Finite Group Databasebeta

← #29707417

Subgroups — #29707417

Order 15616.

1378 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 209 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#21459221C211yesmaximal, characteristic, complemented
2#21448199C213yesmaximal
3#21446986C213yesmaximal, complemented
4#21446984C213yesmaximal
5#21446982C211yesmaximal, characteristic
6#21445550C213yesmaximal, complemented
7C61×#2030C211yesmaximal, characteristic, complemented
8C23×C4×D61C2211yescharacteristic
9#16840391—26no—
10#16840388C2213yes—
11#16840314—26no—
12C2×#14519733C2213yes—
13C2×#14519733C2213yes—
14#16839625C2213yes—
15#16839621—26no—
16#16839617C2213yes—
17#16839617C2213yes—
18#16839609—26no—
19#16839529C2213yes—
20#16839528C2216yes—
21C61×#543C2211yescharacteristic
22C61×#381C2213yes—
23C61×#342C2213yes—
24C24×D61C2311yescharacteristic
25C23×Dic61C2311yescharacteristic
26C22×C4×D61—23no—
27C22×C4×D61—23no—
28C22×C4×D61—26no—
29C22×C4×D61Q814yescomplemented
30#14519776—26no—
31#14519775—26no—
32#14519767—212no—
33#14519756—26no—
34#14519754—212no—
35Q8×Dic61—22no—
36#14519742—26no—
37#14519733—26no—
38#14519733—26no—
39#14519733—212no—
40C2×#11758196—23no—
41C2×#11758196D416yes—
42C2×C22⋊Dic61C2313yes—
43C2×C22⋊Dic61C2313yes—
44C2×#11758198—26no—
45C2×#11758198—26no—
46C2×C4×Dic61—23no—
47C2×C4×Dic61D416yes—
48#14519684—26no—
49C23×C4×C61C2311yescharacteristic
50C22⋊Q8×C61—26no—
51C2×C22⋊C4×C61C2313yes—
52C2×C22⋊C4×C61C2313yes—
53#95×C61—26no—
54C24×C61C2411yescharacteristic
55C23×D61—23no—
56C23×D61—23no—
57C23×D61—44no—
58C23×D61C2×Q812yes—
59C2×Q8×C61—22no—
60C22×C4×C61—23no—
61C22×C4×C61—23no—
62C22×C4×C61—23no—
63C22×C4×C61C2×D416yes—
64C22×C4×C61C2×Q812yes—
65C22×Dic61—23no—
66C22×Dic61C2×D416yes—
67C22×Dic61C2×D413yes—
68C22×Dic61C2×D413yes—
69C22×Dic61C2×Q812yes—
70C22×Dic61#4116yes—
71C2×C4×D61—212no—
72C2×C4×D61—212no—
73C2×C4×D61—41no—
74C2×C4×D61—43no—
75C2×C4×D61—46no—
76C2×C4×D61—46no—
77C2×Dic122—26no—
78C4⋊C4×C61—26no—
79C22⋊C4×C61—26no—
80C22⋊C4×C61—26no—
81C22⋊Dic61—26no—
82C22⋊Dic61—26no—
83#11758198—412no—
84#11758198—412no—
85C4⋊Dic61—26no—
86#11758196—212no—
87#11758196—212no—
88#11758196—46no—
89C4×Dic61—26no—
90C4×Dic61—26no—
91C4×Dic61—46no—
92C23×C61—42no—
93C23×C61C22×D413yes—
94C23×C61C22×Q811yescharacteristic
95C23×C61C2×#4113yes—
96C22×D61—212no—
97C22×D61—26no—
98C22×D61—44no—
99C22×D61—412no—
100C22×D61—46no—
101C22×D61—41no—
102C22×D61—43no—
103Q8×C61—82no—
104C2×C4×C61—26no—
105C2×C4×C61—26no—
106C2×C4×C61—41no—
107C2×C4×C61—43no—
108C2×C4×C61—46no—
109C2×C4×C61—46no—
110C2×C4×C61—43no—
111C2×Dic61—26no—
112C2×Dic61—46no—
113C2×Dic61—46no—
114C2×Dic61—43no—
115C2×Dic61#12011yescharacteristic
116C2×Dic61#12013yes—
117C2×Dic61C22⋊Q816yes—
118C2×Dic61C22⋊Q816yes—
119C2×Dic61#12416yes—
120C2×Dic61#12416yes—
121C4×D61—44no—
122C4×D61—412no—
123Dic122—86no—
124#23467—611nomaximal, Sylow
125C22×C61—26no—
126C22×C61—41no—
127C22×C61—43no—
128C22×C61C2×#12011yescharacteristic
129C22×C61C2×C22⋊Q813yes—
130C22×C61C2×#12413yes—
131C2×D61—212no—
132C2×D61—412no—
133C2×D61—46no—
134C2×D61—42no—
135C2×D61—44no—
136C4×C61—42no—
137C4×C61—46no—
138C4×C61—83no—
139Dic61—46no—
140Dic61—83no—
141Dic61#39312yes—
142Dic61#39316yes—
143#3458—611no—
144#2280—613no—
145#2030—613no—
146#2030D6111yescharacteristic, complemented
