Finite Group Databasebeta

← #24919085

Subgroups — #24919085

Order 10624.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581820C212yesmaximal, complemented
2#19128372C211yesmaximal, characteristic, complemented
3D4×D332C211yesmaximal, characteristic, complemented
4#19128078C211yesmaximal, characteristic, complemented
5#19127954C212yesmaximal, complemented
6#19127902C211yesmaximal, characteristic, complemented
7#19127897C211yesmaximal, characteristic, complemented
8#19127848C212yesmaximal, complemented
9#19127837C212yesmaximal, complemented
10C4×#13050579C211yesmaximal, characteristic, complemented
11#436×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2212yescomplemented
13C22×D332C2212yes—
14C2×D4×D83—21no—
15#14581820—28no—
16C2×#13050579C2211yescharacteristic
17C2×#13050579C2212yescomplemented
18C2×D664C2212yes—
19C2×D664C2212yescomplemented
20C2×D664C2212yescomplemented
21#14581789C2211yescharacteristic
22#14581789C2211yescharacteristic, complemented
23#14581788—21no—
24#14581767—21no—
25#14581766C2212yes—
26#14581766C2212yescomplemented
27#14581765—22no—
28#14581762—22no—
29C4×Dic166C2211yescharacteristic, complemented
30C4×D332C2211yescharacteristic
31#14581754—21no—
32#14581724C2211yescharacteristic, complemented
33#14581717C2211yescharacteristic, complemented
34#14581714C2211yescharacteristic
35#14581714C2212yescomplemented
36#14581714C2212yescomplemented
37C2×M16×C83C2212yescomplemented
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic, complemented
40C22⋊C8×C83C2212yescomplemented
41C4×C8×C83C2211yescharacteristic, complemented
42C23×D83—22no—
43#13050590—24no—
44#13050590—42no—
45C2×Dic166C2311yescharacteristic
46C2×D332—24no—
47C2×D332—22no—
48C2×D332—21no—
49C2×D332—21no—
50C2×D332—21no—
51C2×D332—21no—
52C2×D332—42no—
53C2×D332C2312yes—
54C2×D332C2312yes—
55C2×D332C2311yescharacteristic
56C2×C22⋊D83—22no—
57C2×C22⋊D83—22no—
58C2×C22⋊D83—22no—
59D4×D83—44no—
60C2×C4×D83—22no—
61C2×C4×D83—21no—
62#13050579—24no—
63#13050579D414yescomplemented
64D664—24no—
65D664—24no—
66D664—42no—
67C4⋊Dic83C2311yescharacteristic
68C4⋊Dic83C2311yescharacteristic
69Dic83⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—22no—
72(C2×D83)⋊C4—22no—
73C4×Dic83—21no—
74C2×D4×C83D411yescharacteristic
75C22×C4×C83C2312yes—
76M16×C83—24no—
77C2×C8×C83C2312yes—
78C2×C8×C83C2311yescharacteristic
79C2×C8×C83C2311yescharacteristic
80C4⋊C4×C83D411yescharacteristic
81C22⋊C4×C83D412yes—
82C42×C83C2311yescharacteristic
83C23×C83C2×D412yes—
84C22×D83—21no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—44no—
88C22×D83—44no—
89C22×D83—82no—
90D4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D412yes—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2411yescharacteristic
97C22⋊D83—44no—
98C22⋊D83—44no—
99C22⋊D83—82no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103D332—22no—
104D332—22no—
105D332—21no—
106D332—22no—
107D332—22no—
108D332—44no—
109D332—42no—
110D332—42no—
111D332—81no—
112D332—81no—
113D332C2×D412yes—
114C4×D83—44no—
115C4×D83—42no—
116Dic166—21no—
117Dic166C2×D412yescomplemented
118C8×C83—22no—
119C8×C83—21no—
120C8×C83C2×D412yescomplemented
121C22×C83—22no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83C22×D411yescharacteristic
125C2×D83—22no—
126C2×D83—41no—
127C2×D83—41no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—88no—
135C4×C83—22no—
136C4×C83—21no—
137C4×C83AGL(1,ℤ/8)12yescomplemented
138C4×C83C22×D411yescharacteristic
139C4×C83H(5,𝔽2)11yescharacteristic
140Dic83—22no—
141Dic83—41no—
142Dic83—41no—
143C2×C83—42no—
144C2×C83D4211yescharacteristic
145C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C83#57611yescharacteristic
147D83—42no—
148D83—82no—
149D83—82no—
150#3274—831nomaximal, Sylow
151C83#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—832no—
153#550—831no—
154D42—831no—
155#493—831no—
156#469—832no—
157#460—831no—
158#459—831no—
159#449—832no—
160#447—832no—
161C4×SD16—831no—
