Finite Group Databasebeta

← #24918403

Subgroups — #24918403

Order 10624.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#19128255C211yesmaximal, characteristic, complemented
2#127×D83C211yesmaximal, characteristic
3#19128072C212yesmaximal, complemented
4#19128046C212yesmaximal, complemented
5#19127902C211yesmaximal, characteristic, complemented
6#19127901C211yesmaximal, characteristic, complemented
7#19127898C211yesmaximal, characteristic, complemented
8#19127853C212yesmaximal, complemented
9#19127850C212yesmaximal, complemented
10#19127753C211yesmaximal, characteristic, complemented
11#460×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—21no—
13C2×D4×D83—21no—
14C2×D4×D83C2212yes—
15C2×#13050579C2212yescomplemented
16C2×#13050578C2212yescomplemented
17C2×#13050577C2212yescomplemented
18#14581791C2211yescharacteristic, complemented
19#14581790C2211yescharacteristic, complemented
20#14581789C2211yescharacteristic
21#14581788—21no—
22#14581788—21no—
23#14581788C2212yes—
24#14581782—21no—
25#14581772C2211yescharacteristic
26#14581770—21no—
27#14581768C2211yescharacteristic
28C4×Dic166—21no—
29C4×Dic166C2211yescharacteristic, complemented
30#14581749—21no—
31C42×D83C2211yescharacteristic
32C4⋊(C83⋊C8)C2211yescharacteristic, complemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36(C2×D83)⋊C8C2212yes—
37C8×Dic83C2212yescomplemented
38C2×SD16×C83C2212yescomplemented
39#127×C83C2211yescharacteristic
40C4×Q8×C83C2211yescharacteristic, complemented
41C4⋊C8×C83C2211yescharacteristic, complemented
42#102×C83C2212yescomplemented
43C23×D83—22no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×D332—21no—
47C2×D332—21no—
48C2×D332C2312yes—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—22no—
52D4×D83—42no—
53D4×D83—42no—
54D4×D83—44no—
55D4×D83—44no—
56C2×C4×D83D411yescharacteristic
57C2×C4×D83D411yescharacteristic
58C2×C4×D83C2311yescharacteristic
59#13050579—42no—
60#13050578—42no—
61#13050577—42no—
62C2×C83⋊C8C2312yes—
63C4⋊Dic83—21no—
64C4⋊Dic83C2311yescharacteristic
65Dic83⋊C4—21no—
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68Dic83⋊C4—21no—
69Dic83⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72C4×Dic83—21no—
73C4×Dic83D411yescharacteristic
74C4×Dic83D411yescharacteristic
75C4×Dic83C2311yescharacteristic
76C2×Q8×C83C2311yescharacteristic
77C2×D4×C83—21no—
78C2×D4×C83—21no—
79C2×D4×C83C2312yes—
80SD16×C83—42no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2311yescharacteristic
84C42×C83—21no—
85C42×C83C2311yescharacteristic
86C23×C83—22no—
87C22×D83—22no—
88C22×D83—44no—
89C22×D83—44no—
90C22×D83—44no—
91C22×D83C2×D411yescharacteristic
92Q8×C83—21no—
93Q8×C83—21no—
94D4×C83—22no—
95D4×C83—22no—
96D4×C83—42no—
97D4×C83—42no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2411yescharacteristic
103C22⋊D83—44no—
104C22⋊D83—44no—
105C22⋊D83—44no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110C2×Dic83C2×D411yescharacteristic
111D332—22no—
112D332—22no—
113D332—42no—
114D332—42no—
115C4×D83—21no—
116C4×D83—21no—
117C4×D83—22no—
118C4×D83SD1614yescomplemented
119Dic166—21no—
120Dic166—21no—
121Dic166—42no—
122C8×C83—22no—
123C83⋊C8—22no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83C22×D411yescharacteristic
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—88no—
134C2×D83C2×SD1612yes—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83—41no—
138C4×C83C2×SD1612yes—
139C4×C83H(5,𝔽2)11yescharacteristic
140C4×C83H(5,𝔽2)11yescharacteristic
141Dic83—21no—
142Dic83—41no—
143Dic83—41no—
144Dic83C2×SD1612yes—
145Dic83AGL(1,ℤ/8)12yes—
146C2×C83—82no—
147C2×C83#53511yescharacteristic
148C2×C83C22×SD1611yescharacteristic
149C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
150D83—22no—
151D83—82no—
152#3208—831nomaximal, Sylow
153C83#320811yescharacteristic, complemented, Sylow
154#533—831no—
155C2×#127—831no—
