Finite Group Databasebeta

← #29647883

Subgroups — #29647883

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454904C211yesmaximal, characteristic
2#21448784C211yesmaximal, characteristic
3#21444383C212yesmaximal
4#21444370C214yesmaximal
5#21444331C214yesmaximal
6#21444318C212yesmaximal
7C61×#3502C211yesmaximal, characteristic
8#16839893—24no—
9#16839819C2214yes—
10#16839815C2212yes—
11#16839669—21no—
12#16839648—22no—
13#16839630C2212yes—
14#16839628C2211yescharacteristic
15C4⋊C4×Dic61—21no—
16C4×C4⋊Dic61C2211yescharacteristic
17#16839608C2212yes—
18#16839499C2214yes—
19#16839499C2218yes—
20#16839499C2214yes—
21Q82×C61C414yescomplemented
22C61×#557—24no—
23C61×#557C414yescomplemented
24C61×#544—21no—
25C61×#544—22no—
26C61×#533C2214yes—
27C61×#533C2212yes—
28C61×#519—21no—
29C2×C4×Q8×C61C2211yescharacteristic
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—21no—
32C2×C4×Dic61—22no—
33C2×C4×Dic61—21no—
34C4⋊#92418—22no—
35C4⋊#92418—24no—
36C22⋊#92418C2314yes—
37#14519688—24no—
38C4×#92418—22no—
39#14519684—22no—
40#14519684—22no—
41#14519684—21no—
42#14519684—22no—
43#14519684C2311yescharacteristic
44#14519684C2312yes—
45#14519684C2311yescharacteristic
46C22×Q8×C61D411yescharacteristic
47C4⋊Q8×C61—28no—
48C4⋊Q8×C61—24no—
49C4⋊Q8×C61C2×C414yes—
50C4⋊Q8×C61C2×C412yes—
51C4⋊Q8×C61C2×C414yes—
52C4⋊Q8×C61C2×C412yes—
53#125×C61—28no—
54#125×C61—24no—
55#125×C61—22no—
56#125×C61—24no—
57C22⋊Q8×C61—24no—
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61C2314yes—
60C4×Q8×C61—24no—
61C4×Q8×C61—24no—
62C4×Q8×C61—22no—
63C4×Q8×C61—24no—
64C4×Q8×C61C2×C418yes—
65C4×Q8×C61C2×C414yes—
66C4×Q8×C61C2×C414yes—
67C4×Q8×C61D414yes—
68#117×C61—24no—
69#117×C61—22no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61D412yes—
72C2×C42×C61C2311yescharacteristic
73C2×C42×C61C2312yes—
74C2×Q8×C61—22no—
75C2×Q8×C61—24no—
76C2×Q8×C61—22no—
77C2×Q8×C61—22no—
78C2×Q8×C61C22⋊C414yes—
79C2×Q8×C61C22×C414yes—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—22no—
87C22×Dic61—21no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—24no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61—28no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61C22⋊C418yes—
98C4⋊C4×C61C22⋊C414yes—
99C4⋊C4×C61C22×C414yes—
100C22⋊C4×C61—24no—
101C42×C61—28no—
102C42×C61—24no—
103C42×C61C22×C414yes—
104C42×C61C22×C412yes—
105C42×C61C2×D412yes—
106C42×C61C2×D414yes—
107C4⋊Dic61—42no—
108C4⋊Dic61—42no—
109C4×Dic61—44no—
110C4×Dic61—42no—
111C2×#92418—24no—
112C23×C61C22×D411yescharacteristic
113Q8×C61—28no—
114Q8×C61—44no—
115Q8×C61—84no—
116Q8×C61—82no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—24no—
120C2×C4×C61—24no—
121C2×C4×C61—42no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61C2×C22⋊C414yes—
125C2×C4×C61C2×C22⋊C412yes—
126C2×C4×C61C2×C22⋊C414yes—
127C2×C4×C61C2×C22⋊C412yes—
128C2×C4×C61#13712yes—
129C2×C4×C61#13712yes—
130C2×C4×C61C23×C411yescharacteristic
131C2×C4×C61C22×D411yescharacteristic
132C2×Dic61—22no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—42no—
141#92418—44no—
142#17632—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61C22×C22⋊C411yescharacteristic
146C22×C61C2×#13711yescharacteristic
147C22×C61C2×#13711yescharacteristic
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—24no—
151C4×C61—24no—
152C4×C61—22no—
153C4×C61—44no—
154C4×C61—44no—
155C4×C61#41914yes—
156Dic61—42no—
157Dic61—81no—
158Dic61—82no—
159#3502D6111yescharacteristic
160#2983—611no—
161#2356—611no—
162#1518—612no—
163#1515—614no—
164#1506—614no—
