Finite Group Databasebeta

← #24918213

Subgroups — #24918213

Order 10624.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#14581767C211yesmaximal, characteristic, complemented
2#121×D83C211yesmaximal, characteristic, complemented
3#19128016C212yesmaximal, complemented
4#19128014C211yesmaximal, characteristic, complemented
5#19127938C212yesmaximal, complemented
6#19127936C212yesmaximal, complemented
7#19127837C211yesmaximal, characteristic, complemented
8#19127836C212yesmaximal, complemented
9#19127834C211yesmaximal, characteristic, complemented
10C22⋊C8×D83C211yesmaximal, characteristic, complemented
11#480×C83C211yesmaximal, characteristic, complemented
12C22×D332—21no—
13C22×D332C2211yescharacteristic, complemented
14C2×D4×D83—21no—
15C2×D4×D83C2211yescharacteristic, complemented
16C22×C4×D83C2211yescharacteristic
17C2×D664C2212yescomplemented
18C2×#13050575C2212yescomplemented
19C2×C8×D83C2212yescomplemented
20#14581767—21no—
21#14581767—21no—
22#14581767C2212yes—
23#14581767C2211yescharacteristic, complemented
24#14581767C2212yescomplemented
25#14581766C2211yescharacteristic
26#14581765—21no—
27#14581764—21no—
28#14581763C2211yescharacteristic, complemented
29C4⋊C4×D83C2211yescharacteristic, complemented
30C2×(C2×D83)⋊C4—21no—
31C22⋊C4×D83—21no—
32#14581725C2212yescomplemented
33#14581714C2212yescomplemented
34#14581712C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic, complemented
37(C2×D83)⋊C8C2212yescomplemented
38#121×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic, complemented
40#107×C83C2212yescomplemented
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44C23×D83—21no—
45C23×D83D411yescharacteristic
46C2×D332—21no—
47C2×D332—22no—
48C2×D332—22no—
49C2×D332—21no—
50C2×D332—42no—
51C2×D332—44no—
52C2×D332C2311yescharacteristic
53C2×D332C2312yes—
54C2×C22⋊D83—21no—
55C2×C22⋊D83—21no—
56C2×C22⋊D83—21no—
57D4×D83—42no—
58D4×D83—44no—
59C22×Dic83D411yescharacteristic
60C2×C4×D83—21no—
61C2×C4×D83—42no—
62C2×C4×D83D411yescharacteristic
63C2×C4×D83D411yescharacteristic
64C2×C4×D83C2311yescharacteristic
65C2×C4×D83C2311yescharacteristic
66D664—42no—
67#13050575—42no—
68C2×C83⋊C8C2312yes—
69C8×D83—42no—
70C4⋊Dic83C2311yescharacteristic
71Dic83⋊C4—21no—
72C22⋊Dic83—21no—
73(C2×D83)⋊C4—21no—
74(C2×D83)⋊C4—21no—
75(C2×D83)⋊C4—22no—
76(C2×D83)⋊C4—22no—
77C2×D4×C83—21no—
78C2×D4×C83C2311yescharacteristic
79C22×C4×C83—21no—
80C22×C4×C83C2311yescharacteristic
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2312yes—
84C4⋊C4×C83C2311yescharacteristic
85C22⋊C4×C83—21no—
86C23×C83—21no—
87C23×C83C2×D411yescharacteristic
88C22×D83—21no—
89C22×D83—22no—
90C22×D83—41no—
91C22×D83—41no—
92C22×D83—42no—
93C22×D83—42no—
94C22×D83—42no—
95C22×D83—42no—
96C22×D83—42no—
97C22×D83—42no—
98C22×D83D814yescomplemented
