Finite Group Databasebeta

← C2×C3×#72209831 #72277325

Subgroups — C2×C3×#72209831 #72277325

Order 9216.

At least 10452 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has at least 10452 conjugacy classes of subgroups — too many to draw. Only its 144 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×#6499874C311yesmaximal, characteristic, complemented
2C2×C3×#11710730C211yesmaximal, characteristic, complemented
3C2×C3×#62817C2×C311yescharacteristic, complemented
4C2×#72209831C313yesmaximal, direct product factor
5C2×#6499874C3211yescharacteristic, complemented, Sylow
6C3×#72209831C212yesmaximal, direct product factor
7C3×#6499874C2×C312yescomplemented
8C2×#11710730C2×C313yescomplemented
9C3×#11710730C2211yescharacteristic, complemented
10#72209831C2×C316yesdirect product factor
11#11710730C22×C313yescomplemented
12C2×C3×H(7,𝔽2)A411yescharacteristic, complemented
13C3×#62817C22×C311yescharacteristic, complemented
14C2×C3×#3026A411yescharacteristic, complemented
15C2×C3×#2846A411yescharacteristic, complemented
16C2×C3×#2032A412yescomplemented
17C2×#62817C2×C3211yescharacteristic, complemented
18#6499874C2×C3212yescomplemented
19C27×C3C2×A411yescharacteristic, complemented
20C3×H(7,𝔽2)C2×A412yescomplemented
21C22×C3×H(5,𝔽2)C2×A411yescharacteristic
22C22×C3×H(5,𝔽2)C2×A411yescharacteristic, complemented
23C2×C3×#543C2×A412yescomplemented
24C3×#3026C2×A412yescomplemented
25C3×#2846C2×A412yescomplemented
26C3×#2032C2×A414yescomplemented
27C2×H(7,𝔽2)C3×A411yescharacteristic, complemented
28#62817C22×C3211yescharacteristic, complemented
29C2×#3026C3×A411yescharacteristic, complemented
30C2×#2846C3×A411yescharacteristic, complemented
31C2×#2032C3×A412yescomplemented
32C26×C3C22×A411yescharacteristic, complemented
33C2×C3×H(5,𝔽2)C22×A411yescharacteristic
34C2×C3×H(5,𝔽2)C22×A411yescharacteristic, complemented
35C23×C3×D4C22⋊A411yescharacteristic
36C3×#543C22×A412yescomplemented
37C27C2×C3×A411yescharacteristic, complemented
38H(7,𝔽2)C2×C3×A412yescomplemented
39C22×H(5,𝔽2)C2×C3×A411yescharacteristic
40C22×H(5,𝔽2)C2×C3×A411yescharacteristic, complemented
41C2×#543C2×C3×A412yescomplemented
42#3026C2×C3×A412yescomplemented
43#2846C2×C3×A412yescomplemented
44#2032C2×C3×A414yescomplemented
45C25×C3#120⋊C312yescomplemented
46C25×C3#99711yescharacteristic, complemented
47C25×C3C2×C22⋊A411yescharacteristic
48C22×C3×Q8#120⋊C312yescomplemented
49C22×C3×D4#99712yescomplemented
50C22×C3×D4C2×C22⋊A412yes—
51C23×C3×C4#99711yescharacteristic, complemented
52C26C22×C3×A411yescharacteristic, complemented
53C2×H(5,𝔽2)C22×C3×A411yescharacteristic
54C2×H(5,𝔽2)C22×C3×A411yescharacteristic, complemented
55C23×D4C3×C22⋊A411yescharacteristic
56#543C22×C3×A412yescomplemented
57C24×C3C2×#120⋊C312yescomplemented
58C24×C3C2×#99711yescharacteristic
59C24×C3C2×#99711yescharacteristic
60C24×C3C2×#99712yescomplemented
61C24×C3C22×C22⋊A411yescharacteristic
62C2×C3×Q8C2×#120⋊C312yescomplemented
63C22×C3×C4C2×#99712yescomplemented
64C25C3×#120⋊C312yescomplemented
65C25C3×#99711yescharacteristic, complemented
66C25C2×C3×C22⋊A411yescharacteristic
67C22×Q8C3×#120⋊C312yescomplemented
68C22×D4C3×#99712yescomplemented
69C22×D4C2×C3×C22⋊A412yes—
