Finite Group Databasebeta

← #8932778

Subgroups — #8932778

Order 512.

4638 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#47195C213yesmaximal, complemented
2#47180C2112yesmaximal, complemented
3#47106C2124yesmaximal, complemented
4#46998C213yesmaximal, complemented
5#35554C213yesmaximal
6#35452C216yesmaximal, complemented
7#35444C214yesmaximal, complemented
8#35432C218yesmaximal, complemented
9C22×#127C2211yescharacteristic
10C22×#126C2212yes—
11C2×#518C2213yes—
12#2842C2211yescharacteristic
13#2840C2211yescharacteristic
14#2830C22112yescomplemented
15#2800C22124yescomplemented
16#2784C22112yescomplemented
17#2757C22124yescomplemented
18#2737C22124yescomplemented
19#2736C22112yes—
20#2735C22112yes—
21#2724C22112yescomplemented
22#2707C22124yescomplemented
23#2707C22124yescomplemented
24#2611C22112yes—
25#2606C22112yescomplemented
26#2599C22112yes—
27#2593C22112yescomplemented
28#2590C22124yescomplemented
29#2584C22112yescomplemented
30#2579C22112yescomplemented
31#2531C22124yescomplemented
32#2526C22124yescomplemented
33#2486C22112yescomplemented
34#2483C2213yes—
35#2483C2213yes—
36#2480C22124yes—
37#2478C22124yes—
38#2468C22124yescomplemented
39#2466C22124yescomplemented
40#2458C22124yescomplemented
41#2458C22124yescomplemented
42#2445C22124yes—
43#2445C22124yes—
44#2405C22112yes—
45#2405C2216yes—
46#2405C2216yes—
47#2336C22112yescomplemented
48#2335C22124yescomplemented
49#2322C22112yescomplemented
50#2312C2216yes—
51#2309C2216yes—
52#2309C2213yes—
53C4×#123C22112yescomplemented
54#2296C2216yes—
55#2276C2214yescomplemented
56C23×D4C2313yes—
57C23×D4C2313yes—
58C2×#127—44no—
59C2×#127C2318yes—
60C2×#126—48no—
61C2×#126C2318yes—
62C2×#126C2318yes—
63C2×#125C23112yes—
64C2×#123C23112yes—
65C2×#123C23124yes—
66C2×#123C23112yes—
67#518—412no—
68C2×C4×Q8C23112yes—
69C2×C4×D4C23112yes—
70C2×C4×D4C23112yes—
71C2×#117C2316yes—
72C2×#117C2316yes—
73C22×C4⋊C4C2316yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#400C23112yes—
79#399C2314yes—
80#398C23112yes—
81#398C23124yes—
82#398C23112yes—
83#398C23124yes—
84#397C23124yes—
85#397C23124yes—
86#397C23112yes—
87#397C23112yes—
88#397C23124yes—
89#396C23124yes—
90#396C2318yes—
91#396C23124yes—
92#395C23112yes—
93#394C23124yes—
94#394C23124yes—
95#394C23124yes—
96#394C23124yes—
97#393C23124yes—
98#393C2314yes—
99#393C2318yes—
100#390C23112yes—
101#390C23124yes—
102#390C23124yes—
103#390C23112yes—
104#389C23124yes—
105#389C23124yes—
106#389C23124yes—
107#389C23124yes—
108#388C23124yes—
109#388C23124yes—
110#388C23124yes—
111#388C23124yes—
112#388C23124yes—
113#388C23124yes—
114#388C23124yes—
115#388C23124yes—
116#387C23124yes—
117#387C23124yes—
118#387C23124yes—
119#387C23124yes—
120#387C23124yes—
121#387C23124yes—
122#386C23112yes—
123#386C23112yes—
124#386C23124yes—
125#385C23112yes—
126#385C23124yes—
127#385C23124yes—
128#385C23112yes—
129#385C23112yes—
130#385C23112yes—
131#383C23112yes—
132#381C23112yes—
133#381C23112yes—
134#380C23112yes—
135#380C23124yes—
136#379C2313yes—
137#379C2313yes—
138#379C23112yes—
139#379C2316yes—
140#379C23112yes—
141C4×C4⋊C4C23112yes—
142C4×C4⋊C4C23112yes—
143C4×C4⋊C4C23124yes—
144C4×C22⋊C4C23112yes—
145C4×C22⋊C4C23124yes—
146#376C23112yes—
147C25C2413yes—
148C25C2411yescharacteristic
149C22×Q8C2414yes—
150C22×D4—212no—
151C22×D4—412no—
152C22×D4—412no—
153C22×D4—46no—
154C22×D4C24112yes—
155C22×D4C24112yes—
156C23×C4C2413yes—
157C23×C4C2416yes—
158C23×C4C2416yes—
159C23×C4C24112yes—
160#127—416no—
161#126—416no—
162#126—416no—
163#125—424no—
164#123—424no—
165#123—448no—
166#123—424no—
167C4×Q8—424no—
168C4×D4—424no—
169C4×D4—424no—
170#117—412no—
