Finite Group Databasebeta

← C2×C3×#72215851 #72277863

Subgroups — C2×C3×#72215851 #72277863

Order 9216.

At least 9470 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has at least 9470 conjugacy classes of subgroups — too many to draw. Only its 172 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×#11709314C211yesmaximal, characteristic, complemented
2C2×C3×#8541⋊C3C211yesmaximal, characteristic, complemented
3C3×#72215851C214yesmaximal, direct product factor
4C3×#11709314C2212yescomplemented
5C3×#8541⋊C3C2212yescomplemented
6C2×C3×#11710729C211yesmaximal, characteristic, complemented
7C3×#11710729C2212yescomplemented
8C2×C3×#85645C2211yescharacteristic
9C2×C3×#347042C311yesmaximal, characteristic, complemented
10C2×C3×#8541C2×C311yescharacteristic, complemented
11C2×C3×#62817C2×C311yescharacteristic, complemented
12C2×C3×#15947C2×C311yescharacteristic, complemented
13C3×#347042C2×C314yescomplemented
14C2×#72215851C313yesmaximal, direct product factor
15C2×#8541⋊C3C2×C313yescomplemented
16C2×#11709314C2×C313yescomplemented
17C2×#11710729C2×C313yescomplemented
18#72215851C2×C3112yesdirect product factor
19C2×#347042C3211yescharacteristic, complemented, Sylow
20C2×C3×C22⋊(C22⋊A4)D411yescharacteristic, complemented
21C3×#85645D411yescharacteristic, complemented
22C3×#85645C2311yescharacteristic
23#11710729C22×C316yescomplemented
24C2×#85645C22×C313yes—
25#11709314C22×C316yescomplemented
26#8541⋊C3C22×C316yescomplemented
27C3×#62817C22×C312yescomplemented
28C23×C3×#120C22×C311yescharacteristic
29C2×C3×#2846A411yescharacteristic, complemented
30C3×#15947C22×C312yescomplemented
31C3×#8541C22×C312yescomplemented
32C3×C22⋊(C22⋊A4)C2×D411yescharacteristic, complemented
33C2×#62817C2×C3211yescharacteristic, complemented
34C2×#15947C2×C3211yescharacteristic, complemented
35#347042C2×C3214yescomplemented
36C2×#8541C2×C3211yescharacteristic, complemented
37C27×C3C3×D411yescharacteristic, complemented
38C27×C3C2×A411yescharacteristic, complemented
39C2×C22⋊(C22⋊A4)C3×D413yescomplemented
40C22×C3×#120C3×D411yescharacteristic, complemented
41C22×C3×#120C2×A411yescharacteristic
42C22×C3×#120C23×C311yescharacteristic
43#85645C3×D413yescomplemented
44#85645C23×C313yes—
45C3×#2846C2×A414yescomplemented
46C2×C3×#379C2×A411yescharacteristic, complemented
47#62817C22×C3212yescomplemented
48C23×#120C22×C3211yescharacteristic
49C2×#2846C3×A411yescharacteristic, complemented
50#15947C22×C3212yescomplemented
51#8541C22×C3212yescomplemented
52C26×C3C2×C3×D411yescharacteristic, complemented
53C26×C3C22×A411yescharacteristic
54C26×C3C22×A412yescomplemented
55C22⋊(C22⋊A4)C2×C3×D413yescomplemented
56C2×C3×#120C22×A412yes—
57C3×#379C22×A412yescomplemented
58C27C32×D411yescharacteristic, complemented
59C27C2×C3×A411yescharacteristic, complemented
60C22×#120C32×D411yescharacteristic, complemented
61C22×#120C2×C3×A411yescharacteristic
62C22×#120C23×C3211yescharacteristic
63#2846C2×C3×A414yescomplemented
64C2×#379C2×C3×A411yescharacteristic, complemented
65C25×C3#120⋊C311yescharacteristic, complemented
66C25×C3D4×A411yescharacteristic
67C25×C3D4×A411yescharacteristic
68C25×C3C23×A411yescharacteristic
69C26C2×C32×D411yescharacteristic, complemented
