Finite Group Databasebeta

← #29647886

Subgroups — #29647886

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#21459629C211yesmaximal, characteristic
2#21448784C211yesmaximal, characteristic
3#21444384C212yesmaximal
4#21444371C214yesmaximal
5#21444332C214yesmaximal
6#21444319C212yesmaximal
7C61×#2983C211yesmaximal, characteristic
8Q8×Dic122C414yescomplemented
9#16840491C414yescomplemented
10#16840488—24no—
11#16840393—21no—
12#16840393—22no—
13#16840319C2214yes—
14#16840319C2212yes—
15#16840240—21no—
16#16839669—21no—
17#16839648—22no—
18#16839630C2212yes—
19#16839628C2211yescharacteristic
20C4⋊C4×Dic61—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839608C2212yes—
23#16839500C2214yes—
24#16839500C2218yes—
25#16839500C2214yes—
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#446—24no—
28C61×#432C2214yes—
29C61×#431C2212yes—
30#14519777C2×C414yes—
31#14519777C2×C412yes—
32#14519777C2×C414yes—
33#14519777C2×C412yes—
34#14519776—28no—
35#14519776—24no—
36#14519769—22no—
37#14519769—24no—
38#14519768—28no—
39#14519768—24no—
40#14519756—24no—
41#14519756—22no—
42#14519756C2314yes—
43Q8×Dic61—24no—
44Q8×Dic61—24no—
45C4×Dic122—22no—
46C4×Dic122—24no—
47C4×Dic122C2×C418yes—
48C4×Dic122C2×C414yes—
49#14519736—24no—
50#14519736—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4×Dic61—22no—
54C2×C4×Dic61—21no—
55#14519684—22no—
56#14519684—22no—
57#14519684—21no—
58#14519684—22no—
59#14519684C2311yescharacteristic
60#14519684C2312yes—
61#14519684C2311yescharacteristic
62C22×Q8×C61D411yescharacteristic
63C4×Q8×C61C2×C414yes—
64C4×Q8×C61D414yes—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D412yes—
67C2×C42×C61C2311yescharacteristic
68C2×C42×C61C2312yes—
69C4⋊C8×C61—22no—
70C4⋊C8×C61—24no—
71C22⋊C8×C61C2314yes—
72#97×C61—24no—
73C4×C8×C61—22no—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61C22⋊C414yes—
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D412yes—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2411yescharacteristic
82C22×Dic61—21no—
83C22×Dic61—22no—
84C22×Dic61—21no—
85C2×Dic122—22no—
86C2×Dic122—24no—
87C2×Dic122C22×C414yes—
88C2×C8×C61—24no—
89C4⋊C4×C61C22⋊C418yes—
90C4⋊C4×C61C22⋊C414yes—
91C42×C61C22×C414yes—
92C42×C61C22×C412yes—
93C42×C61C2×D412yes—
94C42×C61C2×D414yes—
95C22⋊Dic61—24no—
96C4⋊Dic61—24no—
97C4⋊Dic61—22no—
98C4⋊Dic61—24no—
99C4⋊Dic61—22no—
100C4⋊Dic61—22no—
101C4⋊Dic61—24no—
102C4⋊Dic61—42no—
103C4⋊Dic61—42no—
104C4⋊Dic61C22×C414yes—
105#11758196—24no—
106#11758196—28no—
107#11758196—24no—
108C4×Dic61—28no—
109C4×Dic61—24no—
110C4×Dic61—44no—
111C4×Dic61—42no—
112C23×C61C22×D411yescharacteristic
113Q8×C61—28no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61C2×C22⋊C414yes—
120C2×C4×C61C2×C22⋊C412yes—
121C2×C4×C61C2×C22⋊C414yes—
122C2×C4×C61C2×C22⋊C412yes—
123C2×C4×C61#13712yes—
124C2×C4×C61#13712yes—
125C2×C4×C61C23×C411yescharacteristic
126C2×C4×C61C22×D411yescharacteristic
127C2×Dic61—22no—
128C2×Dic61—24no—
129C2×Dic61—24no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138Dic122—44no—
139Dic122—84no—
140Dic122—82no—
141C8×C61—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#41914yes—
154Dic61—42no—
155Dic61—44no—
156Dic61—44no—
157Dic61—81no—
