Finite Group Databasebeta

← #29574530

Subgroups — #29574530

Order 15616.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#21454920C211yesmaximal, characteristic
2#21446796C211yesmaximal, characteristic
3#21444608C211yesmaximal, characteristic
4#21444607C211yesmaximal, characteristic
5#21444483C211yesmaximal, characteristic
6#21443652C211yesmaximal, characteristic
7C61×#1276C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16839892—21no—
10#16839889—21no—
11C8×Dic122—21no—
12#16839886—21no—
13#16839820—22no—
14#16839815C2211yescharacteristic
15#16839815C2211yescharacteristic
16#16839786C2211yescharacteristic
17#16839522—22no—
18#16839515—21no—
19#16839514—21no—
20#16839513—21no—
21#16839512—21no—
22#16839509C411yescharacteristic, complemented
23#16839509C411yescharacteristic, complemented
24#16839507C411yescharacteristic, complemented
25#16839507C411yescharacteristic, complemented
26C61×#431C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C43×C61C2211yescharacteristic
29C2×#11758194—21no—
30#14519777C2×C411yescharacteristic
31#14519777C2×C411yescharacteristic
32#14519769Q812yes—
33#14519756—21no—
34#14519756—21no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37#14519736D412yes—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418C2×C411yescharacteristic
41C4⋊#92418C2×C411yescharacteristic
42C4⋊#92418D411yescharacteristic
43C4⋊#92418D411yescharacteristic
44#14519706—22no—
45#14519706—22no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48#14519687—22no—
49C4×#92418—21no—
50C4×#92418—21no—
51C8×Dic61—22no—
52C2×C42×C61—21no—
53C2×C42×C61—21no—
54C2×C42×C61—21no—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61C2×C411yescharacteristic
57C4⋊C8×C61C2×C411yescharacteristic
58C4⋊C8×C61D411yescharacteristic
59C4⋊C8×C61D411yescharacteristic
60C22⋊C8×C61—21no—
61C22⋊C8×C61—21no—
62C4×C8×C61—21no—
63C4×C8×C61—21no—
64C22×C4×C61—41no—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61#4111yescharacteristic
67C22×C4×C61#4111yescharacteristic
68C2×Dic122—21no—
69C2×Dic122—21no—
70C2×C8×C61—21no—
71C2×C8×C61—21no—
72C2×C8×C61—21no—
73C2×C8×C61—21no—
74C42×C61—22no—
75C42×C61—22no—
76C42×C61—22no—
77C42×C61—41no—
78C42×C61—41no—
79C42×C61—41no—
80C42×C61C22×C411yescharacteristic
81C42×C61C2×D411yescharacteristic
82C42×C61C2×D411yescharacteristic
83C42×C61C2×Q811yescharacteristic
84C22⋊Dic61—22no—
85C4⋊Dic61—21no—
86C4⋊Dic61—21no—
87C4⋊Dic61C4⋊C412yes—
88C4⋊Dic61C4⋊C412yes—
89#11758196—22no—
90#11758196—22no—
91C4×Dic61—22no—
92#11758194—42no—
93C2×#92418—21no—
94C2×#92418—21no—
95C2×#92418—21no—
96C2×#92418—21no—
97C2×#92418—41no—
98C23×C61#12311yescharacteristic
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61C2×C4⋊C411yescharacteristic
109C2×C4×C61C4×D411yescharacteristic
110C2×C4×C61C4×D411yescharacteristic
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61C22⋊Q811yescharacteristic
113C2×C4×C61C22⋊Q811yescharacteristic
114C2×Dic61—22no—
115C2×Dic61—22no—
116Dic122—41no—
117Dic122—41no—
118Dic122—41no—
119Dic122—41no—
120C8×C61—41no—
121C8×C61—41no—
122C8×C61—41no—
123C8×C61—41no—
124#92418—41no—
125#92418—41no—
126#92418—41no—
127#92418—41no—
128#92418—81no—
129#8936—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61#38511yescharacteristic
134C4×C61—21no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—44no—
138C4×C61#44311yescharacteristic
139C4×C61#44311yescharacteristic
140C4×C61#44411yescharacteristic
141C4×C61#44411yescharacteristic
