Finite Group Databasebeta

← #29590731

Subgroups — #29590731

Order 15616.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#21449087C211yesmaximal, characteristic
2#21445896C211yesmaximal, characteristic
3#21443844C212yesmaximal
4#21443841C212yesmaximal
5C61×#1464C211yesmaximal, characteristic
6C2×C4×Dic122C411yescharacteristic, complemented
7#16839701C411yescharacteristic, complemented
8#16839698C2211yescharacteristic
9#16839695—21no—
10#16839685—21no—
11#16839682C411yescharacteristic, complemented
12#16839674C2211yescharacteristic
13#16839625—21no—
14C4×#11758196—21no—
15C4×C4⋊Dic61C411yescharacteristic, complemented
16C4×C22⋊Dic61—21no—
17#16839529—21no—
18#16839529C2212yes—
19C2×#117×C61C2211yescharacteristic
20C61×#432—21no—
21C61×#324—21no—
22C61×#323C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C4×Dic122—41no—
25C4×Dic122—41no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—22no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—21no—
36#14519684—21no—
37#14519684—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—41no—
41#14519684—41no—
42#14519684C2×C411yescharacteristic
43#14519684C2×C411yescharacteristic
44C23×C4×C61D411yescharacteristic
45#117×C61—41no—
46#117×C61—41no—
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61C2311yescharacteristic
51C2×C42×C61C2×C411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C4⋊C8×C61—41no—
54C22⋊C8×C61—22no—
55#97×C61—41no—
56C24×C61C2×D411yescharacteristic
57C22×C4×C61—22no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—41no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—42no—
72C2×Dic122—41no—
73C2×Dic122—42no—
74C2×C8×C61—42no—
75C4⋊C4×C61—41no—
76C4⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C22⋊C4×C61—41no—
79C42×C61—21no—
80C42×C61—21no—
81C42×C61—21no—
82C42×C61—21no—
83C22⋊Dic61—42no—
84C22⋊Dic61—42no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C4⋊Dic61—42no—
88C4⋊Dic61—41no—
89#11758196—42no—
90#11758196—42no—
91#11758196—42no—
92C4×Dic61—42no—
93C4×Dic61—41no—
94C4×Dic61—41no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61C2×C22⋊C411yescharacteristic
100C23×C61#13712yes—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61#9911yescharacteristic
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—82no—
131C2×Dic61—82no—
132C2×Dic61—82no—
133Dic122—42no—
134Dic122—42no—
135C8×C61—82no—
136#11810—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—22no—
140C22×C61—42no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—81no—
144C22×C61C2×#9911yescharacteristic
145C22×C61#41911yescharacteristic
146C22×C61#42111yescharacteristic
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—41no—
154Dic61—42no—
155Dic61—42no—
156Dic61—81no—
157Dic61—81no—
158Dic61—162no—
159#2396—611no—
160#1820—611no—
161#1464D6111yescharacteristic
162#1352—612no—
163#1351—612no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61—42no—
167C2×C61#151111yescharacteristic
168C2×C61#212011yescharacteristic
169C2×C61#212611yescharacteristic
170C2×C4×Q8—611no—
171C2×#117C2×D6111yescharacteristic
172#432—21no—
173#400—611no—
174#399—611no—
175#398—1221no—
176#396—1221no—
177#395—611no—
178#393—611no—
179#381—1221no—
180C4×C4⋊C4—611no—
181C4×C4⋊C4—1221no—
182C4×C22⋊C4—1221no—
183#342—612no—
184#342—1221no—
185#324—21no—
186#323C2×D6112yes—
187C61#1181011yescharacteristic, complemented, Sylow
188C22×Q8—611no—
189C23×C4D24411yescharacteristic
190C4×Q8—2441no—
191C4×Q8—2441no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—611no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4C4×D6111yescharacteristic
200C2×C4⋊C4C22⋊D6111yescharacteristic
201C2×C22⋊C4—1221no—
202C2×C22⋊C4—1221no—
203C2×C22⋊C4—1222no—
204C2×C22⋊C4D24411yescharacteristic
205C2×C22⋊C4C22×D6111yescharacteristic
206C2×C42—1221no—
207C2×C42—1221no—
208C2×C42C4×D6111yescharacteristic
209C2×C42C22⋊D6111yescharacteristic
210C4⋊C8—41no—
211C22⋊C8—22no—
212#97—41no—
213#95—611no—
214#95—611no—
215#95—1221no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—2441no—
220#95—2441no—
221C24C2×D24411yescharacteristic
222C2×Q8—2441no—
223C2×Q8—2442no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—1221no—
229C22×C4—1221no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4—2442no—
233C22×C4#1175819811yescharacteristic
234C22×C4#1175819811yescharacteristic
235C22×C4#1175819811yescharacteristic
236C22×C4#1175819811yescharacteristic
237C22×C4C2×C4×D6111yescharacteristic
238C22×C4C2×C22⋊D6111yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—2441no—
243C4⋊C4—2442no—
244C4⋊C4—2442no—
245C4⋊C4—2442no—
246C4⋊C4—2442no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—2442no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C22⋊C4—2442no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—2442no—
258C42—2441no—
259C42—2441no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×#1175819811yescharacteristic
265C23#1451981012yes—
266Q8—2442no—
267Q8—2442no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—1222no—
284C2×C4—1222no—
285C2×C4—4882no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4#1451969111yescharacteristic
299C2×C4#1451969111yescharacteristic
300C8—82no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1451969111yescharacteristic
309C22#1683977211yescharacteristic
310C22#1683977811yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—9762no—
319C4—4881no—
320C4—4881no—
321C4—2442no—
322C4—2442no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2144435311yescharacteristic
327C2#2144746111yescharacteristic
328C2#2144748211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1464C2
socleC22×C61C2×#99
prime radicalC24×C61C2×D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1464D61
hypocenterC61#11810
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic122C4semidirect
7. #16839701C4semidirect
11. #16839682C4semidirect
15. C4×C4⋊Dic61C4semidirect
187. C61#11810semidirect

Maximal subgroups

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

subgroupindex
#214490872
#214458962
#214438442
#214438412
C61×#14642
#1181061