Finite Group Databasebeta

← #30601929

Subgroups — #30601929

Order 15616.

418 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 78 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#21461537C211yesmaximal, characteristic, complemented
2#21461382C211yesmaximal, characteristic, complemented
3#21461290C212yesmaximal, complemented
4#21461210C212yesmaximal, complemented
5#21456399C211yesmaximal, characteristic
6#16840784C2212yes—
7#16840721C2211yescharacteristic
8#16840698C2211yescharacteristic
9(C2×#6923)⋊C8—21no—
10(C2×#6923)⋊C8—21no—
11#16840626—21no—
12#16840624—21no—
13#16840618—24no—
14#16840614—24no—
15#16840313C2212yes—
16#16840139C411yescharacteristic, complemented
17#16840110C412yescomplemented
18#16839999—21no—
19#16839999—21no—
20#16839844—21no—
21#16839844—21no—
22C4×C8×D61C2211yescharacteristic
23C61×#500C411yescharacteristic, complemented
24C2×D61⋊C8—21no—
25C2×C4×#6923—21no—
26C4⋊#92420—22no—
27#14519834—22no—
28#14519832—22no—
29#14519832—21no—
30#14519832D412yes—
31(C2×#6923)⋊C4—21no—
32(C2×#6923)⋊C4D412yes—
33C2×#11758226—22no—
34C2×D488—21no—
35C2×D488—21no—
36C2×C8×D61D411yescharacteristic
37C2×C8×D61D411yescharacteristic
38#14519774C2×C412yes—
39#14519766—22no—
40#14519752—22no—
41#14519752—22no—
42#14519739—22no—
43#14519739—22no—
44C42×D61C2311yescharacteristic
45#14519697—22no—
46#14519697—22no—
47C4×#92418C2×C411yescharacteristic
48C8×Dic61D411yescharacteristic
49C8×Dic61D411yescharacteristic
50C4⋊Q8×C61C2×C412yes—
51#107×C61—21no—
52#107×C61—21no—
53C4×C8×C61C2×C411yescharacteristic
54C22×#6923—22no—
55C2×Q8×C61—22no—
56#11758226—84no—
57C2×D244—22no—
58C2×D244—22no—
59C2×D244—22no—
60C2×D244—42no—
61C2×C4×D61—42no—
62C2×C4×D61C2×D411yescharacteristic
63C2×C4×D61C2×D411yescharacteristic
64C2×C4×D61#4111yescharacteristic
65C2×#92420—22no—
66C2×#92420—42no—
67C4×#6923—41no—
68C4×#6923—41no—
69D61⋊C8—41no—
70D61⋊C8—41no—
71C2×C8×C61C22⋊C411yescharacteristic
72C2×C8×C61C22⋊C411yescharacteristic
73C4⋊C4×C61—22no—
74C4⋊C4×C61—22no—
75C42×C61C22×C411yescharacteristic
76#11758198—42no—
77#11758198—42no—
78C4×Dic61C2×D411yescharacteristic
79C4×Dic61C2×D411yescharacteristic
80C4×Dic61#4111yescharacteristic
81C2×#92418C22⋊C411yescharacteristic
82C2×#92418C22⋊C411yescharacteristic
83D488—42no—
84D488—42no—
85C8×D61—41no—
86C8×D61—41no—
87C22×D61—42no—
88C22×D61#12111yescharacteristic
89C2×#6923—42no—
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#12111yescharacteristic
98C2×Dic61#12411yescharacteristic
99C2×Dic61#12711yescharacteristic
100D244—42no—
101D244—42no—
102D244—82no—
103D244—82no—
104D244—84no—
105C4×D61—22no—
106C4×D61—21no—
107C4×D61—21no—
108C4×D61—41no—
109C4×D61—84no—
110#92420—42no—
111#92420—82no—
112C8×C61—21no—
113C8×C61—21no—
114#92418—21no—
115#92418—21no—
116#9764—611nomaximal, Sylow
117C22×C61#39011yescharacteristic
118C2×D61—21no—
119C2×D61—21no—
120C2×D61—82no—
121C2×D61—82no—
122C2×D61—82no—
123C2×D61#47212yes—
124#6923—82no—
125C4×C61—21no—
126C4×C61—21no—
127C4×C61—82no—
128C4×C61#42111yescharacteristic
129C4×C61#42111yescharacteristic
130Dic61—21no—
131Dic61—21no—
132Dic61—21no—
133#3169—611no—
134#1746—611no—
135#1568—611no—
