Finite Group DatabasebetaFinite Group DatabasebetaOrder 1080.
272 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C22×C5 |
| Frattini | C2 | C5×D3×C3⋊D3 |
| Fitting, socle | C2×C33×C5 | C22 |
| hypocenter | C33 | C5×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×C3⋊(C3⋊D3) | C2 | semidirect |
| 2. C2×C32×C5×D3 | C2 | semidirect |
| 3. C3×C5×C3⋊Dic3 | C2 | semidirect |
| 9. #6291 | C5 | direct (N × H) |
| 14. C2×C3×C5×D3 | D3 | semidirect |
| 15. C5×C3⋊Dic3 | D3 | semidirect |
| 17. C33×C5 | D4 | semidirect |
| 20. C2×C3⋊(C3⋊D3) | C2×C5 | semidirect |
| 21. C2×C32×D3 | C2×C5 | semidirect |
| 22. C3×C3⋊Dic3 | C2×C5 | semidirect |
| 37. C2×C5×D3 | C3⋊D3 | semidirect |
| 44. C32×C5 | D12 | semidirect |
| 45. C32×C5 | #67 | semidirect |
| 51. C2×C3×D3 | C5×D3 | semidirect |
| 52. C3⋊Dic3 | C5×D3 | semidirect |
| 62. C33 | C5×D4 | semidirect |
| 77. C3×C5 | #626 | semidirect |
| 78. C3×C5 | #638 | semidirect |
| 83. C2×D3 | C5×C3⋊D3 | semidirect |
| 90. C32 | C5×D12 | semidirect |
| 91. C32 | C5×#67 | semidirect |
| 101. C5 | #6291 | direct (N × H) |
| 106. C3 | C5×#626 | semidirect |
| 107. C3 | C5×#638 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×C3⋊(C3⋊D3) | 2 |
| C2×C32×C5×D3 | 2 |
| C3×C5×C3⋊Dic3 | 2 |
| C5×#638 | 3 |
| C5×#626 | 3 |
| #6291 | 5 |