Finite Group Databasebeta

← #24910821

Subgroups — #24910821

Order 10624.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D83C211yesmaximal, characteristic, complemented
2C2×#14581769C212yesmaximal, complemented
3C2×(C22⋊D83)⋊C4C211yesmaximal, characteristic
4#19127622C211yesmaximal, characteristic, complemented
5#19127620C211yesmaximal, characteristic, complemented
6#19127613C211yesmaximal, characteristic, complemented
7#19127605C212yesmaximal, complemented
8#19127577C212yesmaximal
9#19127575C212yesmaximal, complemented
10#19127560C211yesmaximal, characteristic, complemented
11#393×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×Dic166C2212yes—
14C2×Q8×D83—28no—
15C2×Q8×D83—24no—
16C22×C22⋊D83C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18C22×C4×D83C2212yes—
19C22×C4×D83C2211yescharacteristic
20#14581769—28no—
21(C22⋊D83)⋊C4—24no—
22C2×C4⋊Dic83C2212yes—
23C2×Dic83⋊C4C2212yes—
24C2×Dic83⋊C4C2212yes—
25C2×Dic83⋊C4C2211yescharacteristic
26C2×C22⋊Dic83C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×C4×Dic83C2211yescharacteristic
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C83⋊#95C2212yes—
33C83⋊#95C2212yes—
34C22×Q8×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36C2×C22⋊C4×C83C2211yescharacteristic
37#95×C83C2211yescharacteristic
38#95×C83C2212yes—
39C23×D83C2311yescharacteristic
40C2×Dic166—22no—
41C2×Dic166—24no—
42C2×Dic166—24no—
43C2×Dic166—24no—
44C2×Dic166D418yescomplemented
45Q8×D83—232no—
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48C2×C22⋊D83Q814yescomplemented
49C22×Dic83—24no—
50C22×Dic83—22no—
51C22×Dic83C2311yescharacteristic
52C22×Dic83C2312yes—
53C22×Dic83C2312yes—
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C2×C4×D83—24no—
57C2×C4×D83—24no—
58C2×C4×D83—24no—
59C2×C4×D83—24no—
60C2×C4×D83D414yes—
61C4⋊Dic83—24no—
62Dic83⋊C4—24no—
63Dic83⋊C4—22no—
64Dic83⋊C4—24no—
65C22⋊Dic83—24no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—24no—
68C4×Dic83—22no—
69C24×C83C2311yescharacteristic
70C2×Q8×C83—24no—
71C2×Q8×C83—22no—
72C22×C4×C83C2311yescharacteristic
73C22×C4×C83C2312yes—
74C22×C4×C83C2312yes—
75C22×C4×C83C2311yescharacteristic
76C22⋊C4×C83—24no—
77C22⋊C4×C83—22no—
78C23×C83—22no—
79C23×C83—21no—
80C23×C83—21no—
81C23×C83—22no—
82C23×C83C2×Q812yescomplemented
83C23×C83C2411yescharacteristic
84C22×D83—22no—
85C22×D83—22no—
86C22×D83C2×D412yes—
87C22×D83C2×Q812yes—
88C22×D83#4112yes—
89Q8×C83—28no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83C2×D414yes—
99C2×C4×C83C2×D412yes—
100C22⋊D83—28no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—44no—
111C2×Dic83C2×D414yes—
112C2×Dic83C2×D412yes—
113C2×Dic83C2×D414yes—
114C2×Dic83C2×Q812yescomplemented
115C4×D83—216no—
116C4×D83—28no—
117Dic166—28no—
118Dic166—216no—
119C22×C83—22no—
120C22×C83—24no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—42no—
124C22×C83C22×D411yescharacteristic
125C22×C83C22×D412yes—
126C22×C83C22×Q811yescharacteristic
127C22×C83C2×#4111yescharacteristic
128C22×C832−1+412yes—
129C2×D83—22no—
130C2×D83—22no—
131C2×D83—24no—
132C2×D83—24no—
133C2×D83C22⋊Q814yescomplemented
134C4×C83—22no—
135C4×C83—24no—
136C4×C83—41no—
137C4×C83—42no—
138Dic83—24no—
139Dic83—22no—
140Dic83—41no—
141Dic83—42no—
142Dic83#12014yes—
143C2×C83—42no—
144C2×C83C2×#12011yescharacteristic
145C2×C83C2×C22⋊Q811yescharacteristic
146C2×C83#53611yescharacteristic
147C2×C83#54812yes—
148C2×C83D4×Q812yes—
149D83—24no—
150#2409—831nomaximal, Sylow