165#1503—612no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61#288411yescharacteristic
169C2×C61C2×#41911yescharacteristic
170C2×C61#290111yescharacteristic
171Q82Dic6114yescomplemented
172#557—24no—
173#557Dic6114yescomplemented
174#544—21no—
175#544—22no—
176#533C2×D6112yes—
177#533C2×D6114yes—
178#519—21no—
179C2×C4×Q8C2×D6111yescharacteristic
180#446—1224no—
181#432—614no—
182#431—612no—
183#391—1221no—
184#387—1222no—
185#383—612no—
186#382—611no—
187C4×C4⋊C4—611no—
188C4×C4⋊C4—1221no—
189#376—612no—
190#328—614no—
191#328—614no—
192#328—618no—
193C61#1763211yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D6111yescharacteristic
195C4⋊Q8—28no—
196C4⋊Q8—24no—
197C4⋊Q8C2×Dic6112yes—
198C4⋊Q8C2×Dic6114yes—
199C4⋊Q8C2×Dic6112yes—
200C4⋊Q8C2×Dic6114yes—
201#125—28no—
202#125—24no—
203#125—22no—
204#125—24no—
205C22⋊Q8—24no—
206C22⋊Q8—22no—
207C22⋊Q8C22×D6114yes—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—24no—
212C4×Q8C2×Dic6114yes—
213C4×Q8C2×Dic6118yes—
214C4×Q8C2×Dic6114yes—
215C4×Q8C22⋊D6114yes—
216#117—24no—
217#117—22no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4C22⋊D6111yescharacteristic
221C2×C4⋊C4C22⋊D6112yes—
222C2×C42—1222no—
223C2×C42—1221no—
224C2×C42C22×D6112yes—
225C2×C42C22×D6111yescharacteristic
226C4⋊C8—1222no—
227C4⋊C8—1224no—
228C22⋊C8—614no—
229#97—1224no—
230C4×C8—1222no—
231#95—611no—
232#95—612no—
233#95—611no—
234#95—1222no—
235#95—1222no—
236#95—1221no—
237#95—1222no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8C22⋊Dic6114yes—
243C2×Q8C22×Dic6114yes—
244C22×C4—1221no—
245C22×C4—1222no—
246C22×C4—1221no—
247C22×C4C2×C22⋊D6112yes—
248C22×C4C2×C22⋊D6112yes—
249C22×C4C2×C22⋊D6111yescharacteristic
250C22×C4C2×C22⋊D6111yescharacteristic
251C22×C4C23×D6111yescharacteristic
252C2×C8—1224no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—28no—
262C4⋊C4—2442no—
263C4⋊C4—2442no—
264C4⋊C4C22⋊Dic6118yes—
265C4⋊C4C22⋊Dic6114yes—
266C4⋊C4C22×Dic6114yes—
267C22⋊C4—24no—
268C42—28no—
269C42—24no—
270C42—2444no—
271C42—2442no—
272C42C22×Dic6114yes—
273C42C22×Dic6112yes—
274C42C2×C22⋊D6112yes—
275C42C2×C22⋊D6114yes—
276C23C22×C22⋊D6111yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—84no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—1222no—
289C2×C4—2442no—
290C2×C4—2441no—
291C2×C4—2442no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2442no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4C2×C22⋊Dic6114yes—
298C2×C4C2×C22⋊Dic6112yes—
299C2×C4C2×C22⋊Dic6114yes—
300C2×C4C2×C22⋊Dic6112yes—
301C2×C4#1451980812yes—
302C2×C4#1451980812yes—
303C2×C4C23×Dic6111yescharacteristic
304C2×C4C22×C22⋊D6111yescharacteristic
305C8—2444no—
306C22—21no—
307C22—21no—
308C22C22×C22⋊Dic6111yescharacteristic
309C22C2×#1451980811yescharacteristic
310C22C2×#1451980811yescharacteristic
311C4—24no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—24no—
316C4—44no—
317C4—44no—
318C4—4882no—
319C4—4881no—
320C4—2442no—
321C4#1683977114yes—
322C2—21no—
323C2—21no—
324C2#2145428311yescharacteristic
325C2C2×#1683977111yescharacteristic
326C2#2145436611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenter#3502D61
hypocenterC61#17632
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3502

Complements

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

NH ≅ G/Ntype
21. Q82×C61C4semidirect
23. C61×#557C4semidirect
171. Q82Dic61semidirect
173. #557Dic61semidirect
193. C61#17632semidirect

Maximal subgroups

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

subgroupindex
#214549042
#214487842
#214443832
#214443702
#214443312
#214443182
C61×#35022
#1763261