Finite Group Databasebeta

← #24918451

Subgroups — #24918451

Order 10624.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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#19128252C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic, complemented
3#19128091C211yesmaximal, characteristic, complemented
4#19128090C211yesmaximal, characteristic, complemented
5#19128063C211yesmaximal, characteristic, complemented
6#19127919C212yesmaximal, complemented
7#19127915C212yesmaximal, complemented
8#19127841C212yesmaximal, complemented
9#19127836C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic, complemented
11#489×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—21no—
13C2×Q8×D83C2211yescharacteristic
14C2×D4×D83C2211yescharacteristic
15C2×#13050579C2212yescomplemented
16C2×#13050577C2212yescomplemented
17C2×D664C2212yes—
18C2×#13050575C2212yes—
19#14581789C2211yescharacteristic, complemented
20#14581781—21no—
21#14581781C2211yescharacteristic
22#14581780C2211yescharacteristic
23#14581779C2211yescharacteristic
24#14581778—21no—
25#14581778—21no—
26#14581767—21no—
27#14581767C2212yes—
28#14581754—21no—
29#14581754C2212yescomplemented
30C22⋊C4×D83—21no—
31C22⋊C4×D83—21no—
32C42×D83C2211yescharacteristic
33Dic83⋊C8C2212yescomplemented
34#14581715C2212yescomplemented
35#14581712C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic, complemented
38C4⋊Q8×C83C2211yescharacteristic, complemented
39#124×C83C2211yescharacteristic
40#103×C83C2212yescomplemented
41#102×C83C2212yes—
42#97×C83C2211yescharacteristic, complemented
43C23×D83—21no—
44#13050591—44no—
45C2×Dic166—21no—
46C2×Dic166C2311yescharacteristic
47Q8×D83—42no—
48C2×D332—22no—
49C2×D332—21no—
50C2×D332—21no—
51C2×D332C2312yes—
52C2×D332C2311yescharacteristic
53C2×C22⋊D83—21no—
54D4×D83—42no—
55C2×C4×D83—21no—
56C2×C4×D83—22no—
57C2×C4×D83D411yescharacteristic
58C2×C4×D83D411yescharacteristic
59C2×C4×D83C2311yescharacteristic
60#13050579—42no—
61#13050577—42no—
62D664—42no—
63#13050575—42no—
64C2×C83⋊C8C2312yes—
65C22⋊Dic83—21no—
66C22⋊Dic83—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73C4×Dic83D411yescharacteristic
74C4×Dic83D411yescharacteristic
75C4×Dic83C2311yescharacteristic
76C2×Q8×C83—21no—
77C2×Q8×C83C2311yescharacteristic
78C2×D4×C83C2311yescharacteristic
79C2×C8×C83C2312yes—
80C4⋊C4×C83—21no—
81C4⋊C4×C83C2312yes—
82C22⋊C4×C83—21no—
83C22⋊C4×C83—21no—
84C42×C83C2311yescharacteristic
85C23×C83—21no—
86C22×D83—21no—
87C22×D83—22no—
88C22×D83—42no—
89C22×D83—42no—
90C22×D83—42no—
91C22×D83C2×D411yescharacteristic
92Q8×C83—22no—
93Q8×C83—42no—
94D4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—42no—
101C2×Dic83—21no—
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104C2×Dic83C2×D411yescharacteristic
105D332—22no—
106D332—22no—
107D332—22no—
108D332—42no—
109D332—42no—
110D332—44no—
111C4×D83—22no—
112C4×D83—22no—
113C4×D83—41no—
114C4×D83—44no—
115C4×D83—42no—
116Dic166—22no—
117Dic166—42no—
118C8×C83—22no—
119C83⋊C8—22no—
120C22×C83—41no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83C22×D411yescharacteristic
124C2×D83—21no—
125C2×D83—21no—
126C2×D83—41no—
127C2×D83—42no—
128C2×D83—42no—
129C2×D83—41no—
130C2×D83—42no—
131C2×D83—41no—
132C2×D83—84no—
133C2×D83AGL(1,ℤ/8)12yes—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83—42no—
138C4×C83H(5,𝔽2)12yes—
139Dic83—21no—
140Dic83—21no—
141Dic83—21no—
142Dic83—41no—
143C2×C83—81no—
144C2×C83#53511yescharacteristic
145C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C83#57511yescharacteristic
147D83—22no—
148D83—82no—
149D83—81no—
150#3213—831nomaximal, Sylow
151C83#321311yescharacteristic, complemented, Sylow
