Finite Group Databasebeta

← #29583196

Subgroups — #29583196

Order 15616.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#21451462C211yesmaximal, characteristic
2#21444370C211yesmaximal, characteristic
3#21444331C211yesmaximal, characteristic
4#21443713C211yesmaximal, characteristic
5#21443684C211yesmaximal, characteristic
6#21443675C211yesmaximal, characteristic
7C61×#1986C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9#16839819C2211yescharacteristic
10#16839701—21no—
11#16839682—21no—
12#16839668—21no—
13#16839666—21no—
14#16839646—21no—
15#16839645—21no—
16#16839630C2211yescharacteristic
17#16839608C2211yescharacteristic
18#16839499—21no—
19#16839499—21no—
20#16839499—21no—
21#16839499—21no—
22C61×#533C2211yescharacteristic
23C4×M16×C61C2211yescharacteristic
24C4×C4⋊C4×C61C2211yescharacteristic
25C61×#339—21no—
26C61×#339—21no—
27C22×Dic122—21no—
28C2×C4⋊Dic61—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×C4×Dic61—21no—
32C4⋊#92418—21no—
33C4⋊#92418—21no—
34C22⋊#92418—21no—
35C22⋊#92418—21no—
36#14519688—21no—
37#14519688—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684D411yescharacteristic
46#14519684D411yescharacteristic
47#14519684D411yescharacteristic
48#14519684D411yescharacteristic
49C2×M16×C61—21no—
50C4⋊Q8×C61Q811yescharacteristic, complemented
51C4⋊Q8×C61Q811yescharacteristic, complemented
52#125×C61—21no—
53C22⋊Q8×C61—21no—
54C22⋊Q8×C61—21no—
55C4×Q8×C61—21no—
56C4×Q8×C61—21no—
57#117×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C42×C61D411yescharacteristic
60C2×C42×C61D411yescharacteristic
61C2×C42×C61C2311yescharacteristic
62#97×C61—21no—
63C4×C8×C61—21no—
64#95×C61—21no—
65C2×Q8×C61—21no—
66C2×Q8×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C2×Dic122—42no—
77C2×Dic122—42no—
78C2×Dic122—81no—
79M16×C61—22no—
80M16×C61—41no—
81C2×C8×C61—41no—
82C4⋊C4×C61—21no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C42×C61—41no—
91C42×C61—41no—
92C42×C61—41no—
93C42×C61C2×Q811yescharacteristic
94C42×C61#4111yescharacteristic
95C42×C61#4111yescharacteristic
96C42×C61#4111yescharacteristic
97C4⋊Dic61—42no—
98#11758196—42no—
99#11758196—81no—
100C4×Dic61—42no—
101C2×#92418—41no—
102C2×#92418—41no—
103C23×C61#12011yescharacteristic
104Q8×C61—81no—
105Q8×C61—81no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
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×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C22⋊Q811yescharacteristic
119C2×C4×C61C22⋊Q811yescharacteristic
120C2×C4×C61C22⋊Q811yescharacteristic
121C2×C4×C61#12411yescharacteristic
122C2×C4×C61#12411yescharacteristic
123C2×C4×C61#12411yescharacteristic
124C2×Dic61—22no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136Dic122—84no—
137C8×C61—42no—
138#92418—81no—
139#92418—81no—
140#10379—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61#39311yescharacteristic
144C22×C61#45511yescharacteristic
145C22×C61#45611yescharacteristic
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—21no—
151C4×C61—21no—
152C4×C61—42no—
153C4×C61—81no—
154C4×C61—81no—
155C4×C61—81no—
156Dic61—42no—
157Dic61—81no—
158Dic61—81no—
159Dic61—81no—
160#2605—611no—
161#1986D6111yescharacteristic
162#1515—611no—
163#1506—611no—
164#1300—611no—
165#1288—611no—
166#1285—611no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61#202811yescharacteristic
170C2×C61#203111yescharacteristic
171C2×C61#203111yescharacteristic
172#533C2×D6111yescharacteristic
173#432—611no—
174#432—611no—
175C4×M16C2×D6111yescharacteristic
176#400—1221no—
177#395—1221no—
178#391—1221no—
179#391—1221no—
180#387—1221no—
181#387—1221no—
182#383—611no—
183C4×C4⋊C4C2×D6111yescharacteristic
184#376—611no—
185#339—21no—
186#339—21no—
187#328—1221no—
188#328—1221no—
189#328—1221no—
190#328—1221no—
191C61#1037911yescharacteristic, complemented, Sylow
192C22×Q8—1221no—
193C2×M16—21no—
194C4⋊Q8Dic12211yescharacteristic, complemented
195C4⋊Q8Dic12211yescharacteristic, complemented
196#125—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201#117—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C42—1221no—
207C2×C42C22⋊D6111yescharacteristic
208C2×C42C22⋊D6111yescharacteristic
209C2×C42C22×D6111yescharacteristic
210C4⋊C8—1221no—
211C4⋊C8—1221no—
212C22⋊C8—1221no—
213C22⋊C8—1221no—
214#97—21no—
215#97—1221no—
216#97—1221no—
217C4×C8—21no—
218#95—21no—
219#95—611no—
220#95—611no—
221#95—611no—
222#95—611no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—4881no—
233C2×Q8—2442no—
234C2×Q8—2442no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4D4×D6111yescharacteristic
242C22×C4D4×D6111yescharacteristic
243C22×C4C2×C22⋊D6111yescharacteristic
244M16—22no—
245M16—41no—
246C2×C8—41no—
247C2×C8—2441no—
248C2×C8—2441no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—4881no—
257C4⋊C4—2442no—
258C4⋊C4—2442no—
259C22⋊C4—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—2442no—
264C42C2×Dic12211yescharacteristic
265C42#1175822211yescharacteristic
266C42#1175822411yescharacteristic
267C42#1175822411yescharacteristic
268C23#1451974511yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—4884no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
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—1222no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4#1451975411yescharacteristic
297C2×C4#1451975411yescharacteristic
298C2×C4#1451975611yescharacteristic
299C2×C4#1451976311yescharacteristic
300C2×C4#1451976311yescharacteristic
301C2×C4#1451976411yescharacteristic
302C8—42no—
303C8—4881no—
304C8—4881no—
305C22—21no—
306C22—21no—
307C22#1683967411yescharacteristic
308C22#1683993911yescharacteristic
309C22#1683994411yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—4881no—
321C4—4881no—
322C4—4881no—
323C4—2442no—
324C2—21no—
325C2—21no—
326C2#2144698011yescharacteristic
327C2#2144698911yescharacteristic
328C2#2144699111yescharacteristic
subgroupG/H
centerC22#16839674
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1986C2
socleC22×C61#393
prime radicalC23×C61#120
hypercenter#1986D61
hypocenterC61#10379
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
50. C4⋊Q8×C61Q8semidirect
51. C4⋊Q8×C61Q8semidirect
191. C61#10379semidirect
194. C4⋊Q8Dic122semidirect
195. C4⋊Q8Dic122semidirect

Maximal subgroups

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

subgroupindex
#214514622
#214443702
#214443312
#214437132
#214436842
#214436752
C61×#19862
#1037961