Finite Group Databasebeta

← #29654870

Subgroups — #29654870

Order 15616.

910 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 191 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#21459629C211yesmaximal, characteristic
2#21456769C212yesmaximal
3#21451024C211yesmaximal, characteristic
4#21444795C218yesmaximal
5#21444319C212yesmaximal
6C61×#1461C211yesmaximal, characteristic
7Q8×Dic122—22no—
8#16840491—22no—
9#16840488—24no—
10#16840393—21no—
11#16840393C2214yes—
12#16840319—22no—
13#16840319C2212yes—
14C2×#14519777C2211yescharacteristic
15#16840240—21no—
16#16840142—22no—
17#16840113—22no—
18#16839973—24no—
19#16839920C2218yes—
20#16839682C2214yes—
21#16839669—21no—
22#16839660—22no—
23#16839631—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839500C2218yes—
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#431C2212yes—
28C61×#324C2214yes—
29C22×Dic122C2312yes—
30C2×Dic244—28no—
31#14519777—22no—
32#14519777—22no—
33#14519777—22no—
34#14519777—22no—
35#14519777D412yes—
36#14519777D412yes—
37#14519776—24no—
38#14519776—28no—
39#14519769—24no—
40#14519769—22no—
41#14519768—24no—
42#14519768—28no—
43#14519756—22no—
44#14519756—24no—
45#14519756C2314yes—
46Q8×Dic61—24no—
47Q8×Dic61—24no—
48C4×Dic122—24no—
49C4×Dic122—24no—
50C4×Dic122—22no—
51C4×Dic122D414yes—
52#14519736—24no—
53#14519736—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×#11758196—24no—
58C2×C4×Dic61—21no—
59#14519702—28no—
60#14519684—24no—
61#14519684C2312yes—
62C22×Q8×C61D411yescharacteristic
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61C2312yes—
67C2×C42×C61D412yes—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—22no—
70C22⋊C8×C61C2314yes—
71C4×C8×C61—22no—
72C2×Q8×C61—22no—
73C2×Q8×C61—22no—
74C2×Q8×C61Q1614yescomplemented
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D412yes—
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2411yescharacteristic
80C22×Dic61—22no—
81C22×Dic61—22no—
82C2×Dic122—22no—
83C2×Dic122—24no—
84C2×Dic122—24no—
85C2×Dic122—82no—
86C2×Dic122C2×D414yes—
87C2×C8×C61—24no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—24no—
90C42×C61—22no—
91C42×C61—22no—
92C42×C61C2×D412yes—
93C42×C61C2×D412yes—
94C22⋊Dic61—24no—
95C4⋊Dic61—24no—
96C4⋊Dic61—24no—
97C4⋊Dic61—24no—
98C4⋊Dic61—24no—
99C4⋊Dic61—22no—
100C4⋊Dic61—22no—
101C4⋊Dic61—22no—
102C4⋊Dic61—41no—
103C4⋊Dic61—42no—
104C4⋊Dic61—41no—
105C4⋊Dic61C2×D414yes—
106#11758196—28no—
107#11758196—24no—
108#11758196—24no—
109#11758196—84no—
110C4×Dic61—28no—
111C4×Dic61—24no—
112C4×Dic61—42no—
113Dic244—88no—
114C23×C61C22×D411yescharacteristic
115Q8×C61—28no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61#12011yescharacteristic
123C2×C4×C61#12012yes—
124C2×C4×C61#12011yescharacteristic
125C2×C4×C61C2×Q1614yes—
126C2×C4×C61C2×Q1612yes—
127C2×C4×C61#13712yes—
128C2×C4×C61C22×D412yes—
129C2×Dic61—24no—
130C2×Dic61—24no—
131C2×Dic61—24no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—44no—
135C2×Dic61—42no—
136Dic122—44no—
137Dic122—44no—
138Dic122—48no—
139Dic122—84no—
140C8×C61—44no—
141#18404—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#12011yescharacteristic
145C22×C61C22×Q1611yescharacteristic
146C22×C61C2×#13711yescharacteristic
147C4×C61—24no—
148C4×C61—21no—
149C4×C61—22no—
150C4×C61—42no—
151C4×C61#45114yes—
152C4×C61#45612yes—
153Dic61—44no—
154Dic61—44no—
155Dic61—44no—
156Dic61—82no—
157#3502—611no—
158#3209—612no—
159#2567—611no—
160#1605—618no—
161#1503—612no—
162#1461D6111yescharacteristic
