Finite Group Databasebeta

← #8932767

Subgroups — #8932767

Order 512.

4654 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#47563C211yesmaximal, characteristic
2#47196C212yesmaximal, complemented
3#47174C2112yesmaximal, complemented
4#47104C2124yesmaximal, complemented
5#46998C213yesmaximal, complemented
6#35626C213yesmaximal, complemented
7#35451C216yesmaximal, complemented
8#35428C2112yesmaximal, complemented
9C22×#124C2213yes—
10C2×#518C2213yes—
11#2840C2211yescharacteristic
12#2840C2211yescharacteristic
13#2832C22112yescomplemented
14#2811C2214yescomplemented
15#2797C22124yescomplemented
16#2760C2218yescomplemented
17#2759C22124yescomplemented
18#2745C22124yescomplemented
19#2737C22124yescomplemented
20#2736C22112yescomplemented
21#2728C22112yescomplemented
22#2700C22124yescomplemented
23#2670C22112yes—
24#2610C22112yes—
25#2605C22112yescomplemented
26#2599C22112yescomplemented
27#2594C22124yescomplemented
28#2592C22112yescomplemented
29#2586C22112yescomplemented
30#2534C22112yescomplemented
31#2527C22124yescomplemented
32#2526C22124yescomplemented
33#2483C2213yes—
34#2483C2216yes—
35#2483C22112yescomplemented
36#2474C22124yescomplemented
37#2472C22124yescomplemented
38#2465C22124yescomplemented
39#2461C22124yes—
40#2459C22124yes—
41#2458C22124yescomplemented
42#2447C22124yescomplemented
43#2447C22124yescomplemented
44#2409C22112yescomplemented
45#2405C2216yes—
46#2403C2213yes—
47#2337C22112yescomplemented
48#2332C22124yescomplemented
49#2321C22112yescomplemented
50#2312C2216yes—
51#2310C2216yescomplemented
52#2309C2213yes—
53C4×#123C22112yescomplemented
54#2296C2216yescomplemented
55#2276C2214yescomplemented
56C23×Q8C2313yes—
57C23×D4C2313yes—
58C2×#124—412no—
59C2×#124C23112yes—
60C2×#124C23124yes—
61C2×#123C23112yes—
62C2×C22⋊Q8C23124yes—
63C2×#120C23112yes—
64#518—412no—
65C2×C4×Q8C23112yes—
66C2×C4×D4C23112yes—
67C2×C4×D4C23112yes—
68C2×#117C2316yes—
69C2×#117C2316yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2316yes—
73C22×C42C2313yes—
74#401C2318yes—
75#401C2318yes—
76#400C23124yes—
77#398C23112yes—
78#397C23124yes—
79#397C23124yes—
80#397C23112yes—
81#397C23124yes—
82#397C23124yes—
83#396C23124yes—
84#395C23112yes—
85#395C2314yes—
86#395C23124yes—
87#395C2318yes—
88#394C23112yes—
89#394C23124yes—
90#394C23124yes—
91#393C23124yes—
92#393C23124yes—
93#393C23112yes—
94#393C23112yes—
95#392C23112yes—
96#392C2318yes—
97#391C23112yes—
98#391C23124yes—
99#391C23112yes—
100#391C23124yes—
101#391C23124yes—
102#390C23124yes—
103#390C23124yes—
104#389C23124yes—
105#388C23124yes—
106#388C23124yes—
107#388C23124yes—
108#388C23124yes—
109#388C23124yes—
110#388C23124yes—
111#388C23124yes—
112#388C23124yes—
113#388C23124yes—
114#387C23124yes—
115#387C23124yes—
116#387C23124yes—
117#387C23124yes—
118#387C23124yes—
119#386C23112yes—
120#386C23124yes—
121#386C23124yes—
122#386C23112yes—
123#385C23112yes—
124#385C23124yes—
125#385C23112yes—
126#385C23112yes—
127#385C23112yes—
128#383C23112yes—
129#381C23112yes—
130#381C23112yes—
131#381C23124yes—
132#381C23112yes—
133#380C23112yes—
134#379C2311yescharacteristic
135#379C2313yes—
136#379C2316yes—
137#379C2312yes—
138#379C23112yes—
139C4×C4⋊C4C23124yes—
140C4×C4⋊C4C23112yes—
141C4×C4⋊C4C23112yes—
142C4×C22⋊C4C23112yes—
143C4×C22⋊C4C23124yes—
144#376C2314yes—
145#376C2318yes—
146C25C2413yes—
147C25C2411yescharacteristic
148C22×Q8—46no—
149C22×Q8—412no—
150C22×Q8C24112yes—
151C22×Q8C2414yes—
152C22×Q8C2414yes—
153C22×D4—212no—
154C22×D4—412no—
155C22×D4C24112yes—
156C22×D4C24112yes—
157C23×C4C2413yes—
158C23×C4C2416yes—
159C23×C4C2416yes—
160C23×C4C24112yes—
161#124—424no—
162#124—448no—
163#123—424no—
164C22⋊Q8—448no—
165#120—424no—
166C4×Q8—424no—
167C4×D4—424no—
168C4×D4—424no—
169#117—412no—
