Finite Group Databasebeta

← #29669343

Subgroups — #29669343

Order 15616.

736 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 163 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#21456560C212yesmaximal, complemented
2#21456195C212yesmaximal
3#21456194C211yesmaximal, characteristic
4#21445482C212yesmaximal, complemented
5#21445477C212yesmaximal, complemented
6#21445476C212yesmaximal, complemented
7#21445432C212yesmaximal, complemented
8#21444122C211yesmaximal, characteristic, complemented
9C61×#3185C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11C2×#14519774C2211yescharacteristic
12#16840158C2212yescomplemented
13#16840141C2212yescomplemented
14#16840128—24no—
15#16840127—22no—
16#16840114C2212yescomplemented
17#16840112C2212yescomplemented
18#16840110C2212yes—
19#16840110C2212yes—
20#16840110C2212yes—
21#16840110C2212yes—
22#16840109C2212yes—
23#16840109C2212yes—
24#16840018—24no—
25#16839867—24no—
26#16839768—22no—
27#16839766—24no—
28C4×#11758194C2212yes—
29#16839484C2212yescomplemented
30#16839483C2211yescharacteristic, complemented
31C8×#92418C2211yescharacteristic, complemented
32C61×#533C2212yes—
33C61×#498C2211yescharacteristic, complemented
34C61×#494C2211yescharacteristic, complemented
35C61×#487C2212yescomplemented
36C61×#470—22no—
37C61×#441—22no—
38C4×M16×C61C2211yescharacteristic
39C22×D244—22no—
40C22×D244—21no—
41C2×#11758202—24no—
42C2×#11758194D414yes—
43#14519777C2312yes—
44#14519774—21no—
45#14519774—22no—
46#14519774C2312yes—
47#14519769—21no—
48#14519756—22no—
49C4×Dic122—22no—
50#14519736—21no—
51#14519708—24no—
52#14519701—24no—
53#14519700—24no—
54#14519699—24no—
55#14519699—24no—
56#14519697—24no—
57#14519697—24no—
58#14519697—24no—
59#14519697—24no—
60#14519688D414yes—
61C4×#92418C2312yes—
62C4×#92418C2312yes—
63C2×Q16×C61—22no—
64C2×M16×C61D412yes—
65C4⋊Q8×C61C2312yes—
66C4⋊Q8×C61C2312yes—
67#125×C61—22no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72#117×C61—22no—
73C2×C42×C61C2311yescharacteristic
74#106×C61—22no—
75#103×C61—24no—
76#97×C61D412yes—
77C4×C8×C61C2311yescharacteristic
78C4×C8×C61C2311yescharacteristic
79C23×D61—41no—
80C2×Q8×C61—22no—
81C2×Q8×C61—22no—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C2×D244—22no—
85C2×D244—22no—
86C2×D244—22no—
87C2×D244—48no—
88C2×D244—44no—
89C2×D244—41no—
90C2×D244—42no—
91C2×Dic122—22no—
92Q16×C61—44no—
93M16×C61—42no—
94C2×C8×C61C2×D412yes—
95C2×C8×C61C2×D412yes—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—42no—
101C4⋊C4×C61—42no—
102C22⋊C4×C61—42no—
103C42×C61—42no—
104C42×C61C2×D412yes—
105C42×C61C2×D411yescharacteristic
106C42×C61C2411yescharacteristic
107C22⋊Dic61—41no—
108#11758202—48no—
109C4⋊Dic61—22no—
110C4⋊Dic61—22no—
111#11758196—42no—
112C4×Dic61—41no—
113#11758194—44no—
114C2×#92418C2×D414yes—
115C2×#92418C2×D414yes—
116C23×C61#12711yescharacteristic
117C22×D61—42no—
118C22×D61—84no—
119C22×D61—82no—
120Q8×C61—44no—
121Q8×C61—44no—
122C2×C4×C61—24no—
123C2×C4×C61—22no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61#12711yescharacteristic
127C2×C4×C61#12712yes—
128C2×C4×C61C22×D412yes—
129C2×C4×C61C22×D411yescharacteristic
130C2×Dic61—42no—
131D244—44no—
132D244—44no—
133D244—44no—
134D244—84no—
135D244—82no—
136Dic122—44no—
137C8×C61—22no—
138C8×C61—22no—
139#92418—24no—
140#92418—24no—
141#19753—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61C2×#12711yescharacteristic
145C2×D61—82no—
146C2×D61—82no—
147C2×D61—82no—
148C2×D61—164no—
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—82no—
152C4×C61—82no—
153C4×C61#57512yes—
154C4×C61#57514yes—
155Dic61—82no—
156#3185—612no—
157#3185D6111yescharacteristic, complemented
158#3138—611no—
159#3138—612no—
160#1720—612no—
161#1720—612no—
162#1720—612no—
163#1711—612no—
164#1455—611no—