163C2×C61—21no—
164C2×C61—21no—
165C2×C61C2×#45111yescharacteristic
166C2×C61C2×#45611yescharacteristic
167C2×C61#302711yescharacteristic
168Q82—1222no—
169#557—1224no—
170#557—1222no—
171#544—614no—
172#544—1221no—
173#533—612no—
174#533—1222no—
175C2×C4⋊Q8—611no—
176#519—1221no—
177C2×C4×Q8C2×D6111yescharacteristic
178#494—1222no—
179#487—1222no—
180#462—1224no—
181#451—618no—
182#431C2×D6112yes—
183#395—614no—
184#391—1221no—
185#389—1222no—
186#384—1221no—
187C4×C4⋊C4—611no—
188#328—618no—
189#324C2×D6114yes—
190C61#1840411yescharacteristic, complemented, Sylow
191C22×Q8—612no—
192C22×Q8D24411yescharacteristic
193C2×Q16—1228no—
194C4⋊Q8—612no—
195C4⋊Q8—612no—
196C4⋊Q8—1222no—
197C4⋊Q8—1222no—
198C4⋊Q8—1228no—
199C4⋊Q8—1224no—
200C4⋊Q8—1222no—
201C4⋊Q8—1222no—
202#125—1224no—
203#125—1228no—
204#125—1224no—
205#125—1222no—
206C22⋊Q8—614no—
207C22⋊Q8—1222no—
208C22⋊Q8—1224no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—614no—
212C4×Q8—1224no—
213C4×Q8—1224no—
214C4×Q8—1224no—
215C4×Q8—1224no—
216C4×Q8—1222no—
217#117—1222no—
218#117—1224no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—1224no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4D24411yescharacteristic
224C2×C4⋊C4C22×D6112yes—
225C2×C42—1221no—
226C2×C42D24412yes—
227C2×C42C22×D6111yescharacteristic
228C4⋊C8—22no—
229#103—1228no—
230C22⋊C8C22×D6114yes—
231C4×C8—22no—
232#95—612no—
233#95—1224no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—614no—
237C2×Q8—1224no—
238C2×Q8—1222no—
239C2×Q8—1224no—
240C2×Q8—4882no—
241C2×Q8Dic24414yescomplemented
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4C2×D24411yescharacteristic
245C22×C4C2×D24412yes—
246C22×C4C2×D24412yes—
247C22×C4C2×D24411yescharacteristic
248C22×C4C23×D6111yescharacteristic
249Q16—4888no—
250C2×C8—24no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—614no—
254C4⋊C4—1224no—
255C4⋊C4—1228no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1224no—
259C4⋊C4—1224no—
260C4⋊C4—1224no—
261C4⋊C4—1222no—
262C4⋊C4—1222no—
263C4⋊C4—1222no—
264C4⋊C4—4884no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C4⋊C4—2442no—
268C22⋊C4—1224no—
269C42—22no—
270C42—22no—
271C42—1228no—
272C42—1224no—
273C42—2442no—
274C42D4×D6112yes—
275C42D4×D6112yes—
276C23C22×D24411yescharacteristic
277Q8—28no—
278Q8—4884no—
279Q8—2448no—
280Q8—2444no—
281Q8—2444no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—1224no—
289C2×C4—1224no—
290C2×C4—1224no—
291C2×C4—2444no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4#1451974711yescharacteristic
296C2×C4#1451974711yescharacteristic
297C2×C4#1451974712yes—
298C2×C4C2×Dic24414yes—
299C2×C4C2×Dic24412yes—
300C2×C4#1451981012yes—
301C2×C4C2×D4×D6112yes—
302C8—44no—
303C22—21no—
304C22—21no—
305C22C2×#1451974711yescharacteristic
306C22C22×Dic24411yescharacteristic
307C22C2×#1451981011yescharacteristic
308C4—21no—
309C4—22no—
310C4—24no—
311C4—42no—
312C4—4882no—
313C4—2444no—
314C4—2444no—
315C4—2444no—
316C4#1683992014yes—
317C4#1683994712yes—
318C2—21no—
319C2—21no—
320C2C2×#1683992011yescharacteristic
321C2C2×#1683994711yescharacteristic
322C2#2145522211yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1461C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#1461D61
hypocenterC61#18404
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
74. C2×Q8×C61Q16semidirect
190. C61#18404semidirect
241. C2×Q8Dic244semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214567692
#214510242
#214447952
#214443192
C61×#14612
#1840461