Finite Group Databasebeta

← #30599199

Subgroups — #30599199

Order 15616.

384 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C4×#14519845C211yesmaximal, characteristic, complemented
2#21461382C211yesmaximal, characteristic, complemented
3#21461221C212yesmaximal
4#21460671C212yesmaximal, complemented
5#21454791C211yesmaximal, characteristic
6#16840784C2212yes—
7C2×#14519845—21no—
8C4×C4⋊#6923C2212yes—
9#16840636—21no—
10#16840626—21no—
11#16840624—21no—
12#16840615—22no—
13#16840613—22no—
14#16840294C2211yescharacteristic
15#16839889C411yescharacteristic, complemented
16C8×Dic122C411yescharacteristic, complemented
17#16839840C411yescharacteristic, complemented
18#16839836—21no—
19C8×C22⋊D61—21no—
20#16839820—21no—
21#16839818C2211yescharacteristic
22#16839717—21no—
23C4×C8×D61C2211yescharacteristic
24C8×D4×C61C411yescharacteristic, complemented
25C2×C4⋊#6923—22no—
26C2×C4×#6923—22no—
27#14519845—24no—
28C4⋊#92420—22no—
29#14519834—22no—
30#14519832—22no—
31(C2×#6923)⋊C4—22no—
32C2×#11758194—21no—
33C2×C8×D61D411yescharacteristic
34C2×C8×D61D411yescharacteristic
35#14519770—21no—
36#14519768Q811yescharacteristic
37#14519764Q811yescharacteristic
38#14519759—21no—
39#14519757D411yescharacteristic
40#14519748D411yescharacteristic
41C4×Dic122C2×C411yescharacteristic
42C4×C22⋊D61—21no—
43#14519736—21no—
44C42×D61C2311yescharacteristic
45C4⋊#92418C2×C411yescharacteristic
46#14519706—21no—
47#14519706—21no—
48C22⋊#92418—21no—
49#14519689—21no—
50#14519687—21no—
51#14519687—21no—
52C4×#92418C2×C411yescharacteristic
53C8×Dic61—21no—
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C22×C8×C61—21no—
57C4×D4×C61C2×C411yescharacteristic
58C4⋊C8×C61C2×C411yescharacteristic
59C22⋊C8×C61—21no—
60C4×C8×C61C2×C411yescharacteristic
61C22×#6923—42no—
62C2×D4×C61C4⋊C411yescharacteristic
63C22×C4×C61—21no—
64C2×C22⋊D61—21no—
65C2×C4×D61C2×D411yescharacteristic
66C2×C4×D61C2×D411yescharacteristic
67C2×C4×D61#4111yescharacteristic
68C2×Dic122C4⋊C411yescharacteristic
69C2×#92420—42no—
70C4⋊#6923—24no—
71C4⋊#6923—42no—
72C4×#6923—82no—
73C2×C8×C61—21no—
74C2×C8×C61—42no—
75C2×C8×C61C22⋊C411yescharacteristic
76C2×C8×C61C22⋊C411yescharacteristic
77C4⋊C4×C61C4⋊C411yescharacteristic
78C22⋊C4×C61—21no—
79C42×C61C22×C411yescharacteristic
80C22⋊Dic61—21no—
81C22⋊Dic61—21no—
82C22⋊Dic61—21no—
83#11758198—21no—
84C4⋊Dic61—21no—
85C4⋊Dic61—21no—
86C4⋊Dic61C4⋊C411yescharacteristic
87#11758196—21no—
88#11758196—21no—
89C4×Dic61—21no—
90C4×Dic61C2×D411yescharacteristic
91C4×Dic61C2×Q811yescharacteristic
92C4×Dic61#4111yescharacteristic
93#11758194—42no—
94C2×#92418—21no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97C8×D61—22no—
98C8×D61—41no—
99C23×C61—21no—
100C22×D61#12111yescharacteristic
101C2×#6923—44no—
102C2×#6923—82no—
103C2×#6923—82no—
104D4×C61—41no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61C2×C4⋊C411yescharacteristic
110C2×C4×C61#11711yescharacteristic
111C22⋊D61—42no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61#12111yescharacteristic
115C2×Dic61C22⋊Q811yescharacteristic
116C2×Dic61C22⋊Q811yescharacteristic
117C4×D61—22no—
118C4×D61—21no—
119C4×D61—21no—
120C4×D61—41no—
121Dic122—41no—
122#92420—82no—
123C8×C61—21no—
124C8×C61—21no—
125C8×C61—41no—
126#92418—21no—
127#92418—41no—
128#92418#10412yes—
129#8696—611nomaximal, Sylow
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—41no—
133C22×C61#38011yescharacteristic
134C2×D61—21no—
135C2×D61—21no—
136C2×D61#47112yes—
137#6923—84no—
138C4×C61—21no—
139C4×C61—21no—
140C4×C61—41no—
141C4×C61C2×#10411yescharacteristic
142C4×C61#42111yescharacteristic
143Dic61—21no—
