Finite Group Databasebeta

← #24919699

Subgroups — #24919699

Order 10624.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D4×D332C211yesmaximal, characteristic, complemented
2#19128049C211yesmaximal, characteristic, complemented
3#19128025C212yesmaximal, complemented
4#19127988C211yesmaximal, characteristic, complemented
5#19127952C212yesmaximal, complemented
6#19127896C211yesmaximal, characteristic, complemented
7#19127834C212yesmaximal, complemented
8C4×#13050575C211yesmaximal, characteristic, complemented
9#19127779C211yesmaximal, characteristic, complemented
10#19127705C212yesmaximal, complemented
11#555×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C2×D4×D83—21no—
14C2×#13050575C2211yescharacteristic
15C2×#13050575C2212yescomplemented
16C2×#13050574C2212yescomplemented
17#14581789C2211yescharacteristic
18#14581767—21no—
19#14581766C2212yes—
20#14581762—22no—
21C4×D332C2211yescharacteristic
22#14581725C2211yescharacteristic, complemented
23#14581723C2212yescomplemented
24C4⋊(C83⋊C8)C2211yescharacteristic, complemented
25#14581713C2211yescharacteristic, complemented
26#14581712C2212yes—
27#14581712C2211yescharacteristic
28#14581712C2212yescomplemented
29#14581712C2212yescomplemented
30#14581712C2212yescomplemented
31C22⋊(C83⋊C8)C2212yescomplemented
32C4×C83⋊C8C2211yescharacteristic, complemented
33C4⋊Q8×C83C2211yescharacteristic, complemented
34#124×C83—22no—
35#121×C83—21no—
36#121×C83C2212yescomplemented
37C4×D4×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic, complemented
39#117×C83—21no—
40#117×C83C2212yescomplemented
41C23×D83—22no—
42C2×D332—24no—
43C2×D332—22no—
44C2×D332—21no—
45C2×D332—21no—
46C2×D332—42no—
47C2×D332C2312yes—
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—22no—
50D4×D83—44no—
51C2×C4×D83—21no—
52#13050575—22no—
53#13050575—42no—
54#13050574—42no—
55C2×C83⋊C8C2311yescharacteristic
56C2×C83⋊C8C2312yes—
57C2×C83⋊C8C2311yescharacteristic
58C4⋊Dic83C2311yescharacteristic
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—22no—
61C2×Q8×C83—21no—
62C2×D4×C83—22no—
63C2×D4×C83D411yescharacteristic
64C2×D4×C83C2311yescharacteristic
65C22×C4×C83—21no—
66C22×C4×C83C2312yes—
67C4⋊C4×C83—21no—
68C4⋊C4×C83D411yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2312yes—
71C22⋊C4×C83—22no—
72C22⋊C4×C83—22no—
73C22⋊C4×C83—21no—
74C22⋊C4×C83D412yes—
75C42×C83—22no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83C2×D412yes—
79C22×D83—21no—
80C22×D83—22no—
81C22×D83—44no—
82C22×D83—44no—
83C22×D83—82no—
84Q8×C83—81no—
85D4×C83—22no—
86D4×C83—21no—
87D4×C83—21no—
88D4×C83—82no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—42no—
94C2×C4×C83C2×D412yes—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—44no—
99C2×Dic83—21no—
100D332—22no—
101D332—21no—
102D332—22no—
103D332—44no—
104D332—42no—
105D332—81no—
106D332#4112yes—
107C4×D83—42no—
108C83⋊C8—21no—
109C83⋊C8—21no—
110C83⋊C8—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83—41no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83C22×D411yescharacteristic
118C2×D83—22no—
119C2×D83—41no—
120C2×D83—41no—
121C2×D83—42no—
122C2×D83—42no—
123C2×D83—42no—
124C2×D83—88no—
125C4×C83—22no—
126C4×C83—21no—
127C4×C83—41no—
128C4×C83—42no—
129C4×C83AGL(1,ℤ/8)12yes—
130C4×C83C2×#4111yescharacteristic
131C4×C83H(5,𝔽2)11yescharacteristic
132Dic83—41no—
133C2×C83—42no—
134C2×C83—81no—
135C2×C83#54611yescharacteristic
136C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C83#57611yescharacteristic
138D83—42no—
139D83—82no—
140#3325—831nomaximal, Sylow
141C83#332511yescharacteristic, complemented, Sylow
142#555D8311yescharacteristic, complemented
143D42—831no—