170#117—412no—
171C2×C4⋊C4C24124yes—
172C2×C4⋊C4C24112yes—
173C2×C4⋊C4C24112yes—
174C2×C4⋊C4C24112yes—
175C2×C4⋊C4C24124yes—
176C2×C4⋊C4C24124yes—
177C2×C22⋊C4—224no—
178C2×C22⋊C4—412no—
179C2×C22⋊C4—412no—
180C2×C22⋊C4C24112yes—
181C2×C22⋊C4C24112yes—
182C2×C22⋊C4C24124yes—
183C2×C22⋊C4C2416yes—
184C2×C22⋊C4C2416yes—
185C2×C22⋊C4C24124yes—
186C2×C22⋊C4C2416yes—
187C2×C22⋊C4C2416yes—
188C2×C22⋊C4C24112yes—
189C2×C22⋊C4C24124yes—
190C2×C22⋊C4C24112yes—
191C2×C22⋊C4C24124yes—
192C2×C22⋊C4C24124yes—
193C2×C42—46no—
194C2×C42C24112yes—
195C2×C42C2416yes—
196C2×C42C2416yes—
197C2×C42C24124yes—
198C2×C42C24112yes—
199#95C2414yes—
200#95C24124yes—
201#95C24112yes—
202#95C24124yes—
203#95C24124yes—
204#95C24124yes—
205#95C24124yes—
206#95C24124yes—
207#95C24112yes—
208#95C24124yes—
209#95C24112yes—
210#95C2418yes—
211C24—26no—
212C24—46no—
213C24—46no—
214C24—412no—
215C24H(5,𝔽2)14yes—
216C24C2516yes—
217C24C2513yes—
218C2×Q8—224no—
219C2×Q8—412no—
220C2×Q8—412no—
221C2×Q8—412no—
222C2×Q8—412no—
223C2×Q8—824no—
224C2×D4—224no—
225C2×D4—224no—
226C2×D4—412no—
227C2×D4—448no—
228C2×D4—424no—
229C2×D4—412no—
230C22×C4—26no—
231C22×C4—212no—
232C22×C4—212no—
233C22×C4—212no—
234C22×C4—412no—
235C22×C4—412no—
236C22×C4—412no—
237C22×C4—424no—
238C22×C4C25112yes—
239C22×C4C25112yes—
240C22×C4C2514yes—
241C22×C4C2518yes—
242C22×C4C25112yes—
243C22×C4C2516yes—
244C4⋊C4—248no—
245C4⋊C4—412no—
246C4⋊C4—412no—
247C4⋊C4—412no—
248C4⋊C4—424no—
249C4⋊C4—424no—
250C22⋊C4—224no—
251C22⋊C4—224no—
252C22⋊C4—248no—
253C22⋊C4—46no—
254C22⋊C4—46no—
255C22⋊C4—424no—
256C22⋊C4—46no—
257C22⋊C4—46no—
258C22⋊C4—412no—
259C22⋊C4—424no—
260C22⋊C4—412no—
261C22⋊C4—424no—
262C22⋊C4—424no—
263C42—224no—
264C42—412no—
265C42—46no—
266C42—46no—
267C42—424no—
268C23—26no—
269C23—212no—
270C23—26no—
271C23—224no—
272C23—212no—
273C23—412no—
274C23—412no—
275C23—812no—
276C23—164no—
277C23—1612no—
278C23C2×H(5,𝔽2)16yes—
279C23C2611yescharacteristic
280Q8—416no—
281Q8—824no—
282Q8—164no—
283D4—448no—
284D4—1612no—
285C2×C4—224no—
286C2×C4—26no—
287C2×C4—26no—
288C2×C4—212no—
289C2×C4—212no—
290C2×C4—212no—
291C2×C4—212no—
292C2×C4—224no—
293C2×C4—224no—
294C2×C4—224no—
295C2×C4—224no—
296C2×C4—812no—
297C2×C4—812no—
298C2×C4—1612no—
299C2×C4#585112yes—
300C22—26no—
301C22—212no—
302C22—412no—
303C22—412no—
304C22—46no—
305C22—424no—
306C22—164no—
307C22—1612no—
308C22C22×H(5,𝔽2)11yescharacteristic
309C22C2×#58516yes—
310C4—212no—
311C4—412no—
312C4—412no—
313C4—412no—
314C4—44no—
315C4—48no—
316C2—46no—
317C2—86no—
318C2#6282013yes—
319C2#6284114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radical#47563C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #47196C2semidirect
3. #47174C2semidirect
4. #47104C2semidirect
5. #46998C2semidirect
6. #35626C2semidirect
7. #35451C2semidirect
8. #35428C2semidirect
13. #2832C22semidirect
14. #2811C22semidirect
15. #2797C22semidirect
16. #2760C22semidirect
17. #2759C22semidirect
18. #2745C22semidirect
19. #2737C22semidirect
20. #2736C22semidirect
21. #2728C22semidirect
22. #2700C22semidirect
25. #2605C22semidirect
26. #2599C22semidirect
27. #2594C22semidirect
28. #2592C22semidirect
29. #2586C22semidirect
30. #2534C22semidirect
31. #2527C22semidirect
32. #2526C22semidirect
35. #2483C22semidirect
36. #2474C22semidirect
37. #2472C22semidirect
38. #2465C22semidirect
41. #2458C22semidirect
42. #2447C22semidirect
43. #2447C22semidirect
44. #2409C22semidirect
47. #2337C22semidirect
48. #2332C22semidirect
49. #2321C22semidirect
51. #2310C22semidirect
53. C4×#123C22semidirect
54. #2296C22semidirect
55. #2276C22semidirect

Maximal subgroups

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

subgroupindex
#475632
#471962
#471742
#471042
#469982
#356262
#354512
#354282