Finite Group Databasebeta

← #29508302

Subgroups — #29508302

Order 15424.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#21440635C211yesmaximal, characteristic, complemented
2#21440635C211yesmaximal, characteristic, complemented
3#21440629C211yesmaximal, characteristic, complemented
4#21440628C211yesmaximal, characteristic, complemented
5#21440624C211yesmaximal, characteristic, complemented
6#21440623C211yesmaximal, characteristic, complemented
7#21440619C211yesmaximal, characteristic, complemented
8#21440614C211yesmaximal, characteristic, complemented
9#21440614C211yesmaximal, characteristic, complemented
10#21440613C211yesmaximal, characteristic, complemented
11#21440611C211yesmaximal, characteristic, complemented
12D4×Dic241C211yesmaximal, characteristic, complemented
13#21440602C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic241C211yesmaximal, characteristic, complemented
15#120×C241C211yesmaximal, characteristic, complemented
16C2×Dic482C2211yescharacteristic, complemented
17C2×C22⋊D241C2211yescharacteristic
18C2×C22⋊D241C2211yescharacteristic
19C2×C22⋊D241C2211yescharacteristic
20C2×C22⋊D241C2211yescharacteristic, complemented
21C22×Dic241C2211yescharacteristic
22C22×Dic241C2211yescharacteristic
23C22×Dic241C2211yescharacteristic, complemented
24C2×C4×D241C2211yescharacteristic, complemented
25C4⋊Dic241C2211yescharacteristic, complemented
26C4⋊Dic241C2211yescharacteristic, complemented
27#16838749C2211yescharacteristic, complemented
28#16838749C2211yescharacteristic, complemented
29#16838749C2211yescharacteristic, complemented
30#16838749C2211yescharacteristic, complemented
31C22⋊Dic241C2211yescharacteristic
32C22⋊Dic241C2211yescharacteristic
33C22⋊Dic241C2211yescharacteristic
34C22⋊Dic241C2211yescharacteristic
35C22⋊Dic241C2211yescharacteristic
36C22⋊Dic241C2211yescharacteristic
37C22⋊Dic241C2211yescharacteristic
38C22⋊Dic241C2211yescharacteristic, complemented
39C22⋊Dic241C2211yescharacteristic, complemented
40#16838747C2211yescharacteristic, complemented
41#16838747C2211yescharacteristic, complemented
42C4×Dic241C2211yescharacteristic, complemented
43C4×Dic241C2211yescharacteristic, complemented
44C24×C241C2211yescharacteristic
45C2×D4×C241C2211yescharacteristic
46C2×D4×C241C2211yescharacteristic
47C2×D4×C241C2211yescharacteristic
48C22⋊C4×C241C2211yescharacteristic
49C22⋊C4×C241C2211yescharacteristic
50C22⋊C4×C241C2211yescharacteristic
51C22×D241C2311yescharacteristic
52Dic482—41no—
53C22⋊D241—22no—
54C22⋊D241—41no—
55C22⋊D241—41no—
56C22⋊D241—41no—
57C2×Dic241—41no—
58C2×Dic241—41no—
59C2×Dic241—41no—
60C2×Dic241C2311yescharacteristic
61C2×Dic241C2311yescharacteristic
62C2×Dic241C2311yescharacteristic
63C2×Dic241C2311yescharacteristic
64C2×Dic241C2311yescharacteristic
65C2×Dic241C2311yescharacteristic
66C2×Dic241C2311yescharacteristic
67C4×D241—41no—
68C23×C241—22no—
69C23×C241—41no—
70C23×C241—41no—
71C23×C241C2311yescharacteristic
72C23×C241C2311yescharacteristic
73C23×C241C2311yescharacteristic
74C23×C241C2311yescharacteristic
75D4×C241—22no—
76D4×C241—41no—
77D4×C241—41no—
78C2×C4×C241C2311yescharacteristic
79C2×C4×C241C2311yescharacteristic
80C2×C4×C241C2311yescharacteristic
81C22×C241—21no—
82C22×C241—21no—
83C22×C241—21no—
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—44no—
93C22×C241#4112yes—
94C22×C241C2411yescharacteristic
95C2×D241—21no—
96C2×D241—21no—
97C2×D241—21no—
98C4×C241—21no—
99C4×C241—21no—
100C4×C241—21no—
101Dic241—21no—
102Dic241—21no—
103Dic241—21no—
104Dic241—21no—
105Dic241—21no—
106Dic241—21no—
107Dic241—21no—
108C2×C241—22no—