151C83#240911yescharacteristic, complemented, Sylow
152C23×Q8—831no—
153C2×C22⋊Q8—832no—
154C2×C4×D4—831no—
155#395—831no—
156#395—831no—
157#393—831no—
158#393D8311yescharacteristic, complemented
159#391—832no—
160#386—832no—
161#385—832no—
162#380—831no—
163C22×Q8—832no—
164C22×Q8—831no—
165C22×Q8—1664no—
166C22×Q8—1668no—
167C22×Q8C2×D8311yescharacteristic
168C22×D4—831no—
169C23×C4—831no—
170C23×C4—831no—
171C23×C4—832no—
172C22⋊Q8—1668no—
173C4×D4—1664no—
174C2×C4⋊C4—831no—
175C2×C4⋊C4—832no—
176C2×C4⋊C4—832no—
177C2×C4⋊C4—832no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—831no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4C2×D8311yescharacteristic
182C2×C22⋊C4C2×D8312yes—
183C2×C42—831no—
184#95—831no—
185#95—832no—
186#95—832no—
187#95—832no—
188#95C2×D8311yescharacteristic
189#95C2×D8312yes—
190C24—831no—
191C24C22×D8311yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—838no—
195C2×Q8—1664no—
196C2×Q8—1664no—
197C2×Q8—1664no—
198C2×Q8—16632no—
199C2×Q8—1662no—
200C2×D4—834no—
201C2×D4—1662no—
202C2×D4—1662no—
203C22×C4—831no—
204C22×C4—831no—
205C22×C4—832no—
206C22×C4—834no—
207C22×C4—832no—
208C22×C4—831no—
209C22×C4—1664no—
210C22×C4—1664no—
211C22×C4—1664no—
212C22×C4—1662no—
213C22×C4—1664no—
214C22×C4—1664no—
215C22×C4C22×D8311yescharacteristic
216C22×C4C22×D8312yes—
217C22×C4C22×D8311yescharacteristic
218C22×C4C22×D8312yes—
219C4⋊C4—1664no—
220C4⋊C4—1664no—
221C4⋊C4—1662no—
222C4⋊C4—1664no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1662no—
226C22⋊C4—1664no—
227C22⋊C4—1664no—
228C42—1662no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—832no—
234C23—832no—
235C23—832no—
236C23—1662no—
237C23—1662no—
238C23Q8×D8312yescomplemented
239C23C23×D8311yescharacteristic
240Q8—28no—
241Q8—16616no—
242Q8—1668no—
243D4—1668no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—3324no—
253C2×C4—832no—
254C2×C4—834no—
255C2×C4—834no—
256C2×C4—832no—
257C2×C4—1668no—
258C2×C4—16616no—
259C2×C4—1662no—
260C2×C4—1661no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1661no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1661no—
268C2×C4D4×D8312yes—
269C2×C4D4×D8314yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—834no—
276C22—1664no—
277C22—1664no—
278C22—1662no—
279C22—1662no—
280C22C2×D4×D8312yes—
281C22C2×D4×D8311yescharacteristic
282C22C2×Q8×D8311yescharacteristic
283C22#41×D8311yescharacteristic
284C22#1458183912yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—3322no—
290C4—3321no—
291C4—834no—
292C4—1664no—
293C4—1662no—
294C2—42no—
295C2—1664no—
296C2#120×D8311yescharacteristic
297C2C22⋊Q8×D8311yescharacteristic
298C2#1912827111yescharacteristic
299C2#1912835912yes—
300C2#1912836412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#393D83
hypocenterC83#2409
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D83C2semidirect
2. C2×#14581769C2semidirect
4. #19127622C2semidirect
5. #19127620C2semidirect
6. #19127613C2semidirect
7. #19127605C2semidirect
9. #19127575C2semidirect
10. #19127560C2semidirect
11. #393×C83C2semidirect
44. C2×Dic166D4semidirect
48. C2×C22⋊D83Q8semidirect
82. C23×C83C2×Q8semidirect
114. C2×Dic83C2×Q8semidirect
133. C2×D83C22⋊Q8semidirect
151. C83#2409semidirect
158. #393D83semidirect
238. C23Q8×D83semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D832
C2×#145817692
C2×(C22⋊D83)⋊C42
#191276222
#191276202
#191276132
#191276052
#191275772
#191275752
#191275602
#393×C832
#240983