Finite Group Databasebeta

← #24917255

Subgroups — #24917255

Order 10624.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#19128256C211yesmaximal, characteristic, complemented
2C4×D4×D83C211yesmaximal, characteristic
3#19128004C212yesmaximal, complemented
4#19128003C212yesmaximal, complemented
5#19127992C211yesmaximal, characteristic, complemented
6#19127991C211yesmaximal, characteristic, complemented
7#19127978C211yesmaximal, characteristic, complemented
8#19127742C212yesmaximal, complemented
9#19127723C211yesmaximal, characteristic, complemented
10#19127704C212yesmaximal
11#474×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83C2211yescharacteristic
13C22×C4×D83—21no—
14C2×#13050573C2212yes—
15#14581792C2211yescharacteristic, complemented
16#14581783C2212yescomplemented
17#14581772—21no—
18#14581768—21no—
19Q8×Dic83—21no—
20#14581757—21no—
21C4×D332C2211yescharacteristic
22D4×Dic83C2211yescharacteristic
23#14581754C2211yescharacteristic
24C4×C22⋊D83—21no—
25(C22⋊D83)⋊C4—21no—
26#14581750C2212yes—
27C4⋊C4×D83C2211yescharacteristic
28C22⋊C4×D83—21no—
29C42×D83C2211yescharacteristic
30#14581725C2212yescomplemented
31#14581724C2212yescomplemented
32#14581723C2212yescomplemented
33C4⋊(C83⋊C8)C2211yescharacteristic, complemented
34Dic83⋊C8C2212yescomplemented
35#14581714C2212yes—
36#14581713C2212yes—
37#14581712C2212yes—
38C4⋊Q8×C83C2211yescharacteristic, complemented
39C4×D4×C83C2211yescharacteristic
40#107×C83C2212yescomplemented
41C4⋊C8×C83C2211yescharacteristic, complemented
42#102×C83C2212yes—
43C23×D83—21no—
44C2×Dic166—21no—
45C2×D332C2311yescharacteristic
46C2×C22⋊D83—21no—
47D4×D83—22no—
48D4×D83Q814yescomplemented
49C22×Dic83—21no—
50C2×C4×D83—21no—
51C2×C4×D83—21no—
52C2×C4×D83—21no—
53C2×C4×D83—44no—
54C2×C4×D83D411yescharacteristic
55C2×C4×D83D411yescharacteristic
56C2×C4×D83C2311yescharacteristic
57#13050573—24no—
58C2×C83⋊C8C2312yes—
59C4⋊Dic83—21no—
60C4⋊Dic83—22no—
61C4⋊Dic83C2312yes—
62C4⋊Dic83C2311yescharacteristic
63Dic83⋊C4—22no—
64Dic83⋊C4—21no—
65C22⋊Dic83—21no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69C4×Dic83—22no—
70C4×Dic83D411yescharacteristic
71C4×Dic83D411yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83—21no—
74C2×D4×C83C2311yescharacteristic
75C22×C4×C83—21no—
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2312yes—
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C22×D83—21no—
84C22×D83—24no—
85C22×D83—42no—
86C22×D83—42no—
87C22×D83C2×D411yescharacteristic
88Q8×C83—42no—
89D4×C83—21no—
90D4×C83C2×Q812yes—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—42no—
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2411yescharacteristic
97C22⋊D83—24no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—42no—
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104D332—21no—
105D332C2×Q812yes—
106C4×D83—22no—
107C4×D83—41no—
108C4×D83—42no—
109C4×D83—42no—
110C4×D83—42no—
111C4×D83C2×Q812yes—
112C4×D83#4112yes—
113Dic166—42no—
114C8×C83—22no—
115C83⋊C8—22no—
116C22×C83—22no—
117C22×C83—41no—
118C22×C83—41no—
119C22×C83C22×D411yescharacteristic
120C2×D83—21no—
121C2×D83—21no—
122C2×D83—22no—
123C2×D83—48no—
124C2×D83—41no—
125C2×D83—41no—
126C2×D83C22⋊Q812yes—
127C4×C83—21no—
128C4×C83—41no—
129C4×C83—42no—
130C4×C83AGL(1,ℤ/8)12yes—
131C4×C83C22×Q811yescharacteristic
132C4×C83C2×#4111yescharacteristic
133Dic83—21no—
134Dic83—21no—
135Dic83—41no—
136Dic83—42no—
137Dic83C22⋊Q812yes—
138C2×C83—42no—
139C2×C83C2×C22⋊Q811yescharacteristic
140C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C83#57511yescharacteristic
142D83—22no—
143D83—42no—