109C2×C241—41no—
110C2×C241—41no—
111C2×C241—41no—
112C2×C241C2×#4111yescharacteristic
113C2×C241H(5,𝔽2)11yescharacteristic
114C2×C241H(5,𝔽2)11yescharacteristic
115D241—41no—
116C241#53811yescharacteristic, complemented, Sylow
117#538—2411nomaximal, Sylow
118#126—2411no—
119#126—2411no—
120#124—2411no—
121#124—2411no—
122#123—2411no—
123#123—2411no—
124C22⋊Q8—2411no—
125#121—2411no—
126#121—2411no—
127#121—2411no—
128#120—2411no—
129#120D24111yescharacteristic, complemented
130C4×D4—2411no—
131C4×D4—2411no—
132C2×C22⋊C4—2411no—
133C24C2×D24111yescharacteristic
134C2×Q8—2411no—
135C2×D4—2411no—
136C2×D4—2411no—
137C2×D4—2411no—
138C2×D4—2411no—
139C2×D4C2×D24111yescharacteristic
140C2×D4C2×D24111yescharacteristic
141C2×D4C2×D24111yescharacteristic
142C22×C4—2411no—
143C22×C4—2411no—
144C22×C4—2411no—
145C22×C4—2411no—
146C4⋊C4—2411no—
147C4⋊C4—2411no—
148C4⋊C4—2411no—
149C4⋊C4—2411no—
150C4⋊C4—2411no—
151C4⋊C4—2411no—
152C22⋊C4—2411no—
153C22⋊C4—2411no—
154C22⋊C4—2411no—
155C22⋊C4—2411no—
156C22⋊C4—2411no—
157C22⋊C4—2411no—
158C22⋊C4—2411no—
159C22⋊C4—2411no—
160C22⋊C4—2411no—
161C22⋊C4—2411no—
162C22⋊C4—2411no—
163C22⋊C4C2×D24111yescharacteristic
164C22⋊C4C2×D24111yescharacteristic
165C22⋊C4C2×D24111yescharacteristic
166C42—2411no—
167C42—2411no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—2411no—
172C23C22×D24111yescharacteristic
173C23C22×D24111yescharacteristic
174C23C22×D24111yescharacteristic
175C23C22×D24111yescharacteristic
176Q8—9641no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—9641no—
181D4—9641no—
182D4—9641no—
183D4—4822no—
184C2×C4—9641no—
185C2×C4—9641no—
186C2×C4—9641no—
187C2×C4—9641no—
188C2×C4—2411no—
189C2×C4—2411no—
190C2×C4—2411no—
191C2×C4—2411no—
192C2×C4—2411no—
193C2×C4—2411no—
194C2×C4—2411no—
195C2×C4C22×D24111yescharacteristic
196C2×C4C22×D24111yescharacteristic
197C2×C4C22×D24111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—4821no—
211C22—4821no—
212C22—4821no—
213C22#1683876912yes—
214C22C23×D24111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—4821no—
219C4—4821no—
220C4—4821no—
221C4—4821no—
222C4—4821no—
223C4—4821no—
224C4—4821no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—9641no—
230C2C2×#1683876911yescharacteristic
231C2#2144068711yescharacteristic
232C2#2144068711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
Fitting#120×C241C2
hypercenter#120D241
hypocenterC241#538
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #21440635C2semidirect
2. #21440635C2semidirect
3. #21440629C2semidirect
4. #21440628C2semidirect
5. #21440624C2semidirect
6. #21440623C2semidirect
7. #21440619C2semidirect
8. #21440614C2semidirect
9. #21440614C2semidirect
10. #21440613C2semidirect
11. #21440611C2semidirect
12. D4×Dic241C2semidirect
13. #21440602C2semidirect
14. C2×C22⋊Dic241C2semidirect
15. #120×C241C2semidirect
16. C2×Dic482C22semidirect
20. C2×C22⋊D241C22semidirect
23. C22×Dic241C22semidirect
24. C2×C4×D241C22semidirect
25. C4⋊Dic241C22semidirect
26. C4⋊Dic241C22semidirect
27. #16838749C22semidirect
28. #16838749C22semidirect
29. #16838749C22semidirect
30. #16838749C22semidirect
38. C22⋊Dic241C22semidirect
39. C22⋊Dic241C22semidirect
40. #16838747C22semidirect
41. #16838747C22semidirect
42. C4×Dic241C22semidirect
43. C4×Dic241C22semidirect
116. C241#538semidirect
129. #120D241semidirect

Maximal subgroups

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

subgroupindex
#214406352
#214406292
#214406282
#214406242
#214406232
#214406192
#214406142
#214406132
#214406112
D4×Dic2412
#214406022
C2×C22⋊Dic2412
#120×C2412
#538241