144#486—831no—
145#482—832no—
146#476—831no—
147#469—832no—
148#459—831no—
149#447—832no—
150C4×D8—831no—
151#436—831no—
152#418—832no—
153C22×D4—832no—
154C22×D4—1661no—
155C2×D8—832no—
156C2×D8—831no—
157C2×M16—832no—
158C4⋊Q8C2×D8311yescharacteristic, complemented
159#127—831no—
160#124—22no—
161#121—21no—
162#121—832no—
163#121—1661no—
164#121C2×D8312yescomplemented
165#120—1662no—
166C4×D4—831no—
167C4×D4C2×D8311yescharacteristic
168C4×D4C2×D8311yescharacteristic, complemented
169#117—21no—
170#117C2×D8312yescomplemented
171#107—831no—
172#106—832no—
173C4⋊C8—831no—
174#102—831no—
175#102—831no—
176#102—832no—
177#102—832no—
178#102—832no—
179#102—832no—
180C22⋊C8—832no—
181C4×C8—831no—
182C24—1662no—
183C2×Q8—21no—
184C2×D4—22no—
185C2×D4—3322no—
186C2×D4—3324no—
187C2×D4—831no—
188C2×D4—832no—
189C2×D4—1664no—
190C2×D4—1662no—
191C2×D4—1662no—
192C2×D4—1661no—
193C2×D4—1661no—
194C2×D4C22⋊D8311yescharacteristic
195C2×D4C22×D8311yescharacteristic
196C22×C4—21no—
197C22×C4—1661no—
198C22×C4C22×D8312yes—
199D8—3322no—
200D8—1662no—
201M16—3322no—
202C2×C8—831no—
203C2×C8—831no—
204C2×C8—832no—
205C4⋊C4—21no—
206C4⋊C4—831no—
207C4⋊C4C22⋊D8311yescharacteristic
208C4⋊C4C22×D8312yes—
209C4⋊C4C22×D8311yescharacteristic
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—21no—
213C22⋊C4—1661no—
214C22⋊C4—1662no—
215C22⋊C4C22⋊D8312yes—
216C42—22no—
217C42C22×D8311yescharacteristic
218C23—21no—
219C23—3324no—
220C23—3324no—
221C23—6642no—
222C23—1662no—
223C23—1661no—
224C23C2×C22⋊D8312yes—
225Q8—81no—
226D4—22no—
227D4—21no—
228D4—21no—
229D4—82no—
230D4—3324no—
231D4—3324no—
232D4—3322no—
233D4—832no—
234D4—6641no—
235D4—1662no—
236D4—1662no—
237D4—1661no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—3322no—
244C2×C4—1661no—
245C2×C4C2×C22⋊D8312yes—
246C2×C4C2×C22⋊D8311yescharacteristic
247C2×C4C2×C22⋊D8311yescharacteristic
248C2×C4C23×D8311yescharacteristic
249C8—1661no—
250C8—1661no—
251C8—1662no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—3321no—
259C22—3321no—
260C22—3322no—
261C22—3322no—
262C22—3322no—
263C22—6648no—
264C22—1662no—
265C22C22×C22⋊D8311yescharacteristic
266C4—22no—
267C4—21no—
268C4—41no—
269C4—42no—
270C4—3321no—
271C4#1458181712yes—
272C4#41×D8311yescharacteristic
273C4#1458183811yescharacteristic
274C2—42no—
275C2—81no—
276C2—3322no—
277C2—6642no—
278C2#1912833911yescharacteristic
279C2C2×#1458181711yescharacteristic
280C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#555×C83C2
socleC22×C83C22×D4
hypercenter#555D83
hypocenterC83#3325
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. D4×D332C2semidirect
2. #19128049C2semidirect
3. #19128025C2semidirect
4. #19127988C2semidirect
5. #19127952C2semidirect
6. #19127896C2semidirect
7. #19127834C2semidirect
8. C4×#13050575C2semidirect
9. #19127779C2semidirect
10. #19127705C2semidirect
11. #555×C83C2semidirect
15. C2×#13050575C22semidirect
16. C2×#13050574C22semidirect
22. #14581725C22semidirect
23. #14581723C22semidirect
24. C4⋊(C83⋊C8)C22semidirect
25. #14581713C22semidirect
28. #14581712C22semidirect
29. #14581712C22semidirect
30. #14581712C22semidirect
31. C22⋊(C83⋊C8)C22semidirect
32. C4×C83⋊C8C22semidirect
33. C4⋊Q8×C83C22semidirect
36. #121×C83C22semidirect
38. C4×D4×C83C22semidirect
40. #117×C83C22semidirect
141. C83#3325semidirect
142. #555D83semidirect
158. C4⋊Q8C2×D83semidirect
164. #121C2×D83semidirect
168. C4×D4C2×D83semidirect
170. #117C2×D83semidirect

Maximal subgroups

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

subgroupindex
D4×D3322
#191280492
#191280252
#191279882
#191279522
#191278962
#191278342
C4×#130505752
#191277792
#191277052
#555×C832
#332583