136#1530—612no—
137#1505—612no—
138C2×C61#196311yescharacteristic
139C2×C61#197611yescharacteristic
140C2×C61#197811yescharacteristic
141D61—22no—
142D61—162no—
143#532—612no—
144#500#692311yescharacteristic, complemented
145#493—611no—
146#486—612no—
147#466—1221no—
148#466—1221no—
149C4×D8—1221no—
150C4×D8—1221no—
151#432—612no—
152C2×#97—611no—
153C2×C4×C8—611no—
154#376—611no—
155#336—1221no—
156#336—1221no—
157#335—1221no—
158#334—1221no—
159#329—1224no—
160#327—1224no—
161C61#976411yescharacteristic, complemented, Sylow
162C2×#41—1222no—
163C2×D8—1221no—
164C2×D8—1221no—
165C22×C8—611no—
166C22×C8—611no—
167C22×C8—1221no—
168C4⋊Q8C2×#692312yes—
169#127—612no—
170#124—1222no—
171#121—1222no—
172#121—1222no—
173C4×D4—1222no—
174C4×D4—1222no—
175C2×C42—611no—
176C2×C42—1221no—
177#107—21no—
178#107—21no—
179C4⋊C8—1222no—
180#102—1222no—
181#102—1222no—
182C22⋊C8—1222no—
183#97—612no—
184#97—1221no—
185#97—1222no—
186C4×C8—611no—
187C4×C8—611no—
188C4×C8—611no—
189C4×C8C2×#692311yescharacteristic
190#95—612no—
191#95—1221no—
192#41—4884no—
193C2×Q8—22no—
194C2×D4—1222no—
195C2×D4—1222no—
196C2×D4—1222no—
197C2×D4—2442no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—611no—
201C22×C4—1222no—
202C22×C4—2442no—
203D8—2442no—
204D8—2442no—
205C2×C8—611no—
206C2×C8—611no—
207C2×C8—1222no—
208C2×C8—2441no—
209C2×C8—2441no—
210C2×C8—2442no—
211C2×C8—2441no—
212C2×C8—2441no—
213C2×C8C22⋊#692311yescharacteristic
214C2×C8C22⋊#692311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—22no—
217C22⋊C4—2442no—
218C22⋊C4—2442no—
219C42—611no—
220C42—611no—
221C42—611no—
222C42—2441no—
223C42—2441no—
224C42C22×#692311yescharacteristic
225C23—611no—
226C23—2442no—
227Q8—82no—
228D4—4882no—
229D4—4882no—
230D4—4884no—
231D4—2442no—
232D4—2442no—
233C2×C4—42no—
234C2×C4—611no—
235C2×C4—611no—
236C2×C4—611no—
237C2×C4—1222no—
238C2×C4—1221no—
239C2×C4—1221no—
240C2×C4—4884no—
241C2×C4—2441no—
242C2×C4—2442no—
243C2×C4—2442no—
244C2×C4—2442no—
245C2×C4C2×C22⋊#692311yescharacteristic
246C2×C4D4×#692311yescharacteristic
247C2×C4D4×#692311yescharacteristic
248C8—21no—
249C8—21no—
250C8—1221no—
251C8—1221no—
252C8—4882no—
253C8—2442no—
254C22—612no—
255C22—1221no—
256C22—1221no—
257C22—4882no—
258C22—4882no—
259C22—4882no—
260C22#1684071411yescharacteristic
261C4—21no—
262C4—21no—
263C4—82no—
264C4—1221no—
265C4—1221no—
266C4—1221no—
267C4—4882no—
268C4#1684076111yescharacteristic
269C4#1684076111yescharacteristic
270C2—1222no—
271C2—9762no—
272C2#2146195511yescharacteristic
273C2#2146197811yescharacteristic
274C2#2146198011yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#500C4
socleC22×C61#390
prime radical#21456399C2
cosocleC42×D61C23
hypercenter#500#6923
hypocenterC61#9764
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #21461537C2semidirect
2. #21461382C2semidirect
3. #21461290C2semidirect
4. #21461210C2semidirect
16. #16840139C4semidirect
17. #16840110C4semidirect
23. C61×#500C4semidirect
144. #500#6923semidirect
161. C61#9764semidirect

Maximal subgroups

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

subgroupindex
#214615372
#214613822
#214612902
#214612102
#214563992
#976461