Finite Group Databasebeta

← #30601100

Subgroups — #30601100

Order 15616.

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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#21462354C211yesmaximal, characteristic, complemented
2#21461302C211yesmaximal, characteristic, complemented
3#21461208C212yesmaximal
4#21460708C212yesmaximal, complemented
5#21456218C211yesmaximal, characteristic
6#16840784C2212yes—
7#16840711—22no—
8#16840711—22no—
9C4⋊C4×#6923—22no—
10#16840698C2212yes—
11#16840619—22no—
12#16840613—22no—
13#16840613—22no—
14#16840310C2211yescharacteristic
15#125×D61C2211yescharacteristic
16#16840132C411yescharacteristic, complemented
17#16840131C411yescharacteristic, complemented
18#16840126C411yescharacteristic, complemented
19#16840115—22no—
20#16839760—22no—
21#16839714C2211yescharacteristic
22C61×#490C411yescharacteristic, complemented
23C2×C4⋊#6923—22no—
24C2×C4×#6923—22no—
25C2×C4×#6923—22no—
26C4⋊#92420—22no—
27#14519834—22no—
28#14519832—22no—
29(C2×#6923)⋊C4—22no—
30(C2×#6923)⋊C4—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4D412yes—
33(C2×#6923)⋊C4D412yes—
34C2×#11758224—21no—
35C2×#11758189—21no—
36#14519777C2×C411yescharacteristic
37#14519769C2×C411yescharacteristic
38#14519768D412yes—
39#14519767—22no—
40#14519764—21no—
41#14519748—22no—
42D4×Dic61—22no—
43C4⋊C4×D61—22no—
44C4⋊C4×D61D412yes—
45C42×D61C2311yescharacteristic
46#14519702—22no—
47#14519700—22no—
48#14519698—22no—
49#14519688C2×C411yescharacteristic
50C8×Dic61—21no—
51#127×C61C2×C411yescharacteristic
52#125×C61C2×C411yescharacteristic
53#102×C61—22no—
54#97×C61C2×C411yescharacteristic
55C22×#6923—22no—
56C22×#6923—22no—
57C2×D4×C61—21no—
58C2×D4×C61—22no—
59C2×C22⋊D61—41no—
60C22×Dic61—41no—
61#11758224—44no—
62C2×C4×D61—22no—
63C2×C4×D61C2×D411yescharacteristic
64C2×C4×D61C2×D412yes—
65C2×Dic122—21no—
66C2×#92420—42no—
67C4⋊#6923—44no—
68C4×#6923—44no—
69C4×#6923—44no—
70C2×C8×C61—21no—
71C4⋊C4×C61—22no—
72C4⋊C4×C61C22⋊C412yes—
73C42×C61C22×C411yescharacteristic
74C22⋊Dic61—42no—
75C4⋊Dic61—22no—
76C4⋊Dic61—22no—
77C4⋊Dic61C22⋊C412yes—
78#11758196—22no—
79#11758196—22no—
80#11758196—22no—
81C4×Dic61C2×D411yescharacteristic
82C4×Dic61#4112yes—
83C2×#92418—21no—
84#11758189—42no—
85C23×C61—41no—
86C22×D61#12011yescharacteristic
87C2×#6923—24no—
88C2×#6923—42no—
89C2×#6923—42no—
90C2×#6923—42no—
91C2×#6923—42no—
92C2×#6923—42no—
93D4×C61—42no—
94D4×C61—82no—
95C2×C4×C61—22no—
96C2×C4×C61C2×C22⋊C411yescharacteristic
97C2×C4×C61C4×D412yes—
98C22⋊D61—82no—
99C2×Dic61—22no—
100C2×Dic61—41no—
101C2×Dic61—82no—
102C2×Dic61#12112yes—
103C2×Dic61#12311yescharacteristic
104C4×D61—22no—
105C4×D61—42no—
106C4×D61—42no—
107C4×D61—42no—
108Dic122—42no—
109#92420—82no—
110C8×C61—41no—
111#92418—41no—
112#9442—611nomaximal, Sylow
113C22×C61—81no—
114C22×C61—82no—
115C22×C61#38611yescharacteristic
116C2×D61—22no—
117C2×D61#45412yes—
118#6923—44no—
119#6923—82no—
120C4×C61—21no—
121C4×C61—22no—
122C4×C61—42no—