99C22×D83C2×D411yescharacteristic
100C22×D83C2×D411yescharacteristic
101D4×C83—22no—
102D4×C83—42no—
103C2×C4×C83—22no—
104C2×C4×C83—21no—
105C2×C4×C83—22no—
106C2×C4×C83—42no—
107C2×C4×C83C2×D411yescharacteristic
108C2×C4×C83C2411yescharacteristic
109C22⋊D83—42no—
110C22⋊D83—42no—
111C22⋊D83—42no—
112C2×Dic83—21no—
113C2×Dic83—41no—
114C2×Dic83C2×D411yescharacteristic
115C2×Dic83C2×D411yescharacteristic
116D332—22no—
117D332—22no—
118D332—22no—
119D332—42no—
120D332—42no—
121D332—44no—
122D332—44no—
123C4×D83—22no—
124C4×D83—22no—
125C4×D83—41no—
126C4×D83—41no—
127C4×D83—42no—
128C8×C83—22no—
129C83⋊C8—22no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83—41no—
133C22×C83—41no—
134C22×C83—41no—
135C22×C83C2×D812yes—
136C22×C83C22×D411yescharacteristic
137C2×D83—21no—
138C2×D83—21no—
139C2×D83—21no—
140C2×D83—21no—
141C2×D83—44no—
142C2×D83—42no—
143C2×D83—41no—
144C2×D83—42no—
145C2×D83—41no—
146C2×D83—41no—
147C2×D83—42no—
148C2×D83—84no—
149C2×D83—84no—
150C2×D83C2×D812yes—
151C2×D83C2×D812yes—
152C4×C83—21no—
153C4×C83—41no—
154C4×C83—42no—
155C4×C83H(5,𝔽2)12yes—
156Dic83—21no—
157Dic83—21no—
158Dic83—41no—
159C2×C83—22no—
160C2×C83—81no—
161C2×C83#53411yescharacteristic
162C2×C83C22×D811yescharacteristic
163C2×C83#57511yescharacteristic
164D83—22no—
165D83—22no—
166D83—82no—
167D83—81no—
168#3186—831nomaximal, Sylow
169C83#318611yescharacteristic, complemented, Sylow
170C2×#121—831no—
171C2×#121—831no—
172#480—831no—
173#480—832no—
174#480D8311yescharacteristic, complemented
175#466—832no—
176#466—832no—
177#447—831no—
178#447—832no—
179#447—831no—
180C2×C22⋊C8—831no—
181C22×D4—831no—
182C22×D4—831no—
183C22×D4—1661no—
184C22×D4—1661no—
185C23×C4—831no—
186C2×D8—832no—
187C2×D8—832no—
188C22×C8—832no—
189#121—831no—
190#121—831no—
191#121—832no—
192#121—831no—
193#121—832no—
194#121—1661no—
195#121—1661no—
196#121—1661no—
197#121—1661no—
198#121C2×D8311yescharacteristic
199C2×C4⋊C4—831no—
200C2×C4⋊C4C2×D8311yescharacteristic, complemented
201C2×C22⋊C4—1661no—
202C2×C22⋊C4—1661no—
203#107—832no—
204#107C2×D8312yescomplemented
205#102—832no—
206#102—832no—
207#102—832no—
208#102C2×D8312yescomplemented
209C22⋊C8—831no—
210C22⋊C8—832no—
211C22⋊C8C2×D8311yescharacteristic, complemented
212C24—831no—
213C24—1661no—
214C24—1661no—
215C2×D4—21no—
216C2×D4—3322no—
217C2×D4—3324no—
218C2×D4—3324no—
219C2×D4—3322no—
220C2×D4—832no—
221C2×D4—831no—
222C2×D4—1661no—
223C2×D4—1662no—
224C2×D4—1662no—
225C2×D4—1661no—
226C2×D4—1661no—
227C2×D4—1661no—
228C2×D4—1661no—
229C2×D4C22×D8311yescharacteristic
230C22×C4—21no—
231C22×C4—3322no—
232C22×C4—831no—
233C22×C4—831no—
234C22×C4—831no—
235C22×C4—831no—
236C22×C4—831no—
237C22×C4—1661no—
238C22×C4C22×D8311yescharacteristic
239D8—3322no—
240D8—3322no—
241C2×C8—3322no—
242C2×C8—832no—
243C2×C8C22×D8312yes—
