Finite Group Databasebeta

← #469×D83 #24917055

Subgroups — #469×D83 #24917055

Order 10624.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D83C211yesmaximal, characteristic, complemented
2C2×M16×D83C211yesmaximal, characteristic, complemented
3#121×D83C212yesmaximal, complemented
4#19127955C211yesmaximal, characteristic, complemented
5#19127954C211yesmaximal, characteristic, complemented
6#19127953C212yesmaximal, complemented
7#19127952C212yesmaximal, complemented
8#19127951C211yesmaximal, characteristic, complemented
9#106×D83C211yesmaximal, characteristic, complemented
10#102×D83C212yesmaximal, complemented
11#469×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—22no—
13C2×D4×D83C2212yes—
14C22×C4×D83C2211yescharacteristic
15C2×D664C2211yescharacteristic, complemented
16C2×#13050575C2212yescomplemented
17D8×D83—24no—
18C2×#13050574C2211yescharacteristic, complemented
19C2×#13050573C2211yescharacteristic, complemented
20C2×#13050573C2211yescharacteristic, complemented
21M16×D83—24no—
22C2×C8×D83C2211yescharacteristic
23C2×C8×D83C2211yescharacteristic
24#14581767C2212yescomplemented
25#14581766C2212yescomplemented
26#14581765—22no—
27#14581764—22no—
28#14581763C2212yescomplemented
29C4⋊C4×D83C2212yes—
30C22⋊C4×D83—22no—
31#14581724C2211yescharacteristic, complemented
32#14581723C2212yescomplemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36C2×D8×C83C2211yescharacteristic, complemented
37C2×M16×C83C2211yescharacteristic, complemented
38#121×C83C2212yescomplemented
39#106×C83C2211yescharacteristic, complemented
40#102×C83C2212yescomplemented
41C23×D83—22no—
42C23×D83D411yescharacteristic
43C2×D332—22no—
44C2×D332C2312yes—
45C2×C22⋊D83—22no—
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48D4×D83—24no—
49D4×D83—24no—
50D4×D83—48no—
51C22×Dic83D411yescharacteristic
52C2×C4×D83—22no—
53C2×C4×D83—22no—
54C2×C4×D83—22no—
55C2×C4×D83D411yescharacteristic
56C2×C4×D83D411yescharacteristic
57C2×C4×D83C2311yescharacteristic
58C2×C4×D83C2311yescharacteristic
59D664—22no—
60#13050575—24no—
61#13050574—22no—
62#13050573—22no—
63#13050573—22no—
64C2×C83⋊C8C2311yescharacteristic
65C2×C83⋊C8C2311yescharacteristic
66C8×D83—22no—
67C8×D83D414yescomplemented
68C4⋊Dic83C2312yes—
69Dic83⋊C4—22no—
70C22⋊Dic83—22no—
71(C2×D83)⋊C4—22no—
72(C2×D83)⋊C4—22no—
73C2×D4×C83—22no—
74C2×D4×C83C2312yes—
75C22×C4×C83C2311yescharacteristic
76D8×C83—22no—
77M16×C83—22no—
78C2×C8×C83C2311yescharacteristic
79C2×C8×C83C2311yescharacteristic
80C4⋊C4×C83C2312yes—
81C22⋊C4×C83—22no—
82C23×C83—22no—
83C23×C83C2×D411yescharacteristic
84C22×D83—22no—
85C22×D83—24no—
86C22×D83—22no—
87C22×D83—44no—
88C22×D83—44no—
89C22×D83—44no—
90C22×D83C2×D411yescharacteristic
91C22×D83C2×D411yescharacteristic
92D4×C83—22no—
93D4×C83—22no—
94D4×C83—44no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—22no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—44no—
101C22⋊D83—44no—
102C22⋊D83—44no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107D332—22no—
108D332—22no—
109D332—44no—
110C4×D83—22no—
111C4×D83—21no—
112C4×D83—21no—
113C4×D83—22no—
114C4×D83—44no—
115C4×D83C2×D412yes—
116C4×D83#4112yes—
117C8×C83—21no—
118C8×C83C2×D412yescomplemented
119C83⋊C8—21no—
120C83⋊C8C2×D412yescomplemented
121C22×C83—21no—
122C22×C83—22no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83C22×D411yescharacteristic
127C2×D83—21no—
128C2×D83—22no—
129C2×D83—44no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—88no—
134C2×D83#12112yes—
135C2×D83AGL(1,ℤ/8)14yes—
136C4×C83—21no—
137C4×C83—42no—
138C4×C83C22×D411yescharacteristic
139C4×C83C2×#4111yescharacteristic
140Dic83—21no—
141Dic83—42no—
142Dic83#12112yes—
143C2×C83—41no—
144C2×C83—82no—
145C2×C83C2×#12111yescharacteristic
146C2×C83C2×AGL(1,ℤ/8)12yes—
147D83—41no—
148D83—82no—
149D83#46914yesdirect product factor
150C2×#469—831nomaximal, Sylow
151C83C2×#46911yescharacteristic, complemented, Sylow
152C22×D8—831no—
153C22×M16—831no—
154C2×#121—832no—
