Finite Group Databasebeta

← #8351

Subgroups — #8351

Order 256.

417 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 89 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#3501C211yesmaximal, characteristic
2#1892C212yesmaximal, complemented
3#1869C211yesmaximal, characteristic
4#1799C211yesmaximal, characteristic
5#1520C212yesmaximal, complemented
6C23×C8C2211yescharacteristic
7#554—24no—
8#554—22no—
9D4×Q8—22no—
10#544C412yescomplemented
11#543—21no—
12#543C412yescomplemented
13#533—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8C2212yes—
16#518—21no—
17C2×C4×D4C2211yescharacteristic
18C2×#103—22no—
19C2×#99—21no—
20C2×C22⋊C8—21no—
21C2×C22⋊C8C2211yescharacteristic
22#380C2212yes—
23#342—21no—
24#340—22no—
25#336—21no—
26#328—24no—
27C22×Q8—21no—
28C22×Q8D412yes—
29C22×D4D411yescharacteristic
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C22×C8—21no—
33C22×C8—22no—
34C22×C8—21no—
35C22×C8—42no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—24no—
39#125—22no—
40#125—24no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43C22⋊Q8—24no—
44C22⋊Q8—24no—
45C22⋊Q8—24no—
46C22⋊Q8—22no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49C22⋊Q8—22no—
50C22⋊Q8—24no—
51C22⋊Q8C2×C414yes—
52C22⋊Q8D414yes—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—22no—
58C4×D4—22no—
59#117—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2×C411yescharacteristic
65C2×C4⋊C4D412yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4C2×C411yescharacteristic
70C2×C22⋊C4D411yescharacteristic
71C2×C42D411yescharacteristic
72#103—44no—
73#99—42no—
74C22⋊C8—22no—
75C22⋊C8—42no—
76#95—22no—
77C24C22⋊C411yescharacteristic
78C24C2×D411yescharacteristic
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—42no—
83C2×Q8—42no—
84C2×Q8—81no—
85C2×Q8—81no—
86C2×Q8C22⋊C414yes—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4SD1612yes—
90C2×D4Q1612yes—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—42no—
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C411yescharacteristic
104C22×C4C22⋊C411yescharacteristic
105C22×C4C22×C411yescharacteristic
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C2×C8—22no—
110C2×C8—24no—
111C2×C8—42no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—24no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4C22⋊C414yes—
130C22⋊C4—24no—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C42—24no—
138C42—21no—
139C42—21no—
140C23—21no—
141C23—21no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23#10312yescomplemented
147C23C2×C22⋊C411yescharacteristic
148C23#12012yes—
149C23C2×SD1611yescharacteristic
150C23C2×Q1611yescharacteristic
151C23#13712yes—
152Q8—44no—
153Q8—42no—
154Q8—42no—
155Q8—82no—
156Q8—82no—
157D4—24no—
158D4—24no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4#10312yescomplemented
184C2×C4C2×C22⋊C412yes—
185C2×C4#12011yescharacteristic
186C2×C4#12011yescharacteristic
187C8—42no—
188C8—81no—
189C22—21no—
190C22—22no—
191C22—21no—
192C22—22no—
193C22—22no—
194C22—41no—
195C22—41no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22#37911yescharacteristic
202C22C2×#10311yescharacteristic
203C22#41911yescharacteristic
204C22#44812yes—
205C22#45112yes—
206C4—22no—
207C4—21no—
208C4—22no—
209C4—41no—
210C4—44no—
211C4—41no—
212C4—44no—
213C4—162no—
214C4B(4,𝔽2)11yescharacteristic
215C4#45811yescharacteristic
216C2—22no—
217C2—21no—
218C2—21no—
219C2—22no—
220C2—42no—
221C2#178711yescharacteristic
222C2#179011yescharacteristic
223C2#179411yescharacteristic
subgroupG/H
center, socleC22#379
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
2. #1892C2semidirect
5. #1520C2semidirect
10. #544C4semidirect
12. #543C4semidirect
146. C23#103semidirect
183. C2×C4#103semidirect

Maximal subgroups

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

subgroupindex
#35012
#18922
#18692
#17992
#15202