Finite Group Databasebeta

← #29707415

Subgroups — #29707415

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#21448197C213yesmaximal
3#21446987C213yesmaximal, complemented
4#21446985C211yesmaximal, characteristic, complemented
5#21446983C213yesmaximal
6#21445550C213yesmaximal, complemented
7C61×#2029C211yesmaximal, characteristic
8C24×Dic61C2211yescharacteristic
9#16840391—26no—
10#16840388C2213yes—
11#16840314—26no—
12C2×#14519733C2213yes—
13C2×#14519733C2213yes—
14#16839623C2213yes—
15#16839623C2213yes—
16#16839620—26no—
17#16839618C2213yes—
18C22⋊C4×Dic61—26no—
19#16839529C2213yes—
20#16839528C2216yes—
21C61×#543C2211yescharacteristic
22C61×#379C2213yes—
23C61×#342C2213yes—
24C23×Dic61—23no—
25C23×Dic61—23no—
26C23×Dic61—26no—
27C23×Dic61Q814yescomplemented
28C23×Dic61C2311yescharacteristic
29C23×Dic61C2311yescharacteristic
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⋊Dic61—26no—
43C2×C22⋊Dic61C2313yes—
44C2×C22⋊Dic61C2313yes—
45C2×C4×Dic61—23no—
46C2×C4×Dic61D416yes—
47#14519684—26no—
48#14519684—26no—
49C25×C61C2311yescharacteristic
50C22⋊Q8×C61—26no—
51C2×C22⋊C4×C61—26no—
52C2×C22⋊C4×C61C2313yes—
53C2×C22⋊C4×C61C2313yes—
54C24×C61—23no—
55C24×C61—23no—
56C24×C61—44no—
57C24×C61C2×Q812yes—
58C24×C61C2411yescharacteristic
59C2×Q8×C61—22no—
60C22×C4×C61—23no—
61C22×C4×C61C2×D416yes—
62C22×Dic61—23no—
63C22×Dic61—23no—
64C22×Dic61—212no—
65C22×Dic61—212no—
66C22×Dic61—23no—
67C22×Dic61—41no—
68C22×Dic61—43no—
69C22×Dic61—46no—
70C22×Dic61—46no—
71C22×Dic61C2×D416yes—
72C22×Dic61C2×D413yes—
73C22×Dic61C2×D413yes—
74C22×Dic61C2×Q812yes—
75C22×Dic61C2×Q812yes—
76C22×Dic61#4116yes—
77C2×Dic122—26no—
78C4⋊C4×C61—26no—
79C22⋊C4×C61—26no—
80C22⋊C4×C61—26no—
81C22⋊C4×C61—412no—
82C22⋊Dic61—26no—
83C22⋊Dic61—26no—
84C22⋊Dic61—412no—
85C4⋊Dic61—26no—
86#11758196—212no—
87#11758196—212no—
88#11758196—46no—
89C4×Dic61—26no—
90C4×Dic61—26no—
91C4×Dic61—46no—
92C23×C61—212no—
93C23×C61—26no—
94C23×C61—44no—
95C23×C61—42no—
96C23×C61—46no—
97C23×C61—412no—
98C23×C61—41no—
99C23×C61—43no—
100C23×C61C22×D413yes—
101C23×C61C22×Q811yescharacteristic
102C23×C61C2×#4113yes—
103Q8×C61—82no—
104C2×C4×C61—26no—
105C2×C4×C61—46no—
106C2×C4×C61—46no—
107C2×C4×C61—43no—
108C2×Dic61—26no—
109C2×Dic61—26no—
110C2×Dic61—41no—
111C2×Dic61—43no—
112C2×Dic61—44no—
113C2×Dic61—412no—
114C2×Dic61—46no—
115C2×Dic61—46no—
116C2×Dic61—43no—
117C2×Dic61#12011yescharacteristic
118C2×Dic61#12013yes—
119C2×Dic61C22⋊Q816yes—
120C2×Dic61C22⋊Q816yes—
121C2×Dic61#12416yes—
122C2×Dic61#12416yes—
123Dic122—86no—
124#23467—611nomaximal, Sylow
125C22×C61—26no—
126C22×C61—212no—
127C22×C61—41no—
128C22×C61—43no—
129C22×C61—412no—
130C22×C61—46no—
131C22×C61—42no—
132C22×C61—44no—
133C22×C61C2×#12011yescharacteristic
134C22×C61C2×C22⋊Q813yes—
135C22×C61C2×#12413yes—
136C4×C61—46no—
137C4×C61—83no—
138Dic61—42no—
139Dic61—46no—
140Dic61—83no—
141Dic61#39312yes—
142Dic61#39316yes—
143#3458—611no—
144#2280—613no—
145#2030—613no—
146#2030—611no—
