Finite Group Databasebeta

← #7630650

Subgroups — #7630650

Order 512.

5276 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35432C218yesmaximal, complemented
2#35425C2124yesmaximal, complemented
3#35336C213yesmaximal, complemented
4#35287C216yesmaximal, complemented
5#35278C213yesmaximal, complemented
6#35271C216yesmaximal, complemented
7#35259C216yesmaximal, complemented
8C4×#540C216yesmaximal, complemented
9#33999C211yesmaximal, characteristic, complemented
10C2×#540C2216yes—
11C2×C4×#41C2213yes—
12#2867C2211yescharacteristic
13#2860C2214yes—
14#2859C2213yes—
15#2856C2216yes—
16#2854C22112yes—
17#2851C2216yes—
18#2647C2216yes—
19#2644C2214yes—
20#2641C2216yes—
21#2629C2216yes—
22#2619C2216yes—
23#2615C2214yes—
24#2593C22124yes—
25#2590C22124yes—
26#2587C22124yes—
27#2554C22124yes—
28#2531C22124yes—
29#2526C22124yes—
30#2468C22124yes—
31#2464C22124yes—
32#2458C22124yes—
33#2457C22124yes—
34#2456C22124yes—
35#2456C22124yes—
36#2445C22124yes—
37#2444C22124yes—
38#2428C22124yes—
39#2425C22124yes—
40#2335C22124yes—
41#2333C22124yes—
42#2330C2216yes—
43#2328C2216yes—
44#2324C22112yes—
45#2319C22112yes—
46#2314C2216yes—
47C4×#127C2216yes—
48C4×#125C2216yes—
49C4×#123C22124yes—
50C4×#121C22124yes—
51C4×#121C22112yes—
52#2276C2214yes—
53#2275C22112yes—
54C42×D4C2214yes—
55C4×#117C2213yes—
56C4×#117C2213yes—
57C4×#117C2213yes—
58C2×#376C2213yes—
59C22×#41C2311yescharacteristic
60#540—248no—
61C2×#127C2316yes—
62C2×#127C2318yes—
63C2×#126C2318yes—
64C2×#125C2316yes—
65C2×#123C23124yes—
66C2×#123C23124yes—
67C2×#121C23112yes—
68C2×#121C23124yes—
69C2×#121C23124yes—
70C4×#41—224no—
71C2×C4×Q8C2316yes—
72C2×C4×D4C23112yes—
73C2×C4×D4C2316yes—
74C2×C4×D4C2316yes—
75C2×C4×D4C2316yes—
76C2×C4×D4C23124yes—
77C2×C4×D4C23124yes—
78C2×#117C2313yes—
79C2×#117C2313yes—
80C2×#117C2313yes—
81C22×C42C2313yes—
82C22×C42C2313yes—
83C22×C42C2313yes—
84C22×C42C2318yes—
85#400C23124yes—
86#398C23124yes—
87#398C23124yes—
88#397C23124yes—
89#396C23124yes—
90#396C2318yes—
91#394C23124yes—
92#394C23124yes—
93#394C23124yes—
94#393C2318yes—
95#392C23124yes—
96#392C23124yes—
97#390C23112yes—
98#390C23124yes—
99#390C23112yes—
100#390C23124yes—
101#390C23124yes—
102#389C23112yes—
103#389C23112yes—
104#389C23124yes—
105#389C23124yes—
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#386C23124yes—
119#386C23124yes—
120#386C23124yes—
121#386C23124yes—
122#386C23124yes—
123#385C23124yes—
124#385C23124yes—
125#384C2316yes—
126#382C2316yes—
127#381C23124yes—
128#380C23112yes—
129#380C23124yes—
130C4×C4⋊C4C23112yes—
131C4×C4⋊C4C23124yes—
132C4×C4⋊C4C23112yes—
133C4×C4⋊C4C23112yes—
134C4×C4⋊C4C23112yes—
135C4×C22⋊C4C23112yes—
136C4×C22⋊C4C23124yes—
137C4×C22⋊C4C23112yes—
138C4×C22⋊C4C23112yes—
139C4×C22⋊C4C23124yes—
140#376C2313yes—
141#376C2313yes—
142#376C2313yes—
143#376C2311yescharacteristic
144#376C23112yes—
145C2×#95C23112yes—
146C43C2313yes—
147C43C2311yescharacteristic
148C43C2311yescharacteristic
149C43C2311yescharacteristic
150C2×#41—212no—
151C22×Q8C2411yescharacteristic
152C22×D4C2413yes—
153C22×D4C2414yes—
154C22×D4C24124yes—
155C22×D4C24124yes—
156C23×C4C2413yes—
157C23×C4C2416yes—
158C23×C4C24124yes—
159C23×C4C2416yes—
160C23×C4C2416yes—
161#127—224no—
162#127—88no—
163#126—88no—
164#125—224no—
165#123—296no—
166#123—824no—
167#121—248no—
168#121—296no—
169#121—824no—
