Finite Group Databasebeta

← #24919708

Subgroups — #24919708

Order 10624.

460 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128416C211yesmaximal, characteristic, complemented
2#19128336C211yesmaximal, characteristic, complemented
3#19128047C211yesmaximal, characteristic, complemented
4#19128024C212yesmaximal, complemented
5#19127992C211yesmaximal, characteristic, complemented
6#19127951C212yesmaximal, complemented
7#19127833C212yesmaximal, complemented
8C4×#13050575C211yesmaximal, characteristic, complemented
9#19127778C211yesmaximal, characteristic, complemented
10#19127704C212yesmaximal, complemented
11#459×C83C211yesmaximal, characteristic, complemented
12C2×D4×D83—21no—
13C2×D4×D83C2212yes—
14C2×#13050575C2211yescharacteristic
15C2×#13050573C2212yescomplemented
16#14581792C2211yescharacteristic, complemented
17#14581778—22no—
18#14581766—21no—
19#14581763—21no—
20#14581763C2212yes—
21#14581763C2212yescomplemented
22#14581760—22no—
23C4×D332C2211yescharacteristic
24C4×D332C2211yescharacteristic
25#14581751—21no—
26#14581750C2212yescomplemented
27#14581725C2211yescharacteristic, complemented
28#14581724C2212yescomplemented
29#14581714C2212yes—
30#14581713C2212yescomplemented
31#14581713C2211yescharacteristic, complemented
32#14581713C2212yescomplemented
33#14581712C2211yescharacteristic
34(C2×D83)⋊C8C2212yescomplemented
35C4×C83⋊C8C2211yescharacteristic, complemented
36C2×D8×C83C2212yescomplemented
37#127×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic, complemented
39C4⋊C8×C83C2211yescharacteristic, complemented
40#102×C83C2212yescomplemented
41C23×D83—22no—
42C2×Dic166—21no—
43C2×D332D411yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—22no—
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48D4×D83—24no—
49D4×D83—42no—
50D4×D83—44no—
51C2×C4×D83—21no—
52C2×C4×D83C2312yes—
53#13050575—22no—
54#13050573—42no—
55C2×C83⋊C8C2311yescharacteristic
56C2×C83⋊C8C2311yescharacteristic
57C4⋊Dic83—21no—
58C4⋊Dic83D411yescharacteristic
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2312yes—
61C22⋊Dic83—22no—
62C22⋊Dic83—22no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4D412yes—
66C4×Dic83—22no—
67C2×D4×C83—21no—
68C2×D4×C83—21no—
69C2×D4×C83C2312yes—
70C2×D4×C83C2311yescharacteristic
71C22×C4×C83—21no—
72D8×C83—42no—
73C2×C8×C83C2312yes—
74C4⋊C4×C83C2311yescharacteristic
75C22⋊C4×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83—22no—
79C22×D83—21no—
80C22×D83—44no—
81C22×D83—44no—
82C22×D83—82no—
83C22×D83C2×D412yes—
84D4×C83—21no—
85D4×C83—22no—
86D4×C83—21no—
87D4×C83—22no—
88D4×C83—42no—
89D4×C83—81no—
90C2×C4×C83—21no—
91C2×C4×C83—42no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2411yescharacteristic
95C22⋊D83—44no—
96C22⋊D83—44no—
97C22⋊D83—82no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83C2×D412yes—
101D332—22no—
102D332—21no—
103D332#4112yes—
104C4×D83—22no—
105C4×D83—22no—
106C4×D83—42no—
107Dic166—81no—
108C8×C83—22no—
109C83⋊C8—21no—
110C83⋊C8—21no—
111C22×C83—42no—
112C22×C83—41no—
113C22×C83—42no—
114C22×C83—41no—
115C22×C83—41no—
116C22×C83—42no—
117C22×C83C22×D411yescharacteristic
118C2×D83—22no—
119C2×D83—22no—
120C2×D83—22no—
121C2×D83—22no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—88no—
125C4×C83—21no—
126C4×C83—41no—
127C4×C83AGL(1,ℤ/8)12yes—
128C4×C83C2×#4111yescharacteristic
129C4×C83H(5,𝔽2)11yescharacteristic
130Dic83—22no—
131Dic83—41no—
132Dic83—42no—
133C2×C83—82no—
134C2×C83—81no—
135C2×C83#54611yescharacteristic
136C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C83#57611yescharacteristic
138D83—42no—
139D83—42no—
140#3325—831nomaximal, Sylow