152#532—831no—
153C2×#124—831no—
154#496—831no—
155#496—831no—
156#489—831no—
157#489D8311yescharacteristic, complemented
158#463—832no—
159#463—832no—
160#448—832no—
161#447—832no—
162#432—831no—
163C2×#41—1661no—
164C22×Q8—831no—
165C22×D4—831no—
166C2×SD16—832no—
167C2×SD16—832no—
168C2×D8—832no—
169C2×D8—832no—
170C4⋊Q8C2×D8311yescharacteristic, complemented
171#127—831no—
172#124—831no—
173#124—831no—
174#124—831no—
175#124—1661no—
176#124—1661no—
177#124—1661no—
178#124C2×D8311yescharacteristic
179#121—832no—
180#121—1661no—
181C4×D4—832no—
182C4×D4—1661no—
183C2×C22⋊C4—1661no—
184C2×C22⋊C4—1661no—
185C2×C42—831no—
186C4⋊C8—832no—
187#103—832no—
188#103C2×D8312yescomplemented
189#102—832no—
190#102C2×D8312yes—
191C22⋊C8—832no—
192#97—831no—
193#97C2×D8311yescharacteristic, complemented
194C24—1661no—
195#41—3324no—
196C2×Q8—21no—
197C2×Q8—3322no—
198C2×Q8—831no—
199C2×Q8—1661no—
200C2×Q8C22×D8311yescharacteristic
201C2×D4—3322no—
202C2×D4—831no—
203C2×D4—832no—
204C2×D4—1662no—
205C2×D4—1661no—
206C2×D4—1661no—
207C2×D4—1661no—
208C2×D4C22×D8311yescharacteristic
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—831no—
212C22×C4—1662no—
213C22×C4—1661no—
214SD16—3322no—
215SD16—3322no—
216D8—3322no—
217D8—3322no—
218C2×C8—832no—
219C2×C8C22×D8312yes—
220C4⋊C4—21no—
221C4⋊C4C22×D8312yes—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—1661no—
225C22⋊C4—1661no—
226C22⋊C4—1662no—
227C22⋊C4—1662no—
228C22⋊C4—1661no—
229C22⋊C4—1661no—
230C22⋊C4—1661no—
231C22⋊C4—1661no—
232C42—831no—
233C42—831no—
234C42—831no—
235C42C22×D8311yescharacteristic
236C23—21no—
237C23—3322no—
238C23—3322no—
239C23—3322no—
240C23—831no—
241C23—1661no—
242C23—1662no—
243Q8—22no—
244Q8—42no—
245Q8—3322no—
246Q8—1662no—
247D4—22no—
248D4—3322no—
249D4—3322no—
250D4—3324no—
251D4—3322no—
252D4—1662no—
253D4—1662no—
254D4—1662no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—3321no—
258C2×C4—3322no—
259C2×C4—3324no—
260C2×C4—831no—
261C2×C4—831no—
262C2×C4—831no—
263C2×C4—1662no—
264C2×C4—1661no—
265C2×C4—1662no—
266C2×C4D4×D8311yescharacteristic
267C2×C4D4×D8311yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—22no—
270C8—1662no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3322no—
275C22—3321no—
276C22—3322no—
277C22—3321no—
278C22—3321no—
279C22—3322no—
280C22—832no—
281C22—6644no—
282C22—1661no—
283C22—1661no—
284C22C2×D4×D8311yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—3321no—
290C4—1661no—
291C4—1661no—
292C4—1661no—
293C4#1458183812yes—
294C2—81no—
295C2—6641no—
296C2—6642no—
297C2—1662no—
298C2#1912826911yescharacteristic
299C2AGL(1,ℤ/8)×D8311yescharacteristic
300C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#489×C83C2
socleC22×C83C22×D4
hypercenter#489D83
hypocenterC83#3213
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #19128252C2semidirect
2. #124×D83C2semidirect
3. #19128091C2semidirect
4. #19128090C2semidirect
5. #19128063C2semidirect
6. #19127919C2semidirect
7. #19127915C2semidirect
8. #19127841C2semidirect
9. #19127836C2semidirect
10. #19127756C2semidirect
11. #489×C83C2semidirect
15. C2×#13050579C22semidirect
16. C2×#13050577C22semidirect
19. #14581789C22semidirect
29. #14581754C22semidirect
33. Dic83⋊C8C22semidirect
34. #14581715C22semidirect
37. #14581703C22semidirect
38. C4⋊Q8×C83C22semidirect
40. #103×C83C22semidirect
42. #97×C83C22semidirect
151. C83#3213semidirect
157. #489D83semidirect
170. C4⋊Q8C2×D83semidirect
188. #103C2×D83semidirect
193. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282522
#124×D832
#191280912
#191280902
#191280632
#191279192
#191279152
#191278412
#191278362
#191277562
#489×C832
#321383