171#117—412no—
172C2×C4⋊C4—412no—
173C2×C4⋊C4C24124yes—
174C2×C4⋊C4C24124yes—
175C2×C4⋊C4C24124yes—
176C2×C4⋊C4C24112yes—
177C2×C4⋊C4C24112yes—
178C2×C4⋊C4C24112yes—
179C2×C4⋊C4C24124yes—
180C2×C4⋊C4C24112yes—
181C2×C4⋊C4C24112yes—
182C2×C22⋊C4—224no—
183C2×C22⋊C4—412no—
184C2×C22⋊C4C24112yes—
185C2×C22⋊C4C24112yes—
186C2×C22⋊C4C2416yes—
187C2×C22⋊C4C2416yes—
188C2×C22⋊C4C2416yes—
189C2×C22⋊C4C2416yes—
190C2×C22⋊C4C24112yes—
191C2×C22⋊C4C24112yes—
192C2×C22⋊C4C24124yes—
193C2×C22⋊C4C24124yes—
194C2×C22⋊C4C24124yes—
195C2×C22⋊C4C24124yes—
196C2×C22⋊C4C24124yes—
197C2×C42—44no—
198C2×C42—42no—
199C2×C42C24112yes—
200C2×C42C2416yes—
201C2×C42C2416yes—
202C2×C42C24124yes—
203C2×C42C2418yes—
204C2×C42C2414yes—
205#95C2414yes—
206#95C24112yes—
207#95C24124yes—
208#95C24124yes—
209#95C24124yes—
210#95C24124yes—
211#95C24124yes—
212#95C24124yes—
213#95C24112yes—
214#95C24112yes—
215C24—26no—
216C24—46no—
217C24—46no—
218C24—412no—
219C24H(5,𝔽2)14yes—
220C24C2516yes—
221C24C2511yescharacteristic
222C24C2512yes—
223C2×Q8—412no—
224C2×D4—224no—
225C2×D4—224no—
226C2×D4—224no—
227C2×D4—412no—
228C2×D4—448no—
229C2×D4—412no—
230C2×D4—412no—
231C2×D4—824no—
232C22×C4—26no—
233C22×C4—212no—
234C22×C4—212no—
235C22×C4—212no—
236C22×C4—412no—
237C22×C4—412no—
238C22×C4—412no—
239C22×C4—424no—
240C22×C4C25112yes—
241C22×C4C25112yes—
242C22×C4C25112yes—
243C22×C4C25112yes—
244C22×C4C2516yes—
245C4⋊C4—248no—
246C4⋊C4—248no—
247C4⋊C4—424no—
248C4⋊C4—412no—
249C4⋊C4—412no—
250C4⋊C4—412no—
251C4⋊C4—412no—
252C4⋊C4—412no—
253C4⋊C4—424no—
254C22⋊C4—224no—
255C22⋊C4—224no—
256C22⋊C4—46no—
257C22⋊C4—46no—
258C22⋊C4—46no—
259C22⋊C4—46no—
260C22⋊C4—412no—
261C22⋊C4—424no—
262C22⋊C4—424no—
263C22⋊C4—424no—
264C22⋊C4—424no—
265C22⋊C4—412no—
266C22⋊C4—424no—
267C42—224no—
268C42—48no—
269C42—44no—
270C42—46no—
271C42—46no—
272C42—424no—
273C23—26no—
274C23—212no—
275C23—26no—
276C23—212no—
277C23—224no—
278C23—412no—
279C23—412no—
280C23—812no—
281C23—164no—
282C23—1612no—
283C23C2×H(5,𝔽2)12yes—
284C23C2×H(5,𝔽2)14yes—
285C23C2611yescharacteristic
286Q8—416no—
287D4—448no—
288D4—824no—
289C2×C4—224no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—212no—
293C2×C4—212no—
294C2×C4—212no—
295C2×C4—212no—
296C2×C4—224no—
297C2×C4—224no—
298C2×C4—224no—
299C2×C4—224no—
300C2×C4—812no—
301C2×C4—812no—
302C2×C4—1612no—
303C2×C4#585112yes—
304C22—26no—
305C22—212no—
306C22—48no—
307C22—44no—
308C22—412no—
309C22—46no—
310C22—424no—
311C22—164no—
312C22—1612no—
313C22C22×H(5,𝔽2)11yescharacteristic
314C22C2×#58516yes—
315C4—212no—
316C4—412no—
317C4—412no—
318C4—412no—
319C4—412no—
320C2—44no—
321C2—42no—
322C2—86no—
323C2#6282013yes—
324C2#6283914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #47195C2semidirect
2. #47180C2semidirect
3. #47106C2semidirect
4. #46998C2semidirect
6. #35452C2semidirect
7. #35444C2semidirect
8. #35432C2semidirect
14. #2830C22semidirect
15. #2800C22semidirect
16. #2784C22semidirect
17. #2757C22semidirect
18. #2737C22semidirect
21. #2724C22semidirect
22. #2707C22semidirect
23. #2707C22semidirect
25. #2606C22semidirect
27. #2593C22semidirect
28. #2590C22semidirect
29. #2584C22semidirect
30. #2579C22semidirect
31. #2531C22semidirect
32. #2526C22semidirect
33. #2486C22semidirect
38. #2468C22semidirect
39. #2466C22semidirect
40. #2458C22semidirect
41. #2458C22semidirect
47. #2336C22semidirect
48. #2335C22semidirect
49. #2322C22semidirect
53. C4×#123C22semidirect
55. #2276C22semidirect

Maximal subgroups

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

subgroupindex
#471952
#471802
#471062
#469982
#355542
#354522
#354442
#354322