Finite Group Databasebeta

← #30605270

Subgroups — #30605270

Order 15616.

390 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 64 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#21461717C211yesmaximal, characteristic
2#21461053C211yesmaximal, characteristic
3#21460813C212yesmaximal
4#21460799C212yesmaximal
5#21451399C211yesmaximal, characteristic
6#16840779—21no—
7#16840705—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×#14519742C411yescharacteristic, complemented
13C2×#14519735C2211yescharacteristic
14#16839688C2211yescharacteristic
15#16839682C411yescharacteristic, complemented
16#16839675C2211yescharacteristic
17#16839651—21no—
18#16839648—21no—
19#16839644—21no—
20#16839633—21no—
21#16839629C411yescharacteristic, complemented
22C61×#398C411yescharacteristic, complemented
23C2×C22⋊#6923—22no—
24#14519844—41no—
25C22⋊#92420—22no—
26C4×#92420—41no—
27(C2×#6923)⋊C4—41no—
28(C2×#6923)⋊C4—41no—
29C22×Dic122C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31#14519742—41no—
32#14519742—41no—
33#14519735—41no—
34#14519735—41no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198D411yescharacteristic
41C2×#11758198C2311yescharacteristic
42C2×C4×Dic61—21no—
43C2×C4×Dic61C2×C411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684C2×C411yescharacteristic
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61C2×C411yescharacteristic
54#95×C61—21no—
55#95×C61C2×C411yescharacteristic
56C23×D61C2×D411yescharacteristic
57C22×#6923—42no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—21no—
64C22×Dic61—21no—
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×C4×D61—22no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—41no—
72C2×Dic122—41no—
73C2×Dic122—42no—
74C22⋊#6923—42no—
75C22⋊#6923—42no—
76C2×#92420—42no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—41no—
80#11758198—41no—
81#11758198—41no—
82#11758198—42no—
83#11758198—42no—
84C4⋊Dic61—41no—
85#11758196—42no—
86#11758196—41no—
87#11758196—41no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C4×Dic61—21no—
91C4×Dic61—21no—
92C4×Dic61—42no—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—41no—
97C22×D61—41no—
98C22×D61#13512yes—
99C2×#6923—82no—
100C2×#6923—82no—
101C2×#6923—82no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—22no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61#9911yescharacteristic
128C2×Dic61#9911yescharacteristic
129C4×D61—42no—
130C4×D61—42no—
131Dic122—42no—
132Dic122—42no—
133#92420—82no—
134#11842—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61C2×#9911yescharacteristic
138C22×C61#41611yescharacteristic
139C22×C61#42111yescharacteristic
140C2×D61—22no—
141C2×D61—41no—
142C2×D61—41no—
143C2×D61—42no—
144C2×D61—81no—
145#6923—162no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61—81no—
150C4×C61—81no—
151Dic61—21no—
152Dic61—21no—
153Dic61—42no—
154Dic61—41no—
155Dic61—41no—
156Dic61—81no—
157#2599—611no—
158#1820—611no—
159#1458—611no—
160#1351—612no—
161#1344—612no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61#151611yescharacteristic
165C2×C61#212011yescharacteristic
166C2×C61#212911yescharacteristic
167D61—42no—
168C2×C4×Q8—611no—
169C2×#117—611no—
170#431—1221no—
171#398#692311yescharacteristic, complemented
172#397—611no—
173#395—611no—
174#393—611no—
175#388—1221no—
176#387—1221no—
177#387—1221no—
178#385—1221no—
179#383—611no—
180#381—1221no—
181#342—612no—
182#342—1221no—
183#324—1221no—
184#323—612no—
185C61#1184211yescharacteristic, complemented, Sylow
186C22×Q8—611no—
187C23×C4—611no—
188C4×Q8—2441no—
189C4×Q8—2441no—
190#117—2441no—
191#117—2441no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—611no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4C2×#692311yescharacteristic
197C2×C4⋊C4C2×#692311yescharacteristic
198C2×C22⋊C4—611no—
199C2×C22⋊C4—611no—
200C2×C22⋊C4—1221no—
201C2×C22⋊C4—1221no—
202C2×C22⋊C4—1222no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—1221no—
206C4⋊C8—2441no—
207C22⋊C8—1222no—
208C4×C8—2441no—
209#95—21no—
210#95—611no—
211#95—1221no—
212#95—1221no—
213#95—1221no—
214#95—1221no—
215#95—1221no—
216#95—2441no—
217#95—2441no—
218#95C2×#692311yescharacteristic
219C24—611no—
220C2×Q8—2441no—
221C2×Q8—2442no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4—1222no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4—2441no—
233C22×C4—2442no—
234C22×C4C22⋊#692311yescharacteristic
235C22×C4C22⋊#692311yescharacteristic
236C22×C4C22×#692311yescharacteristic
237C2×C8—2442no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—2441no—
242C4⋊C4—2441no—
243C4⋊C4—2441no—
244C4⋊C4—2442no—
245C22⋊C4—2441no—
246C22⋊C4—2441no—
247C22⋊C4—2442no—
248C22⋊C4—2442no—
249C22⋊C4—2442no—
250C22⋊C4—2442no—
251C42—1221no—
252C42—1221no—
253C42—1221no—
254C42—1221no—
255C42—2442no—
256C23—612no—
257C23—1221no—
258C23—1221no—
259C23—2441no—
260C23—2441no—
261C23C2×C22⋊#692311yescharacteristic
262Q8—2442no—
263Q8—2442no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—611no—
278C2×C4—611no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1222no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—4882no—
285C2×C4—4882no—
286C2×C4—4882no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C8—4882no—
297C22—21no—
298C22—21no—
299C22—1222no—
300C22—4881no—
301C22—2442no—
302C22—2441no—
303C22—2441no—
304C22D61⋊#9911yescharacteristic
305C22#1684075211yescharacteristic
306C22#1684076011yescharacteristic
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—1221no—
313C4—1221no—
314C4—9762no—
315C4—4881no—
316C4—2442no—
317C4—2441no—
318C4—2441no—
319C2—21no—
320C2—21no—
321C2—2442no—
322C2#2146124411yescharacteristic
323C2#2146207311yescharacteristic
324C2#2146210211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenter#398#6923
hypocenterC61#11842
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #398

Complements

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

NH ≅ G/Ntype
12. C2×#14519742C4semidirect
15. #16839682C4semidirect
21. #16839629C4semidirect
22. C61×#398C4semidirect
171. #398#6923semidirect
185. C61#11842semidirect

Maximal subgroups

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

subgroupindex
#214617172
#214610532
#214608132
#214607992
#214513992
#1184261