144#3077—831nomaximal, Sylow
145C83#307711yescharacteristic, complemented, Sylow
146#533—831no—
147C2×C4×D4—831no—
148#478—832no—
149#478—832no—
150#476—831no—
151#476—831no—
152#474—831no—
153#474D8311yescharacteristic, complemented
154#428—832no—
155#423—831no—
156#418—832no—
157C22×D4—831no—
158C23×C4—1661no—
159C2×M16—832no—
160C4⋊Q8—831no—
161C4⋊Q8C2×D8311yescharacteristic, complemented
162#125—832no—
163C22⋊Q8—1661no—
164C22⋊Q8—1661no—
165C4×Q8—1661no—
166C4×Q8—1661no—
167C4×D4—831no—
168C4×D4—831no—
169C4×D4—831no—
170C4×D4—1661no—
171C4×D4—1661no—
172C4×D4C2×D8311yescharacteristic
173#117—832no—
174C2×C4⋊C4—831no—
175C2×C22⋊C4—1661no—
176C2×C42—831no—
177#107—832no—
178#107C2×D8312yescomplemented
179#106—832no—
180#106—832no—
181C4⋊C8—832no—
182C4⋊C8—831no—
183C4⋊C8C2×D8311yescharacteristic, complemented
184#102—832no—
185#102—832no—
186#102—832no—
187#102C2×D8312yes—
188C24—1661no—
189C2×Q8—21no—
190C2×Q8—1661no—
191C2×D4—834no—
192C2×D4—831no—
193C2×D4—1662no—
194C2×D4—1661no—
195C2×D4C22×D8311yescharacteristic
196C22×C4—21no—
197C22×C4—3324no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—831no—
201C22×C4—1661no—
202C22×C4—1661no—
203C22×C4—1661no—
204C22×C4—1661no—
205M16—1664no—
206C2×C8—832no—
207C2×C8C22×D8312yes—
208C4⋊C4—21no—
209C4⋊C4—832no—
210C4⋊C4—831no—
211C4⋊C4—1661no—
212C4⋊C4—1662no—
213C4⋊C4—1662no—
214C4⋊C4—1661no—
215C4⋊C4C22×D8312yes—
216C4⋊C4C22×D8311yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1661no—
219C22⋊C4—1662no—
220C22⋊C4—1661no—
221C22⋊C4—1661no—
222C42—831no—
223C42—831no—
224C42—831no—
225C42—1662no—
226C42C22×D8311yescharacteristic
227C23—21no—
228C23—3322no—
229C23—3322no—
230C23—831no—
231C23—1664no—
232C23—1661no—
233Q8—42no—
234Q8—3322no—
235D4—21no—
236D4—832no—
237D4—1664no—
238D4—1661no—
239D4Q8×D8312yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—3321no—
244C2×C4—3322no—
245C2×C4—3322no—
246C2×C4—3322no—
247C2×C4—3322no—
248C2×C4—831no—
249C2×C4—832no—
250C2×C4—832no—
251C2×C4—831no—
252C2×C4—831no—
253C2×C4—1662no—
254C2×C4—1661no—
255C2×C4—1662no—
256C2×C4D4×D8311yescharacteristic
257C2×C4D4×D8311yescharacteristic
258C2×C4C23×D8311yescharacteristic
259C8—22no—
260C8—1662no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—3328no—
265C22—3321no—
266C22—3321no—
267C22—832no—
268C22—1662no—
269C22—1661no—
270C22—1661no—
271C22C2×D4×D8311yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—3321no—
276C4—3322no—
277C4—832no—
278C4—1661no—
279C4—1661no—
280C4#1458181612yes—
281C4C2×Q8×D8311yescharacteristic
282C4#41×D8311yescharacteristic
283C2—42no—
284C2—3322no—
285C2—1662no—
286C2C22⋊Q8×D8311yescharacteristic
287C2C2×#1458181611yescharacteristic
288C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
hypercenter#474D83
hypocenterC83#3077
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128004C2semidirect
4. #19128003C2semidirect
5. #19127992C2semidirect
6. #19127991C2semidirect
7. #19127978C2semidirect
8. #19127742C2semidirect
9. #19127723C2semidirect
11. #474×C83C2semidirect
15. #14581792C22semidirect
16. #14581783C22semidirect
30. #14581725C22semidirect
31. #14581724C22semidirect
32. #14581723C22semidirect
33. C4⋊(C83⋊C8)C22semidirect
34. Dic83⋊C8C22semidirect
38. C4⋊Q8×C83C22semidirect
40. #107×C83C22semidirect
41. C4⋊C8×C83C22semidirect
48. D4×D83Q8semidirect
145. C83#3077semidirect
153. #474D83semidirect
161. C4⋊Q8C2×D83semidirect
178. #107C2×D83semidirect
183. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
C4×D4×D832
#191280042
#191280032
#191279922
#191279912
#191279782
#191277422
#191277232
#191277042
#474×C832
#307783