Finite Group Databasebeta

← #24918282

Subgroups — #24918282

Order 10624.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128214C211yesmaximal, characteristic, complemented
2C2×#14581767C211yesmaximal, characteristic, complemented
3#19128027C212yesmaximal, complemented
4#19128014C211yesmaximal, characteristic, complemented
5#19127955C212yesmaximal, complemented
6#19127950C212yesmaximal, complemented
7#19127851C211yesmaximal, characteristic, complemented
8#19127849C212yesmaximal, complemented
9#19127837C211yesmaximal, characteristic, complemented
10#19127663C211yesmaximal, characteristic, complemented
11#483×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic, complemented
13C22×D332—21no—
14C22×D332C2211yescharacteristic
15C22×C4×D83C2211yescharacteristic
16C2×#13050577C2212yescomplemented
17C2×D664C2212yescomplemented
18C2×#13050573C2212yescomplemented
19#14581777—21no—
20#14581768C2211yescharacteristic, complemented
21#14581767—21no—
22#14581767—21no—
23#14581767C2212yes—
24#14581767C2212yes—
25#14581767C2211yescharacteristic, complemented
26#14581766C2211yescharacteristic
27#14581762—21no—
28#14581754—21no—
29(C22⋊D83)⋊C4—21no—
30#14581750C2211yescharacteristic, complemented
31C2×(C2×D83)⋊C4—21no—
32#14581725C2212yescomplemented
33#14581714C2212yescomplemented
34#14581712C2212yescomplemented
35#14581712C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yescomplemented
38C22⋊Q8×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yescomplemented
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C23×D83D411yescharacteristic
45#13050591—42no—
46C2×D332—21no—
47C2×D332—21no—
48C2×D332—22no—
49C2×D332—22no—
50C2×D332—21no—
51C2×D332—42no—
52C2×D332—44no—
53C2×D332C2311yescharacteristic
54C2×D332C2312yes—
55C2×C22⋊D83—21no—
56C22×Dic83D411yescharacteristic
57C2×C4×D83—21no—
58C2×C4×D83—42no—
59C2×C4×D83D411yescharacteristic
60C2×C4×D83D411yescharacteristic
61C2×C4×D83C2311yescharacteristic
62C2×C4×D83C2311yescharacteristic
63#13050577—42no—
64D664—42no—
65#13050573—42no—
66C2×C83⋊C8C2312yes—
67C4⋊Dic83C2311yescharacteristic
68Dic83⋊C4—21no—
69(C2×D83)⋊C4—21no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—22no—
73(C2×D83)⋊C4—22no—
74(C2×D83)⋊C4—21no—
75C4×Dic83—21no—
76C2×Q8×C83C2311yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83C2311yescharacteristic
79C2×C8×C83C2312yes—
80C4⋊C4×C83—21no—
81C4⋊C4×C83—21no—
82C4⋊C4×C83C2312yes—
83C4⋊C4×C83C2311yescharacteristic
84C22⋊C4×C83—21no—
85C23×C83C2×D411yescharacteristic
86C22×D83—22no—
87C22×D83—21no—
88C22×D83—22no—
89C22×D83—41no—
90C22×D83—41no—
91C22×D83—42no—
92C22×D83—42no—
93C22×D83—42no—
94C22×D83C2×D411yescharacteristic
95C22×D83C2×D411yescharacteristic
96Q8×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—22no—
101C2×C4×C83—42no—
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—42no—
105C2×Dic83—21no—
106C2×Dic83—41no—
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109D332—22no—
110D332—22no—
111D332—22no—
112D332—42no—
113D332—42no—
114D332—44no—
115D332—44no—
116D332—42no—
117C4×D83—22no—
118C4×D83—22no—
119C4×D83—41no—
120C4×D83—41no—
121C4×D83—42no—
122C8×C83—22no—
123C83⋊C8—22no—
124C22×C83—21no—
125C22×C83—21no—
126C22×C83AGL(1,ℤ/8)12yes—
127C22×C83C22×D411yescharacteristic
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—21no—
132C2×D83—21no—
133C2×D83—21no—
134C2×D83—44no—
135C2×D83—41no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—41no—
139C2×D83—41no—
140C2×D83—42no—
141C2×D83—84no—
142C4×C83—21no—
143C4×C83—41no—
144C4×C83—42no—
145C4×C83—41no—
146C4×C83H(5,𝔽2)12yes—
147Dic83—21no—
148Dic83—21no—
149Dic83—41no—
150C2×C83—22no—
151C2×C83#53411yescharacteristic
152C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C83#57511yescharacteristic
154D83—41no—
155D83—41no—
156D83—81no—
157D83—82no—
158#3194—831nomaximal, Sylow