147#2029—613no—
148#2029D6111yescharacteristic
149#1740—613no—
150C2×C61—23no—
151C2×C61—23no—
152C2×C61—44no—
153C2×C61C2×#39311yescharacteristic
154C2×C61#302512yes—
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#381—613no—
164#380—1226no—
165#379—613no—
166#379C2×D6113yes—
167C4×C22⋊C4—1226no—
168#342—616no—
169#342—613no—
170#342C2×D6113yes—
171C61#2346711yescharacteristic, complemented, Sylow
172C25C22×D6111yescharacteristic
173C23×C4—611no—
174C23×C4—614no—
175C23×C4—611no—
176C23×C4—1226no—
177C23×C4—1223no—
178C23×C4—1223no—
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—26no—
193C2×C22⋊C4—613no—
194C2×C22⋊C4—613no—
195C2×C22⋊C4—1226no—
196C2×C22⋊C4C22×D6113yes—
197C2×C22⋊C4C22×D6113yes—
198C2×C42—616no—
199C2×C42—1223no—
200#95—1226no—
201#95—1226no—
202C24—23no—
203C24—23no—
204C24—44no—
205C24Q8×D6112yes—
206C24C23×D6111yescharacteristic
207C2×Q8—22no—
208C2×Q8—1226no—
209C22×C4—23no—
210C22×C4—613no—
211C22×C4—613no—
212C22×C4—612no—
213C22×C4—616no—
214C22×C4—612no—
215C22×C4—616no—
216C22×C4—12212no—
217C22×C4—12212no—
218C22×C4—1223no—
219C22×C4—1223no—
220C22×C4—1223no—
221C22×C4—2441no—
222C22×C4—2443no—
223C22×C4—2446no—
224C22×C4—2446no—
225C22×C4D4×D6116yes—
226C4⋊C4—26no—
227C4⋊C4—12212no—
228C4⋊C4—12212no—
229C4⋊C4—1226no—
230C4⋊C4—2446no—
231C22⋊C4—26no—
232C22⋊C4—26no—
233C22⋊C4—412no—
234C22⋊C4—1226no—
235C22⋊C4—1226no—
236C22⋊C4—24412no—
237C42—1226no—
238C42—1226no—
239C42—2446no—
240C23—212no—
241C23—26no—
242C23—44no—
243C23—42no—
244C23—46no—
245C23—412no—
246C23—41no—
247C23—43no—
248C23C2×D4×D6113yes—
249C23C2×Q8×D6111yescharacteristic
250C23#41×D6113yes—
251Q8—82no—
252Q8—4886no—
253C2×C4—26no—
254C2×C4—46no—
255C2×C4—46no—
256C2×C4—43no—
257C2×C4—611no—
258C2×C4—613no—
259C2×C4—616no—
260C2×C4—616no—
261C2×C4—616no—
262C2×C4—616no—
263C2×C4—1226no—
264C2×C4—1226no—
265C2×C4—2444no—
266C2×C4—24412no—
267C2×C4—2446no—
268C2×C4—2443no—
269C2×C4—2441no—
270C2×C4—2443no—
271C2×C4—2446no—
272C22—26no—
273C22—212no—
274C22—41no—
275C22—43no—
276C22—412no—
277C22—46no—
278C22—42no—
279C22—44no—
280C22#120×D6111yescharacteristic
281C22C22⋊Q8×D6113yes—
282C22#124×D6113yes—
283C4—46no—
284C4—83no—
285C4—612no—
286C4—616no—
287C4—4883no—
288C4—2442no—
289C4—2446no—
290C2—23no—
291C2—23no—
292C2—44no—
293C2#393×D6111yescharacteristic
294C2#2145520912yes—
subgroupG/H
centerC22#120×D61
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2029C2
socleC22×C61C2×#120
prime radicalC25×C61C23
hypercenter#2029D61
hypocenterC61#23467
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #2029

Complements

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

NH ≅ G/Ntype
1. #21459221C2semidirect
3. #21446987C2semidirect
4. #21446985C2semidirect
6. #21445550C2semidirect
27. C23×Dic61Q8semidirect
171. C61#23467semidirect

Maximal subgroups

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

subgroupindex
#214592212
#214481972
#214469872
#214469852
#214469832
#214455502
C61×#20292
#2346761