Finite Group Databasebeta

← #30604813

Subgroups — #30604813

Order 15616.

468 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461519C211yesmaximal, characteristic
2#21461053C211yesmaximal, characteristic
3#21460816C212yesmaximal
4#21460813C212yesmaximal
5#21449092C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840779—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×#14519742C411yescharacteristic, complemented
13C2×#14519735C2211yescharacteristic
14#16839697C2211yescharacteristic
15#16839696—21no—
16#16839686—21no—
17#16839682C411yescharacteristic, complemented
18#16839675C2211yescharacteristic
19#16839613—21no—
20#16839613C411yescharacteristic, complemented
21#16839609—21no—
22C61×#400C411yescharacteristic, complemented
23#14519856—24no—
24#14519856—42no—
25C2×C4⋊#6923—41no—
26C2×C22⋊#6923—22no—
27C2×C4×#6923—41no—
28#14519844—41no—
29C22⋊#92420—22no—
30C4×#92420—41no—
31C22×Dic122C2×C411yescharacteristic
32C22×C4×D61D411yescharacteristic
33#14519742—41no—
34#14519742—41no—
35#14519735—41no—
36#14519735—41no—
37C2×C4⋊Dic61D411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198D411yescharacteristic
45C2×#11758198C2311yescharacteristic
46C2×C4×Dic61—21no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684C2×C411yescharacteristic
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C4⋊C4×C61C2×C411yescharacteristic
55#95×C61—21no—
56#95×C61—21no—
57#95×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—22no—
73C2×C4×D61SD1612yes—
74C2×C4×D61Q1612yes—
75C2×Dic122—41no—
76C2×Dic122—42no—
77C22⋊#6923—24no—
78C22⋊#6923—42no—
79C2×#92420—42no—
80C4⋊#6923—44no—
81C4×#6923—44no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84#11758198—41no—
85#11758198—41no—
86#11758198—42no—
87#11758198—42no—
88C4⋊Dic61—41no—
89#11758196—42no—
90#11758196—42no—
91#11758196—42no—
92#11758196—42no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—41no—
98C4×Dic61—41no—
99C4×Dic61—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61C2×SD1611yescharacteristic
106C22×D61C2×Q1611yescharacteristic
107C2×#6923—44no—
108C2×#6923—82no—
109C2×#6923—82no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61#10312yes—
123C2×Dic61—22no—
124C2×Dic61—22no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61#9911yescharacteristic
134C2×Dic61#9911yescharacteristic
135C2×Dic61#10312yes—
136C2×Dic61#10412yes—
137C4×D61—22no—
138C4×D61—22no—
139C4×D61—22no—
140C4×D61—22no—
141Dic122—42no—
142Dic122—42no—
143#92420—82no—
144#11715—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#10311yescharacteristic
149C22×C61C2×#10411yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—41no—
153C2×D61—41no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—81no—
157#6923—84no—
158C4×C61—22no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—81no—
162C4×C61—81no—
163Dic61—42no—
164Dic61—42no—
165Dic61—41no—
166Dic61—41no—
167Dic61#32812yes—
168Dic61#32812yes—
169#2396—611no—
170#1740—611no—
171#1458—611no—
172#1352—612no—
173#1351—612no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61C2×#32811yescharacteristic
177C2×C61#212011yescharacteristic
178C2×C61#212611yescharacteristic
179D61—41no—
180D61—41no—
181C2×C4×Q8—611no—
182C2×#117—611no—
183C2×#117—1221no—
184#431—1221no—
185#400#692311yescharacteristic, complemented
186#399—611no—
187#398—1221no—
188#396—1221no—
189#395—611no—
190#393—611no—
191C4×C4⋊C4—611no—
192C4×C4⋊C4—1221no—
193C4×C22⋊C4—1221no—
194#342—612no—
195#342—1221no—
196#324—1221no—
197#323—612no—
198C61#1171511yescharacteristic, complemented, Sylow
199C22×Q8—611no—
200C23×C4—611no—
201C4×Q8—2441no—
202C4×Q8—2441no—
203#117—1224no—
204#117—2441no—
205#117—2441no—
206#117—2442no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—2441no—
213C2×C4⋊C4C2×#692311yescharacteristic
214C2×C4⋊C4C2×#692311yescharacteristic
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C42—611no—
221C2×C42—611no—
222C2×C42—1221no—
223C2×C42—1221no—
224C2×C42—2441no—
225C4⋊C8—2441no—
226C22⋊C8—1222no—
227C4×C8—2441no—
228#95—21no—
229#95—21no—
230#95—611no—
231#95—1221no—
232#95—1221no—
233#95C2×#692311yescharacteristic
234C24—611no—
235C2×Q8—2441no—
236C2×Q8—2442no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1222no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—2442no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22×#692311yescharacteristic
253C2×C8—2442no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2441no—
257C4⋊C4—2442no—
258C4⋊C4—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2444no—
262C22⋊C4—1224no—
263C22⋊C4—2441no—
264C22⋊C4—2441no—
265C22⋊C4—2442no—
266C22⋊C4—2442no—
267C22⋊C4—2442no—
268C42—1221no—
269C42—1221no—
270C42—1221no—
271C42—1221no—
272C42—2442no—
273C42—2441no—
274C42—2441no—
275C42—2444no—
276C23—611no—
277C23—611no—
278C23—1221no—
279C23—1221no—
280C23—2441no—
281C23—2441no—
282C23C2×C22⋊#692311yescharacteristic
283Q8—2442no—
284Q8—2442no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—611no—
298C2×C4—611no—
299C2×C4—612no—
300C2×C4—612no—
301C2×C4—1222no—
302C2×C4—1222no—
303C2×C4—1222no—
304C2×C4—1222no—
305C2×C4—1222no—
306C2×C4—1222no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—4882no—
310C2×C4—4882no—
311C2×C4—2444no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4#1451984012yes—
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—1221no—
323C22—1221no—
324C22—4881no—
325C22—2441no—
326C22—2441no—
327C22—2441no—
328C22—2441no—
329C22D61⋊#9911yescharacteristic
330C22C2×#1451984011yescharacteristic
331C22D61⋊#10411yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—612no—
338C4—612no—
339C4—4884no—
340C4—2442no—
341C4—2442no—
342C4—2441no—
343C4—2441no—
344C2—21no—
345C2—21no—
346C2—2441no—
347C2—2441no—
348C2D61⋊#32811yescharacteristic
349C2#2146207311yescharacteristic
350C2#2146209111yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#400C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenter#400#6923
hypocenterC61#11715
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #400

Complements

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

NH ≅ G/Ntype
12. C2×#14519742C4semidirect
17. #16839682C4semidirect
20. #16839613C4semidirect
22. C61×#400C4semidirect
185. #400#6923semidirect
198. C61#11715semidirect

Maximal subgroups

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

subgroupindex
#214615192
#214610532
#214608162
#214608132
#214490922
#1171561