Finite Group Databasebeta

← #30604764

Subgroups — #30604764

Order 15616.

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

normal subgroups only

This group has 452 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#21461525C211yesmaximal, characteristic
2#21461051C211yesmaximal, characteristic
3#21460804C212yesmaximal, complemented
4#21460803C212yesmaximal, complemented
5#21451584C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840779—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840605—21no—
12C2×#14519776C411yescharacteristic
13C2×#14519754—21no—
14C2×#14519733C2211yescharacteristic
15#16839696—21no—
16#16839694C2211yescharacteristic
17#16839693C2211yescharacteristic
18#16839690—21no—
19#16839687C411yescharacteristic
20#16839666—21no—
21#16839626C411yescharacteristic
22C61×#397C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C22×#92420—41no—
28#14519844—41no—
29C22⋊#92420—22no—
30C4×#92420—41no—
31C22×Dic122—21no—
32C23×Dic61D411yescharacteristic
33#14519776—41no—
34#14519776—41no—
35#14519754—42no—
36#14519754—42no—
37#14519733—41no—
38#14519733—41no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196D411yescharacteristic
45C2×#11758196C2311yescharacteristic
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61D411yescharacteristic
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684C2×C411yescharacteristic
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61C2×C411yescharacteristic
58#95×C61—21no—
59#95×C61C2×C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61—22no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61D812yes—
73C22×Dic61SD1612yes—
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×Dic122—42no—
79C2×Dic122—42no—
80#11758217—24no—
81#11758217—44no—
82#11758217—42no—
83C2×#92420—42no—
84C2×#92420—44no—
85C2×#92420—42no—
86C4⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊C4×C61—42no—
89C22⋊Dic61—41no—
90C22⋊Dic61—42no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94#11758196—41no—
95#11758196—41no—
96#11758196—42no—
97#11758196—42no—
98#11758196—41no—
99#11758196—41no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61#10212yes—
109C23×C61C2×C22⋊C411yescharacteristic
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—41no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61#9911yescharacteristic
137C2×Dic61#9911yescharacteristic
138C2×Dic61#10212yes—
139C2×Dic61#10412yes—
140C2×Dic61C2×D811yescharacteristic
141C2×Dic61C2×SD1611yescharacteristic
142Dic122—82no—
143Dic122—82no—
144#92420—44no—
145#92420—82no—
146#11700—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—22no—
150C22×C61—42no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61C2×#9911yescharacteristic
155C22×C61C2×#10211yescharacteristic
156C22×C61C2×#10411yescharacteristic
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61—81no—
161Dic61—21no—
162Dic61—21no—
163Dic61—41no—
164Dic61—41no—
165Dic61—81no—
166Dic61—81no—
167Dic61#32712yes—
168Dic61#32712yes—
169#2621—611no—
170#1742—611no—
171#1458—611no—
172#1346—612no—
173#1346—612no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61—42no—
177C2×C61C2×#32711yescharacteristic
178C2×C61#212311yescharacteristic
179C2×C61#213511yescharacteristic
180C22×M16—1221no—
181C2×C4⋊Q8—611no—
182C2×C22⋊Q8—1221no—
183C2×#117—611no—
184#431—1221no—
185#398—611no—
186#398—611no—
187#398—1221no—
188#397—611no—
189#397—1221no—
190#397#692311yescharacteristic
191#391—1221no—
192#382—611no—
193#343—612no—
194#343—1221no—
195#324—612no—
196#323—1221no—
197C61#1170011yescharacteristic, complemented, Sylow
198C22×Q8—1221no—
199C23×C4—611no—
200C2×M16—1222no—
201C2×M16—1224no—
202C2×M16—2442no—
203C2×M16—2441no—
204C22×C8—2441no—
205C4⋊Q8—2441no—
206C4⋊Q8—2441no—
207C22⋊Q8—2442no—
208C22⋊Q8—2442no—
209#117—2441no—
210#117—2441no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4C2×#692311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—611no—
221C2×C22⋊C4—1221no—
222C2×C22⋊C4C2×#692311yescharacteristic
223C2×C42—611no—
224C2×C42—611no—
225C4⋊C8—2441no—
226C22⋊C8—1222no—
227C4×C8—2441no—
228#95—21no—
229#95—611no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95C2×#692311yescharacteristic
235C24C22⋊#692311yescharacteristic
236C2×Q8—2442no—
237C2×Q8—2441no—
238C2×Q8—2441no—
239C2×Q8—2442no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—611no—
243C22×C4—612no—
244C22×C4—612no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1222no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255M16—1224no—
256M16—2442no—
257M16—2444no—
258C2×C8—2442no—
259C2×C8—2444no—
260C2×C8—2442no—
261C4⋊C4—41no—
262C4⋊C4—2441no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2442no—
266C4⋊C4—2441no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C4⋊C4—2442no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—2441no—
274C22⋊C4—2442no—
275C42—1221no—
276C42—1221no—
277C42—1221no—
278C42—1221no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23#1451983912yes—
284C23C2×C22⋊#692311yescharacteristic
285Q8—4882no—
286Q8—4882no—
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—611no—
302C2×C4—611no—
303C2×C4—1222no—
304C2×C4—1222no—
305C2×C4—1222no—
306C2×C4—1222no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—1221no—
310C2×C4—1221no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C8—4882no—
320C8—2444no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22D61⋊#9911yescharacteristic
329C22C2×#1451983911yescharacteristic
330C22D61⋊#10411yescharacteristic
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—612no—
336C4—612no—
337C4—1221no—
338C4—1221no—
339C4—4881no—
340C4—4881no—
341C4—2441no—
342C4—2441no—
343C2—21no—
344C2—21no—
345C2—42no—
346C2D61⋊#32711yescharacteristic
347C2#2146208211yescharacteristic
348C2#2146212211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#397C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×#11758196C23
hypercenter#397#6923
hypocenterC61#11700
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #21460804C2semidirect
4. #21460803C2semidirect
197. C61#11700semidirect

Maximal subgroups

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

subgroupindex
#214615252
#214610512
#214608042
#214608032
#214515842
#1170061