Finite Group Databasebeta

← #30601885

Subgroups — #30601885

Order 15616.

406 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C8×C4⋊#6923C211yesmaximal, characteristic
2#21461369C211yesmaximal, characteristic
3#21461203C212yesmaximal
4#21461129C212yesmaximal
5#21456330C211yesmaximal, characteristic
6#16840781C2212yes—
7D61⋊(C4⋊C8)C2211yescharacteristic
8C2×C8×#6923—21no—
9C4×C4⋊#6923C2211yescharacteristic
10(C2×#6923)⋊C8—21no—
11#16840625—21no—
12#16840624—21no—
13#16840616—24no—
14#16840612—24no—
15#16840317C2212yes—
16#16840167C411yescharacteristic, complemented
17#16840163—21no—
18#16840163—21no—
19#16840158C412yescomplemented
20#16839799—21no—
21#16839799—21no—
22C4×C8×D61C2211yescharacteristic
23C61×#500C411yescharacteristic, complemented
24C2×D61⋊C8—21no—
25C2×C4⋊#6923—21no—
26C2×C4×#6923—21no—
27#14519844—21no—
28C4⋊#92420—22no—
29C4⋊#92420D412yes—
30#14519834—22no—
31C4×#92420—22no—
32C8×#6923—42no—
33(C2×#6923)⋊C4D412yes—
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36#14519777C2×C412yes—
37#14519768—22no—
38#14519757—22no—
39#14519753—22no—
40Q8×Dic61—22no—
41#14519742—22no—
42#14519735—22no—
43C42×D61C2311yescharacteristic
44#14519711—21no—
45#14519711—21no—
46#14519708—22no—
47#14519708—22no—
48C4×#92418C2×C411yescharacteristic
49C8×Dic61D411yescharacteristic
50C8×Dic61D411yescharacteristic
51C4⋊Q8×C61C2×C412yes—
52#107×C61—21no—
53#107×C61—21no—
54C4×C8×C61C2×C411yescharacteristic
55C22×#6923—22no—
56C2×Q8×C61—22no—
57C2×C4×D61C2×D411yescharacteristic
58C2×C4×D61C2×D411yescharacteristic
59C2×C4×D61#4111yescharacteristic
60C2×Dic122—22no—
61C2×#92420—22no—
62C2×#92420—42no—
63C4⋊#6923—24no—
64C4×#6923—41no—
65C4×#6923—41no—
66D61⋊C8—41no—
67D61⋊C8—41no—
68C2×C8×C61C22⋊C411yescharacteristic
69C2×C8×C61C22⋊C411yescharacteristic
70C4⋊C4×C61—22no—
71C4⋊C4×C61—22no—
72C42×C61C22×C411yescharacteristic
73#11758198—42no—
74C4⋊Dic61—22no—
75C4⋊Dic61—22no—
76C4⋊Dic61—42no—
77#11758196—42no—
78C4×Dic61—42no—
79C4×Dic61C2×D411yescharacteristic
80C4×Dic61C2×D411yescharacteristic
81C4×Dic61#4111yescharacteristic
82C2×#92418C22⋊C411yescharacteristic
83C2×#92418C22⋊C411yescharacteristic
84C8×D61—21no—
85C8×D61—21no—
86C8×D61SD1612yes—
87C8×D61Q1612yes—
88C22×D61#12111yescharacteristic
89C2×#6923—24no—
90C2×#6923—42no—
91C2×#6923—42no—
92Q8×C61—82no—
93C2×C4×C61—42no—
94C2×C4×C61C2×C22⋊C411yescharacteristic
95C2×C4×C61C4×D411yescharacteristic
96C2×C4×C61C4×D411yescharacteristic
97C2×Dic61—42no—
98C2×Dic61#12111yescharacteristic
99C2×Dic61#12411yescharacteristic
100C2×Dic61#12711yescharacteristic
101C4×D61—22no—
102C4×D61—41no—
103C4×D61C2×SD1611yescharacteristic
104C4×D61C2×Q1611yescharacteristic
105C4×D61#13711yescharacteristic
106C4×D61#13711yescharacteristic
107Dic122—44no—
108#92420—42no—
109#92420—82no—
110C8×C61—21no—
111C8×C61#10312yes—
112#92418—21no—
113#92418#10312yes—
114#9749—611nomaximal, Sylow
115C22×C61#39011yescharacteristic
116C2×D61—21no—
117C2×D61—21no—
118C2×D61#46111yescharacteristic
119C2×D61#46211yescharacteristic
120#6923—44no—
121C4×C61—21no—
122C4×C61—82no—
123C4×C61C2×#10311yescharacteristic
124C4×C61#41911yescharacteristic
125C4×C61#44312yes—
126Dic61—21no—
