Finite Group Databasebeta

← #29508455

Subgroups — #29508455

Order 15424.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#21440638C211yesmaximal, characteristic, complemented
2#21440637C211yesmaximal, characteristic, complemented
3#21440635C211yesmaximal, characteristic, complemented
4#21440627C211yesmaximal, characteristic, complemented
5#21440626C211yesmaximal, characteristic, complemented
6#21440625C211yesmaximal, characteristic, complemented
7#21440616C211yesmaximal, characteristic, complemented
8#21440615C211yesmaximal, characteristic, complemented
9#21440614C211yesmaximal, characteristic, complemented
10#21440614C211yesmaximal, characteristic, complemented
11#21440613C211yesmaximal, characteristic, complemented
12#21440612C211yesmaximal, characteristic, complemented
13#21440610C211yesmaximal, characteristic, complemented
14#21440610C211yesmaximal, characteristic, complemented
15#121×C241C211yesmaximal, characteristic, complemented
16C23×D241C2211yescharacteristic
17C2×D964C2211yescharacteristic
18C2×D964C2211yescharacteristic
19C2×C22⋊D241C2211yescharacteristic
20C2×C22⋊D241C2211yescharacteristic
21C2×C22⋊D241C2211yescharacteristic
22C2×C22⋊D241C2211yescharacteristic
23C2×C22⋊D241C2211yescharacteristic
24C2×C22⋊D241C2211yescharacteristic, complemented
25C2×C22⋊D241C2211yescharacteristic, complemented
26C22×Dic241C2211yescharacteristic, complemented
27C2×C4×D241C2211yescharacteristic, complemented
28C4⋊Dic241C2211yescharacteristic, complemented
29C4⋊Dic241C2211yescharacteristic, complemented
30#16838749C2211yescharacteristic, complemented
31#16838749C2211yescharacteristic, complemented
32#16838749C2211yescharacteristic, complemented
33C22⋊Dic241C2211yescharacteristic
34C22⋊Dic241C2211yescharacteristic
35C22⋊Dic241C2211yescharacteristic, complemented
36#16838747C2211yescharacteristic
37#16838747C2211yescharacteristic
38#16838747C2211yescharacteristic
39#16838747C2211yescharacteristic
40#16838747C2211yescharacteristic
41#16838747C2211yescharacteristic
42#16838747C2211yescharacteristic, complemented
43C4×Dic241C2211yescharacteristic, complemented
44C2×D4×C241C2211yescharacteristic
45C2×D4×C241C2211yescharacteristic
46C2×D4×C241C2211yescharacteristic, complemented
47C22×C4×C241C2211yescharacteristic, complemented
48C4⋊C4×C241C2211yescharacteristic, complemented
49C22⋊C4×C241C2211yescharacteristic
50C22⋊C4×C241C2211yescharacteristic, complemented
51C22×D241—41no—
52C22×D241—41no—
53C22×D241—41no—
54C22×D241C2311yescharacteristic
55C22×D241C2311yescharacteristic
56C22×D241C2311yescharacteristic
57D964—41no—
58D964—41no—
59C22⋊D241—41no—
60C22⋊D241—41no—
61C22⋊D241—41no—
62C22⋊D241—41no—
63C22⋊D241—41no—
64C22⋊D241—41no—
65C22⋊D241—41no—
66C2×Dic241—41no—
67C2×Dic241C2311yescharacteristic
68C2×Dic241C2311yescharacteristic
69C2×Dic241C2311yescharacteristic
70C2×Dic241C2311yescharacteristic
71C2×Dic241C2311yescharacteristic
72C4×D241—41no—
73C23×C241C2311yescharacteristic
74C23×C241C2311yescharacteristic
75C23×C241C2311yescharacteristic
76D4×C241—41no—
77D4×C241—41no—
78D4×C241—41no—
79C2×C4×C241—41no—
80C2×C4×C241C2311yescharacteristic
81C2×C4×C241C2311yescharacteristic
82C2×C4×C241C2311yescharacteristic
83C2×C4×C241C2311yescharacteristic
84C22×C241—21no—
85C22×C241—21no—
86C22×C241—21no—
87C22×C241—21no—
88C22×C241—21no—
89C22×C241—21no—
90C22×C241—21no—
91C22×C241—21no—
92C22×C241—21no—
93C22×C241C2411yescharacteristic
94C2×D241—21no—
95C2×D241—21no—
96C2×D241—21no—
97C2×D241—21no—
98C2×D241—21no—
99C2×D241—21no—
100C2×D241—21no—
101C2×D241—21no—
102C2×D241—21no—
103C2×D241—44no—
104C4×C241—21no—
105C4×C241—21no—
106C4×C241—21no—
107C4×C241—21no—
108Dic241—21no—
109Dic241—21no—
110Dic241—21no—
111Dic241—21no—
112Dic241—21no—
113C2×C241—41no—
114C2×C241—41no—
115C2×C241—41no—