159C83#319411yescharacteristic, complemented, Sylow
160#525—831no—
161C2×#121—831no—
162#483D8311yescharacteristic, complemented
163#482—832no—
164#480—831no—
165#469—832no—
166#468—832no—
167#450—831no—
168#449—832no—
169#447—831no—
170#407—831no—
171C2×#41—831no—
172C22×D4—831no—
173C22×D4—1661no—
174C23×C4—831no—
175C2×SD16—832no—
176C2×D8—832no—
177C2×M16—832no—
178#123—1661no—
179C22⋊Q8—831no—
180C22⋊Q8C2×D8311yescharacteristic
181#121—831no—
182#121—832no—
183#121—832no—
184#121—831no—
185#121—1661no—
186#121—1661no—
187#120—1661no—
188C4×D4—1661no—
189C4×D4—1661no—
190#117—831no—
191C2×C4⋊C4C2×D8311yescharacteristic
192C2×C22⋊C4—1661no—
193#107—832no—
194#106C2×D8312yescomplemented
195#103C2×D8312yescomplemented
196#102—832no—
197#102—832no—
198#102—832no—
199C22⋊C8—831no—
200C22⋊C8—832no—
201C22⋊C8C2×D8311yescharacteristic
202C24—831no—
203C24—1661no—
204#41—3322no—
205C2×Q8C22×D8311yescharacteristic
206C2×D4—3322no—
207C2×D4—3324no—
208C2×D4—831no—
209C2×D4—832no—
210C2×D4—1661no—
211C2×D4—1661no—
212C2×D4—1662no—
213C2×D4—1662no—
214C2×D4—1661no—
215C2×D4—1661no—
216C22×C4—21no—
217C22×C4—3322no—
218C22×C4—831no—
219C22×C4—831no—
220C22×C4—831no—
221C22×C4—831no—
222C22×C4—831no—
223C22×C4—1661no—
224C22×C4C22×D8311yescharacteristic
225SD16—3322no—
226D8—3322no—
227M16—3322no—
228C2×C8—832no—
229C2×C8C22×D8312yes—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—831no—
233C4⋊C4—1661no—
234C4⋊C4C22×D8311yescharacteristic
235C4⋊C4C22×D8312yes—
236C22⋊C4—21no—
237C22⋊C4—1662no—
238C22⋊C4—1662no—
239C22⋊C4—1661no—
240C22⋊C4—1661no—
241C22⋊C4—1661no—
242C22⋊C4—1661no—
243C42—1661no—
244C23—3321no—
245C23—3321no—
246C23—3322no—
247C23—3322no—
248C23—3322no—
249C23—831no—
250C23—831no—
251C23—1661no—
252C23—1662no—
253C23—1662no—
254C23D4×D8311yescharacteristic
255Q8—22no—
256D4—3322no—
257D4—3324no—
258D4—3324no—
259D4—3322no—
260D4—3322no—
261D4—3322no—
262D4—1662no—
263D4—1662no—
264D4—1662no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—3321no—
271C2×C4—3321no—
272C2×C4—3322no—
273C2×C4—3321no—
274C2×C4—831no—
275C2×C4—831no—
276C2×C4—1661no—
277C2×C4—1662no—
278C2×C4—1662no—
279C2×C4D4×D8311yescharacteristic
280C2×C4C23×D8311yescharacteristic
281C8—22no—
282C8—1662no—
283C22—21no—
284C22—21no—
285C22—3324no—
286C22—3322no—
287C22—3321no—
288C22—3321no—
289C22—3322no—
290C22—3321no—
291C22—3322no—
292C22—6644no—
293C22—1661no—
294C22—1661no—
295C22—1661no—
296C22—1661no—
297C22—1661no—
298C22—1661no—
299C22#1458181912yes—
300C22C2×D4×D8311yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—3321no—
306C4—1661no—
307C4—1661no—
308C4#1458183812yes—
309C2—22no—
310C2—3321no—
311C2—3321no—
312C2—6642no—
313C2—6641no—
314C2#1912826211yescharacteristic
315C2C2×#1458181911yescharacteristic
316C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#483×C83C2
socleC22×C83C22×D4
hypercenter#483D83
hypocenterC83#3194
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #19128214C2semidirect
2. C2×#14581767C2semidirect
3. #19128027C2semidirect
4. #19128014C2semidirect
5. #19127955C2semidirect
6. #19127950C2semidirect
7. #19127851C2semidirect
8. #19127849C2semidirect
9. #19127837C2semidirect
10. #19127663C2semidirect
11. #483×C83C2semidirect
12. C2×#13050591C22semidirect
16. C2×#13050577C22semidirect
17. C2×D664C22semidirect
18. C2×#13050573C22semidirect
20. #14581768C22semidirect
25. #14581767C22semidirect
30. #14581750C22semidirect
32. #14581725C22semidirect
33. #14581714C22semidirect
34. #14581712C22semidirect
35. #14581712C22semidirect
37. (C2×D83)⋊C8C22semidirect
40. #106×C83C22semidirect
41. #103×C83C22semidirect
159. C83#3194semidirect
162. #483D83semidirect
194. #106C2×D83semidirect
195. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
C2×#145817672
#191280272
#191280142
#191279552
#191279502
#191278512
#191278492
#191278372
#191276632
#483×C832
#319483