Finite Group Databasebeta

← #29590971

Subgroups — #29590971

Order 15616.

336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 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#21456390C211yesmaximal, characteristic
2#21447619C211yesmaximal, characteristic, complemented
3#21445945C211yesmaximal, characteristic, complemented
4#21443831C211yesmaximal, characteristic, complemented
5#21443829C211yesmaximal, characteristic
6#21443827C211yesmaximal, characteristic
7C61×#1348C211yesmaximal, characteristic, complemented
8#16840298—21no—
9#16840163C411yescharacteristic, complemented
10#16840147C411yescharacteristic, complemented
11#16840118—21no—
12#16839998C2211yescharacteristic
13#16839993—21no—
14Q16×Dic61C411yescharacteristic, complemented
15#16839847—21no—
16#16839844C411yescharacteristic, complemented
17C2×C8×Dic61C2211yescharacteristic
18#16839631—21no—
19#16839580—21no—
20C22⋊#11758185C2211yescharacteristic
21#16839537—21no—
22#16839525C2211yescharacteristic
23#16839524—21no—
24C61×#467C2211yescharacteristic
25C61×C22⋊C16C2211yescharacteristic
26C61×#340C2211yescharacteristic
27C2×#11758201—21no—
28C2×D488C2×C411yescharacteristic
29C22×#92418D411yescharacteristic
30#14519772—41no—
31#14519768—21no—
32#14519766—21no—
33#14519761—41no—
34#14519751—21no—
35Q8×Dic61—21no—
36#14519739—21no—
37#14519737—41no—
38#14519735—41no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—41no—
41C2×#11758196—41no—
42C2×C4×Dic61D411yescharacteristic
43C2×#11758185—21no—
44#14519711C2×C411yescharacteristic
45#14519708—21no—
46#14519701—21no—
47#14519699—21no—
48#14519697—21no—
49C8×Dic61C2×C411yescharacteristic
50C8×Dic61C2×C411yescharacteristic
51C8×Dic61D411yescharacteristic
52C8×Dic61D411yescharacteristic
53C2×Q16×C61C2×C411yescharacteristic
54C22×C8×C61C2311yescharacteristic
55C22⋊Q8×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C16×C61—21no—
58#107×C61C2×C411yescharacteristic
59#103×C61—21no—
60C2×Q8×C61—21no—
61C22×C4×C61—41no—
62C22×C4×C61C2×D411yescharacteristic
63C2×C22⋊D61—41no—
64C22×Dic61—41no—
65C22×Dic61D811yescharacteristic
66C22×Dic61D811yescharacteristic
67C2×D244—21no—
68C2×C4×D61—41no—
69Q16×C61—41no—
70C2×C8×C61—41no—
71C2×C8×C61C22×C411yescharacteristic
72C2×C8×C61C2×D411yescharacteristic
73C4⋊C4×C61—21no—
74C4⋊C4×C61—41no—
75C4⋊C4×C61—41no—
76C4⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78#11758201—42no—
79#11758198—41no—
80#11758198—41no—
81#11758198—41no—
82#11758198—41no—
83C4⋊Dic61—21no—
84C4⋊Dic61—41no—
85C4⋊Dic61—41no—
86C4⋊Dic61—41no—
87C4⋊Dic61—41no—
88#11758196—41no—
89#11758196—82no—
90#11758196—82no—
91C4×Dic61—41no—
92C4×Dic61C22⋊C411yescharacteristic
93C4×Dic61C22⋊C411yescharacteristic
94C4×Dic61SD1611yescharacteristic
95C4×Dic61SD1611yescharacteristic
96C2×#92418—41no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C22⋊C411yescharacteristic
99D488—41no—
100C16×C61—41no—
101#11758185—41no—
102C23×C61C2×D811yescharacteristic
103C22×D61—41no—
104Q8×C61—41no—
105Q8×C61—41no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—81no—
111C2×C4×C61—81no—
112C2×C4×C61—81no—
113C2×C4×C61C2×C22⋊C411yescharacteristic
114C2×C4×C61C2×SD1611yescharacteristic
115C22⋊D61—81no—
116C22⋊D61—81no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61—81no—
122C2×Dic61—81no—
123C2×Dic61—81no—
124C2×Dic61#10211yescharacteristic
125C2×Dic61#10211yescharacteristic
126C2×Dic61#10211yescharacteristic
127C2×Dic61#10211yescharacteristic
128D244—41no—
129D244—41no—
130C4×D61—82no—
131C8×C61—21no—
132C8×C61—21no—
133#92418—21no—
134#92418—21no—
135#11860—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61C2×#10211yescharacteristic
139C22×C61#50911yescharacteristic
140C22×C61#51011yescharacteristic
141C2×D61—81no—
142C2×D61—81no—
143C2×D61—81no—
144C4×C61—21no—
145C4×C61—81no—
146C4×C61—81no—
147C4×C61—161no—
148C4×C61#41011yescharacteristic
