Finite Group Databasebeta

← #29707422

Subgroups — #29707422

Order 15616.

1506 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1506 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#543×D61C211yesmaximal, characteristic, complemented
2#21448396C213yesmaximal
3#21446986C213yesmaximal, complemented
4#21446984C213yesmaximal, complemented
5#21446982C211yesmaximal, characteristic, complemented
6#342×D61C213yesmaximal, complemented
7C61×#2030C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9#16840388C2213yes—
10#16840387—26no—
11C22⋊Q8×D61—26no—
12C2×C22⋊C4×D61C2213yes—
13C2×C22⋊C4×D61C2213yes—
14#16839640—26no—
15#16839640—26no—
16#16839625C2213yes—
17#16839617C2213yes—
18#16839617C2213yes—
19#16839529C2213yes—
20#16839528C2216yes—
21C61×#543C2211yescharacteristic
22C61×#381C2213yes—
23C61×#342C2213yes—
24C24×D61C2311yescharacteristic
25C24×D61C2311yescharacteristic
26C2×Q8×D61—22no—
27C22×D244—23no—
28C22×D244—23no—
29C22×D244—26no—
30C22×D244Q814yescomplemented
31C22×C4×D61—23no—
32C22×C4×D61D416yes—
33#14519757—26no—
34#14519756—26no—
35#14519755—212no—
36#14519754—212no—
37#14519753—26no—
38C4⋊C4×D61—26no—
39C2×C22⋊Dic61C2313yes—
40C2×C22⋊Dic61C2313yes—
41C2×#11758198—23no—
42C2×#11758198—26no—
43C2×#11758198—26no—
44C2×#11758198D416yes—
45C22⋊C4×D61—26no—
46C22⋊C4×D61—26no—
47C22⋊C4×D61—212no—
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—82no—
58C23×D61—82no—
59C23×D61C2×D413yes—
60C23×D61C2×D413yes—
61C23×D61C2×Q812yes—
62C23×D61C2×Q812yescomplemented
63C23×D61#4116yes—
64C2×Q8×C61—22no—
65C22×C4×C61—23no—
66C22×C4×C61—23no—
67C22×C4×C61—23no—
68C22×C4×C61C2×D416yes—
69C22×C4×C61C2×Q812yescomplemented
70C22×Dic61—23no—
71C22×Dic61C2×D416yes—
72Q8×D61—84no—
73C2×D244—212no—
74C2×D244—212no—
75C2×D244—43no—
76C2×D244—41no—
77C2×D244—46no—
78C2×D244—46no—
79C2×C4×D61—26no—
80C2×C4×D61—26no—
81C2×C4×D61—46no—
82C2×C4×D61—86no—
83C2×C4×D61—86no—
84C2×Dic122—26no—
85C4⋊C4×C61—26no—
86C22⋊C4×C61—26no—
87C22⋊C4×C61—26no—
88C22⋊Dic61—26no—
89C22⋊Dic61—26no—
90#11758198—212no—
91#11758198—46no—
92#11758198—86no—
93#11758198—86no—
94C4⋊Dic61—26no—
95#11758196—212no—
96C23×C61—42no—
97C23×C61C22×D413yes—
98C23×C61C22×Q811yescharacteristic
99C23×C61C2×#4113yes—
100C22×D61—26no—
101C22×D61—46no—
102C22×D61—46no—
103C22×D61—46no—
104C22×D61—46no—
105C22×D61—41no—
106C22×D61—43no—
107C22×D61—82no—
108C22×D61—82no—
109C22×D61—86no—
110C22×D61—86no—
111C22×D61#12013yes—
112C22×D61#12011yescharacteristic
113C22×D61C22⋊Q816yes—
114C22×D61C22⋊Q816yes—
115C22×D61#12416yes—
116C22×D61#12416yes—
117Q8×C61—82no—
118C2×C4×C61—26no—
119C2×C4×C61—26no—
120C2×C4×C61—41no—
121C2×C4×C61—43no—
122C2×C4×C61—46no—
123C2×C4×C61—46no—
124C2×C4×C61—43no—
125C2×Dic61—26no—
126C2×Dic61—46no—
127C2×Dic61—46no—
128C2×Dic61—43no—
129D244—412no—
130D244—44no—
131C4×D61—412no—
132C4×D61—86no—
133C4×D61—812no—
134Dic122—86no—
135#23468—611nomaximal, Sylow
136C22×C61—26no—
137C22×C61—41no—
138C22×C61—43no—
139C22×C61C2×#12011yescharacteristic
140C22×C61C2×C22⋊Q813yes—
141C22×C61C2×#12413yes—
142C2×D61—22no—
143C2×D61—26no—
144C2×D61—46no—
145C2×D61—46no—
146C2×D61—46no—
147C2×D61—46no—
148C2×D61—46no—
149C2×D61—46no—
150C2×D61—42no—
151C2×D61—86no—
152C2×D61—82no—
153C2×D61#39316yes—
154C2×D61#39312yes—
155C4×C61—42no—
156C4×C61—46no—
157C4×C61—83no—
158Dic61—46no—
159Dic61—83no—
160C2×#543—611no—
161#2308—613no—
162#2030—613no—
163#2030D6111yescharacteristic, complemented
164#2029—613no—
165#2029—611no—
166C2×#342—613no—
167C2×C61—23no—
168C2×C61—23no—
