—
150C22—22no—
151C22—21no—
152C22—22no—
153C22—22no—
154C22—41no—
155C22—41no—
156C22—42no—
157C22—42no—
158C22—41no—
159C22—41no—
160C22—81no—
161C22#38611yescharacteristic
162C22C2×#10311yescharacteristic
163C22#41611yescharacteristic
164C22#46512yes—
165C22#46712yes—
166C4—21no—
167C4—22no—
168C4—22no—
169C4—44no—
170C4—41no—
171C4—41no—
172C4—82no—
173C4—162no—
174C4B(4,𝔽2)12yes—
175C2—21no—
176C2—22no—
177C2—21no—
178C2—22no—
179C2—42no—
180C2#188911yescharacteristic
181C2#189211yescharacteristic
182C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #1787C2semidirect
5. #1289C2semidirect
15. #395C4semidirect
16. #393C4semidirect
110. C23#103semidirect
144. C2×C4#103semidirect

Maximal subgroups

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

subgroupindex
#29722
#24092
#17992
#17872
#12892