149C4×C61#42111yescharacteristic
150Dic61—21no—
151Dic61—21no—
152Dic61—21no—
153Dic61—81no—
154Dic61—161no—
155Dic61#36012yes—
156#3168—611no—
157#2163—611no—
158#1834—611no—
159#1348—611no—
160#1348D6111yescharacteristic, complemented
161#1347—611no—
162#1347—611no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61#151711yescharacteristic
166C2×C61#213911yescharacteristic
167C2×C61C2×#36011yescharacteristic
168D61—161no—
169#529—1221no—
170#499—611no—
171#495—611no—
172#488—1221no—
173#467C2×D6111yescharacteristic
174#466—611no—
175#465—1221no—
176C4×Q16—611no—
177C4×SD16—1221no—
178C4×D8—611no—
179C2×C4×C8—611no—
180#384—1221no—
181C4⋊C16—1221no—
182C22⋊C16—611no—
183C22⋊C16C2×D6111yescharacteristic
184#346—1221no—
185#340—611no—
186#340—1221no—
187#340C2×D6111yescharacteristic
188C61#1186011yescharacteristic, complemented, Sylow
189C2×Q16C4×D6111yescharacteristic
190C2×SD16—1221no—
191C2×D8—611no—
192C22×C8—611no—
193C22×C8C22×D6111yescharacteristic
194#126—2441no—
195#125—1221no—
196#124—1221no—
197#123—2441no—
198C22⋊Q8—21no—
199#121—1221no—
200C4×Q8—1221no—
201C4×D4—1221no—
202C4×D4—2441no—
203#117—2441no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—2441no—
207C2×C4⋊C4—2441no—
208C2×C42—611no—
209C2×C16—21no—
210C2×C16—1221no—
211#107—611no—
212#107C4×D6111yescharacteristic
213#106—1221no—
214#103—21no—
215#103—1221no—
216#102—1221no—
217#102—1221no—
218C4×C8—611no—
219C4×C8—611no—
220C4×C8—611no—
221C4×C8—611no—
222C2×Q8—21no—
223C2×D4—1221no—
224C2×D4—2441no—
225C22×C4—41no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—2441no—
229C22×C4—2441no—
230C22×C4D4×D6111yescharacteristic
231Q16—41no—
232SD16—2442no—
233D8—2441no—
234C2×C8—41no—
235C2×C8—611no—
236C2×C8—611no—
237C2×C8—2441no—
238C2×C8C2×C4×D6111yescharacteristic
239C2×C8D4×D6111yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—1221no—
245C4⋊C4—4882no—
246C4⋊C4—4882no—
247C4⋊C4—2441no—
248C4⋊C4—2441no—
249C4⋊C4—2441no—
250C4⋊C4—2441no—
251C4⋊C4—2441no—
252C22⋊C4—41no—
253C22⋊C4—2441no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C22⋊C4—2441no—
257C42—611no—
258C42—611no—
259C42—611no—
260C42—611no—
261C42—2441no—
262C16—41no—
263C16—2441no—
264C23—2441no—
265C23D8×D6111yescharacteristic
266Q8—41no—
267Q8—41no—
268D4—4881no—
269D4—4881no—
270D4—2441no—
271D4—2441no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—611no—
280C2×C4—611no—
281C2×C4—611no—
282C2×C4—611no—
283C2×C4—1222no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—4882no—
287C2×C4—4881no—
288C2×C4—4881no—
289C2×C4—4881no—
290C2×C4—2441no—
291C2×C4C22⋊C4×D6111yescharacteristic
292C2×C4SD16×D6111yescharacteristic
293C8—21no—
294C8—21no—
295C8—1221no—
296C8—1221no—
297C22—21no—
298C22—21no—
299C22—4881no—
300C22—4881no—
301C22—4881no—
302C22#102×D6111yescharacteristic
303C22#1684019911yescharacteristic
304C22#1684020411yescharacteristic
305C4—21no—
306C4—81no—
307C4—81no—
308C4—161no—
309C4—612no—
310C4—1221no—
311C4—1221no—
312C4—1221no—
313C4—9761no—
314C4—4881no—
315C4#1683973411yescharacteristic
316C4#1683977711yescharacteristic
317C2—21no—
318C2—21no—
319C2—9761no—
320C2#2144437711yescharacteristic
321C2#2144753011yescharacteristic
322C2#360×D6111yescharacteristic
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#1348C2
socleC22×C61C2×#102
prime radical#16839998C22
cosocleC22×C8×C61C23
hypercenter#1348D61
hypocenterC61#11860
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #21447619C2semidirect
3. #21445945C2semidirect
4. #21443831C2semidirect
7. C61×#1348C2semidirect
9. #16840163C4semidirect
10. #16840147C4semidirect
14. Q16×Dic61C4semidirect
16. #16839844C4semidirect
160. #1348D61semidirect
188. C61#11860semidirect

Maximal subgroups

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

subgroupindex
#214563902
#214476192
#214459452
#214438312
#214438292
#214438272
C61×#13482
#1186061