Finite Group Databasebeta

← #30601060

Subgroups — #30601060

Order 15616.

402 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 72 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#21462733C211yesmaximal, characteristic, complemented
2#21461308C211yesmaximal, characteristic
3#21461242C211yesmaximal, characteristic, complemented
4#21461239C211yesmaximal, characteristic
5#21460686C211yesmaximal, characteristic, complemented
6#21460679C211yesmaximal, characteristic, complemented
7#21456536C211yesmaximal, characteristic
8#16840814—22no—
9#16840813—22no—
10#16840787—22no—
11#16840785C2211yescharacteristic
12#16840785C2211yescharacteristic
13#16840785C2211yescharacteristic
14#16840707C2211yescharacteristic
15#16840617—22no—
16#16840615—22no—
17#16840613—22no—
18#16840317C2211yescharacteristic
19#16840310C2211yescharacteristic
20#16840165C411yescharacteristic, complemented
21#16840134C411yescharacteristic, complemented
22#16840113C411yescharacteristic, complemented
23#16840007—22no—
24#16839859—22no—
25C4×#11758189C2211yescharacteristic
26C61×#486C411yescharacteristic, complemented
27#14519844—22no—
28#14519844—21no—
29#14519844—21no—
30#14519844—21no—
31C4⋊#92420—22no—
32#14519834—21no—
33#14519834—21no—
34#14519834D412yes—
35#14519832—22no—
36#14519832—21no—
37#14519832—21no—
38#14519832D412yes—
39C4×#92420—22no—
40(C2×#6923)⋊C4—21no—
41(C2×#6923)⋊C4—21no—
42(C2×#6923)⋊C4—21no—
43C2×#11758224—21no—
44C2×#11758200—22no—
45C2×#11758189—21no—
46#14519777C2×C411yescharacteristic
47#14519777C2×C411yescharacteristic
48#14519768—21no—
49#14519764—21no—
50#14519757D412yes—
51#14519753—21no—
52#14519748—21no—
53#14519748—21no—
54Q8×Dic61—21no—
55#14519742D412yes—
56D4×Dic61—21no—
57D4×Dic61—21no—
58#14519735—21no—
59C42×D61C2311yescharacteristic
60#14519710—22no—
61#14519702—22no—
62#14519687—21no—
63C4×#92418C2×C411yescharacteristic
64C4⋊Q8×C61C2×C411yescharacteristic
65#127×C61C2×C411yescharacteristic
66#102×C61—22no—
67C4×C8×C61C2×C411yescharacteristic
68C22×#6923—41no—
69C2×Q8×C61—21no—
70C2×D4×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—21no—
73C2×C22⋊D61—41no—
74C22×Dic61—41no—
75#11758224—82no—
76C2×C4×D61C2×D411yescharacteristic
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61#4111yescharacteristic
79C2×Dic122—21no—
80C2×Dic122C22⋊C412yes—
81C2×#92420—22no—
82C2×#92420—22no—
83C2×#92420—41no—
84C2×C8×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61C22⋊C412yes—
87C42×C61C22×C411yescharacteristic
88C22⋊Dic61—41no—
89C22⋊Dic61—41no—
90#11758200—44no—
91#11758198—22no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—22no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97#11758196—22no—
98C4×Dic61—22no—
99C4×Dic61C2×D411yescharacteristic
100C4×Dic61C2×D411yescharacteristic
101C4×Dic61#4111yescharacteristic
102C2×#92418—21no—
103#11758189—42no—
104C23×C61—41no—
105C22×D61#12111yescharacteristic
106C2×#6923—81no—
107C2×#6923—81no—
108C2×#6923—81no—
109Q8×C61—81no—
110D4×C61—42no—
111D4×C61—81no—
112D4×C61—81no—
113C2×C4×C61—22no—
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C2×C4×C61C4×D411yescharacteristic
116C2×C4×C61C4×D411yescharacteristic
117C22⋊D61—82no—
118C2×Dic61—22no—
119C2×Dic61—41no—
120C2×Dic61—82no—
121C2×Dic61#12011yescharacteristic
122C2×Dic61#12111yescharacteristic
123C2×Dic61#12311yescharacteristic
124C4×D61—41no—
125C4×D61—41no—
126C4×D61—41no—
127Dic122—42no—
128Dic122—81no—
129#92420—42no—
130#92420—42no—
131#92420—81no—
132C8×C61—41no—
133#92418—22no—
134#9418—611nomaximal, Sylow
135C22×C61—81no—
136C22×C61—81no—
137C22×C61—81no—
138C22×C61#38611yescharacteristic
139C2×D61—21no—