244C4⋊C4—21no—
245C4⋊C4—831no—
246C4⋊C4—1661no—
247C4⋊C4C22×D8311yescharacteristic
248C4⋊C4C22×D8312yes—
249C22⋊C4—21no—
250C22⋊C4—1661no—
251C22⋊C4—1662no—
252C22⋊C4—1662no—
253C22⋊C4—1661no—
254C22⋊C4—1661no—
255C23—21no—
256C23—3321no—
257C23—3321no—
258C23—3322no—
259C23—3322no—
260C23—3322no—
261C23—3322no—
262C23—3322no—
263C23—3322no—
264C23—834no—
265C23—831no—
266C23—831no—
267C23—1662no—
268C23—1661no—
269C23D4×D8311yescharacteristic
270D4—22no—
271D4—42no—
272D4—3322no—
273D4—3324no—
274D4—3324no—
275D4—3322no—
276D4—3322no—
277D4—3322no—
278D4—3322no—
279D4—1662no—
280D4—1662no—
281D4—1662no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—3321no—
287C2×C4—3321no—
288C2×C4—3322no—
289C2×C4—3321no—
290C2×C4—831no—
291C2×C4—831no—
292C2×C4—1661no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4D4×D8311yescharacteristic
296C2×C4C23×D8311yescharacteristic
297C8—22no—
298C8—1662no—
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—3324no—
305C22—3322no—
306C22—3321no—
307C22—3322no—
308C22—3321no—
309C22—3322no—
310C22—3321no—
311C22—832no—
312C22—832no—
313C22—6644no—
314C22—6644no—
315C22—1661no—
316C22—1661no—
317C22—1661no—
318C22—1661no—
319C22D8×D8312yes—
320C22C2×D4×D8311yescharacteristic
321C4—21no—
322C4—42no—
323C4—41no—
324C4—3321no—
325C4—1661no—
326C4—1661no—
327C4#1458183812yes—
328C2—22no—
329C2—81no—
330C2—6642no—
331C2—6641no—
332C2—1662no—
333C2—1662no—
334C2#1912826211yescharacteristic
335C2C2×D8×D8311yescharacteristic
336C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#480×C83C2
socleC22×C83C22×D4
hypercenter#480D83
hypocenterC83#3186
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. C2×#14581767C2semidirect
2. #121×D83C2semidirect
3. #19128016C2semidirect
4. #19128014C2semidirect
5. #19127938C2semidirect
6. #19127936C2semidirect
7. #19127837C2semidirect
8. #19127836C2semidirect
9. #19127834C2semidirect
10. C22⋊C8×D83C2semidirect
11. #480×C83C2semidirect
13. C22×D332C22semidirect
15. C2×D4×D83C22semidirect
17. C2×D664C22semidirect
18. C2×#13050575C22semidirect
19. C2×C8×D83C22semidirect
23. #14581767C22semidirect
24. #14581767C22semidirect
28. #14581763C22semidirect
29. C4⋊C4×D83C22semidirect
32. #14581725C22semidirect
33. #14581714C22semidirect
34. #14581712C22semidirect
35. #14581712C22semidirect
36. C22⋊(C83⋊C8)C22semidirect
37. (C2×D83)⋊C8C22semidirect
39. C2×C4⋊C4×C83C22semidirect
40. #107×C83C22semidirect
41. #102×C83C22semidirect
42. C22⋊C8×C83C22semidirect
98. C22×D83D8semidirect
169. C83#3186semidirect
174. #480D83semidirect
200. C2×C4⋊C4C2×D83semidirect
204. #107C2×D83semidirect
208. #102C2×D83semidirect
211. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817672
#121×D832
#191280162
#191280142
#191279382
#191279362
#191278372
#191278362
#191278342
C22⋊C8×D832
#480×C832
#318683