116C2×C241H(5,𝔽2)11yescharacteristic
117C2×C241H(5,𝔽2)11yescharacteristic
118C2×C241H(5,𝔽2)11yescharacteristic
119D241—41no—
120D241—41no—
121D241—41no—
122C241#56011yescharacteristic, complemented, Sylow
123#560—2411nomaximal, Sylow
124#127—2411no—
125#126—2411no—
126#126—2411no—
127#123—2411no—
128#123—2411no—
129#123—2411no—
130#121—2411no—
131#121—2411no—
132#121—2411no—
133#121—2411no—
134#121—2411no—
135#121D24111yescharacteristic, complemented
136#120—2411no—
137#120—2411no—
138#120—2411no—
139C24—2411no—
140C2×D4—2411no—
141C2×D4—2411no—
142C2×D4—2411no—
143C2×D4—2411no—
144C2×D4—2411no—
145C2×D4—2411no—
146C2×D4—2411no—
147C2×D4—2411no—
148C2×D4—2411no—
149C2×D4C2×D24111yescharacteristic
150C2×D4C2×D24111yescharacteristic
151C2×D4C2×D24111yescharacteristic, complemented
152C22×C4—2411no—
153C22×C4—2411no—
154C22×C4C2×D24111yescharacteristic, complemented
155C4⋊C4—2411no—
156C4⋊C4—2411no—
157C4⋊C4—2411no—
158C4⋊C4—2411no—
159C4⋊C4—2411no—
160C4⋊C4C2×D24111yescharacteristic, complemented
161C22⋊C4—2411no—
162C22⋊C4—2411no—
163C22⋊C4—2411no—
164C22⋊C4—2411no—
165C22⋊C4—2411no—
166C22⋊C4—2411no—
167C22⋊C4—2411no—
168C22⋊C4—2411no—
169C22⋊C4—2411no—
170C22⋊C4—2411no—
171C22⋊C4C2×D24111yescharacteristic
172C22⋊C4C2×D24111yescharacteristic, complemented
173C42—2411no—
174C23—9641no—
175C23—9641no—
176C23—9641no—
177C23—2411no—
178C23—2411no—
179C23—2411no—
180C23C22×D24111yescharacteristic
181C23C22×D24111yescharacteristic
182C23C22×D24111yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—9641no—
187D4—9641no—
188D4—9641no—
189D4—9641no—
190D4—9641no—
191D4—9641no—
192D4—9641no—
193D4—9641no—
194D4—9641no—
195C2×C4—41no—
196C2×C4—9641no—
197C2×C4—9641no—
198C2×C4—2411no—
199C2×C4—2411no—
200C2×C4—2411no—
201C2×C4—2411no—
202C2×C4—2411no—
203C2×C4C22×D24111yescharacteristic
204C2×C4C22×D24111yescharacteristic
205C2×C4C22×D24111yescharacteristic
206C2×C4C22×D24111yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—9644no—
217C22—4821no—
218C22—4821no—
219C22—4821no—
220C22—4821no—
221C22—4821no—
222C22—4821no—
223C22—4821no—
224C22—4821no—
225C22—4821no—
226C22C23×D24111yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—4821no—
232C4—4821no—
233C4—4821no—
234C4—4821no—
235C4—4821no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—9641no—
240C2—9641no—
241C2—9641no—
242C2#2144068711yescharacteristic
243C2#2144068711yescharacteristic
244C2#2144068811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
Fitting#121×C241C2
hypercenter#121D241
hypocenterC241#560
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #21440638C2semidirect
2. #21440637C2semidirect
3. #21440635C2semidirect
4. #21440627C2semidirect
5. #21440626C2semidirect
6. #21440625C2semidirect
7. #21440616C2semidirect
8. #21440615C2semidirect
9. #21440614C2semidirect
10. #21440614C2semidirect
11. #21440613C2semidirect
12. #21440612C2semidirect
13. #21440610C2semidirect
14. #21440610C2semidirect
15. #121×C241C2semidirect
24. C2×C22⋊D241C22semidirect
25. C2×C22⋊D241C22semidirect
26. C22×Dic241C22semidirect
27. C2×C4×D241C22semidirect
28. C4⋊Dic241C22semidirect
29. C4⋊Dic241C22semidirect
30. #16838749C22semidirect
31. #16838749C22semidirect
32. #16838749C22semidirect
35. C22⋊Dic241C22semidirect
42. #16838747C22semidirect
43. C4×Dic241C22semidirect
46. C2×D4×C241C22semidirect
47. C22×C4×C241C22semidirect
48. C4⋊C4×C241C22semidirect
50. C22⋊C4×C241C22semidirect
122. C241#560semidirect
135. #121D241semidirect
151. C2×D4C2×D241semidirect
154. C22×C4C2×D241semidirect
160. C4⋊C4C2×D241semidirect
172. C22⋊C4C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406382
#214406372
#214406352
#214406272
#214406262
#214406252
#214406162
#214406152
#214406142
#214406132
#214406122
#214406102
#121×C2412
#560241