Finite Group Databasebeta

← #30598991

Subgroups — #30598991

Order 15616.

550 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461695C212yesmaximal, complemented
2#21461679C212yesmaximal
3#21460682C212yesmaximal, complemented
4C2×C8×C22⋊D61C211yesmaximal, characteristic
5C2×#14519845—22no—
6C2×#14519835—22no—
7#16840704C2214yes—
8#16840640—22no—
9#16840636—22no—
10C22×C8×D61C2211yescharacteristic
11C2×C4×C22⋊D61C2211yescharacteristic
12C8×C22⋊D61—22no—
13C8×C22⋊D61—22no—
14C2×#14519706C411yescharacteristic, complemented
15C2×C22⋊#92418C411yescharacteristic, complemented
16C2×#14519689C2211yescharacteristic
17C2×C8×Dic61C411yescharacteristic, complemented
18C23×C8×C61C411yescharacteristic, complemented
19C2×C4⋊#6923—24no—
20C2×C22⋊#6923—24no—
21#14519845—44no—
22#14519835—44no—
23(C2×#6923)⋊C4—24no—
24C22×C22⋊D61D411yescharacteristic
25C22×C4×D61C2311yescharacteristic
26C22×#92418C2×C411yescharacteristic
27C22×#92418C2×C411yescharacteristic
28C2×C8×D61—42no—
29C2×C8×D61D412yes—
30C2×C8×D61D412yes—
31C4×C22⋊D61—22no—
32C4×C22⋊D61—22no—
33C2×#11758196Q811yescharacteristic
34C2×C22⋊Dic61Q811yescharacteristic
35C2×#11758198D411yescharacteristic
36C2×C4×Dic61C2×C411yescharacteristic
37#14519706—22no—
38C22⋊#92418—22no—
39#14519689—22no—
40C8×Dic61—21no—
41C8×Dic61—21no—
42C23×C4×C61C2×C411yescharacteristic
43C22×C8×C61—21no—
44C22×C8×C61—21no—
45C22×C8×C61—42no—
46C22×C8×C61C2×C411yescharacteristic
47C22×C8×C61C2×C411yescharacteristic
48C24×C61C4⋊C411yescharacteristic
49C23×D61C2×D411yescharacteristic
50C22×#6923—44no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61C4⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C2×C22⋊D61—22no—
58C2×C22⋊D61—22no—
59C2×C22⋊D61SD1614yes—
60C22×Dic61C4⋊C411yescharacteristic
61C22×Dic61C4⋊C411yescharacteristic
62C22×Dic61C2×Q811yescharacteristic
63C2×C4×D61—41no—
64C2×C4×D61—41no—
65C2×C4×D61C2×D412yes—
66C2×C4×D61#4112yes—
67C22⋊#6923—44no—
68C22⋊#6923—44no—
69C4⋊#6923—44no—
70C4⋊#6923—44no—
71C2×C8×C61—22no—
72C2×C8×C61—21no—
73C2×C8×C61—42no—
74C2×C8×C61—42no—
75C2×C8×C61—41no—
76C2×C8×C61C22⋊C411yescharacteristic
77C2×C8×C61C22⋊C411yescharacteristic
78C22⋊Dic61—22no—
79#11758198—22no—
80#11758196—22no—
81C4×Dic61—21no—
82C4×Dic61—21no—
83C2×#92418—21no—
84C2×#92418—41no—
85C2×#92418—41no—
86C2×#92418C22⋊C412yes—
87C8×D61—24no—
88C8×D61—42no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61#10612yescomplemented
94C23×C61C2×C4⋊C411yescharacteristic
95C22×D61—41no—
96C22×D61—41no—
97C22×D61#12112yes—
98C22×D61C2×SD1612yes—
99C22×D61#13512yes—
100C2×#6923—84no—
101C2×#6923—84no—
102C2×#6923—84no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—42no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61#11711yescharacteristic
112C22⋊D61—28no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61#10612yescomplemented
119C2×Dic61C22⋊Q812yes—
120C4×D61—22no—
121C4×D61—22no—
122C4×D61—42no—
123C8×C61—21no—
124C8×C61—41no—
125C8×C61#9912yescomplemented
126#92418—22no—
127#92418—41no—
128#8608—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—22no—
132C22×C61—42no—
133C22×C61—42no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61#38011yescharacteristic
137C22×C61C2×#10611yescharacteristic
138C22×C61#42711yescharacteristic
139C2×D61—22no—
140C2×D61—22no—
141C2×D61—42no—
142C2×D61—42no—
143C2×D61#46514yes—
144#6923—164no—
145C4×C61—21no—
146C4×C61—41no—
147C4×C61C2×#9911yescharacteristic
148C4×C61#41011yescharacteristic
149Dic61—22no—
150Dic61—22no—
151Dic61—41no—
152C2×C8×D4—611no—
153#1807—612no—
154#1799—612no—
155#1289—612no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—42no—
