Finite Group DatabasebetaFinite Group DatabasebetaOrder 486.
144 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×C27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C3⋊(C3⋊D9) | — | 3 | 1 | no | maximal |
| 3 | C3⋊D27 | — | 3 | 12 | no | maximal |
| 4 | C32×C9 | D3 | 1 | 1 | yes | characteristic |
| 5 | C3×C27 | D3 | 1 | 12 | yes | complemented |
| 6 | C3⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 7 | C3⋊D9 | — | 9 | 8 | no | — |
| 8 | C3⋊D9 | — | 9 | 4 | no | — |
| 9 | D27 | — | 9 | 9 | no | — |
| 10 | C33 | D9 | 1 | 1 | yes | characteristic |
| 11 | C3×C9 | D9 | 1 | 8 | yes | — |
| 12 | C3×C9 | C3⋊D3 | 1 | 4 | yes | — |
| 13 | C27 | C3⋊D3 | 1 | 9 | yes | complemented |
| 14 | C3⋊D3 | — | 27 | 9 | no | — |
| 15 | C3⋊D3 | — | 27 | 4 | no | — |
| 16 | D9 | — | 27 | 8 | no | — |
| 17 | D9 | — | 27 | 1 | no | — |
| 18 | C32 | D27 | 1 | 9 | yes | complemented |
| 19 | C32 | C3⋊D9 | 1 | 4 | yes | — |
| 20 | C9 | C3⋊D9 | 1 | 8 | yes | — |
| 21 | C9 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 22 | D3 | — | 81 | 12 | no | — |
| 23 | D3 | — | 81 | 1 | no | — |
| 24 | C3 | C3⋊D27 | 1 | 12 | yes | complemented |
| 25 | C3 | C3⋊(C3⋊D9) | 1 | 1 | yes | characteristic |
| 26 | C2 | — | 243 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C27 | C2 |
| Frattini | C9 | C3⋊(C3⋊D3) |
| socle | C33 | D9 |
| LCS | G ▷ C32×C27 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×C27 | C2 | semidirect |
| 5. C3×C27 | D3 | semidirect |
| 13. C27 | C3⋊D3 | semidirect |
| 18. C32 | D27 | semidirect |
| 24. C3 | C3⋊D27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C27 | 2 |
| C3⋊(C3⋊D9) | 3 |
| C3⋊D27 | 3 |