169C2×C61C2×#39311yescharacteristic
170C2×C61#302612yes—
171D61—22no—
172D61—26no—
173D61—82no—
174C23×D4—611no—
175#543—613no—
176#543—1226no—
177#543C2×D6111yescharacteristic
178C2×C22⋊Q8—1226no—
179C22×C22⋊C4—613no—
180C22×C22⋊C4—613no—
181#386—1226no—
182#386—1226no—
183#381—613no—
184#381C2×D6113yes—
185#379—613no—
186#379—613no—
187#342—616no—
188#342—613no—
189#342C2×D6113yes—
190C61#2346811yescharacteristic, complemented, Sylow
191C25—611no—
192C25—611no—
193C22×Q8—1222no—
194C22×D4—614no—
195C22×D4—1226no—
196C22×D4—1223no—
197C22×D4—1223no—
198C23×C4—616no—
199C23×C4—1223no—
200C23×C4C22×D6111yescharacteristic
201C22⋊Q8—26no—
202C22⋊Q8—12212no—
203C22⋊Q8—12212no—
204C22⋊Q8—1226no—
205C22⋊Q8—1226no—
206C22⋊Q8—1226no—
207C2×C4⋊C4—1226no—
208C2×C22⋊C4—613no—
209C2×C22⋊C4—616no—
210C2×C22⋊C4—613no—
211C2×C22⋊C4—1226no—
212C2×C22⋊C4—1226no—
213C2×C22⋊C4—12212no—
214C2×C22⋊C4—1226no—
215C2×C22⋊C4—1226no—
216C2×C22⋊C4—1223no—
217C2×C22⋊C4C22×D6113yes—
218C2×C22⋊C4C22×D6113yes—
219#95—26no—
220#95—1226no—
221C24—612no—
222C24—616no—
223C24—613no—
224C24—613no—
225C24—612no—
226C24—1223no—
227C24—1223no—
228C24—4882no—
229C24—4882no—
230C24C23×D6111yescharacteristic
231C2×Q8—22no—
232C2×Q8—1226no—
233C2×Q8—4884no—
234C2×D4—12212no—
235C2×D4—12212no—
236C2×D4—2443no—
237C2×D4—2441no—
238C2×D4—2446no—
239C2×D4—2446no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—616no—
244C22×C4—1226no—
245C22×C4—1226no—
246C22×C4—1223no—
247C22×C4—4886no—
248C22×C4—4886no—
249C22×C4—2446no—
250C22×C4D4×D6116yes—
251C22×C4Q8×D6112yescomplemented
252C4⋊C4—26no—
253C4⋊C4—12212no—
254C4⋊C4—1226no—
255C22⋊C4—26no—
256C22⋊C4—26no—
257C22⋊C4—12212no—
258C22⋊C4—1226no—
259C22⋊C4—1226no—
260C22⋊C4—4886no—
261C22⋊C4—4886no—
262C22⋊C4—2446no—
263C23—42no—
264C23—613no—
265C23—611no—
266C23—616no—
267C23—616no—
268C23—616no—
269C23—616no—
270C23—1226no—
271C23—4886no—
272C23—4882no—
273C23—4882no—
274C23—4886no—
275C23—2446no—
276C23—2446no—
277C23—2446no—
278C23—2446no—
279C23—2441no—
280C23—2443no—
281C23C2×D4×D6113yes—
282C23C2×Q8×D6111yescharacteristic
283C23#41×D6113yes—
284Q8—82no—
285Q8—4886no—
286D4—24412no—
287D4—2444no—
288C2×C4—26no—
289C2×C4—26no—
290C2×C4—41no—
291C2×C4—43no—
292C2×C4—46no—
293C2×C4—46no—
294C2×C4—43no—
295C2×C4—1226no—
296C2×C4—4886no—
297C2×C4—48812no—
298C2×C4—24412no—
299C2×C4—2446no—
300C2×C4—2443no—
301C2×C4—2446no—
302C22—26no—
303C22—41no—
304C22—43no—
305C22—616no—
306C22—612no—
307C22—1226no—
308C22—1222no—
309C22—4886no—
310C22—4882no—
311C22—2446no—
312C22—2446no—
313C22—2446no—
314C22—2446no—
315C22—2446no—
316C22—2446no—
317C22—2442no—
318C22#120×D6111yescharacteristic
319C22C22⋊Q8×D6113yes—
320C22#124×D6113yes—
321C4—42no—
322C4—46no—
323C4—83no—
324C4—4883no—
325C4—2446no—
326C2—23no—
327C2—23no—
328C2—1226no—
329C2—1222no—
330C2—4882no—
331C2#393×D6111yescharacteristic
332C2#2145521712yes—
subgroupG/H
centerC22#120×D61
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2030C2
socleC22×C61C2×#120
prime radicalC23×D244C22
hypercenter#2030D61
hypocenterC61#23468
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. #543×D61C2semidirect
3. #21446986C2semidirect
4. #21446984C2semidirect
5. #21446982C2semidirect
6. #342×D61C2semidirect
7. C61×#2030C2semidirect
30. C22×D244Q8semidirect
62. C23×D61C2×Q8semidirect
69. C22×C4×C61C2×Q8semidirect
163. #2030D61semidirect
190. C61#23468semidirect
251. C22×C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#543×D612
#214483962
#214469862
#214469842
#214469822
#342×D612
C61×#20302
#2346861