123Dic61—21no—
124Dic61—22no—
125Dic61—42no—
126Dic61—81no—
127#3142—611no—
128#2371—611no—
129#1534—611no—
130#1505—612no—
131#1300—612no—
132C2×C61—161no—
133C2×C61#188811yescharacteristic
134C2×C61#190612yes—
135D61—22no—
136#532—611no—
137C2×#125—611no—
138#491—611no—
139#491—611no—
140#490—611no—
141#490#692311yescharacteristic, complemented
142#488—1222no—
143#432—612no—
144#417—1222no—
145C4×M16—611no—
146#387—1222no—
147#387—1222no—
148C4×C4⋊C4—1222no—
149#376—612no—
150#330—1222no—
151#327—1222no—
152#327—1222no—
153C61#944211yescharacteristic, complemented, Sylow
154C2×#41—1221no—
155C2×M16—1221no—
156C4⋊Q8—611no—
157#127C2×#692311yescharacteristic
158#125—612no—
159#125—611no—
160#125—1222no—
161#125C2×#692311yescharacteristic
162#124—1221no—
163#121—1222no—
164C4×D4—1222no—
165C2×C4⋊C4—612no—
166C2×C4⋊C4—1222no—
167C2×C4⋊C4—1222no—
168C2×C42—611no—
169C2×C42—1222no—
170C2×C42—1222no—
171C4⋊C8—1222no—
172#103—1222no—
173#103—1222no—
174#102—22no—
175#102—1222no—
176C22⋊C8—1222no—
177#97—611no—
178#97—1222no—
179#97C2×#692311yescharacteristic
180C4×C8—1221no—
181#95—612no—
182#95—612no—
183#95—1222no—
184#95—1222no—
185#95—1222no—
186#41—2444no—
187C2×Q8—1221no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—2441no—
191C22×C4—611no—
192C22×C4—612no—
193C22×C4—1222no—
194C22×C4—1222no—
195C22×C4—1222no—
196C22×C4—2441no—
197M16—2442no—
198C2×C8—21no—
199C2×C8—1221no—
200C2×C8—2442no—
201C4⋊C4—22no—
202C4⋊C4—612no—
203C4⋊C4—1222no—
204C4⋊C4—1222no—
205C4⋊C4—1222no—
206C4⋊C4—1222no—
207C4⋊C4—1222no—
208C4⋊C4—2444no—
209C4⋊C4C22⋊#692312yes—
210C22⋊C4—2442no—
211C42—611no—
212C42—612no—
213C42—2444no—
214C42—2444no—
215C42C22×#692311yescharacteristic
216C23—41no—
217C23—611no—
218Q8—2442no—
219D4—42no—
220D4—82no—
221D4—4882no—
222C2×C4—22no—
223C2×C4—612no—
224C2×C4—611no—
225C2×C4—1222no—
226C2×C4—1224no—
227C2×C4—1222no—
228C2×C4—4882no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4—2441no—
236C2×C4—2442no—
237C2×C4—2442no—
238C2×C4C2×C22⋊#692311yescharacteristic
239C2×C4D4×#692312yes—
240C8—41no—
241C8—4882no—
242C8—2441no—
243C22—81no—
244C22—82no—
245C22—612no—
246C22—1222no—
247C22#1684071011yescharacteristic
248C4—21no—
249C4—22no—
250C4—42no—
251C4—1221no—
252C4—1222no—
253C4—4882no—
254C4—4881no—
255C4—2444no—
256C4—2442no—
257C2—161no—
258C2—1222no—
259C2#2146182711yescharacteristic
260C2#2146185612yes—
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#490C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#490#6923
hypocenterC61#9442
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #21462354C2semidirect
2. #21461302C2semidirect
4. #21460708C2semidirect
16. #16840132C4semidirect
17. #16840131C4semidirect
18. #16840126C4semidirect
22. C61×#490C4semidirect
141. #490#6923semidirect
153. C61#9442semidirect

Maximal subgroups

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

subgroupindex
#214623542
#214613022
#214612082
#214607082
#214562182
#944261