Finite Group Databasebeta

← #7664847

Subgroups — #7664847

Order 512.

4968 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39623C2112yesmaximal, complemented
2#39620C2124yesmaximal, complemented
3#36564C2112yesmaximal, complemented
4#35002C216yesmaximal
5#35002C213yesmaximal
6C2×#2345C213yesmaximal
7C4×#543C211yesmaximal, characteristic
8C4×#543C212yesmaximal
9C2×#543C2216yes—
10C2×#543C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12#2857C22124yes—
13#2857C22112yes—
14#2806C22124yescomplemented
15#2792C22148yescomplemented
16#2734C22124yescomplemented
17#2659C22112yes—
18#2659C22124yes—
19#2586C22148yescomplemented
20#2567C22112yescomplemented
21#2551C22112yescomplemented
22#2549C22124yes—
23#2549C22148yes—
24#2507C22124yescomplemented
25#2507C22124yescomplemented
26#2451C22124yescomplemented
27#2440C22148yes—
28#2440C22124yes—
29#2369C22112yescomplemented
30Q8×C4⋊C4C2214yescomplemented
31#2345C22124yes—
32#2345C22112yes—
33#2345C22112yes—
34#2329C22112yes—
35#2329C22124yes—
36#2315C2216yes—
37#2315C2213yes—
38C4×C22⋊Q8C22124yes—
39C4×C22⋊Q8C22112yes—
40C2×#389C2216yes—
41C2×#387C22112yes—
42C2×#380C2213yes—
43C2×#380C2216yes—
44C2×C4×C4⋊C4C2216yes—
45C2×C4×C22⋊C4C2213yes—
46C2×C4×C22⋊C4C2216yes—
47C24×C4C2311yescharacteristic
48C24×C4C2312yes—
49#543—424no—
50#543Q8116yescomplemented
51C2×C4⋊Q8C23112yes—
52C2×C4⋊Q8C23124yes—
53C2×C22⋊Q8C23148yes—
54C2×C22⋊Q8C23124yes—
55C2×C22⋊Q8C23112yes—
56C2×C4×Q8C2314yes—
57C2×C4×Q8C2318yes—
58C22×C4⋊C4Q8116yescomplemented
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C23112yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2313yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2313yes—
66C22×C22⋊C4C2313yes—
67C22×C42C2316yes—
68C22×C42C23112yes—
69#398C23148yes—
70#398C23148yes—
71#398C23148yes—
72#395C23148yes—
73#395C23148yes—
74#395C23148yes—
75#391C23124yes—
76#391C23112yes—
77#391C23148yes—
78#391C23124yes—
79#389C23124yes—
80#389C23124yes—
81#389C23112yes—
82#389C23124yes—
83#389C23124yes—
84#389C23148yes—
85#389C23124yes—
86#389C23124yes—
87#389C23124yes—
88#387C23124yes—
89#387C23148yes—
90#387C23148yes—
91#387C23148yes—
92#385C23148yes—
93#385C23124yes—
94#384C23124yes—
95#384C23112yes—
96#380C23124yes—
97#380C23112yes—
98#380C23112yes—
99#380C23124yes—
100#380C23124yes—
101#380C23112yes—
102C4×C4⋊C4C23112yes—
103C4×C4⋊C4C23124yes—
104C4×C4⋊C4C23124yes—
105C4×C4⋊C4C23112yes—
106C4×C4⋊C4C23112yes—
107C4×C22⋊C4C23124yes—
108C4×C22⋊C4C23112yes—
109C4×C22⋊C4C23112yes—
110C4×C22⋊C4C23124yes—
111C2×#95C2316yes—
112C2×#95C23112yes—
113C2×#95C23112yes—
114C2×#95C2316yes—
115C25C2411yescharacteristic
116C22×Q8C24124yes—
117C22×Q8C2414yes—
118C23×C4—44no—
119C23×C4—84no—
120C23×C4C2×Q818yes—
121C23×C4C2×Q8116yes—
122C23×C4C2413yes—
123C23×C4C2416yes—
124C23×C4C24112yes—
125C23×C4C2416yes—
126C23×C4C2416yes—
127C23×C4C2413yes—
128C23×C4C2416yes—
129C4⋊Q8—424no—
130C4⋊Q8—824no—
131C22⋊Q8—496no—
132C22⋊Q8—448no—
133C22⋊Q8C2×Q8196yes—
134C4×Q8—48no—
135C4×Q8—88no—
136C2×C4⋊C4—412no—
137C2×C4⋊C4—812no—
138C2×C4⋊C4C2×Q8124yes—
139C2×C4⋊C4C2×Q8124yes—
140C2×C4⋊C4C2×Q8148yes—
141C2×C4⋊C4C24112yes—
142C2×C4⋊C4C24148yes—
143C2×C4⋊C4C24112yes—
144C2×C4⋊C4C24124yes—
145C2×C4⋊C4C24112yes—
146C2×C4⋊C4C24148yes—
