Finite Group Databasebeta

← C3×#88251 #11929276

Subgroups — C3×#88251 #11929276

Order 1248.

332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×C3×D4×C13C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊D13C214yesmaximal, complemented
3C3×#6129C214yesmaximal, complemented
4C3×D4×D13C214yesmaximal, complemented
5C3×#6127C212yesmaximal, complemented
6#88251C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×C13C2212yescomplemented
8C22×C3×D13C2214yescomplemented
9C3×D4×C13C2214yescomplemented
10C2×C3×C4×C13C2211yescharacteristic, complemented
11C3×C22⋊D13C2214yescomplemented
12C3×C22⋊D13C2218yescomplemented
13C2×C3×Dic13C2214yescomplemented
14C3×D52C2212yescomplemented
15C3×C4×D13C2214yescomplemented
16C3×Dic26C2212yescomplemented
17C2×D4×C13C2×C311yescharacteristic, complemented
18C2×C22⋊D13C2×C314yescomplemented
19#6129C2×C314yescomplemented
20D4×D13C2×C314yescomplemented
21#6127C2×C312yescomplemented
22C22×C3×C13—42no—
23C22×C3×C13C2314yes—
24C22×C3×C13C2311yescharacteristic
25C2×C3×D13—44no—
26C2×C3×D13C2314yes—
27C3×C4×C13C2312yes—
28C3×Dic13C2314yes—
29C23×C13C22×C312yescomplemented
30C22×D13C22×C314yescomplemented
31D4×C13C22×C314yescomplemented
32C2×C4×C13C22×C311yescharacteristic, complemented
33C22⋊D13C22×C314yescomplemented
34C22⋊D13C22×C318yescomplemented
35C2×Dic13C22×C314yescomplemented
36D52C22×C312yescomplemented
37C4×D13C22×C314yescomplemented
38Dic26C22×C312yescomplemented
39C3×H(5,𝔽2)—131nomaximal, Hall
40C2×C3×C13—24no—
41C2×C3×C13—21no—
42C2×C3×C13C2411yescharacteristic
43C3×D13—24no—
44C22×C13—42no—
45C22×C13C23×C314yes—
46C22×C13C23×C311yescharacteristic
47C2×D13—44no—
48C2×D13C23×C314yes—
49C4×C13C23×C312yes—
50Dic13C23×C314yes—
51C3×#41—134no—
52C3×#41—132no—
53C2×C3×D4—134no—
54C2×C3×D4—134no—
55C2×C3×D4D1311yescharacteristic, complemented
56C3×C13H(5,𝔽2)11yescharacteristic, complemented, Hall
57H(5,𝔽2)—131noSylow
58C2×C13—24no—
59C2×C13—21no—
60C2×C13C24×C311yescharacteristic
61D13—24no—
62C23×C3—134no—
63C23×C3C2×D1312yescomplemented
64C3×Q8—132no—
65C3×D4—138no—
66C3×D4—134no—
67C3×D4—132no—
68C3×D4C2×D1314yescomplemented
69C2×C3×C4—134no—
70C2×C3×C4—134no—
71C2×C3×C4C2×D1311yescharacteristic, complemented
72#41—134no—
73#41—132no—
74C2×D4—134no—
75C2×D4—134no—
76C2×D4C3×D1311yescharacteristic, complemented
77C13C3×H(5,𝔽2)11yescharacteristic, complemented, Sylow
78C22×C3—42no—
79C22×C3—134no—
80C22×C3—524no—
81C22×C3C22×D1314yes—
82C22×C3C22×D1311yescharacteristic
83C3×C4—134no—
84C3×C4C22×D1312yes—
85C23—134no—
86C23C2×C3×D1312yescomplemented
87Q8—132no—
88D4—138no—
89D4—134no—
90D4—132no—
91D4C2×C3×D1314yescomplemented
92C2×C4—134no—
93C2×C4—134no—
94C2×C4C2×C3×D1311yescharacteristic, complemented
95C2×C3—21no—
96C2×C3—24no—
97C2×C3—264no—
98C2×C3C23×D1311yescharacteristic
99C22—42no—
100C22—134no—
101C22—524no—
102C22C22×C3×D1314yes—
103C22C22×C3×D1311yescharacteristic
104C4—134no—
105C4C22×C3×D1312yes—
106C3#8825111yescharacteristic, direct product factor, Sylow
107C2—21no—
108C2—24no—
109C2—264no—
110C2C23×C3×D1311yescharacteristic
subgroupG/H
centerC2×C3C23×D13
commutator, cosocleC2×C13C24×C3
FrattiniC2C23×C3×D13
FittingC2×C3×D4×C13C2
socleC2×C3×C13C24
hypercenterC2×C3×D4D13
hypocenterC13C3×H(5,𝔽2)
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C13C2semidirect
2. C2×C3×C22⋊D13C2semidirect
3. C3×#6129C2semidirect
4. C3×D4×D13C2semidirect
5. C3×#6127C2semidirect
6. #88251C3direct (N × H)
7. C23×C3×C13C22semidirect
8. C22×C3×D13C22semidirect
9. C3×D4×C13C22semidirect
10. C2×C3×C4×C13C22semidirect
11. C3×C22⋊D13C22semidirect
12. C3×C22⋊D13C22semidirect
13. C2×C3×Dic13C22semidirect
14. C3×D52C22semidirect
15. C3×C4×D13C22semidirect
16. C3×Dic26C22semidirect
17. C2×D4×C13C2×C3semidirect
18. C2×C22⋊D13C2×C3semidirect
19. #6129C2×C3semidirect
20. D4×D13C2×C3semidirect
21. #6127C2×C3semidirect
29. C23×C13C22×C3semidirect
30. C22×D13C22×C3semidirect
31. D4×C13C22×C3semidirect
32. C2×C4×C13C22×C3semidirect
33. C22⋊D13C22×C3semidirect
34. C22⋊D13C22×C3semidirect
35. C2×Dic13C22×C3semidirect
36. D52C22×C3semidirect
37. C4×D13C22×C3semidirect
38. Dic26C22×C3semidirect
55. C2×C3×D4D13semidirect
56. C3×C13H(5,𝔽2)semidirect
63. C23×C3C2×D13semidirect
68. C3×D4C2×D13semidirect
71. C2×C3×C4C2×D13semidirect
76. C2×D4C3×D13semidirect
77. C13C3×H(5,𝔽2)semidirect
86. C23C2×C3×D13semidirect
91. D4C2×C3×D13semidirect
94. C2×C4C2×C3×D13semidirect
106. C3#88251direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×C132
C2×C3×C22⋊D132
C3×#61292
C3×D4×D132
C3×#61272
#882513
C3×H(5,𝔽2)13