165C2×C61—41no—
166C2×C61#315312yes—
167C2×C61#315311yescharacteristic
168D61—162no—
169#533—611no—
170#533C2×D6112yes—
171C2×#127—611no—
172#498—612no—
173#498C2×D6111yescharacteristic, complemented
174#494—612no—
175#494C2×D6111yescharacteristic, complemented
176#490—1224no—
177#490—1222no—
178#487—612no—
179#487—612no—
180#487C2×D6112yescomplemented
181#486—612no—
182#486—612no—
183#486—612no—
184#486—612no—
185#486—612no—
186#486—612no—
187#470—22no—
188#470—1224no—
189#441—22no—
190#441—1224no—
191#418—1224no—
192#418—1222no—
193C4×M16—612no—
194C4×M16C2×D6111yescharacteristic
195#322—611no—
196#322—612no—
197C82—611no—
198C61#1975311yescharacteristic, complemented, Sylow
199C22×D4—1222no—
200C22×D4—1221no—
201C2×Q16—22no—
202C2×Q16—1224no—
203C2×M16—614no—
204C2×M16C22⋊D6112yes—
205C4⋊Q8—612no—
206C4⋊Q8C22×D6112yes—
207C4⋊Q8C22×D6112yes—
208#127—612no—
209#127—1222no—
210#127—1221no—
211#125—22no—
212#125—1221no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—1222no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—1222no—
219#117—22no—
220#117—1221no—
221C2×C42C22×D6111yescharacteristic
222#106—22no—
223#106—1224no—
224#103—24no—
225#103—1224no—
226#103—1224no—
227#102—1224no—
228#102—1224no—
229#102—1224no—
230#102—1224no—
231#102—1224no—
232#102—1224no—
233#97—614no—
234#97C22⋊D6112yes—
235C4×C8—612no—
236C4×C8—612no—
237C4×C8C22×D6111yescharacteristic
238C4×C8C22×D6111yescharacteristic
239C24—2441no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—1222no—
243C2×D4—1222no—
244C2×D4—1222no—
245C2×D4—1222no—
246C2×D4—2444no—
247C2×D4—2448no—
248C2×D4—2442no—
249C2×D4—2441no—
250C22×C4D4×D6112yes—
251C22×C4C2×C22⋊D6111yescharacteristic
252Q16—44no—
253Q16—2448no—
254M16—42no—
255M16—2444no—
256C2×C8—614no—
257C2×C8—614no—
258C2×C8C2×C22⋊D6112yes—
259C2×C8C2×C22⋊D6112yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—1222no—
267C4⋊C4—1222no—
268C4⋊C4—2442no—
269C22⋊C4—42no—
270C22⋊C4—2441no—
271C42—42no—
272C42—2441no—
273C42D4×D6112yes—
274C42C2×C22⋊D6111yescharacteristic
275C42C23×D6111yescharacteristic
276C23—4884no—
277C23—4882no—
278C23—2442no—
279C23#1451977311yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—2444no—
283D4—4884no—
284D4—4882no—
285D4—2444no—
286D4—2444no—
287D4—2444no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—2442no—
293C2×C4#1451977312yes—
294C2×C4#1451977311yescharacteristic
295C2×C4C2×D4×D6112yes—
296C2×C4C22×C22⋊D6111yescharacteristic
297C8—22no—
298C8—22no—
299C8—1224no—
300C8—1224no—
301C22—22no—
302C22—21no—
303C22—9764no—
304C22—4882no—
305C22—4882no—
306C22—4882no—
307C22C2×#1451977311yescharacteristic
308C4—22no—
309C4—21no—
310C4—82no—
311C4—82no—
312C4—4882no—
313C4#1684056012yes—
314C4#1684056114yes—
315C2—41no—
316C2—9762no—
317C2#2145628612yes—
318C2#2145628711yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3185C2
socleC22×C61C2×#127
prime radicalC2×#14519774C22
hypercenter#3185D61
hypocenterC61#19753
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
1. #21456560C2semidirect
4. #21445482C2semidirect
5. #21445477C2semidirect
6. #21445476C2semidirect
7. #21445432C2semidirect
8. #21444122C2semidirect
9. C61×#3185C2semidirect
12. #16840158C22semidirect
13. #16840141C22semidirect
16. #16840114C22semidirect
17. #16840112C22semidirect
29. #16839484C22semidirect
30. #16839483C22semidirect
31. C8×#92418C22semidirect
33. C61×#498C22semidirect
34. C61×#494C22semidirect
35. C61×#487C22semidirect
157. #3185D61semidirect
173. #498C2×D61semidirect
175. #494C2×D61semidirect
180. #487C2×D61semidirect
198. C61#19753semidirect

Maximal subgroups

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

subgroupindex
#214565602
#214561952
#214561942
#214454822
#214454772
#214454762
#214454322
#214441222
C61×#31852
#1975361