147C2×C4⋊C4C24124yes—
148C2×C4⋊C4C24112yes—
149C2×C4⋊C4C24124yes—
150C2×C4⋊C4C2411yescharacteristic
151C2×C4⋊C4C2416yes—
152C2×C4⋊C4C2413yes—
153C2×C4⋊C4C2416yes—
154C2×C4⋊C4C24112yes—
155C2×C22⋊C4—224no—
156C2×C22⋊C4—212no—
157C2×C22⋊C4C2×Q8124yes—
158C2×C22⋊C4C24124yes—
159C2×C22⋊C4C24112yes—
160C2×C22⋊C4C24112yes—
161C2×C42—412no—
162C2×C42—812no—
163C2×C42C24112yes—
164C2×C42C24124yes—
165C2×C42C24124yes—
166C2×C42C24112yes—
167#95C24124yes—
168#95C24148yes—
169#95C24124yes—
170#95C24124yes—
171#95C24124yes—
172#95C24148yes—
173#95C24124yes—
174C24—42no—
175C24—42no—
176C24C22×Q814yes—
177C24C22×Q814yes—
178C24H(5,𝔽2)14yes—
179C24C2513yes—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—424no—
183C2×Q8—424no—
184C2×Q8—424no—
185C2×Q8C22×Q8116yes—
186C22×C4—212no—
187C22×C4—212no—
188C22×C4—224no—
189C22×C4—416no—
190C22×C4—46no—
191C22×C4—46no—
192C22×C4—412no—
193C22×C4—412no—
194C22×C4—46no—
195C22×C4—46no—
196C22×C4—412no—
197C22×C4—412no—
198C22×C4—46no—
199C22×C4—816no—
200C22×C4C22×Q8112yes—
201C22×C4C22×Q8124yes—
202C22×C4C22×Q8112yes—
203C22×C4C22×Q8124yes—
204C22×C4C22×Q8124yes—
205C22×C4C25112yes—
206C22×C4C25124yes—
207C22×C4C2513yes—
208C22×C4C2512yes—
209C22×C4C25112yes—
210C22×C4C2516yes—
211C22×C4C2511yescharacteristic
212C4⋊C4—224no—
213C4⋊C4—412no—
214C4⋊C4—448no—
215C4⋊C4—412no—
216C4⋊C4—424no—
217C4⋊C4—448no—
218C4⋊C4—424no—
219C4⋊C4—424no—
220C4⋊C4—412no—
221C4⋊C4C22×Q8124yes—
222C4⋊C4C22×Q8124yes—
223C4⋊C4C22×Q814yes—
224C4⋊C4C22×Q8112yes—
225C4⋊C4C22×Q8148yes—
226C22⋊C4—248no—
227C22⋊C4—224no—
228C22⋊C4C22×Q8148yes—
229C42—224no—
230C42—412no—
231C42—424no—
232C42—424no—
233C23—26no—
234C23—26no—
235C23—44no—
236C23—44no—
237C23—48no—
238C23—48no—
239C23Q82116yes—
240C23C23×Q816yes—
241C23C23×Q816yes—
242C23C2×H(5,𝔽2)16yes—
243C23C2611yescharacteristic
244Q8—416no—
245Q8—1624no—
246C2×C4—212no—
247C2×C4—26no—
248C2×C4—24no—
249C2×C4—212no—
250C2×C4—212no—
251C2×C4—224no—
252C2×C4—212no—
253C2×C4—224no—
254C2×C4—212no—
255C2×C4—224no—
256C2×C4—224no—
257C2×C4—212no—
258C2×C4—412no—
259C2×C4—424no—
260C2×C4—424no—
261C2×C4—424no—
262C2×C4—812no—
263C2×C4—824no—
264C2×C4C23×Q8124yes—
265C2×C4C23×Q814yes—
266C2×C4C23×Q8112yes—
267C2×C4C23×Q816yes—
268C2×C4C23×Q812yes—
269C2×C4#58514yes—
270C22—212no—
271C22—26no—
272C22—212no—
273C22—26no—
274C22—416no—
275C22C2×Q82112yes—
276C22C24×Q811yescharacteristic
277C22C24×Q811yescharacteristic
278C22C22×H(5,𝔽2)11yescharacteristic
279C22C2×#58512yes—
280C22H(7,𝔽2)12yes—
281C4—224no—
282C4—24no—
283C4—212no—
284C4—26no—
285C4—412no—
286C4—424no—
287C4#345814yes—
288C2—212no—
289C2C2×#345811yescharacteristic
290C2C22×Q8211yescharacteristic
291C2#6284311yescharacteristic
292C2#6290512yes—
293C2#6298912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #39623C2semidirect
2. #39620C2semidirect
3. #36564C2semidirect
14. #2806C22semidirect
15. #2792C22semidirect
16. #2734C22semidirect
19. #2586C22semidirect
20. #2567C22semidirect
21. #2551C22semidirect
24. #2507C22semidirect
25. #2507C22semidirect
26. #2451C22semidirect
29. #2369C22semidirect
30. Q8×C4⋊C4C22semidirect
50. #543Q8semidirect
58. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#396232
#396202
#365642
#350022
C2×#23452
C4×#5432