170C4×Q8—224no—
171C4×D4—224no—
172C4×D4—224no—
173C4×D4—224no—
174C4×D4—248no—
175C4×D4—824no—
176C4×D4—824no—
177#117—212no—
178#117—212no—
179#117#41124yes—
180C2×C4⋊C4C2416yes—
181C2×C4⋊C4C2416yes—
182C2×C4⋊C4C24112yes—
183C2×C4⋊C4C24124yes—
184C2×C4⋊C4C24124yes—
185C2×C4⋊C4C24124yes—
186C2×C4⋊C4C24124yes—
187C2×C4⋊C4C2416yes—
188C2×C22⋊C4C2416yes—
189C2×C22⋊C4C2416yes—
190C2×C22⋊C4C24124yes—
191C2×C22⋊C4C24124yes—
192C2×C22⋊C4C24124yes—
193C2×C22⋊C4C24124yes—
194C2×C22⋊C4C24124yes—
195C2×C22⋊C4C24124yes—
196C2×C22⋊C4C24124yes—
197C2×C22⋊C4C2416yes—
198C2×C42—212no—
199C2×C42—212no—
200C2×C42—212no—
201C2×C42—88no—
202C2×C42C2413yes—
203C2×C42C2416yes—
204C2×C42C2416yes—
205C2×C42C2416yes—
206C2×C42C2413yes—
207C2×C42C24112yes—
208C2×C42C24124yes—
209C2×C42C2414yes—
210C2×C42C2414yes—
211C2×C42C2414yes—
212C2×C42C2413yes—
213C2×C42C2413yes—
214#95C2414yes—
215#95C24112yes—
216#95C24112yes—
217#95C24112yes—
218#95C24124yes—
219#95C24124yes—
220#95C24124yes—
221#95C24124yes—
222#95C24124yes—
223C24C2518yes—
224C24C2513yes—
225#41—232no—
226C2×Q8—26no—
227C2×D4—26no—
228C2×D4—26no—
229C2×D4—248no—
230C2×D4—26no—
231C2×D4—224no—
232C2×D4—248no—
233C2×D4—424no—
234C2×D4—424no—
235C2×D4—424no—
236C2×D4—424no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—212no—
240C22×C4—212no—
241C22×C4—212no—
242C22×C4—248no—
243C22×C4—424no—
244C22×C4—412no—
245C22×C4—424no—
246C22×C4—412no—
247C22×C4—412no—
248C22×C4C2×#41112yes—
249C22×C4C2516yes—
250C22×C4C25124yes—
251C22×C4C2516yes—
252C22×C4C2516yes—
253C22×C4C2513yes—
254C22×C4C2516yes—
255C22×C4C2511yescharacteristic
256C4⋊C4—212no—
257C4⋊C4—212no—
258C4⋊C4—224no—
259C4⋊C4—248no—
260C4⋊C4—424no—
261C4⋊C4—424no—
262C4⋊C4—424no—
263C4⋊C4C2×#41124yes—
264C22⋊C4—212no—
265C22⋊C4—212no—
266C22⋊C4—248no—
267C22⋊C4—248no—
268C22⋊C4—424no—
269C22⋊C4—424no—
270C22⋊C4—424no—
271C22⋊C4—424no—
272C22⋊C4—424no—
273C22⋊C4C2×#41124yes—
274C42—26no—
275C42—26no—
276C42—212no—
277C42—212no—
278C42—212no—
279C42—44no—
280C42—424no—
281C42—412no—
282C42—44no—
283C42—44no—
284C42C2×#41112yes—
285C42C2×#41112yes—
286C23—23no—
287C23—23no—
288C23—212no—
289C23—224no—
290C23—28no—
291C23—224no—
292C23C22×#4116yes—
293C23C2611yescharacteristic
294Q8—28no—
295D4—224no—
296D4—416no—
297D4—848no—
298D4—848no—
299C2×C4—23no—
300C2×C4—23no—
301C2×C4—212no—
302C2×C4—212no—
303C2×C4—224no—
304C2×C4—26no—
305C2×C4—26no—
306C2×C4—224no—
307C2×C4—212no—
308C2×C4—224no—
309C2×C4—212no—
310C2×C4—224no—
311C2×C4—812no—
312C2×C4—848no—
313C2×C4—812no—
314C2×C4—812no—
315C2×C4C22×#4116yes—
316C2×C4C22×#41112yes—
317C2×C4C22×#41112yes—
318C2×C4C22×#4116yes—
319C22—212no—
320C22—26no—
321C22—46no—
322C22—46no—
323C22—424no—
324C22—424no—
325C22—48no—
326C22C23×#4113yes—
327C22H(7,𝔽2)14yes—
328C4—212no—
329C4—212no—
330C4—26no—
331C4—46no—
332C4—46no—
333C4—424no—
334C4#344414yes—
335C2—46no—
336C2—88no—
337C2C2×#344411yescharacteristic
338C2#6285816yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35432C2semidirect
2. #35425C2semidirect
3. #35336C2semidirect
4. #35287C2semidirect
5. #35278C2semidirect
6. #35271C2semidirect
7. #35259C2semidirect
8. C4×#540C2semidirect
9. #33999C2semidirect

Maximal subgroups

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

subgroupindex
#354322
#354252
#353362
#352872
#352782
#352712
#352592
C4×#5402
#339992