Finite Group Databasebeta

← #30604742

Subgroups — #30604742

Order 15616.

568 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461518C211yesmaximal, characteristic, complemented
2#21461052C211yesmaximal, characteristic, complemented
3#21460798C214yesmaximal, complemented
4#21451572C211yesmaximal, characteristic
5C2×#14519856—21no—
6#16840778—21no—
7#16840640—21no—
8#16840640C2212yes—
9#16840608—21no—
10#16840606C2212yes—
11C2×#14519774C411yescharacteristic, complemented
12C2×#14519752—21no—
13C2×#14519734C2211yescharacteristic
14#16839680—21no—
15#16839680C412yescomplemented
16#16839672—21no—
17#16839672C2212yes—
18#16839663—21no—
19C61×#382C411yescharacteristic, complemented
20#14519856—24no—
21#14519856—42no—
22C2×C4⋊#6923—41no—
23C2×C22⋊#6923—22no—
24C2×C4×#6923—41no—
25C4⋊#92420—41no—
26C22⋊#92420—22no—
27C4×#92420—41no—
28C22×D244—21no—
29C22×D244—22no—
30C22×D244—21no—
31C22×D244C2×C412yes—
32C22×C4×D61D411yescharacteristic
33#14519774—41no—
34#14519774—41no—
35#14519752—42no—
36#14519752—42no—
37#14519734—41no—
38#14519734—41no—
39C2×#11758196C2×C411yescharacteristic
40C2×#11758196D411yescharacteristic
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198D411yescharacteristic
45C2×#11758198C2311yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4⋊C4×C61—21no—
48C2×C42×C61C2×C411yescharacteristic
49#95×C61—21no—
50#95×C61C2×C412yes—
51C23×D61—22no—
52C23×D61C2×D411yescharacteristic
53C22×#6923—42no—
54C22×C4×C61—22no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22×C411yescharacteristic
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C2×D411yescharacteristic
61C2×D244—42no—
62C2×D244—42no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—41no—
66C2×D244—41no—
67C2×D244—42no—
68C2×D244—42no—
69C2×D244—82no—
70C2×D244—82no—
71C2×D244—82no—
72C2×D244—82no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—22no—
76C2×C4×D61D812yes—
77C2×C4×D61SD1612yes—
78C22⋊#6923—24no—
79C22⋊#6923—42no—
80C2×#92420—42no—
81C4⋊#6923—44no—
82C4×#6923—44no—
83C4⋊C4×C61—42no—
84C42×C61—21no—
85C42×C61—21no—
86#11758198—41no—
87#11758198—41no—
88#11758198—44no—
89#11758198—82no—
90#11758198—82no—
91#11758196—41no—
92#11758196—41no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—42no—
101C22×D61—42no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—42no—
107C22×D61C2×D811yescharacteristic
108C22×D61C2×SD1611yescharacteristic
109C2×#6923—44no—
110C2×#6923—82no—
111C2×#6923—82no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61#9911yescharacteristic
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61#10212yes—
122C2×C4×C61#10412yes—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61#10212yes—
130D244—84no—
131D244—84no—
132D244—84no—
133D244—84no—
134D244—82no—
135D244—82no—
136C4×D61—22no—
137C4×D61—22no—
138C4×D61—22no—
139C4×D61—22no—
140#92420—82no—
141#11693—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#9911yescharacteristic
145C22×C61C2×#10211yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—41no—
150C2×D61—41no—
151C2×D61—41no—
152C2×D61—41no—
153C2×D61—81no—
154C2×D61—82no—
155C2×D61—82no—
156C2×D61—82no—
157C2×D61—82no—
158C2×D61—82no—
159C2×D61—82no—
160C2×D61—82no—
161#6923—84no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—82no—
165C4×C61#32712yes—
166C4×C61#32712yes—
167Dic61—22no—
168Dic61—41no—
169Dic61—41no—
170#2619—611no—
171#1740—611no—
172#1458—611no—
173#1343—614no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61C2×#32711yescharacteristic
