Finite Group Databasebeta

← #24918402

Subgroups — #24918402

Order 10624.

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

normal subgroups only

This group has 488 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
1#19128253C211yesmaximal, characteristic, complemented
2C2×#14581788C211yesmaximal, characteristic
3#19128072C212yesmaximal, complemented
4#19128046C212yesmaximal, complemented
5#19127900C212yesmaximal, complemented
6#19127899C212yesmaximal, complemented
7#19127854C211yesmaximal, characteristic, complemented
8#19127852C211yesmaximal, characteristic, complemented
9#19127851C211yesmaximal, characteristic, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#450×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83—21no—
14C22×C22⋊D83—21no—
15C2×#13050579C2212yescomplemented
16C2×#13050578C2212yescomplemented
17C2×#13050577C2212yescomplemented
18#14581791C2211yescharacteristic, complemented
19#14581790C2211yescharacteristic, complemented
20#14581788—21no—
21#14581788—21no—
22#14581788C2212yes—
23#14581788C2212yes—
24#14581782—21no—
25#14581771C2211yescharacteristic
26#14581769—21no—
27Q8×Dic83C2211yescharacteristic, complemented
28#14581757—21no—
29#14581757C2211yescharacteristic, complemented
30#14581748—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581714C2212yescomplemented
34#14581713C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×SD16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#102×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×D332—21no—
47C2×D332—42no—
48C2×D332C2312yes—
49C2×C22⋊D83—22no—
50C2×C22⋊D83—22no—
51C2×C22⋊D83—22no—
52C2×C22⋊D83—22no—
53C2×C22⋊D83—44no—
54C2×C22⋊D83—44no—
55C22×Dic83D411yescharacteristic
56C22×Dic83D411yescharacteristic
57#13050579—42no—
58#13050578—42no—
59#13050577—42no—
60C2×C83⋊C8C2312yes—
61C4⋊Dic83—21no—
62C4⋊Dic83C2311yescharacteristic
63Dic83⋊C4—21no—
64Dic83⋊C4—21no—
65Dic83⋊C4—21no—
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68C22⋊Dic83—21no—
69C4×Dic83—21no—
70C4×Dic83—21no—
71C4×Dic83D411yescharacteristic
72C4×Dic83D411yescharacteristic
73C4×Dic83C2311yescharacteristic
74C4×Dic83C2311yescharacteristic
75C24×C83—21no—
76C2×Q8×C83C2311yescharacteristic
77C2×D4×C83—21no—
78C2×D4×C83—42no—
79C2×D4×C83C2312yes—
80C22×C4×C83C2311yescharacteristic
81SD16×C83—42no—
82C2×C8×C83C2312yes—
83C4⋊C4×C83—21no—
84C4⋊C4×C83C2311yescharacteristic
85C22⋊C4×C83—21no—
86C23×C83—22no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83—42no—
90C23×C83C2×D411yescharacteristic
91C22×D83—22no—
92C22×D83—42no—
93C22×D83—42no—
94C22×D83—42no—
95Q8×C83—22no—
96D4×C83—22no—
97D4×C83—22no—
98D4×C83—42no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—44no—
105C22⋊D83—44no—
106C22⋊D83—44no—
107C22⋊D83—44no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—21no—
111C2×Dic83SD1614yescomplemented
112C2×Dic83C2×D411yescharacteristic
113C2×Dic83C2×D411yescharacteristic
114C2×Dic83C2×D411yescharacteristic
115C2×Dic83C2×D411yescharacteristic
116D332—22no—
117D332—22no—
118D332—42no—
119Dic166—22no—
120Dic166—42no—
121C8×C83—22no—
122C83⋊C8—22no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—84no—
129C22×C83C2×SD1612yes—
130C22×C83C22×D411yescharacteristic
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—84no—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83—41no—
138C4×C83H(5,𝔽2)12yes—
139Dic83—21no—
140Dic83—41no—
141Dic83—41no—
142Dic83C2×SD1612yes—
143Dic83C2×SD1612yes—
144Dic83AGL(1,ℤ/8)12yes—
145C2×C83—22no—
146C2×C83—82no—
147C2×C83#53511yescharacteristic
148C2×C83C22×SD1611yescharacteristic
149C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
150D83—82no—
151#3208—831nomaximal, Sylow
152C83#320811yescharacteristic, complemented, Sylow