70C23×C4C3×#99711yescharacteristic, complemented
71C23×C3#8564511yescharacteristic
72C23×C3C22×#99711yescharacteristic
73C24C2×C3×#120⋊C312yescomplemented
74C24C2×C3×#99711yescharacteristic
75C24C2×C3×#99711yescharacteristic
76C24C2×C3×#99712yescomplemented
77C24C22×C3×C22⋊A411yescharacteristic
78C2×Q8C2×C3×#120⋊C312yescomplemented
79C22×C4C2×C3×#99712yescomplemented
80C22×C3#35346⋊C311yescharacteristic
81C22×C3#35346⋊C311yescharacteristic
82C22×C3C2×#8564511yescharacteristic
83C23C3×#8564511yescharacteristic
84C23C22×C3×#99711yescharacteristic
85C2×C3#7220983112yesdirect product factor
86C2×C3C2×#35346⋊C311yescharacteristic
87C22C3×#35346⋊C311yescharacteristic
88C22C3×#35346⋊C311yescharacteristic
89C22C2×C3×#8564511yescharacteristic
90C3C2×#7220983111yescharacteristic, direct product factor
91C2C3×#7220983112yesdirect product factor
92C2C2×C3×#35346⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3#35346⋊C3
commutator, cosocle, hypocenter#62817C22×C32
FrattiniC24C22×C3×C22⋊A4
FittingC2×C3×#6499874C3
hypercenterC23×C3#85645
LCSG ▷ #62817
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#6499874C3semidirect
2. C2×C3×#11710730C2semidirect
3. C2×C3×#62817C2×C3semidirect
4. C2×#72209831C3direct (N × H)
5. C2×#6499874C32semidirect
6. C3×#72209831C2direct (N × H)
7. C3×#6499874C2×C3semidirect
8. C2×#11710730C2×C3semidirect
9. C3×#11710730C22semidirect
10. #72209831C2×C3direct (N × H)
11. #11710730C22×C3semidirect
12. C2×C3×H(7,𝔽2)A4semidirect
13. C3×#62817C22×C3semidirect
14. C2×C3×#3026A4semidirect
15. C2×C3×#2846A4semidirect
16. C2×C3×#2032A4semidirect
17. C2×#62817C2×C32semidirect
18. #6499874C2×C32semidirect
19. C27×C3C2×A4semidirect
20. C3×H(7,𝔽2)C2×A4semidirect
22. C22×C3×H(5,𝔽2)C2×A4semidirect
23. C2×C3×#543C2×A4semidirect
24. C3×#3026C2×A4semidirect
25. C3×#2846C2×A4semidirect
26. C3×#2032C2×A4semidirect
27. C2×H(7,𝔽2)C3×A4semidirect
28. #62817C22×C32semidirect
29. C2×#3026C3×A4semidirect
30. C2×#2846C3×A4semidirect
31. C2×#2032C3×A4semidirect
32. C26×C3C22×A4semidirect
34. C2×C3×H(5,𝔽2)C22×A4semidirect
36. C3×#543C22×A4semidirect
37. C27C2×C3×A4semidirect
38. H(7,𝔽2)C2×C3×A4semidirect
40. C22×H(5,𝔽2)C2×C3×A4semidirect
41. C2×#543C2×C3×A4semidirect
42. #3026C2×C3×A4semidirect
43. #2846C2×C3×A4semidirect
44. #2032C2×C3×A4semidirect
45. C25×C3#120⋊C3semidirect
46. C25×C3#997semidirect
48. C22×C3×Q8#120⋊C3semidirect
49. C22×C3×D4#997semidirect
51. C23×C3×C4#997semidirect
52. C26C22×C3×A4semidirect
54. C2×H(5,𝔽2)C22×C3×A4semidirect
56. #543C22×C3×A4semidirect
57. C24×C3C2×#120⋊C3semidirect
60. C24×C3C2×#997semidirect
62. C2×C3×Q8C2×#120⋊C3semidirect
63. C22×C3×C4C2×#997semidirect
64. C25C3×#120⋊C3semidirect
65. C25C3×#997semidirect
67. C22×Q8C3×#120⋊C3semidirect
68. C22×D4C3×#997semidirect
70. C23×C4C3×#997semidirect
73. C24C2×C3×#120⋊C3semidirect
76. C24C2×C3×#997semidirect
78. C2×Q8C2×C3×#120⋊C3semidirect
79. C22×C4C2×C3×#997semidirect
85. C2×C3#72209831direct (N × H)
90. C3C2×#72209831direct (N × H)
91. C2C3×#72209831direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#64998743
C2×C3×#3026⋊C34
C2×C3×#2032⋊C34
C2×C3×#875474
C2×C3×AΣL(1,𝔽64)4
C2×C3×#117107302
C2×#722098313
C3×#722098312