142C4×C61#45411yescharacteristic
143C4×C61#45611yescharacteristic
144Dic61—42no—
145Dic61—42no—
146#2986—611no—
147#1994—611no—
148#1574—611no—
149#1574—611no—
150#1544—611no—
151#1276—611no—
152#1276D6111yescharacteristic
153C2×C61—41no—
154C2×C61#185711yescharacteristic
155C2×C61#186611yescharacteristic
156C2×C61#186611yescharacteristic
157#533—611no—
158#446—1221no—
159#446—1221no—
160C8×Q8—1221no—
161C8×Q8—1221no—
162#433—1222no—
163#431—611no—
164#431—611no—
165#431C2×D6111yescharacteristic
166#431C2×D6111yescharacteristic
167#423—611no—
168C43C2×D6111yescharacteristic
169#339—1222no—
170#335—1221no—
171#335—1221no—
172#334—1221no—
173#334—1221no—
174#332—611no—
175#332—611no—
176#332—611no—
177#332—611no—
178C61#893611yescharacteristic, complemented, Sylow
179C2×M16—1221no—
180C4⋊Q8—611no—
181C4⋊Q8—611no—
182#125—612no—
183C22⋊Q8—1221no—
184C22⋊Q8—1221no—
185C4×Q8—1221no—
186C4×Q8—1221no—
187#117—612no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C22×D6111yescharacteristic
192C4⋊C8—611no—
193C4⋊C8—611no—
194C4⋊C8—611no—
195C4⋊C8—611no—
196C4⋊C8—1222no—
197C4⋊C8—1222no—
198C4⋊C8—1221no—
199C4⋊C8—1221no—
200C4⋊C8C4×D6111yescharacteristic
201C4⋊C8C4×D6111yescharacteristic
202C4⋊C8D24411yescharacteristic
203C4⋊C8D24411yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—1221no—
207C22⋊C8—1221no—
208#97—1222no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—1222no—
212C4×C8—1221no—
213C4×C8—1221no—
214C2×Q8—1221no—
215C2×Q8—1221no—
216C22×C4—41no—
217C22×C4#1175822211yescharacteristic
218C22×C4D4×D6111yescharacteristic
219C22×C4#1175822411yescharacteristic
220M16—2442no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—1221no—
226C2×C8—1221no—
227C2×C8—1221no—
228C2×C8—1221no—
229C2×C8—2441no—
230C4⋊C4—612no—
231C4⋊C4—612no—
232C4⋊C4—1221no—
233C4⋊C4—1221no—
234C4⋊C4—1222no—
235C4⋊C4—1222no—
236C22⋊C4—1222no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—1222no—
244C42C2×C4×D6111yescharacteristic
245C42C2×D24411yescharacteristic
246C42D4×D6111yescharacteristic
247C42Q8×D6111yescharacteristic
248C23#1451975911yescharacteristic
249Q8—2441no—
250Q8—2441no—
251Q8—2441no—
252Q8—2441no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4C4⋊C4×D6111yescharacteristic
265C2×C4#1451973711yescharacteristic
266C2×C4C4×D24411yescharacteristic
267C2×C4#1451974711yescharacteristic
268C2×C4#1451975511yescharacteristic
269C2×C4#1451975711yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—4881no—
275C8—2441no—
276C8—2441no—
277C8—2441no—
278C8—2441no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1683963611yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—44no—
287C4—2442no—
288C4—2442no—
289C4#1683987611yescharacteristic
290C4#1683988011yescharacteristic
291C4#1683988111yescharacteristic
292C4#1683988511yescharacteristic
293C4#1683993611yescharacteristic
294C4#1683994611yescharacteristic
295C2—41no—
296C2#2144603511yescharacteristic
297C2#2144609511yescharacteristic
298C2#2144609711yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1276C2
socleC22×C61#385
prime radicalC23×C61#123
cosocleC2×C42×C61C23
hypercenter#1276D61
hypocenterC61#8936
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
22. #16839509C4semidirect
23. #16839509C4semidirect
24. #16839507C4semidirect
25. #16839507C4semidirect
178. C61#8936semidirect

Maximal subgroups

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

subgroupindex
#214549202
#214467962
#214446082
#214446072
#214444832
#214436522
C61×#12762
#893661