Finite Group Databasebeta

← #30601080

Subgroups — #30601080

Order 15616.

444 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 74 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#21462364C211yesmaximal, characteristic, complemented
2#21461349C211yesmaximal, characteristic, complemented
3#21461239C211yesmaximal, characteristic
4#21461239C211yesmaximal, characteristic
5#21460679C211yesmaximal, characteristic, complemented
6#21460679C211yesmaximal, characteristic, complemented
7#21456209C211yesmaximal, characteristic
8#16840785C2211yescharacteristic
9#16840785C2211yescharacteristic
10(C2×#6923)⋊Q8—22no—
11#16840711—22no—
12#16840711—22no—
13#16840707C2211yescharacteristic
14#16840707C2211yescharacteristic
15#16840621—22no—
16#16840613—22no—
17#16840613—22no—
18#16840310C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#16840120—22no—
21#16840114C411yescharacteristic, complemented
22#16840113C411yescharacteristic, complemented
23#16840108C411yescharacteristic, complemented
24#16839760—22no—
25C4×#11758189C2211yescharacteristic
26C61×#486C411yescharacteristic, complemented
27C2×C4⋊#6923—22no—
28C2×C4×#6923—22no—
29#14519844—21no—
30#14519844—21no—
31#14519834—21no—
32#14519834—21no—
33#14519832—21no—
34#14519832—21no—
35(C2×#6923)⋊C4—24no—
36(C2×#6923)⋊C4—21no—
37(C2×#6923)⋊C4—21no—
38(C2×#6923)⋊C4—21no—
39(C2×#6923)⋊C4—21no—
40(C2×#6923)⋊C4D412yes—
41(C2×#6923)⋊C4D412yes—
42C2×#11758224—21no—
43C2×Q8×D61—21no—
44C2×#11758189—21no—
45#14519777C2×C411yescharacteristic
46#14519777C2×C411yescharacteristic
47#14519776—21no—
48#14519776D412yes—
49#14519775—21no—
50#14519764—21no—
51#14519748—21no—
52#14519748—21no—
53D4×Dic61—21no—
54D4×Dic61—21no—
55C4⋊C4×D61—21no—
56C4⋊C4×D61D412yes—
57C42×D61C2311yescharacteristic
58#14519702—22no—
59#14519700—22no—
60#14519698—22no—
61#14519687—21no—
62C4×#92418C2×C411yescharacteristic
63C4⋊Q8×C61C2×C411yescharacteristic
64#127×C61C2×C411yescharacteristic
65#102×C61—22no—
66C4×C8×C61C2×C411yescharacteristic
67C22×#6923—22no—
68C22×#6923—22no—
69C2×Q8×C61—21no—
70C2×D4×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—21no—
73C2×C22⋊D61—41no—
74C22×Dic61—41no—
75Q8×D61—81no—
76Q8×D61—81no—
77#11758224—44no—
78C2×C4×D61—22no—
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×D411yescharacteristic
81C2×C4×D61C2×D411yescharacteristic
82C2×Dic122—21no—
83C2×Dic122—21no—
84C2×Dic122—22no—
85C2×#92420—41no—
86C2×#92420—41no—
87C4⋊#6923—82no—
88C4×#6923—82no—
89C2×C8×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61C22⋊C412yes—
92C42×C61C22×C411yescharacteristic
93C22⋊Dic61—41no—
94C22⋊Dic61—41no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61C22⋊C412yes—
99#11758196—22no—
100#11758196—22no—
101C4×Dic61C2×D411yescharacteristic
102C4×Dic61#4111yescharacteristic
103C4×Dic61#4111yescharacteristic
104C2×#92418—21no—
105#11758189—42no—
106C23×C61—41no—
107C22×D61#12011yescharacteristic
108C2×#6923—42no—
109C2×#6923—42no—
110C2×#6923—42no—
111C2×#6923—42no—
112C2×#6923—42no—
113C2×#6923—42no—
114Q8×C61—81no—
115D4×C61—42no—
116D4×C61—81no—
117D4×C61—81no—
118C2×C4×C61—22no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61C4×D411yescharacteristic
121C2×C4×C61C4×D411yescharacteristic
122C22⋊D61—82no—
123C2×Dic61—22no—
124C2×Dic61—41no—
125C2×Dic61—82no—
126C2×Dic61#12111yescharacteristic
127C2×Dic61#12111yescharacteristic
128C2×Dic61#12311yescharacteristic
129C4×D61—22no—
130C4×D61—41no—
131C4×D61—41no—
132C4×D61—42no—
133C4×D61—42no—
134Dic122—42no—
135Dic122—82no—
136Dic122—81no—
137#92420—81no—
138#92420—81no—
139C8×C61—41no—
140#92418—41no—
141#9430—611nomaximal, Sylow
142C22×C61—81no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61#38611yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61#45412yes—