155#469—831no—
156#469—831no—
157#469—832no—
158#469—831no—
159#469—832no—
160#469D8311yescharacteristic, direct product factor
161C2×#106—831no—
162C2×#102—832no—
163C22×D4—832no—
164C22×D4—1662no—
165C23×C4—831no—
166C2×D8—832no—
167C2×D8—831no—
168C2×D8—1664no—
169C2×D8C2×D8311yescharacteristic, complemented
170C2×M16—831no—
171C2×M16—831no—
172C2×M16—831no—
173C2×M16—1664no—
174C2×M16C2×D8311yescharacteristic, complemented
175C22×C8—831no—
176C22×C8—831no—
177#121—832no—
178#121—832no—
179#121—832no—
180#121—1662no—
181#121—1662no—
182#121C2×D8312yescomplemented
183C2×C4⋊C4—832no—
184C2×C22⋊C4—1662no—
185#106—831no—
186#106—832no—
187#106C2×D8311yescharacteristic, complemented
188#102—832no—
189#102—832no—
190#102—832no—
191#102C2×D8312yescomplemented
192C24—831no—
193C24—1662no—
194C2×D4—22no—
195C2×D4—3328no—
196C2×D4—832no—
197C2×D4—1662no—
198C2×D4—1664no—
199C2×D4—1664no—
200C2×D4—1662no—
201C2×D4—1662no—
202C2×D4—1662no—
203C2×D4C22×D8312yes—
204C22×C4—831no—
205C22×C4—831no—
206C22×C4—831no—
207C22×C4—831no—
208C22×C4—831no—
209C22×C4—1662no—
210C22×C4—1662no—
211C22×C4—1662no—
212C22×C4C22×D8311yescharacteristic
213D8—22no—
214D8—1664no—
215D8—1662no—
216M16—22no—
217M16—1662no—
218M16—1662no—
219M16—1662no—
220C2×C8—831no—
221C2×C8—834no—
222C2×C8—831no—
223C2×C8—1662no—
224C2×C8C22×D8311yescharacteristic
225C2×C8C22×D8311yescharacteristic
226C4⋊C4—832no—
227C4⋊C4—1662no—
228C4⋊C4C22×D8312yes—
229C22⋊C4—22no—
230C22⋊C4—1662no—
231C22⋊C4—1662no—
232C22⋊C4—1662no—
233C23—22no—
234C23—3324no—
235C23—3324no—
236C23—3324no—
237C23—831no—
238C23—831no—
239C23—1662no—
240C23—1664no—
241C23—1662no—
242C23D4×D8311yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—3324no—
247D4—3324no—
248D4—3324no—
249D4—3324no—
250D4—1662no—
251D4—1662no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—3324no—
256C2×C4—831no—
257C2×C4—832no—
258C2×C4—832no—
259C2×C4—831no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1661no—
265C2×C4—1661no—
266C2×C4D4×D8311yescharacteristic
267C2×C4C23×D8311yescharacteristic
268C8—21no—
269C8—832no—
270C8—1661no—
271C8D4×D8312yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—3324no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—832no—
282C22—834no—
283C22—6648no—
284C22—1661no—
285C22—1662no—
286C22C2×D4×D8311yescharacteristic
287C4—21no—
288C4—42no—
289C4—3322no—
290C4—832no—
291C4—1661no—
292C4C2×D4×D8311yescharacteristic
293C4#41×D8311yescharacteristic
294C2—41no—
295C2—82no—
296C2—3321no—
297C2—834no—
298C2—6642no—
299C2#121×D8311yescharacteristic
300C2AGL(1,ℤ/8)×D8312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#469×C83C2
socleC22×C83C22×D4
hypercenter#469D83
hypocenterC83C2×#469
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D83C2semidirect
2. C2×M16×D83C2semidirect
3. #121×D83C2semidirect
4. #19127955C2semidirect
5. #19127954C2semidirect
6. #19127953C2semidirect
7. #19127952C2semidirect
8. #19127951C2semidirect
9. #106×D83C2semidirect
10. #102×D83C2semidirect
11. #469×C83C2semidirect
15. C2×D664C22semidirect
16. C2×#13050575C22semidirect
18. C2×#13050574C22semidirect
19. C2×#13050573C22semidirect
20. C2×#13050573C22semidirect
24. #14581767C22semidirect
25. #14581766C22semidirect
28. #14581763C22semidirect
31. #14581724C22semidirect
32. #14581723C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
36. C2×D8×C83C22semidirect
37. C2×M16×C83C22semidirect
38. #121×C83C22semidirect
39. #106×C83C22semidirect
40. #102×C83C22semidirect
67. C8×D83D4semidirect
118. C8×C83C2×D4semidirect
120. C83⋊C8C2×D4semidirect
149. D83#469direct (N × H)
151. C83C2×#469semidirect
160. #469D83direct (N × H)
169. C2×D8C2×D83semidirect
174. C2×M16C2×D83semidirect
182. #121C2×D83semidirect
187. #106C2×D83semidirect
191. #102C2×D83semidirect
271. C8D4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D832
C2×M16×D832
#121×D832
#191279552
#191279542
#191279532
#191279522
#191279512
#106×D832
#102×D832
#469×C832
C2×#46983