Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C4×C9 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊(C3⋊C8) | — | 3 | 1 | no | maximal |
| 3 | C9⋊C8 | — | 3 | 3 | no | maximal |
| 4 | C2×C3×C9 | C4 | 1 | 1 | yes | characteristic |
| 5 | C32×C4 | D3 | 1 | 1 | yes | characteristic |
| 6 | C4×C9 | D3 | 1 | 3 | yes | — |
| 7 | C3×C9 | C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C3⋊C8 | — | 9 | 3 | no | — |
| 9 | C3⋊C8 | — | 9 | 1 | no | — |
| 10 | C2×C32 | Dic3 | 1 | 1 | yes | characteristic |
| 11 | C2×C9 | Dic3 | 1 | 3 | yes | — |
| 12 | C3×C4 | D9 | 1 | 3 | yes | — |
| 13 | C3×C4 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 14 | C32 | C3⋊C8 | 1 | 1 | yes | characteristic |
| 15 | C9 | C3⋊C8 | 1 | 3 | yes | complemented |
| 16 | C8 | — | 27 | 1 | no | Sylow |
| 17 | C2×C3 | Dic9 | 1 | 3 | yes | — |
| 18 | C2×C3 | C3⋊Dic3 | 1 | 1 | yes | characteristic |
| 19 | C4 | C3⋊D9 | 1 | 1 | yes | characteristic |
| 20 | C3 | C9⋊C8 | 1 | 3 | yes | complemented |
| 21 | C3 | C3⋊(C3⋊C8) | 1 | 1 | yes | characteristic |
| 22 | C2 | C3⋊Dic9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C8 |
| Frattini | C3×C4 | C3⋊D3 |
| Fitting, cosocle | C3×C4×C9 | C2 |
| socle, prime radical | C2×C32 | Dic3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C3×C9 | C8 | semidirect |
| 15. C9 | C3⋊C8 | semidirect |
| 20. C3 | C9⋊C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×C9 | 2 |
| C3⋊(C3⋊C8) | 3 |
| C9⋊C8 | 3 |