Finite Group Databasebeta

← #24918217

Subgroups — #24918217

Order 10624.

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

normal subgroups only

This group has 468 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
1C22⋊Q8×D83C211yesmaximal, characteristic, complemented
2C2×#14581763C211yesmaximal, characteristic
3#19128032C211yesmaximal, characteristic, complemented
4#19128029C211yesmaximal, characteristic, complemented
5#19128018C212yesmaximal, complemented
6#19127929C212yesmaximal, complemented
7#19127927C212yesmaximal, complemented
8#19127852C211yesmaximal, characteristic, complemented
9#19127838C212yesmaximal, complemented
10C22⋊C8×D83C211yesmaximal, characteristic, complemented
11#450×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2211yescharacteristic, complemented
13C22×C22⋊D83—21no—
14C22×C4×D83C2211yescharacteristic
15C2×#13050578C2212yescomplemented
16C2×C8×D83C2212yescomplemented
17#14581772C2211yescharacteristic, complemented
18#14581771C2211yescharacteristic
19#14581770—21no—
20#14581769—21no—
21#14581768C2211yescharacteristic, complemented
22#14581763—21no—
23#14581763—21no—
24#14581763C2212yes—
25#14581763C2212yes—
26C2×C4⋊Dic83C2211yescharacteristic
27C4⋊C4×D83—21no—
28C4⋊C4×D83C2211yescharacteristic, complemented
29C2×C22⋊Dic83—21no—
30C22⋊C4×D83—21no—
31#14581724C2212yescomplemented
32#14581723C2212yescomplemented
33#14581717C2212yescomplemented
34#14581716C2212yescomplemented
35#14581713C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×SD16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46Q8×D83—42no—
47C2×C22⋊D83—22no—
48C2×C22⋊D83—22no—
49C2×C22⋊D83—44no—
50C22×Dic83—21no—
51C22×Dic83D411yescharacteristic
52C2×C4×D83—21no—
53C2×C4×D83—21no—
54C2×C4×D83—42no—
55C2×C4×D83D411yescharacteristic
56C2×C4×D83D411yescharacteristic
57C2×C4×D83C2311yescharacteristic
58C2×C4×D83C2311yescharacteristic
59#13050578—42no—
60C2×C83⋊C8C2312yes—
61C8×D83—42no—
62C4⋊Dic83—21no—
63C4⋊Dic83—21no—
64C4⋊Dic83C2312yes—
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68Dic83⋊C4—21no—
69C22⋊Dic83—21no—
70C22⋊Dic83—22no—
71C22⋊Dic83—22no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—21no—
74C24×C83—21no—
75C2×Q8×C83C2311yescharacteristic
76C2×D4×C83—21no—
77C2×D4×C83—42no—
78C2×D4×C83C2312yes—
79C22×C4×C83C2311yescharacteristic
80SD16×C83—42no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2311yescharacteristic
84C22⋊C4×C83—21no—
85C23×C83—22no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83C2×D411yescharacteristic
90C22×D83—41no—
91C22×D83—41no—
92C22×D83SD1614yescomplemented
93C22×D83C2×D411yescharacteristic
94C22×D83C2×D411yescharacteristic
95Q8×C83—22no—
96D4×C83—22no—
97D4×C83—22no—
98D4×C83—42no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—44no—
105C22⋊D83—44no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109C2×Dic83—41no—
110C2×Dic83—42no—
111C2×Dic83C2×D411yescharacteristic
112C2×Dic83C2×D411yescharacteristic
113C4×D83—22no—
114C4×D83—22no—
115C4×D83—41no—
116C4×D83—41no—
117C4×D83—42no—
118C4×D83—42no—
119Dic166—22no—
120Dic166—42no—
121C8×C83—22no—
122C83⋊C8—22no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—84no—
129C22×C83C2×SD1612yes—
130C22×C83C22×D411yescharacteristic
131C2×D83—21no—
132C2×D83—21no—
133C2×D83—21no—
134C2×D83—21no—
135C2×D83—44no—
136C2×D83C2×SD1612yes—
137C2×D83C2×SD1612yes—
138C4×C83—21no—
139C4×C83—41no—
140C4×C83—41no—
141C4×C83H(5,𝔽2)12yes—
142Dic83—21no—
143Dic83—21no—
144Dic83—41no—
145Dic83—42no—
146Dic83—41no—
147C2×C83—22no—
148C2×C83—82no—
149C2×C83#53411yescharacteristic
150C2×C83C22×SD1611yescharacteristic
151C2×C83#57511yescharacteristic
152D83—22no—
153D83—22no—
154#3187—831nomaximal, Sylow
155C83#318711yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—831no—
