Finite Group Databasebeta

← #29590906

Subgroups — #29590906

Order 15616.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#21445896C211yesmaximal, characteristic
2#21444133C211yesmaximal, characteristic
3#21443843C212yesmaximal
4#21443819C212yesmaximal
5C61×#2611C211yesmaximal, characteristic
6#16839815—21no—
7#16839625—21no—
8#16839529—21no—
9#16839529C2212yes—
10#16839488—21no—
11#16839486C2212yes—
12C2×#125×C61C411yescharacteristic, complemented
13C2×#117×C61C2211yescharacteristic
14C61×#400C411yescharacteristic, complemented
15C61×#399C2211yescharacteristic
16C61×#398C411yescharacteristic, complemented
17C61×#397C2211yescharacteristic
18C61×#389—21no—
19C61×#388—21no—
20C61×#387—21no—
21C61×#385—21no—
22C61×#376C411yescharacteristic, complemented
23C2×C22⋊Dic61—22no—
24C4⋊#92418—41no—
25C22⋊#92418—22no—
26C4×#92418—41no—
27#14519684—41no—
28#14519684—41no—
29C23×C4×C61D411yescharacteristic
30#125×C61—41no—
31#125×C61—41no—
32#117×C61—41no—
33#117×C61—41no—
34C2×C4⋊C4×C61—21no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61C2×C411yescharacteristic
39C2×C4⋊C4×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61C2311yescharacteristic
46C2×C42×C61—21no—
47C2×C42×C61C2×C411yescharacteristic
48C2×C42×C61D411yescharacteristic
49#95×C61—21no—
50#95×C61—21no—
51#95×C61—21no—
52#95×C61—21no—
53#95×C61—21no—
54#95×C61C2×C411yescharacteristic
55#95×C61C2×C411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C22×C4×C61—21no—
58C22×C4×C61—22no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—41no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—42no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C4⋊C4×C61—42no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—42no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88C42×C61—21no—
89C42×C61—42no—
90C22⋊Dic61—42no—
91C22⋊Dic61—42no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61C2×C22⋊C411yescharacteristic
98C23×C61#13512yes—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61#9911yescharacteristic
127C2×C4×C61#9911yescharacteristic
128C2×Dic61—82no—
129C2×Dic61—82no—
130C2×Dic61—82no—
131#92418—82no—
132#11846—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—22no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61—81no—
140C22×C61C2×#9911yescharacteristic
141C22×C61#41611yescharacteristic
142C22×C61#42111yescharacteristic
143C4×C61—21no—
144C4×C61—21no—
145C4×C61—42no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61—81no—
152C4×C61—81no—
153C4×C61—81no—
154Dic61—162no—
155#2611D6111yescharacteristic
156#1820—611no—
157#1458—611no—
158#1352—612no—
159#1344—612no—
160C2×C61—21no—
161C2×C61—21no—
162C2×C61—42no—
163C2×C61#151611yescharacteristic
164C2×C61#212311yescharacteristic
165C2×C61#212611yescharacteristic
166C2×#125Dic6111yescharacteristic, complemented
167C2×#117C2×D6111yescharacteristic
168#431—1221no—
169#400Dic6111yescharacteristic, complemented
170#399C2×D6111yescharacteristic
171#398Dic6111yescharacteristic, complemented
172#397C2×D6111yescharacteristic
173#389—21no—
174#388—21no—
175#387—21no—
176#385—21no—
177#381—1221no—
178#376Dic6111yescharacteristic, complemented
179#342—612no—
180#342—1221no—
181#324—1221no—
182#323—612no—
183C61#1184611yescharacteristic, complemented, Sylow
184C23×C4C22⋊D6111yescharacteristic
185#125—41no—
186#125—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic6111yescharacteristic
194C2×C4⋊C4C2×Dic6111yescharacteristic
195C2×C4⋊C4C2×Dic6111yescharacteristic
196C2×C4⋊C4C22⋊D6111yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—1222no—
200C2×C22⋊C4C22⋊D6111yescharacteristic
201C2×C22⋊C4C22×D6111yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic6111yescharacteristic
204C2×C42C22⋊D6111yescharacteristic
205C4⋊C8—2441no—
206C22⋊C8—1222no—
207C4×C8—2441no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—2441no—
214#95—2441no—
215#95C2×Dic6111yescharacteristic
216#95C2×Dic6111yescharacteristic
217C24C2×C22⋊D6111yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—2442no—
227C22×C4C22⋊Dic6111yescharacteristic
228C22×C4C22⋊Dic6111yescharacteristic
229C22×C4C22⋊Dic6111yescharacteristic
230C22×C4C22⋊Dic6111yescharacteristic
231C22×C4C22×Dic6111yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233C2×C8—2442no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—2442no—
248C22⋊C4—2442no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic6111yescharacteristic
259C23#1451979712yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—4882no—
288C2×C4—4882no—
289C2×C4—4882no—
290C2×C4#1451969211yescharacteristic
291C2×C4#1451969211yescharacteristic
292C8—4882no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1451969211yescharacteristic
301C22#1683975811yescharacteristic
302C22#1683977911yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—9762no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2144437511yescharacteristic
319C2#2144747311yescharacteristic
320C2#2144748411yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2611C2
socleC22×C61C2×#99
prime radicalC24×C61C2×D4
cosocleC2×C22⋊C4×C61C23
hypercenter#2611D61
hypocenterC61#11846
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C61C4semidirect
14. C61×#400C4semidirect
16. C61×#398C4semidirect
22. C61×#376C4semidirect
166. C2×#125Dic61semidirect
169. #400Dic61semidirect
171. #398Dic61semidirect
178. #376Dic61semidirect
183. C61#11846semidirect

Maximal subgroups

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

subgroupindex
#214458962
#214441332
#214438432
#214438192
C61×#26112
#1184661