177C2×C61#212211yescharacteristic
178C2×C61#212811yescharacteristic
179D61—41no—
180D61—41no—
181D61—162no—
182C2×#127—611no—
183C2×#121—1221no—
184C2×#117—611no—
185C2×#117—1221no—
186#431—1221no—
187#394—612no—
188#394—1221no—
189#392—612no—
190#392—1221no—
191#390—1221no—
192#382#692311yescharacteristic, complemented
193#342—612no—
194#342—1221no—
195#324—1221no—
196#323—612no—
197C61#1169311yescharacteristic, complemented, Sylow
198C22×D4—612no—
199C22×D4—1222no—
200C22×D4—1221no—
201C22×D4—1221no—
202C23×C4—611no—
203#127—2441no—
204#127—2441no—
205#121—2442no—
206#121—2442no—
207#117—1224no—
208#117—2441no—
209#117—2441no—
210#117—2442no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4—2441no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1222no—
220C2×C22⋊C4—1222no—
221C2×C42—611no—
222C2×C42—2441no—
223C2×C42C2×#692311yescharacteristic
224C4⋊C8—2441no—
225C22⋊C8—1222no—
226C4×C8—2441no—
227#95—21no—
228#95C2×#692312yes—
229C24—611no—
230C24—1222no—
231C2×D4—4882no—
232C2×D4—4882no—
233C2×D4—4882no—
234C2×D4—4882no—
235C2×D4—2442no—
236C2×D4—2442no—
237C2×D4—2442no—
238C2×D4—2442no—
239C2×D4—2441no—
240C2×D4—2441no—
241C2×D4—2442no—
242C2×D4—2442no—
243C22×C4—22no—
244C22×C4—611no—
245C22×C4—612no—
246C22×C4—612no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1222no—
252C22×C4—2442no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22⋊#692311yescharacteristic
255C22×C4C22×#692311yescharacteristic
256C2×C8—2442no—
257C4⋊C4—42no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C4⋊C4—2444no—
261C22⋊C4—1224no—
262C22⋊C4—4882no—
263C22⋊C4—4882no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C22⋊C4—2444no—
267C22⋊C4—2442no—
268C42—21no—
269C42—21no—
270C42—1221no—
271C42—1221no—
272C42—2444no—
273C23—611no—
274C23—611no—
275C23—1221no—
276C23—1221no—
277C23—2442no—
278C23—2442no—
279C23—2442no—
280C23—2442no—
281C23—2441no—
282C23—2441no—
283C23—2442no—
284C23—2442no—
285C23—2442no—
286C23C2×C22⋊#692311yescharacteristic
287D4—4884no—
288D4—4884no—
289D4—4884no—
290D4—4884no—
291D4—4882no—
292D4—4882no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—612no—
301C2×C4—1221no—
302C2×C4—1221no—
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#1451983611yescharacteristic
315C2×C4#1451983611yescharacteristic
316C2×C4#1451983912yes—
317C2×C4#1451984212yes—
318C8—4882no—
319C22—21no—
320C22—21no—
321C22—1221no—
322C22—1221no—
323C22—4882no—
324C22—4882no—
325C22—4882no—
326C22—4882no—
327C22—4881no—
328C22—4882no—
329C22—4882no—
330C22—4882no—
331C22—2441no—
332C22—2441no—
333C22—2441no—
334C22—2441no—
335C22C2×#1451983611yescharacteristic
336C22C2×#1451983911yescharacteristic
337C22C2×#1451984211yescharacteristic
338C4—41no—
339C4—41no—
340C4—82no—
341C4—1222no—
342C4—4884no—
343C4—2441no—
344C4—2441no—
345C4#1684061412yes—
346C4#1684061412yes—
347C2—21no—
348C2—21no—
349C2—9762no—
350C2—2441no—
351C2—2441no—
352C2C2×#1684061411yescharacteristic
353C2#2146208011yescharacteristic
354C2#2146210011yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#382C4
socleC22×C61C2×#99
prime radical#21451572C2
cosocleC2×#11758198C23
hypercenter#382#6923
hypocenterC61#11693
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #21461518C2semidirect
2. #21461052C2semidirect
3. #21460798C2semidirect
11. C2×#14519774C4semidirect
15. #16839680C4semidirect
19. C61×#382C4semidirect
192. #382#6923semidirect
197. C61#11693semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610522
#214607982
#214515722
#1169361