Finite Group Databasebeta

← #29508300

Subgroups — #29508300

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#21440637C211yesmaximal, characteristic, complemented
2#21440636C211yesmaximal, characteristic, complemented
3#21440630C211yesmaximal, characteristic, complemented
4#21440628C211yesmaximal, characteristic, complemented
5#21440627C211yesmaximal, characteristic, complemented
6#21440625C211yesmaximal, characteristic, complemented
7#21440619C211yesmaximal, characteristic, complemented
8#21440616C211yesmaximal, characteristic, complemented
9#21440615C211yesmaximal, characteristic, complemented
10#21440614C211yesmaximal, characteristic, complemented
11#21440612C211yesmaximal, characteristic, complemented
12#21440610C211yesmaximal, characteristic, complemented
13C4×C22⋊D241C211yesmaximal, characteristic, complemented
14C2×#16838747C211yesmaximal, characteristic, complemented
15C4×D4×C241C211yesmaximal, characteristic, complemented
16C23×D241C2211yescharacteristic
17C2×Dic482C2211yescharacteristic, complemented
18C2×D964C2211yescharacteristic
19C2×D964C2211yescharacteristic
20C2×C22⋊D241C2211yescharacteristic
21C2×C22⋊D241C2211yescharacteristic
22C2×C22⋊D241C2211yescharacteristic
23C2×C22⋊D241C2211yescharacteristic, complemented
24C22×Dic241C2211yescharacteristic
25C2×C4×D241C2211yescharacteristic, complemented
26C4⋊Dic241C2211yescharacteristic, complemented
27#16838749C2211yescharacteristic, complemented
28#16838749C2211yescharacteristic, complemented
29#16838749C2211yescharacteristic, complemented
30#16838749C2211yescharacteristic, complemented
31C22⋊Dic241C2211yescharacteristic
32C22⋊Dic241C2211yescharacteristic, complemented
33#16838747C2211yescharacteristic
34#16838747C2211yescharacteristic
35#16838747C2211yescharacteristic
36#16838747C2211yescharacteristic
37#16838747C2211yescharacteristic
38#16838747C2211yescharacteristic
39#16838747C2211yescharacteristic
40#16838747C2211yescharacteristic
41#16838747C2211yescharacteristic, complemented
42#16838747C2211yescharacteristic, complemented
43C4×Dic241C2211yescharacteristic, complemented
44C2×D4×C241C2211yescharacteristic
45C22×C4×C241C2211yescharacteristic
46C22×C4×C241C2211yescharacteristic, complemented
47C4⋊C4×C241C2211yescharacteristic, complemented
48C22⋊C4×C241C2211yescharacteristic
49C22⋊C4×C241C2211yescharacteristic, complemented
50C42×C241C2211yescharacteristic, complemented
51C22×D241—22no—
52C22×D241—41no—
53C22×D241—41no—
54C22×D241C2311yescharacteristic
55C22×D241C2311yescharacteristic
56C22×D241C2311yescharacteristic
57Dic482—41no—
58D964—41no—
59D964—41no—
60C22⋊D241—22no—
61C22⋊D241—41no—
62C22⋊D241—41no—
63C22⋊D241—41no—
64C2×Dic241—41no—
65C2×Dic241C2311yescharacteristic
66C2×Dic241C2311yescharacteristic
67C2×Dic241C2311yescharacteristic
68C2×Dic241C2311yescharacteristic
69C2×Dic241C2311yescharacteristic
70C4×D241—41no—
71C23×C241C2311yescharacteristic
72C23×C241C2311yescharacteristic
73D4×C241—22no—
74C2×C4×C241—41no—
75C2×C4×C241—41no—
76C2×C4×C241C2311yescharacteristic
77C2×C4×C241C2311yescharacteristic
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#4112yes—
87C22×C241C2411yescharacteristic
88C2×D241—21no—
89C2×D241—21no—
90C2×D241—21no—
91C2×D241—21no—
92C2×D241—21no—
93C2×D241—21no—
94C2×D241—21no—
95C2×D241—21no—
96C2×D241—21no—
97C2×D241—44no—
98C4×C241—21no—
99C4×C241—21no—
100C4×C241—21no—
101C4×C241—21no—
102C4×C241—21no—
103Dic241—21no—
104Dic241—21no—
105Dic241—21no—
106Dic241—21no—