162#436D8311yescharacteristic, complemented
163C2×#41—832no—
164C22×D4—832no—
165C22×D4—1661no—
166AGL(1,ℤ/8)—1668no—
167C2×SD16—831no—
168C2×SD16—832no—
169C2×D8—832no—
170C2×D8—832no—
171C2×D8—832no—
172C2×M16C2×D8312yescomplemented
173#127—831no—
174#127—831no—
175#127—1661no—
176#121—832no—
177#121—832no—
178#121—1661no—
179#121—1662no—
180#120—1662no—
181C4×Q8—831no—
182C4×D4—831no—
183C4×D4—1661no—
184C4×D4C2×D8311yescharacteristic
185#106—831no—
186C4⋊C8C2×D8311yescharacteristic, complemented
187#103—831no—
188#102—831no—
189#102—832no—
190#102—832no—
191C22⋊C8C2×D8312yescomplemented
192C4×C8C2×D8311yescharacteristic, complemented
193C24—1662no—
194#41—3322no—
195#41—1664no—
196C2×Q8—831no—
197C2×D4—3322no—
198C2×D4—3324no—
199C2×D4—831no—
200C2×D4—832no—
201C2×D4—832no—
202C2×D4—1664no—
203C2×D4—1662no—
204C2×D4—1662no—
205C2×D4—1662no—
206C2×D4—1661no—
207C2×D4—1662no—
208C2×D4—1661no—
209C2×D4—1661no—
210C2×D4—1661no—
211C2×D4D33211yescharacteristic
212C22×C4—1661no—
213C22×C4—1662no—
214C22×C4C22×D8312yes—
215SD16—834no—
216SD16—1664no—
217D8—3322no—
218D8—1664no—
219D8—1664no—
220M16—24no—
221C2×C8C22×D8311yescharacteristic
222C2×C8C22×D8312yes—
223C2×C8C22×D8311yescharacteristic
224C4⋊C4—831no—
225C4⋊C4—831no—
226C4⋊C4—1661no—
227C4⋊C4D33211yescharacteristic
228C22⋊C4—1661no—
229C22⋊C4—1662no—
230C22⋊C4—1662no—
231C22⋊C4D33212yes—
232C42—1661no—
233C42C22×D8311yescharacteristic
234C23—3324no—
235C23—3324no—
236C23—6642no—
237C23—1662no—
238C23—1662no—
239C23—1661no—
240C23C2×D33212yes—
241Q8—832no—
242Q8—1661no—
243D4—22no—
244D4—3324no—
245D4—3324no—
246D4—3324no—
247D4—3322no—
248D4—3322no—
249D4—832no—
250D4—6641no—
251D4—6641no—
252D4—6642no—
253D4—1662no—
254D4—1662no—
255D4—1662no—
256D4—1662no—
257D4—1661no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3324no—
261C2×C4—3322no—
262C2×C4—1661no—
263C2×C4—1661no—
264C2×C4—1661no—
265C2×C4C2×D33211yescharacteristic
266C2×C4C2×D33212yes—
267C2×C4C2×D33211yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D8312yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—3321no—
276C22—3321no—
277C22—3322no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—3322no—
282C22—3322no—
283C22—6648no—
284C22—1662no—
285C22C22×D33211yescharacteristic
286C4—22no—
287C4—21no—
288C4—3321no—
289C4—3321no—
290C4—1662no—
291C4#1458182012yescomplemented
292C4C2×D4×D8311yescharacteristic
293C4#1458183811yescharacteristic
294C2—42no—
295C2—3322no—
296C2—6642no—
297C2—6642no—
298C2D4×D33211yescharacteristic
299C2C2×#1458182011yescharacteristic
300C2#1912850611yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#436×C83C2
socleC22×C83C22×D4
hypercenter#436D83
hypocenterC83#3274
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#14581820C2semidirect
2. #19128372C2semidirect
3. D4×D332C2semidirect
4. #19128078C2semidirect
5. #19127954C2semidirect
6. #19127902C2semidirect
7. #19127897C2semidirect
8. #19127848C2semidirect
9. #19127837C2semidirect
10. C4×#13050579C2semidirect
11. #436×C83C2semidirect
12. C2×#13050590C22semidirect
17. C2×#13050579C22semidirect
19. C2×D664C22semidirect
20. C2×D664C22semidirect
22. #14581789C22semidirect
26. #14581766C22semidirect
29. C4×Dic166C22semidirect
32. #14581724C22semidirect
33. #14581717C22semidirect
35. #14581714C22semidirect
36. #14581714C22semidirect
37. C2×M16×C83C22semidirect
39. C4⋊C8×C83C22semidirect
40. C22⋊C8×C83C22semidirect
41. C4×C8×C83C22semidirect
63. #13050579D4semidirect
117. Dic166C2×D4semidirect
120. C8×C83C2×D4semidirect
137. C4×C83AGL(1,ℤ/8)semidirect
151. C83#3274semidirect
162. #436D83semidirect
172. C2×M16C2×D83semidirect
186. C4⋊C8C2×D83semidirect
191. C22⋊C8C2×D83semidirect
192. C4×C8C2×D83semidirect
271. C8D4×D83semidirect
291. C4#14581820semidirect

Maximal subgroups

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

subgroupindex
C2×#145818202
#191283722
D4×D3322
#191280782
#191279542
#191279022
#191278972
#191278482
#191278372
C4×#130505792
#436×C832
#327483