156#492—832no—
157#486—832no—
158#460—831no—
159#460—831no—
160#460—831no—
161#460D8311yescharacteristic, complemented
162#450—832no—
163#450—832no—
164#431—831no—
165C22×D4—832no—
166C22×D4—1661no—
167C22×D4—1661no—
168C2×SD16—832no—
169C2×SD16—832no—
170C2×SD16—832no—
171C2×SD16C2×D8312yescomplemented
172C4⋊Q8—831no—
173C4⋊Q8—831no—
174#127—832no—
175#127—831no—
176#127—1661no—
177#127—1661no—
178#127C2×D8311yescharacteristic
179#125—1661no—
180C22⋊Q8—831no—
181C22⋊Q8—831no—
182C22⋊Q8—1661no—
183C4×Q8—831no—
184C4×Q8—1661no—
185C4×Q8C2×D8311yescharacteristic, complemented
186#117—1661no—
187C2×C42—831no—
188C4⋊C8—831no—
189C4⋊C8C2×D8311yescharacteristic, complemented
190#102—832no—
191#102—832no—
192#102—832no—
193#102C2×D8312yescomplemented
194C22⋊C8—832no—
195C4×C8—832no—
196C24—1662no—
197C2×Q8—831no—
198C2×Q8—1661no—
199C2×Q8C22×D8311yescharacteristic
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—3324no—
203C2×D4—3322no—
204C2×D4—3322no—
205C2×D4—3324no—
206C2×D4—832no—
207C2×D4—1662no—
208C2×D4—1662no—
209C2×D4—1662no—
210C2×D4—1661no—
211C2×D4—1661no—
212C2×D4C22×D8312yes—
213C22×C4—831no—
214C22×C4—831no—
215C22×C4—831no—
216SD16—42no—
217SD16—3322no—
218SD16—3322no—
219SD16—3322no—
220C2×C8—832no—
221C2×C8C22×D8312yes—
222C4⋊C4—21no—
223C4⋊C4—831no—
224C4⋊C4—1661no—
225C4⋊C4—1661no—
226C4⋊C4—1661no—
227C4⋊C4—1661no—
228C4⋊C4—1661no—
229C4⋊C4—1661no—
230C4⋊C4C22×D8311yescharacteristic
231C22⋊C4—1661no—
232C22⋊C4—1661no—
233C42—21no—
234C42—831no—
235C42—831no—
236C42—831no—
237C42—1661no—
238C42C22×D8311yescharacteristic
239C23—22no—
240C23—3324no—
241C23—3324no—
242C23—3324no—
243C23—831no—
244C23—1662no—
245Q8—21no—
246Q8—21no—
247Q8—3322no—
248Q8—1661no—
249Q8—1661no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—42no—
254D4—3322no—
255D4—3324no—
256D4—3324no—
257D4—3324no—
258D4—3322no—
259D4—1662no—
260D4—1662no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—831no—
264C2×C4—831no—
265C2×C4—834no—
266C2×C4—831no—
267C2×C4—1662no—
268C2×C4—1661no—
269C2×C4—1661no—
270C2×C4—1661no—
271C2×C4—1661no—
272C2×C4D4×D8311yescharacteristic
273C2×C4D4×D8311yescharacteristic
274C2×C4C23×D8311yescharacteristic
275C8—22no—
276C8—1662no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—3322no—
281C22—3322no—
282C22—3322no—
283C22—832no—
284C22—6648no—
285C22—1661no—
286C22—1661no—
287C22C2×D4×D8311yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—3321no—
292C4—3321no—
293C4—832no—
294C4—832no—
295C4—1661no—
296C4SD16×D8312yes—
297C4#1458183711yescharacteristic
298C4#1458183811yescharacteristic
299C2—82no—
300C2—6642no—
301C2—1662no—
302C2#1912826711yescharacteristic
303C2C2×SD16×D8311yescharacteristic
304C2AGL(1,ℤ/8)×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#460×C83C2
socleC22×C83C22×D4
prime radical#127×D83C2
hypercenter#460D83
hypocenterC83#3208
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #19128255C2semidirect
3. #19128072C2semidirect
4. #19128046C2semidirect
5. #19127902C2semidirect
6. #19127901C2semidirect
7. #19127898C2semidirect
8. #19127853C2semidirect
9. #19127850C2semidirect
10. #19127753C2semidirect
11. #460×C83C2semidirect
15. C2×#13050579C22semidirect
16. C2×#13050578C22semidirect
17. C2×#13050577C22semidirect
18. #14581791C22semidirect
19. #14581790C22semidirect
29. C4×Dic166C22semidirect
32. C4⋊(C83⋊C8)C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
37. C8×Dic83C22semidirect
38. C2×SD16×C83C22semidirect
40. C4×Q8×C83C22semidirect
41. C4⋊C8×C83C22semidirect
42. #102×C83C22semidirect
118. C4×D83SD16semidirect
153. C83#3208semidirect
161. #460D83semidirect
171. C2×SD16C2×D83semidirect
185. C4×Q8C2×D83semidirect
189. C4⋊C8C2×D83semidirect
193. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282552
#127×D832
#191280722
#191280462
#191279022
#191279012
#191278982
#191278532
#191278502
#191277532
#460×C832
#320883