Finite Group Databasebeta

← #24919154

Subgroups — #24919154

Order 10624.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D83C212yesmaximal, complemented
2Q8×D332C211yesmaximal, characteristic, complemented
3#19128336C211yesmaximal, characteristic
4#19128073C211yesmaximal, characteristic, complemented
5#19127929C212yesmaximal, complemented
6#19127907C211yesmaximal, characteristic, complemented
7#19127853C212yesmaximal, complemented
8#19127838C212yesmaximal, complemented
9C4×#13050577C211yesmaximal, characteristic, complemented
10#19127766C211yesmaximal, characteristic, complemented
11#460×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2212yescomplemented
13C2×D4×D83—21no—
14C2×D4×D83C2212yes—
15C2×#13050579C2212yes—
16C2×#13050578C2212yescomplemented
17C2×#13050578C2212yescomplemented
18C2×#13050577C2211yescharacteristic
19SD16×D83—28no—
20C2×C8×D83C2212yescomplemented
21#14581792—21no—
22#14581792C2211yescharacteristic, complemented
23#14581772—22no—
24#14581772C2212yescomplemented
25#14581763—21no—
26#14581763C2212yes—
27#14581760—22no—
28C4×D332—21no—
29C4×D332C2211yescharacteristic
30C4×D332C2211yescharacteristic
31#14581723C2211yescharacteristic, complemented
32#14581717C2212yescomplemented
33#14581716C2211yescharacteristic, complemented
34#14581712C2211yescharacteristic
35(C2×D83)⋊C8C2212yescomplemented
36C4×C83⋊C8C2211yescharacteristic, complemented
37C2×SD16×C83C2212yescomplemented
38#127×C83C2211yescharacteristic
39C4×Q8×C83C2211yescharacteristic, complemented
40C4⋊C8×C83C2211yescharacteristic, complemented
41#102×C83C2212yescomplemented
42C23×D83—22no—
43C2×Dic166—22no—
44C2×Dic166C2312yes—
45Q8×D83—24no—
46Q8×D83—42no—
47C2×D332D411yescharacteristic
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51D4×D83—24no—
52D4×D83—42no—
53D4×D83—44no—
54C2×C4×D83—22no—
55C2×C4×D83—21no—
56C2×C4×D83C2312yes—
57#13050579—24no—
58#13050578—24no—
59#13050578—42no—
60#13050577D414yescomplemented
61C2×C83⋊C8C2311yescharacteristic
62C2×C83⋊C8C2311yescharacteristic
63C8×D83—24no—
64C4⋊Dic83—21no—
65C4⋊Dic83—21no—
66C4⋊Dic83—22no—
67C4⋊Dic83D411yescharacteristic
68C4⋊Dic83C2311yescharacteristic
69C22⋊Dic83—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4D412yes—
73C2×Q8×C83C2311yescharacteristic
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83C2312yes—
77SD16×C83—24no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83—21no—
80C4⋊C4×C83C2311yescharacteristic
81C42×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—22no—
84C22×D83—21no—
85C22×D83—44no—
86C22×D83—44no—
87C22×D83—82no—
88C22×D83C2×D412yes—
89Q8×C83—21no—
90Q8×C83C2×D412yescomplemented
91D4×C83—22no—
92D4×C83—22no—
93D4×C83—42no—
94D4×C83—81no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C22⋊D83—44no—
101C22⋊D83—44no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104C2×Dic83C2×D412yes—
105D332—21no—
106D332SD1614yescomplemented
107D332C2×D412yes—
108C4×D83—22no—
109C4×D83—22no—
110C4×D83—44no—
111C4×D83—42no—
112Dic166—22no—
113Dic166—22no—
114Dic166—44no—
115C8×C83—22no—
116C83⋊C8—21no—
117C83⋊C8C2×D412yescomplemented
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83C22×D411yescharacteristic
122C2×D83—22no—
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—41no—
126C2×D83—41no—
127C2×D83—88no—
128C2×D83C2×SD1614yescomplemented
129C4×C83—22no—
130C4×C83—21no—
131C4×C83—41no—
132C4×C83C2×SD1612yescomplemented
133C4×C83C22×D411yescharacteristic
134C4×C83H(5,𝔽2)11yescharacteristic
135Dic83—22no—
136Dic83—42no—
137Dic83—41no—
138C2×C83—82no—
139C2×C83D4211yescharacteristic
140C2×C83C22×SD1611yescharacteristic
141C2×C83#57711yescharacteristic
142D83—24no—
143D83—42no—
144D4×SD16—831nomaximal, Sylow
145C83D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—832no—
147D4×Q8—831no—
148D42—831no—
149#492—831no—
150#465—832no—
151#461—831no—
152#460D8311yescharacteristic, complemented