158Dic61—82no—
159#3502—611no—
160#2983D6111yescharacteristic
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
171Q82—614no—
172#557—614no—
173#557—1224no—
174#544—1221no—
175#544—1222no—
176#533—612no—
177#533—614no—
178#519—1221no—
179C2×C4×Q8C2×D6111yescharacteristic
180#446—24no—
181#432C2×D6114yes—
182#431C2×D6112yes—
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×Q8D24411yescharacteristic
195C4⋊Q8—612no—
196C4⋊Q8—614no—
197C4⋊Q8—612no—
198C4⋊Q8—614no—
199C4⋊Q8—1228no—
200C4⋊Q8—1224no—
201#125—1228no—
202#125—1224no—
203#125—1222no—
204#125—1224no—
205C22⋊Q8—614no—
206C22⋊Q8—1224no—
207C22⋊Q8—1222no—
208C4×Q8—614no—
209C4×Q8—618no—
210C4×Q8—1224no—
211C4×Q8—1224no—
212C4×Q8—1222no—
213C4×Q8—1224no—
214C4×Q8C4×D6114yes—
215C4×Q8C22⋊D6114yes—
216#117—1224no—
217#117—1222no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4D24411yescharacteristic
221C2×C4⋊C4D24412yes—
222C2×C42—1222no—
223C2×C42—1221no—
224C2×C42C22×D6112yes—
225C2×C42C22×D6111yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—24no—
228C22⋊C8C22×D6114yes—
229#97—24no—
230C4×C8—22no—
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—614no—
241C2×Q8—1222no—
242C2×Q8—1224no—
243C2×Q8#1175819814yes—
244C22×C4—1221no—
245C22×C4—1222no—
246C22×C4—1221no—
247C22×C4C2×D24412yes—
248C22×C4C2×D24412yes—
249C22×C4C2×D24411yescharacteristic
250C22×C4C2×D24411yescharacteristic
251C22×C4C23×D6111yescharacteristic
252C2×C8—24no—
253C4⋊C4—614no—
254C4⋊C4—1224no—
255C4⋊C4—1224no—
256C4⋊C4—1228no—
257C4⋊C4—1224no—
258C4⋊C4—1222no—
259C4⋊C4—1224no—
260C4⋊C4—1222no—
261C4⋊C4—1222no—
262C4⋊C4—1224no—
263C4⋊C4—2442no—
264C4⋊C4—2442no—
265C4⋊C4#1175819818yes—
266C4⋊C4#1175819814yes—
267C22⋊C4—1224no—
268C42—1228no—
269C42—1224no—
270C42—2444no—
271C42—2442no—
272C42C2×C4×D6114yes—
273C42C2×C4×D6112yes—
274C42C2×C22⋊D6112yes—
275C42C2×C22⋊D6114yes—
276C23C22×D24411yescharacteristic
277Q8—28no—
278Q8—4884no—
279Q8—4882no—
280Q8—2444no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—1222no—
287C2×C4—1224no—
288C2×C4—1224no—
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×#1175819814yes—
298C2×C4C2×#1175819812yes—
299C2×C4C2×#1175819814yes—
300C2×C4C2×#1175819812yes—
301C2×C4#1451981012yes—
302C2×C4#1451981012yes—
303C2×C4C22×C4×D6111yescharacteristic
304C2×C4C22×C22⋊D6111yescharacteristic
305C8—44no—
306C22—21no—
307C22—21no—
308C22C22×#1175819811yescharacteristic
309C22C2×#1451981011yescharacteristic
310C22C2×#1451981011yescharacteristic
311C4—24no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—24no—
316C4—4882no—
317C4—4881no—
318C4—2442no—
319C4—2444no—
320C4—2444no—
321C4#1683977214yes—
322C2—21no—
323C2—21no—
324C2#2145428211yescharacteristic
325C2C2×#1683977211yescharacteristic
326C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2983C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenter#2983D61
hypocenterC61#17632
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2983

Complements

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

NH ≅ G/Ntype
8. Q8×Dic122C4semidirect
9. #16840491C4semidirect
193. C61#17632semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214487842
#214443842
#214443712
#214443322
#214443192
C61×#29832
#1763261