149#6923—82no—
150#6923—82no—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—42no—
155Dic61—21no—
156Dic61—21no—
157Dic61—21no—
158Dic61—42no—
159Dic61—81no—
160#3141—611no—
161#2381—611no—
162#1556—611no—
163#1514—611no—
164#1514—611no—
165#1288—611no—
166#1288—611no—
167C2×C61—161no—
168C2×C61#188811yescharacteristic
169C2×C61#190411yescharacteristic
170C2×C61#190511yescharacteristic
171D61—22no—
172#532—611no—
173C2×C4⋊Q8—611no—
174#489—1222no—
175#487—611no—
176#487—611no—
177#486—611no—
178#486#692311yescharacteristic, complemented
179#432—611no—
180#432—611no—
181#417—1222no—
182C4×M16—611no—
183#391—1222no—
184#387—1222no—
185#387—1222no—
186#383—611no—
187#383—611no—
188#332—1222no—
189#327—1222no—
190#327—1222no—
191C61#943011yescharacteristic, complemented, Sylow
192C2×#41—1221no—
193C22×Q8—1221no—
194C2×M16—1221no—
195C4⋊Q8—612no—
196C4⋊Q8—611no—
197C4⋊Q8—611no—
198C4⋊Q8—1221no—
199C4⋊Q8—1221no—
200C4⋊Q8C2×#692311yescharacteristic
201#127C2×#692311yescharacteristic
202#124—1221no—
203#121—1221no—
204#121—1221no—
205C4×D4—1221no—
206C4×D4—1221no—
207C2×C4⋊C4—612no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1222no—
210C2×C42—611no—
211C2×C42—1222no—
212C4⋊C8—1221no—
213C4⋊C8—1221no—
214#103—1222no—
215#103—1222no—
216#102—22no—
217#102—1222no—
218C22⋊C8—1221no—
219C22⋊C8—1221no—
220#97—1221no—
221#97—1221no—
222#97—1221no—
223C4×C8—611no—
224C4×C8C2×#692311yescharacteristic
225#95—612no—
226#95—612no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1224no—
232#41—2444no—
233C2×Q8—21no—
234C2×Q8—1222no—
235C2×Q8—1221no—
236C2×Q8—1221no—
237C2×Q8—4881no—
238C2×Q8—4881no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—2441no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—611no—
246C22×C4—1222no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—2441no—
250M16—2442no—
251C2×C8—21no—
252C2×C8—1221no—
253C2×C8—2441no—
254C2×C8—2441no—
255C4⋊C4—21no—
256C4⋊C4—612no—
257C4⋊C4—1222no—
258C4⋊C4—1222no—
259C4⋊C4—1221no—
260C4⋊C4—1221no—
261C4⋊C4—1221no—
262C4⋊C4—4882no—
263C4⋊C4C22⋊#692312yes—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C42—611no—
267C42—611no—
268C42—611no—
269C42—4882no—
270C42C22×#692311yescharacteristic
271C23—41no—
272C23—611no—
273Q8—81no—
274Q8—4881no—
275Q8—4882no—
276Q8—2442no—
277D4—42no—
278D4—81no—
279D4—81no—
280D4—4882no—
281C2×C4—22no—
282C2×C4—611no—
283C2×C4—611no—
284C2×C4—611no—
285C2×C4—1222no—
286C2×C4—1222no—
287C2×C4—4882no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2441no—
299C2×C4C2×C22⋊#692311yescharacteristic
300C2×C4D4×#692311yescharacteristic
301C2×C4D4×#692311yescharacteristic
302C8—41no—
303C8—4881no—
304C8—4881no—
305C8—2441no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—612no—
310C22—1221no—
311C22—1221no—
312C22#1684071011yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—1221no—
318C4—1221no—
319C4—1221no—
320C4—4882no—
321C4—4882no—
322C4—4881no—
323C4—2442no—
324C2—161no—
325C2—1222no—
326C2#2146182711yescharacteristic
327C2#2146185311yescharacteristic
328C2#2146185511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#486C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#486#6923
hypocenterC61#9430
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. #21462364C2semidirect
2. #21461349C2semidirect
5. #21460679C2semidirect
6. #21460679C2semidirect
21. #16840114C4semidirect
22. #16840113C4semidirect
23. #16840108C4semidirect
26. C61×#486C4semidirect
178. #486#6923semidirect
191. C61#9430semidirect

Maximal subgroups

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

subgroupindex
#214623642
#214613492
#214612392
#214606792
#214562092
#943061