157C2×#121—831no—
158#483—831no—
159#483—831no—
160#481—832no—
161#465—832no—
162#465—832no—
163#450—831no—
164#450D8311yescharacteristic, complemented
165#448—832no—
166C2×C22⋊C8—831no—
167C22×Q8—831no—
168C22×D4—1661no—
169C22×D4C2×D8311yescharacteristic
170C23×C4—831no—
171C2×SD16—832no—
172C2×SD16C2×D8312yescomplemented
173C22×C8—832no—
174C22⋊Q8—831no—
175C22⋊Q8—831no—
176C22⋊Q8—831no—
177C22⋊Q8—1661no—
178C22⋊Q8—1661no—
179C22⋊Q8C2×D8311yescharacteristic
180#121—832no—
181#121—832no—
182#121—1661no—
183#121—1661no—
184C2×C4⋊C4—831no—
185C2×C4⋊C4—831no—
186C2×C4⋊C4—1661no—
187C2×C22⋊C4—1661no—
188C2×C22⋊C4—1661no—
189#106—832no—
190#106—832no—
191#103—832no—
192#103—832no—
193#102—832no—
194#102C2×D8312yescomplemented
195C22⋊C8—831no—
196C22⋊C8—832no—
197C22⋊C8C2×D8311yescharacteristic
198C24—21no—
199C24—831no—
200C2×Q8—3322no—
201C2×Q8—831no—
202C2×Q8—1661no—
203C2×Q8C22×D8311yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—3324no—
207C2×D4—1662no—
208C2×D4—1662no—
209C2×D4C22×D8312yes—
210C22×C4—3322no—
211C22×C4—831no—
212C22×C4—831no—
213C22×C4—831no—
214C22×C4—831no—
215C22×C4—831no—
216C22×C4—1661no—
217C22×C4—1661no—
218C22×C4—1661no—
219C22×C4C22×D8311yescharacteristic
220SD16—42no—
221SD16—3322no—
222C2×C8—3322no—
223C2×C8—832no—
224C2×C8C22×D8312yes—
225C4⋊C4—21no—
226C4⋊C4—831no—
227C4⋊C4—832no—
228C4⋊C4—1661no—
229C4⋊C4—1661no—
230C4⋊C4—1661no—
231C4⋊C4—1661no—
232C4⋊C4—1661no—
233C4⋊C4C22×D8311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—1661no—
236C22⋊C4—1661no—
237C22⋊C4—1662no—
238C22⋊C4—1662no—
239C22⋊C4—1661no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—3321no—
245C23—3321no—
246C23—834no—
247C23—831no—
248C23—831no—
249C23D4×D8311yescharacteristic
250Q8—22no—
251Q8—3322no—
252Q8—1662no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—3324no—
257D4—3324no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—3321no—
262C2×C4—3321no—
263C2×C4—3322no—
264C2×C4—3322no—
265C2×C4—3322no—
266C2×C4—3321no—
267C2×C4—831no—
268C2×C4—831no—
269C2×C4—1661no—
270C2×C4—1662no—
271C2×C4—1661no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4D4×D8311yescharacteristic
275C2×C4C23×D8311yescharacteristic
276C8—22no—
277C8—1662no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—3324no—
285C22—832no—
286C22—832no—
287C22—1661no—
288C22—1661no—
289C22—1661no—
290C22—1661no—
291C22SD16×D8312yes—
292C22C2×D4×D8311yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—3322no—
297C4—3321no—
298C4—3321no—
299C4—1661no—
300C4—1661no—
301C4#1458183712yes—
302C2—22no—
303C2—82no—
304C2—1662no—
305C2—1662no—
306C2#1912826111yescharacteristic
307C2C2×SD16×D8311yescharacteristic
308C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581763C2
hypercenter#450D83
hypocenterC83#3187
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D83C2semidirect
3. #19128032C2semidirect
4. #19128029C2semidirect
5. #19128018C2semidirect
6. #19127929C2semidirect
7. #19127927C2semidirect
8. #19127852C2semidirect
9. #19127838C2semidirect
10. C22⋊C8×D83C2semidirect
11. #450×C83C2semidirect
12. C2×Q8×D83C22semidirect
15. C2×#13050578C22semidirect
16. C2×C8×D83C22semidirect
17. #14581772C22semidirect
21. #14581768C22semidirect
28. C4⋊C4×D83C22semidirect
31. #14581724C22semidirect
32. #14581723C22semidirect
33. #14581717C22semidirect
34. #14581716C22semidirect
35. #14581713C22semidirect
37. (C2×D83)⋊C8C22semidirect
39. C2×SD16×C83C22semidirect
41. #102×C83C22semidirect
92. C22×D83SD16semidirect
155. C83#3187semidirect
164. #450D83semidirect
172. C2×SD16C2×D83semidirect
194. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D832
C2×#145817632
#191280322
#191280292
#191280182
#191279292
#191279272
#191278522
#191278382
C22⋊C8×D832
#450×C832
#318783