127Dic61—21no—
128Dic61—82no—
129Dic61#46111yescharacteristic
130Dic61#46211yescharacteristic
131#3160—611no—
132C8×C4⋊C4—611no—
133#1564—611no—
134#1503—612no—
135#1481—612no—
136C2×C61#197411yescharacteristic
137C2×C61#197611yescharacteristic
138C2×C61#198111yescharacteristic
139D61—21no—
140D61—21no—
141#533—612no—
142#500—611no—
143#500#692311yescharacteristic, complemented
144#499—1221no—
145#499—1221no—
146#498—612no—
147#431—612no—
148#427—1221no—
149#427—1221no—
150C2×C4⋊C8—611no—
151C2×C4×C8—611no—
152C2×C4×C8—1221no—
153C4×C4⋊C4—611no—
154#336—1221no—
155#335—1221no—
156#334—1221no—
157#328—1224no—
158Q8⋊C8—1224no—
159C61#974911yescharacteristic, complemented, Sylow
160C22×C8—611no—
161C22×C8—611no—
162C22×C8—1221no—
163C4⋊Q8—612no—
164C4⋊Q8C2×#692312yes—
165#125—1222no—
166C22⋊Q8—1222no—
167C22⋊Q8—1222no—
168C4×Q8—1222no—
169C4×Q8—1222no—
170#117—1222no—
171C2×C4⋊C4—1221no—
172C2×C42—611no—
173C2×C42—1221no—
174#107—21no—
175#107—21no—
176#107—1221no—
177#107—1221no—
178#106—1222no—
179#106—1222no—
180C4⋊C8—612no—
181C4⋊C8—1221no—
182C4⋊C8—1222no—
183C22⋊C8—1222no—
184C4×C8—611no—
185C4×C8—611no—
186C4×C8—611no—
187C4×C8—1222no—
188C4×C8—2442no—
189C4×C8C2×#692311yescharacteristic
190#95—612no—
191C2×Q8—22no—
192C2×Q8—1222no—
193C22×C4—611no—
194C22×C4—611no—
195C22×C4—611no—
196C22×C4—1222no—
197C2×C8—611no—
198C2×C8—612no—
199C2×C8—612no—
200C2×C8—611no—
201C2×C8—1221no—
202C2×C8—1221no—
203C2×C8—1222no—
204C2×C8—2442no—
205C2×C8—2441no—
206C2×C8—2441no—
207C2×C8C22⋊#692311yescharacteristic
208C2×C8C22⋊#692311yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—1224no—
212C4⋊C4—1222no—
213C4⋊C4—1222no—
214C4⋊C4—2442no—
215C4⋊C4—2442no—
216C22⋊C4—2442no—
217C42—611no—
218C42—611no—
219C42—611no—
220C42—2442no—
221C42—2441no—
222C42—2441no—
223C42C22×#692311yescharacteristic
224C23—611no—
225Q8—82no—
226Q8—2444no—
227C2×C4—42no—
228C2×C4—611no—
229C2×C4—611no—
230C2×C4—611no—
231C2×C4—611no—
232C2×C4—611no—
233C2×C4—611no—
234C2×C4—611no—
235C2×C4—1222no—
236C2×C4—1224no—
237C2×C4—2441no—
238C2×C4—2442no—
239C2×C4—2442no—
240C2×C4—2442no—
241C2×C4C2×C22⋊#692311yescharacteristic
242C2×C4D4×#692311yescharacteristic
243C2×C4D4×#692311yescharacteristic
244C8—21no—
245C8—612no—
246C8—1221no—
247C8—4882no—
248C8—2442no—
249C8#1451984012yes—
250C22—611no—
251C22—611no—
252C22—1221no—
253C22—1221no—
254C22#1684071411yescharacteristic
255C4—21no—
256C4—82no—
257C4—611no—
258C4—611no—
259C4—1221no—
260C4—1221no—
261C4—4882no—
262C4—2444no—
263C4C2×#1451984011yescharacteristic
264C4#1684075511yescharacteristic
265C4#1684080612yes—
266C2—1221no—
267C2—1221no—
268C2#2146197011yescharacteristic
269C2#2146197611yescharacteristic
270C2#2146198511yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#500C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#500#6923
hypocenterC61#9749
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
16. #16840167C4semidirect
19. #16840158C4semidirect
23. C61×#500C4semidirect
143. #500#6923semidirect
159. C61#9749semidirect

Maximal subgroups

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

subgroupindex
C8×C4⋊#69232
#214613692
#214612032
#214611292
#214563302
#974961