Finite Group Databasebeta

← #30604821

Subgroups — #30604821

Order 15616.

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

normal subgroups only

This group has 448 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#21460814C212yesmaximal
4#21460813C212yesmaximal
5#21451642C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840779—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×#14519775C411yescharacteristic, complemented
13C2×#14519735C2211yescharacteristic
14#16839704—21no—
15#16839697—21no—
16#16839682C411yescharacteristic, complemented
17#16839675C2211yescharacteristic
18#16839673C2211yescharacteristic
19#16839657—21no—
20#16839626C411yescharacteristic, complemented
21#16839621—21no—
22C61×#395C411yescharacteristic, 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#14519775—41no—
34#14519775—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×Dic61C2×C411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684C2×C411yescharacteristic
53C22×Q8×C61C2×C411yescharacteristic
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56#95×C61—21no—
57#95×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—41no—
61C2×Q8×C61—81no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—22no—
75C2×C4×D61SD1612yes—
76C2×C4×D61Q1612yes—
77C2×Dic122—41no—
78C2×Dic122—81no—
79C22⋊#6923—24no—
80C22⋊#6923—42no—
81C2×#92420—42no—
82C4⋊#6923—44no—
83C4×#6923—44no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—81no—
86#11758198—41no—
87#11758198—41no—
88#11758198—81no—
89#11758198—81no—
90C4⋊Dic61—41no—
91#11758196—42no—
92#11758196—41no—
93#11758196—41no—
94#11758196—42no—
95#11758196—81no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—21no—
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—
110Q8×C61—82no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
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—41no—
123C2×C4×C61#10312yes—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61#9911yescharacteristic
135C2×Dic61#9911yescharacteristic
136C2×Dic61#10312yes—
137C2×Dic61#10412yes—
138C4×D61—22no—
139C4×D61—22no—
140C4×D61—22no—
141C4×D61—22no—
142Dic122—82no—
143#92420—82no—
144#11717—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—41no—
164Dic61—41no—
165Dic61—81no—
166Dic61—81no—
167Dic61#32812yes—
168Dic61#32812yes—
169#2628—611no—
170#1740—611no—
171#1458—611no—
172#1351—612no—
173#1351—612no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61C2×#32811yescharacteristic
177C2×C61#212211yescharacteristic
178C2×C61#212911yescharacteristic
179D61—41no—
180D61—41no—
181C2×C4⋊Q8—611no—
182C2×#117—611no—
183C2×#117—1221no—
184#431—1221no—
185#400—1221no—
186#399—1221no—
187#395—611no—
188#395#692311yescharacteristic, complemented
189#393—611no—
190#393—611no—
191#389—1221no—
192#382—611no—
193#380—1221no—
194#342—612no—
195#342—1221no—
196#324—1221no—
197#323—612no—
198C61#1171711yescharacteristic, complemented, Sylow
199C22×Q8—611no—
200C22×Q8C2×#692311yescharacteristic
201C23×C4—611no—
202C4⋊Q8—2441no—
203C4⋊Q8—2441no—
204#117—1224no—
205#117—2441no—
206#117—2441no—
207#117—2442no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—2441no—
215C2×C4⋊C4C2×#692311yescharacteristic
216C2×C22⋊C4—611no—
217C2×C22⋊C4—611no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C42—611no—
222C2×C42—611no—
223C2×C42—2441no—
224C4⋊C8—2441no—
225C22⋊C8—1222no—
226C4×C8—2441no—
227#95—21no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95C2×#692311yescharacteristic
234C24—611no—
235C2×Q8—41no—
236C2×Q8—81no—
237C2×Q8—4881no—
238C2×Q8—2441no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—611no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—611no—
245C22×C4—611no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1222no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—2442no—
252C22×C4C22⋊#692311yescharacteristic
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255C2×C8—2442no—
256C4⋊C4—41no—
257C4⋊C4—81no—
258C4⋊C4—4881no—
259C4⋊C4—2441no—
260C4⋊C4—2442no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C4⋊C4—2444no—
265C22⋊C4—1224no—
266C22⋊C4—4881no—
267C22⋊C4—4881no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2442no—
271C42—1221no—
272C42—1221no—
273C42—1221no—
274C42—1221no—
275C42—2444no—
276C23—611no—
277C23—611no—
278C23—1221no—
279C23—1221no—
280C23—2441no—
281C23—2441no—
282C23C2×C22⋊#692311yescharacteristic
283Q8—82no—
284Q8—4882no—
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—1221no—
306C2×C4—1221no—
307C2×C4—4882no—
308C2×C4—4882no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2444no—
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—4881no—
340C4—4884no—
341C4—4881no—
342C4—2441no—
343C4—2441no—
344C2—21no—
345C2—21no—
346C2—2441no—
347C2—2441no—
348C2D61⋊#32811yescharacteristic
349C2#2146207911yescharacteristic
350C2#2146210211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#395C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenter#395#6923
hypocenterC61#11717
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
12. C2×#14519775C4semidirect
16. #16839682C4semidirect
20. #16839626C4semidirect
22. C61×#395C4semidirect
188. #395#6923semidirect
198. C61#11717semidirect

Maximal subgroups

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

subgroupindex
#214615192
#214610532
#214608142
#214608132
#214516422
#1171761