107Dic241—21no—
108C2×C241—22no—
109C2×C241—41no—
110C2×C241C2×#4111yescharacteristic
111C2×C241H(5,𝔽2)11yescharacteristic
112C2×C241H(5,𝔽2)11yescharacteristic
113D241—41no—
114D241—41no—
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#120—2411no—
130C4×D4—2411no—
131C4×D4D24111yescharacteristic, complemented
132C2×C22⋊C4—2411no—
133C24—2411no—
134C2×Q8—2411no—
135C2×D4—2411no—
136C2×D4—2411no—
137C2×D4—2411no—
138C2×D4—2411no—
139C2×D4—2411no—
140C2×D4—2411no—
141C2×D4C2×D24111yescharacteristic
142C22×C4—2411no—
143C22×C4—2411no—
144C22×C4C2×D24111yescharacteristic
145C22×C4C2×D24111yescharacteristic, complemented
146C4⋊C4—2411no—
147C4⋊C4—2411no—
148C4⋊C4—2411no—
149C4⋊C4—2411no—
150C4⋊C4—2411no—
151C4⋊C4C2×D24111yescharacteristic, complemented
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⋊C4—2411no—
164C22⋊C4C2×D24111yescharacteristic
165C22⋊C4C2×D24111yescharacteristic, complemented
166C42—2411no—
167C42C2×D24111yescharacteristic, complemented
168C23—9641no—
169C23—9641no—
170C23—4822no—
171C23—2411no—
172C23—2411no—
173C23—2411no—
174C23C22×D24111yescharacteristic
175C23C22×D24111yescharacteristic
176Q8—9641no—
177D4—22no—
178D4—9641no—
179D4—9641no—
180D4—9641no—
181D4—9641no—
182D4—9641no—
183D4—4822no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—9641no—
187C2×C4—9641no—
188C2×C4—2411no—
189C2×C4—2411no—
190C2×C4—2411no—
191C2×C4—2411no—
192C2×C4—2411no—
193C2×C4C22×D24111yescharacteristic
194C2×C4C22×D24111yescharacteristic
195C2×C4C22×D24111yescharacteristic
196C2×C4C22×D24111yescharacteristic
197C2×C4C22×D24111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—9644no—
204C22—4821no—
205C22—4821no—
206C22—4821no—
207C22—4821no—
208C22—4821no—
209C22—4821no—
210C22—4821no—
211C22—4821no—
212C22—4821no—
213C22#1683877012yes—
214C22C23×D24111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—4821no—
221C4—4821no—
222C4—4821no—
223C4—4821no—
224C4—4821no—
225C2—22no—
226C2—41no—
227C2—9641no—
228C2—9641no—
229C2—9641no—
230C2C2×#1683877011yescharacteristic
231C2#2144068711yescharacteristic
232C2#2144068811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
FittingC4×D4×C241C2
hypercenterC4×D4D241
hypocenterC241#538
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #21440637C2semidirect
2. #21440636C2semidirect
3. #21440630C2semidirect
4. #21440628C2semidirect
5. #21440627C2semidirect
6. #21440625C2semidirect
7. #21440619C2semidirect
8. #21440616C2semidirect
9. #21440615C2semidirect
10. #21440614C2semidirect
11. #21440612C2semidirect
12. #21440610C2semidirect
13. C4×C22⋊D241C2semidirect
14. C2×#16838747C2semidirect
15. C4×D4×C241C2semidirect
17. C2×Dic482C22semidirect
23. C2×C22⋊D241C22semidirect
25. C2×C4×D241C22semidirect
26. C4⋊Dic241C22semidirect
27. #16838749C22semidirect
28. #16838749C22semidirect
29. #16838749C22semidirect
30. #16838749C22semidirect
32. C22⋊Dic241C22semidirect
41. #16838747C22semidirect
42. #16838747C22semidirect
43. C4×Dic241C22semidirect
46. C22×C4×C241C22semidirect
47. C4⋊C4×C241C22semidirect
49. C22⋊C4×C241C22semidirect
50. C42×C241C22semidirect
116. C241#538semidirect
131. C4×D4D241semidirect
145. C22×C4C2×D241semidirect
151. C4⋊C4C2×D241semidirect
165. C22⋊C4C2×D241semidirect
167. C42C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406372
#214406362
#214406302
#214406282
#214406272
#214406252
#214406192
#214406162
#214406152
#214406142
#214406122
#214406102
C4×C22⋊D2412
C2×#168387472
C4×D4×C2412
#538241