144Dic61—21no—
145Dic61—21no—
146Dic61—41no—
147Dic61—41no—
148#2966—611no—
149C4×#106—611no—
150#1568—611no—
151#1509—612no—
152#1285—612no—
153C2×C61—81no—
154C2×C61#180611yescharacteristic
155C2×C61#181211yescharacteristic
156C2×C61#181511yescharacteristic
157D61—22no—
158#529—611no—
159#446—611no—
160C8×Q8—611no—
161#436—611no—
162#435—1221no—
163C8×D4—1221no—
164C8×D4#692311yescharacteristic, complemented
165#433—1221no—
166#432—611no—
167#432—612no—
168C2×#106—1221no—
169#405—1221no—
170C2×C4×C8—611no—
171C4×C4⋊C4—612no—
172#340—1221no—
173#335—1221no—
174#334—1221no—
175#328—1222no—
176#327—1222no—
177C61#869611yescharacteristic, complemented, Sylow
178C2×M16—1221no—
179C22×C8—21no—
180C22×C8—611no—
181C22×C8—611no—
182#126—1221no—
183#125—611no—
184#124—611no—
185#123—1221no—
186C22⋊Q8—611no—
187#121—611no—
188C4×Q8—611no—
189C4×D4—1221no—
190C4×D4C2×#692311yescharacteristic
191#117—1221no—
192C2×C4⋊C4—1222no—
193C2×C42—611no—
194C2×C42—1222no—
195#106—1224no—
196C4⋊C8—611no—
197C4⋊C8—1222no—
198C4⋊C8—1221no—
199C4⋊C8—1221no—
200C4⋊C8C2×#692311yescharacteristic
201C22⋊C8—21no—
202C22⋊C8—1221no—
203C22⋊C8—1221no—
204C22⋊C8—1222no—
205#97—1221no—
206#97—1222no—
207#97—1221no—
208C4×C8—611no—
209C4×C8—611no—
210C4×C8—611no—
211C4×C8—1221no—
212C4×C8C2×#692311yescharacteristic
213#95—1222no—
214C2×Q8—611no—
215C2×D4—1221no—
216C2×D4C4⋊#692311yescharacteristic
217C22×C4—21no—
218C22×C4—611no—
219C22×C4—611no—
220C22×C4—611no—
221C22×C4—2442no—
222M16—2442no—
223C2×C8—21no—
224C2×C8—42no—
225C2×C8—611no—
226C2×C8—611no—
227C2×C8—1222no—
228C2×C8—1221no—
229C2×C8—2441no—
230C2×C8—2442no—
231C2×C8C22⋊#692311yescharacteristic
232C2×C8C22⋊#692311yescharacteristic
233C4⋊C4—611no—
234C4⋊C4—1221no—
235C4⋊C4—1221no—
236C4⋊C4—1221no—
237C4⋊C4—1221no—
238C4⋊C4—1224no—
239C4⋊C4—2442no—
240C4⋊C4C4⋊#692311yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—1221no—
243C22⋊C4—1221no—
244C22⋊C4—1221no—
245C22⋊C4—1221no—
246C42—611no—
247C42—611no—
248C42—611no—
249C42—1221no—
250C42—4882no—
251C42C22×#692311yescharacteristic
252C23—21no—
253C23—611no—
254Q8—2441no—
255D4—41no—
256D4—2442no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—611no—
261C2×C4—611no—
262C2×C4—611no—
263C2×C4—1222no—
264C2×C4—1221no—
265C2×C4—1221no—
266C2×C4—1221no—
267C2×C4—1221no—
268C2×C4—4882no—
269C2×C4—4882no—
270C2×C4—2441no—
271C2×C4—2444no—
272C2×C4C2×C22⋊#692311yescharacteristic
273C2×C4C2×C4⋊#692311yescharacteristic
274C2×C4#1451985611yescharacteristic
275C8—21no—
276C8—21no—
277C8—41no—
278C8—612no—
279C8—1221no—
280C8—4882no—
281C8—2441no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—612no—
286C22—1221no—
287C22—1221no—
288C22#1684070411yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—1221no—
293C4—1221no—
294C4—1221no—
295C4—4884no—
296C4—2441no—
297C4—2441no—
298C4D61⋊#10411yescharacteristic
299C4#1684076011yescharacteristic
300C2—81no—
301C2—1222no—
302C2#2146169311yescharacteristic
303C2#2146170111yescharacteristic
304C2#2146170711yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC8×D4×C61C4
socleC22×C61#380
prime radical#14519748D4
cosocleC42×D61C23
hypercenterC8×D4#6923
hypocenterC61#8696
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C4×#14519845C2semidirect
2. #21461382C2semidirect
4. #21460671C2semidirect
15. #16839889C4semidirect
16. C8×Dic122C4semidirect
17. #16839840C4semidirect
24. C8×D4×C61C4semidirect
164. C8×D4#6923semidirect
177. C61#8696semidirect

Maximal subgroups

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

subgroupindex
C4×#145198452
#214613822
#214612212
#214606712
#214547912
#869661