Finite Group Databasebeta

← #30604991

Subgroups — #30604991

Order 15616.

428 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 68 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#21461524C211yesmaximal, characteristic
2#21461072C211yesmaximal, characteristic
3#21460820C212yesmaximal, complemented
4#21460803C212yesmaximal, complemented
5#21451390C211yesmaximal, characteristic
6C2×D61⋊M16—21no—
7#16840782—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840606—21no—
12C2×#14519734C2211yescharacteristic
13#16839693C2211yescharacteristic
14#16839688C411yescharacteristic
15#16839681C2211yescharacteristic
16#16839673C411yescharacteristic
17#16839653—21no—
18#16839646—21no—
19#16839643—21no—
20#16839633—21no—
21#16839630C411yescharacteristic
22C2×C4×Q8×C61C411yescharacteristic
23C2×#11758213—41no—
24D61⋊M16—24no—
25D61⋊M16—42no—
26C2×#11758217—22no—
27C2×D61⋊C8—41no—
28#14519844—41no—
29C22⋊#92420—22no—
30#14519832—41no—
31C22×C4×D61D411yescharacteristic
32#14519734—41no—
33#14519734—41no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196D411yescharacteristic
37C2×#11758196C2311yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198C2×C411yescharacteristic
41C2×#11758198D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51C22×Q8×C61C2×C411yescharacteristic
52C4×Q8×C61—41no—
53C4×Q8×C61—41no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C42×C61—21no—
57C2×C42×C61C2×C411yescharacteristic
58C23×D61C22⋊C411yescharacteristic
59C2×Q8×C61—42no—
60C2×Q8×C61—41no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C2×C4×D61—22no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76#11758217—24no—
77#11758217—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80#11758213—44no—
81D61⋊C8—44no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—42no—
88#11758198—41no—
89#11758198—41no—
90#11758198—42no—
91#11758198—42no—
92#11758196—41no—
93#11758196—41no—
94#11758196—42no—
95#11758196—41no—
96#11758196—41no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61#10412yes—
105Q8×C61—42no—
106Q8×C61—42no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—22no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61#9911yescharacteristic
120C2×C4×C61#9911yescharacteristic
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61AGL(1,ℤ/8)11yescharacteristic
134C2×Dic61#13711yescharacteristic
135C4×D61—22no—
136C4×D61—22no—
137C4×D61—22no—
138C4×D61—22no—
139#92420—44no—
140#92420—82no—
141#11763—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#9911yescharacteristic
145C22×C61#41711yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C2×D61—22no—
148C2×D61—41no—
149C2×D61—41no—
150C2×D61—42no—
151C2×D61—81no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—42no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—81no—
158Dic61—21no—
159Dic61—21no—
160Dic61—41no—
161Dic61—41no—
162Dic61—81no—
163Dic61—81no—
164#2599—611no—
165#1742—611no—
166#1464—611no—
167#1354—612no—
168#1346—612no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61#150411yescharacteristic
172C2×C61#212011yescharacteristic
173C2×C61#213511yescharacteristic
174D61—42no—
175C22×M16—1221no—
176C2×C4×Q8#692311yescharacteristic
177C2×#117—611no—
178#432—1221no—
179#398—611no—
180#397—611no—
181#395—611no—
182#393—611no—
183#388—1221no—
184#387—1221no—
185#387—1221no—
186#385—1221no—
187#383—611no—
188#343—612no—
189#343—1221no—
190#324—612no—
191#323—1221no—
192C61#1176311yescharacteristic, complemented, Sylow
193C22×Q8C2×#692311yescharacteristic
194C23×C4—611no—
195C2×M16—1222no—
196C2×M16—1224no—
197C2×M16—2442no—
198C2×M16—2441no—
199C22×C8—2441no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—2441no—
203#117—2441no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4C2×#692311yescharacteristic
210C2×C22⋊C4—611no—
211C2×C22⋊C4—611no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C42—21no—
215C2×C42—611no—
216C2×C42C2×#692311yescharacteristic
217C4⋊C8—2441no—
218C22⋊C8—1222no—
219#97—2441no—
220#95—611no—
221#95—611no—
222#95—1221no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228C24—611no—
229C2×Q8—42no—
230C2×Q8—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—1222no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22⋊#692311yescharacteristic
245C22×C4C22×#692311yescharacteristic
246M16—1224no—
247M16—2442no—
248M16—2444no—
249C2×C8—2442no—
250C2×C8—2444no—
251C2×C8—2442no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C4⋊C4—2442no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C42—21no—
265C42—21no—
266C42—42no—
267C42—1221no—
268C42—1221no—
269C23—612no—
270C23—1221no—
271C23—1221no—
272C23—2441no—
273C23—2441no—
274C23C2×C22⋊#692311yescharacteristic
275Q8—42no—
276Q8—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—611no—
290C2×C4—611no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451983511yescharacteristic
308C2×C4#1451983511yescharacteristic
309C8—4882no—
310C8—2444no—
311C22—21no—
312C22—21no—
313C22—1222no—
314C22—4881no—
315C22—2442no—
316C22—2441no—
317C22—2441no—
318C22C2×#1451983511yescharacteristic
319C22#1684075311yescharacteristic
320C22D61⋊#10411yescharacteristic
321C4—21no—
322C4—21no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—1221no—
328C4—1221no—
329C4—4881no—
330C4—4881no—
331C4—2441no—
332C4—2441no—
333C2—21no—
334C2—21no—
335C2—2442no—
336C2#2146120511yescharacteristic
337C2#2146207011yescharacteristic
338C2#2146212111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×#11758196C23
hypercenterC2×C4×Q8#6923
hypocenterC61#11763
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #21460820C2semidirect
4. #21460803C2semidirect
192. C61#11763semidirect

Maximal subgroups

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

subgroupindex
#214615242
#214610722
#214608202
#214608032
#214513902
#1176361