159C2×C61#179911yescharacteristic
160C2×C61#180711yescharacteristic
161C2×C61#181211yescharacteristic
162D61—24no—
163C23×C8—611no—
164C23×C8#692311yescharacteristic, complemented
165C2×C4×D4—611no—
166C8×D4—1222no—
167C8×D4—1222no—
168C2×#106—1222no—
169C2×C4⋊C8—611no—
170C2×#99—1222no—
171C2×C22⋊C8—611no—
172C2×C22⋊C8—611no—
173C2×C4×C8—611no—
174#380—614no—
175#342—1222no—
176#340—1222no—
177C61#860811yescharacteristic, complemented, Sylow
178C22×D4—611no—
179C23×C4—611no—
180C23×C4C2×#692311yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—611no—
185C22×C8—612no—
186C22×C8—612no—
187C22×C8—611no—
188C22×C8—2442no—
189C22×C8C2×#692311yescharacteristic
190C22×C8C2×#692311yescharacteristic
191C4×D4—1222no—
192C4×D4—1222no—
193C2×C4⋊C4—611no—
194C2×C4⋊C4—1224no—
195C2×C22⋊C4—611no—
196C2×C22⋊C4—611no—
197C2×C22⋊C4—1224no—
198C2×C42—611no—
199#106—2444no—
200C4⋊C8—1222no—
201#99—2444no—
202C22⋊C8—1222no—
203C22⋊C8—1222no—
204C4×C8—1221no—
205C4×C8—1221no—
206#95—1224no—
207C24—611no—
208C24C4⋊#692311yescharacteristic
209C2×D4—614no—
210C2×D4—1222no—
211C2×D4—1222no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—612no—
219C22×C4—612no—
220C22×C4—611no—
221C22×C4—2441no—
222C22×C4—2441no—
223C22×C4—2444no—
224C22×C4C4⋊#692311yescharacteristic
225C22×C4C22×#692311yescharacteristic
226C2×C8—22no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—612no—
232C2×C8—1224no—
233C2×C8—1221no—
234C2×C8—2442no—
235C2×C8—2441no—
236C2×C8—2441no—
237C2×C8C22⋊#692311yescharacteristic
238C2×C8C22⋊#692311yescharacteristic
239C4⋊C4—1222no—
240C4⋊C4—2444no—
241C4⋊C4—2444no—
242C22⋊C4—1222no—
243C22⋊C4—1222no—
244C22⋊C4—2444no—
245C22⋊C4—2444no—
246C42—1221no—
247C42—1221no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—612no—
253C23—612no—
254C23—612no—
255C23—2441no—
256C23—2441no—
257C23#1451984512yescomplemented
258C23C2×C4⋊#692311yescharacteristic
259D4—1228no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—612no—
268C2×C4—612no—
269C2×C4—1221no—
270C2×C4—1221no—
271C2×C4—1222no—
272C2×C4—1222no—
273C2×C4—1221no—
274C2×C4—4884no—
275C2×C4—4884no—
276C2×C4—4884no—
277C2×C4—2442no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4C2×C22⋊#692311yescharacteristic
281C2×C4#1451985611yescharacteristic
282C8—21no—
283C8—41no—
284C8—1222no—
285C8—2441no—
286C8#1451983512yescomplemented
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—614no—
295C22—1222no—
296C22—1222no—
297C22—2442no—
298C22—2442no—
299C22#1684070411yescharacteristic
300C22C2×#1451984511yescharacteristic
301C22#1684077211yescharacteristic
302C4—21no—
303C4—41no—
304C4—1222no—
305C4—1222no—
306C4—9764no—
307C4—2441no—
308C4C2×#1451983511yescharacteristic
309C4#1684074111yescharacteristic
310C2—21no—
311C2—21no—
312C2—42no—
313C2—1224no—
314C2#2146167911yescharacteristic
315C2#2146169511yescharacteristic
316C2#2146170111yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×C8×C61C4
socleC22×C61#380
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenterC23×C8#6923
hypocenterC61#8608
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #21461695C2semidirect
3. #21460682C2semidirect
14. C2×#14519706C4semidirect
15. C2×C22⋊#92418C4semidirect
17. C2×C8×Dic61C4semidirect
18. C23×C8×C61C4semidirect
93. C23×C61#106semidirect
118. C2×Dic61#106semidirect
125. C8×C61#99semidirect
164. C23×C8#6923semidirect
177. C61#8608semidirect
257. C23#14519845semidirect
286. C8#14519835semidirect

Maximal subgroups

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

subgroupindex
#214616952
#214616792
#214606822
C2×C8×C22⋊D612
#860861