70C26C22×C3×A411yescharacteristic
71C26C22×C3×A412yescomplemented
72C2×#120C22×C3×A412yes—
73#379C22×C3×A412yescomplemented
74C24×C3C2×#120⋊C311yescharacteristic
75C24×C3C2×#120⋊C312yescomplemented
76C24×C3C2×D4×A411yescharacteristic
77C25C3×#120⋊C311yescharacteristic, complemented
78C25C3×D4×A411yescharacteristic
79C25C3×D4×A411yescharacteristic
80C25C23×C3×A411yescharacteristic
81C23×C3AΣL(1,𝔽64)11yescharacteristic
82C23×C3AΣL(1,𝔽64)11yescharacteristic
83C23×C3C22×#120⋊C311yescharacteristic
84C24C2×C3×#120⋊C311yescharacteristic
85C24C2×C3×#120⋊C312yescomplemented
86C24C2×C3×D4×A411yescharacteristic
87C22×C3C2×AΣL(1,𝔽64)11yescharacteristic
88C23C3×AΣL(1,𝔽64)11yescharacteristic
89C23C3×AΣL(1,𝔽64)11yescharacteristic
90C23C22×C3×#120⋊C311yescharacteristic
91C2×C3#7221585111yescharacteristic, direct product factor
92C22C2×C3×AΣL(1,𝔽64)11yescharacteristic
93C3C2×#7221585111yescharacteristic, direct product factor
94C2C3×#7221585111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#72215851
commutator, cosocleC22×#120C23×C32
FrattiniC25C23×C3×A4
FittingC2×C3×#347042C3
socleC23×C3AΣL(1,𝔽64)
hypocenterC26C2×C32×D4
LCSG ▷ C22×#120 ▷ C26
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#11709314C2semidirect
2. C2×C3×#8541⋊C3C2semidirect
3. C3×#72215851C2direct (N × H)
4. C3×#11709314C22semidirect
5. C3×#8541⋊C3C22semidirect
6. C2×C3×#11710729C2semidirect
7. C3×#11710729C22semidirect
9. C2×C3×#347042C3semidirect
10. C2×C3×#8541C2×C3semidirect
11. C2×C3×#62817C2×C3semidirect
12. C2×C3×#15947C2×C3semidirect
13. C3×#347042C2×C3semidirect
14. C2×#72215851C3direct (N × H)
15. C2×#8541⋊C3C2×C3semidirect
16. C2×#11709314C2×C3semidirect
17. C2×#11710729C2×C3semidirect
18. #72215851C2×C3direct (N × H)
19. C2×#347042C32semidirect
20. C2×C3×C22⋊(C22⋊A4)D4semidirect
21. C3×#85645D4semidirect
23. #11710729C22×C3semidirect
25. #11709314C22×C3semidirect
26. #8541⋊C3C22×C3semidirect
27. C3×#62817C22×C3semidirect
29. C2×C3×#2846A4semidirect
30. C3×#15947C22×C3semidirect
31. C3×#8541C22×C3semidirect
32. C3×C22⋊(C22⋊A4)C2×D4semidirect
33. C2×#62817C2×C32semidirect
34. C2×#15947C2×C32semidirect
35. #347042C2×C32semidirect
36. C2×#8541C2×C32semidirect
37. C27×C3C3×D4semidirect
38. C27×C3C2×A4semidirect
39. C2×C22⋊(C22⋊A4)C3×D4semidirect
40. C22×C3×#120C3×D4semidirect
43. #85645C3×D4semidirect
45. C3×#2846C2×A4semidirect
46. C2×C3×#379C2×A4semidirect
47. #62817C22×C32semidirect
49. C2×#2846C3×A4semidirect
50. #15947C22×C32semidirect
51. #8541C22×C32semidirect
52. C26×C3C2×C3×D4semidirect
54. C26×C3C22×A4semidirect
55. C22⋊(C22⋊A4)C2×C3×D4semidirect
57. C3×#379C22×A4semidirect
58. C27C32×D4semidirect
59. C27C2×C3×A4semidirect
60. C22×#120C32×D4semidirect
63. #2846C2×C3×A4semidirect
64. C2×#379C2×C3×A4semidirect
65. C25×C3#120⋊C3semidirect
69. C26C2×C32×D4semidirect
71. C26C22×C3×A4semidirect
73. #379C22×C3×A4semidirect
75. C24×C3C2×#120⋊C3semidirect
77. C25C3×#120⋊C3semidirect
85. C24C2×C3×#120⋊C3semidirect
91. C2×C3#72215851direct (N × H)
93. C3C2×#72215851direct (N × H)
94. C2C3×#72215851direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#117093142
C2×C3×#8541⋊C32
C3×#722158512
C2×C3×#117107292
C2×C3×AΣL(1,𝔽64)4
C2×C3×#3470423
C2×#722158513