153#450—832no—
154#448—832no—
155C4×SD16—831no—
156C8×D4—831no—
157C22×Q8—832no—
158C22×D4—832no—
159C22×D4—1661no—
160C2×SD16—832no—
161C2×SD16—831no—
162C2×SD16—832no—
163C2×SD16—832no—
164C2×SD16—1668no—
165C2×SD16C2×D8312yescomplemented
166C22×C8—832no—
167C4⋊Q8—831no—
168C4⋊Q8—1661no—
169#127C2×D8311yescharacteristic
170C22⋊Q8—832no—
171C22⋊Q8—1662no—
172#121—832no—
173#121—1661no—
174#120—1662no—
175C4×Q8C2×D8311yescharacteristic, complemented
176C4×D4—831no—
177C4×D4—831no—
178C4×D4—1661no—
179#106—831no—
180C4⋊C8C2×D8311yescharacteristic, complemented
181#103—831no—
182#103—832no—
183#102—831no—
184#102C2×D8312yescomplemented
185C22⋊C8—832no—
186C4×C8—831no—
187C24—1662no—
188C2×Q8—3322no—
189C2×Q8—832no—
190C2×Q8—1662no—
191C2×Q8—1664no—
192C2×Q8C22×D8311yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—3324no—
196C2×D4—3322no—
197C2×D4—831no—
198C2×D4—831no—
199C2×D4—1662no—
200C2×D4—1664no—
201C2×D4—1662no—
202C2×D4C22×D8312yes—
203C22×C4—832no—
204C22×C4—1662no—
205C22×C4—1661no—
206SD16—24no—
207SD16—3322no—
208SD16—834no—
209SD16—1664no—
210SD16—1664no—
211C2×C8—831no—
212C2×C8—831no—
213C2×C8—1664no—
214C2×C8C22×D8312yes—
215C4⋊C4—21no—
216C4⋊C4—831no—
217C4⋊C4—831no—
218C4⋊C4—1662no—
219C4⋊C4—1661no—
220C4⋊C4—1661no—
221C4⋊C4C22×D8311yescharacteristic
222C22⋊C4—832no—
223C22⋊C4—1662no—
224C22⋊C4—1662no—
225C22⋊C4—1661no—
226C42—21no—
227C42C22×D8311yescharacteristic
228C23—22no—
229C23—3324no—
230C23—3324no—
231C23—832no—
232C23—6642no—
233C23—1661no—
234Q8—21no—
235Q8—3324no—
236Q8—1662no—
237Q8—1662no—
238Q8D4×D8312yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3324no—
244D4—3324no—
245D4—834no—
246D4—832no—
247D4—1661no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—3324no—
251C2×C4—3322no—
252C2×C4—832no—
253C2×C4—1661no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4—1662no—
257C2×C4D4×D8311yescharacteristic
258C2×C4D4×D8311yescharacteristic
259C2×C4C23×D8311yescharacteristic
260C8—22no—
261C8—832no—
262C8—1661no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—3321no—
267C22—3321no—
268C22—834no—
269C22—6648no—
270C22—1662no—
271C22—1662no—
272C22—1662no—
273C22C2×D4×D8311yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—3322no—
278C4—3321no—
279C4—1662no—
280C4SD16×D8312yescomplemented
281C4C2×D4×D8311yescharacteristic
282C4#1458183811yescharacteristic
283C2—82no—
284C2—3322no—
285C2—1664no—
286C2#1912833511yescharacteristic
287C2C2×SD16×D8311yescharacteristic
288C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#460×C83C2
socleC22×C83C22×D4
prime radical#19128336C2
hypercenter#460D83
hypocenterC83D4×SD16
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D83C2semidirect
2. Q8×D332C2semidirect
4. #19128073C2semidirect
5. #19127929C2semidirect
6. #19127907C2semidirect
7. #19127853C2semidirect
8. #19127838C2semidirect
9. C4×#13050577C2semidirect
10. #19127766C2semidirect
11. #460×C83C2semidirect
12. C2×Q8×D83C22semidirect
16. C2×#13050578C22semidirect
17. C2×#13050578C22semidirect
20. C2×C8×D83C22semidirect
22. #14581792C22semidirect
24. #14581772C22semidirect
31. #14581723C22semidirect
32. #14581717C22semidirect
33. #14581716C22semidirect
35. (C2×D83)⋊C8C22semidirect
36. C4×C83⋊C8C22semidirect
37. C2×SD16×C83C22semidirect
39. C4×Q8×C83C22semidirect
40. C4⋊C8×C83C22semidirect
41. #102×C83C22semidirect
60. #13050577D4semidirect
90. Q8×C83C2×D4semidirect
106. D332SD16semidirect
117. C83⋊C8C2×D4semidirect
128. C2×D83C2×SD16semidirect
132. C4×C83C2×SD16semidirect
145. C83D4×SD16semidirect
152. #460D83semidirect
165. C2×SD16C2×D83semidirect
175. C4×Q8C2×D83semidirect
180. C4⋊C8C2×D83semidirect
184. #102C2×D83semidirect
238. Q8D4×D83semidirect
280. C4SD16×D83semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D832
Q8×D3322
#191283362
#191280732
#191279292
#191279072
#191278532
#191278382
C4×#130505772
#191277662
#460×C832
D4×SD1683