Finite Group Databasebeta

← #23481

Subgroups — #23481

Order 256.

779 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C2×#419C211yesmaximal, characteristic, complemented
4C2×#419C211yesmaximal, characteristic, complemented
5C2×#407C211yesmaximal, characteristic
6#2280C211yesmaximal, characteristic
7#2036C214yesmaximal, complemented
8#2035C214yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10C23×Q8C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C22×M16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×C22⋊Q8—24no—
15C2×C22⋊Q8—22no—
16C2×C22⋊Q8—24no—
17C2×C22⋊Q8C2214yes—
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4—21no—
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4—21no—
23#428—24no—
24C2×#107C2212yescomplemented
25C2×#106C2212yescomplemented
26#419—24no—
27#419—24no—
28C2×#103C2212yescomplemented
29C2×#103C2214yescomplemented
30C2×#103C2212yescomplemented
31#407—24no—
32C2×C22⋊C8C2214yes—
33#381—21no—
34#380C2212yes—
35#380C2212yes—
36#379—21no—
37C4×C22⋊C4—22no—
38#340C2214yescomplemented
39C25D411yescharacteristic
40C22×Q8—22no—
41C22×Q8—21no—
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8D414yes—
47C22×Q8C2312yes—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4D411yescharacteristic
57C23×C4D412yes—
58C23×C4C2312yes—
59C23×C4C2311yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—22no—
65C2×M16D414yes—
66C22×C8C2312yes—
67C22×C8C2312yes—
68C22⋊Q8—216no—
69C22⋊Q8—28no—
70C22⋊Q8—28no—
71#117—22no—
72#117—22no—
73#117—22no—
74#117—22no—
75C2×C4⋊C4—28no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4D414yes—
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C22⋊C4—28no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C42—21no—
91C2×C42—21no—
92#107—24no—
93#106—24no—
94#103—24no—
95#103—24no—
96#103—28no—
97C22⋊C8—28no—
98#95—22no—
99#95—22no—
100C24—22no—
101C24—24no—
102C24—22no—
103C24—22no—
104C24—42no—
105C24C2×D412yes—
106C24C2×D411yescharacteristic
107C2×Q8—24no—
108C2×Q8—22no—
109C2×Q8—28no—
110C2×Q8—28no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—42no—
116C2×Q8—44no—
117C2×Q8—42no—
118C2×Q8—44no—
119C2×Q8C2×D414yes—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4C2×D412yes—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D412yes—
149C22×C4C2×D412yes—
150C22×C4C2×D412yes—
151C22×C4#4112yes—
152C22×C4#4112yes—
153C22×C4#4112yes—
154C22×C4C2411yescharacteristic
155M16—28no—
156M16—44no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8C2×D414yes—
163C4⋊C4—216no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4C2×D414yes—
173C22⋊C4—216no—
174C22⋊C4—44no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—44no—
178C42—42no—
179C42—42no—
180C23—22no—
181C23—22no—
182C23—28no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—48no—
190C23—48no—
191C23—44no—
192C23—44no—
193C23#12012yes—
194C23#12112yes—
195C23#12312yes—
196C23#13714yes—
197C23C22×D411yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—48no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—48no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—48no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4#12012yes—
227C2×C4#12114yes—
228C2×C4#12114yes—
229C2×C4#12112yes—
230C2×C4#12314yes—
231C2×C4#12312yes—
232C2×C4#12414yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4C2×#4111yescharacteristic
239C8—24no—
240C8—42no—
241C22—24no—
242C22—22no—
243C22—22no—
244C22—24no—
245C22—24no—
246C22—24no—
247C22—24no—
248C22—416no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22#39414yes—
254C22C2×#12011yescharacteristic
255C22C2×#12111yescharacteristic
256C22C2×#12311yescharacteristic
257C22C2×#13712yes—
258C22#57512yes—
259C4—42no—
260C4—44no—
261C4—44no—
262C4—81no—
263C4—81no—
264C4—82no—
265C4#39414yes—
266C4C2×#12111yescharacteristic
267C4C2×#12111yescharacteristic
268C4C2×#12311yescharacteristic
269C4C2×#12411yescharacteristic
270C2—24no—
271C2—44no—
272C2C2×#39411yescharacteristic
273C2#300411yescharacteristic
274C2#300511yescharacteristic
275C2#305612yes—
276C2#309512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#428C2semidirect
3. C2×#419C2semidirect
4. C2×#419C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1736C2semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
38. #340C22semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4282
C2×#4192
C2×#4072
#22802
#20362
#20352
#17362