140C2×D61—21no—
141C2×D61—21no—
142#6923—161no—
143C4×C61—21no—
144C4×C61—21no—
145C4×C61—21no—
146C4×C61—42no—
147Dic61—21no—
148Dic61—21no—
149Dic61—21no—
150Dic61—42no—
151Dic61—81no—
152#3182—611no—
153#2987—611no—
154#1536—611no—
155#1515—611no—
156#1514—611no—
157#1291—611no—
158#1288—611no—
159C2×C61—161no—
160C2×C61#188411yescharacteristic
161C2×C61#190311yescharacteristic
162C2×C61#190511yescharacteristic
163D61—41no—
164#533—611no—
165#532—611no—
166#500—611no—
167#492—611no—
168#487—611no—
169#486#692311yescharacteristic, complemented
170#468—1222no—
171#446—1222no—
172#446—1222no—
173#440—1222no—
174#433—1222no—
175#432—611no—
176#432—611no—
177#432—611no—
178C4×M16—611no—
179#383—611no—
180#329—1222no—
181#328—1222no—
182#327—1222no—
183C61#941811yescharacteristic, complemented, Sylow
184C2×#41—1221no—
185C2×SD16—1222no—
186C2×M16—1221no—
187C4⋊Q8—611no—
188C4⋊Q8—611no—
189C4⋊Q8C2×#692311yescharacteristic
190#127C2×#692311yescharacteristic
191#125—1221no—
192#124—1221no—
193C22⋊Q8—612no—
194C22⋊Q8—1221no—
195#121—1221no—
196#121—1221no—
197C4×Q8—612no—
198C4×Q8—1221no—
199C4×D4—1221no—
200C4×D4—1221no—
201#117—1221no—
202C2×C42—611no—
203#107—1222no—
204C4⋊C8—1221no—
205C4⋊C8—1221no—
206C4⋊C8—1221no—
207C4⋊C8—1222no—
208C4⋊C8—1222no—
209#103—1222no—
210#102—22no—
211C22⋊C8—612no—
212C22⋊C8—1221no—
213C22⋊C8—1221no—
214#97—612no—
215#97—1221no—
216#97—1221no—
217#97—1221no—
218#97—1222no—
219C4×C8—611no—
220C4×C8—1222no—
221C4×C8C2×#692311yescharacteristic
222#95—1221no—
223#95—1221no—
224#95—1221no—
225#41—4882no—
226C2×Q8—21no—
227C2×Q8—612no—
228C2×Q8—1221no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—2441no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—2441no—
237C22×C4—2441no—
238SD16—2444no—
239M16—2442no—
240C2×C8—21no—
241C2×C8—1221no—
242C2×C8—1222no—
243C2×C8—1222no—
244C2×C8—2441no—
245C4⋊C4—21no—
246C4⋊C4—1222no—
247C4⋊C4—1222no—
248C4⋊C4—1221no—
249C4⋊C4—1221no—
250C4⋊C4—1221no—
251C4⋊C4—1221no—
252C4⋊C4C22⋊#692312yes—
253C22⋊C4—1222no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C42—611no—
257C42—611no—
258C42—611no—
259C42—1222no—
260C42C22×#692311yescharacteristic
261C23—41no—
262C23—611no—
263Q8—81no—
264Q8—4881no—
265Q8—2442no—
266D4—42no—
267D4—81no—
268D4—81no—
269D4—4882no—
270C2×C4—22no—
271C2×C4—611no—
272C2×C4—611no—
273C2×C4—611no—
274C2×C4—1222no—
275C2×C4—4882no—
276C2×C4—4881no—
277C2×C4—4881no—
278C2×C4—4881no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4C2×C22⋊#692311yescharacteristic
284C2×C4D4×#692311yescharacteristic
285C2×C4D4×#692311yescharacteristic
286C8—41no—
287C8—1222no—
288C8—4881no—
289C8—2442no—
290C8—2442no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—1221no—
295C22—1221no—
296C22—1221no—
297C22#1684071011yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—1221no—
303C4—1221no—
304C4—1221no—
305C4—9761no—
306C4—4881no—
307C4—2442no—
308C2—161no—
309C2—2441no—
310C2#2146182311yescharacteristic
311C2#2146185011yescharacteristic
312C2#2146185511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#486C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#486#6923
hypocenterC61#9418
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #21462733C2semidirect
3. #21461242C2semidirect
5. #21460686C2semidirect
6. #21460679C2semidirect
20. #16840165C4semidirect
21. #16840134C4semidirect
22. #16840113C4semidirect
26. C61×#486C4semidirect
169. #486#6923semidirect
183. C61#9418semidirect

Maximal subgroups

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

subgroupindex
#214627332
#214613082
#214612422
#214612392
#214606862
#214606792
#214565362
#941861