147#2029—613no—
148#2029—611no—
149#1740—613no—
150C2×C61—23no—
151C2×C61—23no—
152C2×C61C2×#39311yescharacteristic
153C2×C61#302512yes—
154D61—44no—
155C24×C4—611no—
156#544—1226no—
157#543—613no—
158#543C2×D6111yescharacteristic
159#533—1226no—
160C2×#117—613no—
161C2×#117—613no—
162#381—613no—
163#381C2×D6113yes—
164#380—1226no—
165#379—613no—
166#379—613no—
167C4×C22⋊C4—1226no—
168#342—616no—
169#342—613no—
170#342C2×D6113yes—
171C61#2346711yescharacteristic, complemented, Sylow
172C25—611no—
173C23×C4—611no—
174C23×C4—614no—
175C23×C4—1226no—
176C23×C4—1223no—
177C23×C4—1223no—
178C23×C4C22×D6111yescharacteristic
179C4⋊Q8—1226no—
180C4⋊Q8—1226no—
181#125—12212no—
182C22⋊Q8—26no—
183C22⋊Q8—12212no—
184C22⋊Q8—1226no—
185C4×Q8—1226no—
186C4×Q8—1222no—
187#117—1226no—
188#117—1226no—
189#117—12212no—
190C2×C4⋊C4—616no—
191C2×C4⋊C4—1223no—
192C2×C22⋊C4—613no—
193C2×C22⋊C4—613no—
194C2×C22⋊C4—1226no—
195C2×C22⋊C4—1226no—
196C2×C22⋊C4C22×D6113yes—
197C2×C22⋊C4C22×D6113yes—
198C2×C42—616no—
199C2×C42—1223no—
200#95—26no—
201#95—1226no—
202C24—612no—
203C24—1223no—
204C24—1223no—
205C24—2444no—
206C24C23×D6111yescharacteristic
207C2×Q8—22no—
208C2×Q8—1226no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—23no—
212C22×C4—613no—
213C22×C4—613no—
214C22×C4—612no—
215C22×C4—616no—
216C22×C4—616no—
217C22×C4—12212no—
218C22×C4—12212no—
219C22×C4—1223no—
220C22×C4—2441no—
221C22×C4—2443no—
222C22×C4—2446no—
223C22×C4—2446no—
224C22×C4D4×D6116yes—
225C22×C4Q8×D6112yes—
226C4⋊C4—26no—
227C4⋊C4—12212no—
228C4⋊C4—12212no—
229C4⋊C4—1226no—
230C4⋊C4—2446no—
231C22⋊C4—26no—
232C22⋊C4—26no—
233C22⋊C4—1226no—
234C22⋊C4—1226no—
235C22⋊C4—24412no—
236C22⋊C4—24412no—
237C42—1226no—
238C42—1226no—
239C42—2446no—
240C23—42no—
241C23—12212no—
242C23—1226no—
243C23—2446no—
244C23—2444no—
245C23—24412no—
246C23—2441no—
247C23—2443no—
248C23C2×D4×D6113yes—
249C23C2×Q8×D6111yescharacteristic
250C23#41×D6113yes—
251Q8—82no—
252Q8—4886no—
253C2×C4—26no—
254C2×C4—26no—
255C2×C4—41no—
256C2×C4—43no—
257C2×C4—46no—
258C2×C4—46no—
259C2×C4—43no—
260C2×C4—611no—
261C2×C4—613no—
262C2×C4—616no—
263C2×C4—616no—
264C2×C4—616no—
265C2×C4—616no—
266C2×C4—1226no—
267C2×C4—2444no—
268C2×C4—24412no—
269C2×C4—2446no—
270C2×C4—2443no—
271C2×C4—2446no—
272C22—26no—
273C22—41no—
274C22—43no—
275C22—12212no—
276C22—24412no—
277C22—2446no—
278C22—2442no—
279C22—2444no—
280C22#120×D6111yescharacteristic
281C22C22⋊Q8×D6113yes—
282C22#124×D6113yes—
283C4—42no—
284C4—46no—
285C4—83no—
286C4—612no—
287C4—616no—
288C4—4883no—
289C4—2446no—
290C2—23no—
291C2—23no—
292C2—2444no—
293C2#393×D6111yescharacteristic
294C2#2145521212yes—
subgroupG/H
centerC22#120×D61
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2030C2
socleC22×C61C2×#120
prime radicalC24×D61C23
hypercenter#2030D61
hypocenterC61#23467
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #2030

Complements

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

NH ≅ G/Ntype
1. #21459221C2semidirect
3. #21446986C2semidirect
6. #21445550C2semidirect
7. C61×#2030C2semidirect
29. C22×C4×D61Q8semidirect
146. #2030D61semidirect
171. C61#23467semidirect

Maximal subgroups

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

subgroupindex
#214592212
#214481992
#214469862
#214469842
#214469822
#214455502
C61×#20302
#2346761