153#533—831no—
154C2×#127—831no—
155#492—832no—
156#486—832no—
157#460—832no—
158#460—832no—
159#450—831no—
160#450—831no—
161#450—831no—
162#450D8311yescharacteristic, complemented
163#431—831no—
164C22×D4—831no—
165C22×D4—1661no—
166C22×D4—1661no—
167C22×D4C2×D8311yescharacteristic
168C2×SD16—832no—
169C2×SD16—832no—
170C2×SD16—832no—
171C2×SD16C2×D8312yescomplemented
172C4⋊Q8—831no—
173C4⋊Q8—831no—
174#127—832no—
175#127—832no—
176#127—1661no—
177#127—1661no—
178#125—1661no—
179C22⋊Q8—831no—
180C22⋊Q8—1661no—
181C22⋊Q8C2×D8311yescharacteristic
182C4×Q8—831no—
183C4×Q8—831no—
184C4×Q8—1661no—
185#117—1661no—
186C2×C42—831no—
187C4⋊C8—832no—
188#102—832no—
189#102—832no—
190#102—832no—
191#102C2×D8312yescomplemented
192C22⋊C8—831no—
193C22⋊C8C2×D8311yescharacteristic
194C4×C8—832no—
195C24—21no—
196C24—1661no—
197C2×Q8—831no—
198C2×Q8—1661no—
199C2×Q8C22×D8311yescharacteristic
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—3322no—
203C2×D4—3324no—
204C2×D4—3324no—
205C2×D4—832no—
206C2×D4—1662no—
207C2×D4—1662no—
208C2×D4—1662no—
209C2×D4—1662no—
210C2×D4—1661no—
211C2×D4C22×D8312yes—
212C22×C4—831no—
213C22×C4—831no—
214C22×C4C22×D8311yescharacteristic
215SD16—42no—
216SD16—3322no—
217SD16—3322no—
218SD16—3322no—
219C2×C8—832no—
220C2×C8C22×D8312yes—
221C4⋊C4—21no—
222C4⋊C4—831no—
223C4⋊C4—1661no—
224C4⋊C4—1661no—
225C4⋊C4—1661no—
226C4⋊C4—1661no—
227C4⋊C4—1661no—
228C4⋊C4—1661no—
229C4⋊C4C22×D8311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—1661no—
232C42—831no—
233C42—831no—
234C42—831no—
235C42—831no—
236C42—1661no—
237C42—1661no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—3322no—
243C23—3322no—
244C23—3322no—
245C23—1662no—
246C23D4×D8311yescharacteristic
247Q8—22no—
248Q8—3322no—
249Q8—1662no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—3322no—
254D4—3324no—
255D4—3324no—
256D4—3324no—
257D4—3324no—
258D4—1662no—
259D4—1662no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—834no—
264C2×C4—831no—
265C2×C4—831no—
266C2×C4—831no—
267C2×C4—831no—
268C2×C4—1661no—
269C2×C4—1661no—
270C2×C4—1662no—
271C2×C4D4×D8311yescharacteristic
272C2×C4C23×D8311yescharacteristic
273C8—22no—
274C8—1662no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—3322no—
282C22—3322no—
283C22—3322no—
284C22—6644no—
285C22SD16×D8312yes—
286C22C2×D4×D8311yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—3321no—
291C4—3321no—
292C4—832no—
293C4—832no—
294C4—832no—
295C4—1661no—
296C4#1458183712yes—
297C2—22no—
298C2—82no—
299C2—6642no—
300C2#1912826611yescharacteristic
301C2C2×SD16×D8311yescharacteristic
302C2AGL(1,ℤ/8)×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581788C2
hypercenter#450D83
hypocenterC83#3208
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #19128253C2semidirect
3. #19128072C2semidirect
4. #19128046C2semidirect
5. #19127900C2semidirect
6. #19127899C2semidirect
7. #19127854C2semidirect
8. #19127852C2semidirect
9. #19127851C2semidirect
10. #19127751C2semidirect
11. #450×C83C2semidirect
15. C2×#13050579C22semidirect
16. C2×#13050578C22semidirect
17. C2×#13050577C22semidirect
18. #14581791C22semidirect
19. #14581790C22semidirect
27. Q8×Dic83C22semidirect
29. #14581757C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581714C22semidirect
34. #14581713C22semidirect
35. #14581712C22semidirect
37. C8×Dic83C22semidirect
39. C2×SD16×C83C22semidirect
41. #102×C83C22semidirect
111. C2×Dic83SD16semidirect
152. C83#3208semidirect
162. #450D83semidirect
171. C2×SD16C2×D83semidirect
191. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282532
C2×#145817882
#191280722
#191280462
#191279002
#191278992
#191278542
#191278522
#191278512
#191277512
#450×C832
#320883