141C83#332511yescharacteristic, complemented, Sylow
142#555—831no—
143D42—831no—
144#486—831no—
145#482—832no—
146#476—831no—
147#469—832no—
148#459D8311yescharacteristic, complemented
149#447—832no—
150C4×D8—831no—
151#436—831no—
152#418—832no—
153C22×D4—832no—
154C22×D4—1661no—
155C2×D8—831no—
156C2×D8C2×D8312yescomplemented
157C2×M16—832no—
158C4⋊Q8—831no—
159#127C2×D8311yescharacteristic
160#124—1662no—
161#121—832no—
162#121—832no—
163#121—1661no—
164#121—1661no—
165#120—1662no—
166C4×D4—831no—
167C4×D4—831no—
168C4×D4C2×D8311yescharacteristic, complemented
169#117—832no—
170#117—1661no—
171#107—831no—
172#106—832no—
173C4⋊C8C2×D8311yescharacteristic, complemented
174#102—832no—
175#102—831no—
176#102—832no—
177#102—832no—
178#102—831no—
179#102C2×D8312yescomplemented
180C22⋊C8—832no—
181C4×C8—831no—
182C24—1662no—
183C2×Q8—1661no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—3324no—
187C2×D4—3322no—
188C2×D4—831no—
189C2×D4—831no—
190C2×D4—1662no—
191C2×D4—1664no—
192C2×D4—1662no—
193C2×D4—1662no—
194C2×D4C22×D8312yes—
195C2×D4C22×D8311yescharacteristic
196C22×C4—21no—
197C22×C4—832no—
198C22×C4—1661no—
199D8—42no—
200D8—1662no—
201M16—3322no—
202C2×C8—831no—
203C2×C8—831no—
204C2×C8C22×D8312yes—
205C4⋊C4—831no—
206C4⋊C4—832no—
207C4⋊C4—831no—
208C4⋊C4—1661no—
209C4⋊C4C22×D8311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—832no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C22⋊C4—1661no—
216C42—1662no—
217C42C22×D8311yescharacteristic
218C23—21no—
219C23—22no—
220C23—3324no—
221C23—3324no—
222C23—832no—
223C23—6642no—
224C23—1661no—
225Q8—6641no—
226D4—21no—
227D4—22no—
228D4—21no—
229D4—22no—
230D4—42no—
231D4—81no—
232D4—3324no—
233D4—3324no—
234D4—832no—
235D4—6642no—
236D4—1662no—
237D4—1661no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—3322no—
241C2×C4—832no—
242C2×C4—1662no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1662no—
246C2×C4D4×D8311yescharacteristic
247C2×C4D4×D8311yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—22no—
250C8—1661no—
251C8—1661no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—3321no—
259C22—3321no—
260C22—6648no—
261C22—1662no—
262C22—1662no—
263C22—1662no—
264C22—1662no—
265C22C2×D4×D8311yescharacteristic
266C4—21no—
267C4—41no—
268C4—3321no—
269C4—3322no—
270C4—1662no—
271C4#1458181612yes—
272C4#41×D8311yescharacteristic
273C4#1458183711yescharacteristic
274C2—82no—
275C2—81no—
276C2—3322no—
277C2—3322no—
278C2#1912834811yescharacteristic
279C2C2×#1458181611yescharacteristic
280C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#459×C83C2
socleC22×C83C22×D4
hypercenter#459D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #19128416C2semidirect
2. #19128336C2semidirect
3. #19128047C2semidirect
4. #19128024C2semidirect
5. #19127992C2semidirect
6. #19127951C2semidirect
7. #19127833C2semidirect
8. C4×#13050575C2semidirect
9. #19127778C2semidirect
10. #19127704C2semidirect
11. #459×C83C2semidirect
15. C2×#13050573C22semidirect
16. #14581792C22semidirect
21. #14581763C22semidirect
26. #14581750C22semidirect
27. #14581725C22semidirect
28. #14581724C22semidirect
30. #14581713C22semidirect
31. #14581713C22semidirect
32. #14581713C22semidirect
34. (C2×D83)⋊C8C22semidirect
35. C4×C83⋊C8C22semidirect
36. C2×D8×C83C22semidirect
38. C4×D4×C83C22semidirect
39. C4⋊C8×C83C22semidirect
40. #102×C83C22semidirect
141. C83#3325semidirect
148. #459D83semidirect
156. C2×D8C2×D83semidirect
168. C4×D4C2×D83semidirect
173. C4⋊C8C2×D83semidirect
179. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284162
#191283362
#191280472
#191280242
#191279922
#191279512
#191278332
C4×#130505752
#191277782
#191277042
#459×C832
#332583