Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
66 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1084 | C2 | 1 | 2 | yes | maximal |
| 3 | C3×Dic6 | — | 3 | 4 | no | maximal |
| 4 | C2×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 5 | C32×C4 | D3 | 1 | 4 | yes | — |
| 6 | C3×Dic3 | — | 3 | 8 | no | — |
| 7 | H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C3×Q8 | — | 9 | 1 | no | — |
| 9 | Dic6 | — | 3 | 4 | no | — |
| 10 | C2×C32 | C2×D3 | 1 | 4 | yes | — |
| 11 | C3×C4 | — | 3 | 4 | no | — |
| 12 | C3×C4 | — | 9 | 2 | no | — |
| 13 | C3×C4 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 14 | Dic3 | — | 3 | 8 | no | — |
| 15 | C32 | Dic6 | 1 | 4 | yes | complemented |
| 16 | Q8 | — | 9 | 1 | no | Sylow |
| 17 | C2×C3 | — | 3 | 4 | no | — |
| 18 | C2×C3 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 19 | C4 | — | 9 | 2 | no | — |
| 20 | C4 | #272 | 1 | 1 | yes | characteristic |
| 21 | C3 | — | 3 | 4 | no | — |
| 22 | C3 | #634 | 1 | 1 | yes | characteristic |
| 23 | C2 | C2×#272 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. H(3,𝔽3) | Q8 | semidirect |
| 15. C32 | Dic6 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×H(3,𝔽3) | 2 |
| #1084 | 2 |
| C3×Dic6 | 3 |