Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
56 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22⋊H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×C3⋊D3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D3×A4 | — | 3 | 1 | no | maximal |
| 4 | (C3⋊D3)⋊C3 | — | 4 | 1 | no | maximal |
| 5 | C22×C32 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 7 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 8 | C3×A4 | — | 2 | 1 | no | — |
| 9 | C3×A4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 11 | C22×D3 | — | 3 | 1 | no | — |
| 12 | C22×D3 | — | 9 | 1 | no | — |
| 13 | C2×A4 | — | 9 | 1 | no | — |
| 14 | C2×C32 | — | 3 | 1 | no | — |
| 15 | C3⋊D3 | — | 3 | 1 | no | — |
| 16 | C3⋊D3 | A4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C3×D3 | — | 12 | 1 | no | — |
| 18 | C22×C3 | — | 3 | 1 | no | — |
| 19 | C22×C3 | C3×D3 | 1 | 1 | yes | characteristic |
| 20 | C2×D3 | — | 9 | 1 | no | — |
| 21 | C2×D3 | — | 9 | 3 | no | — |
| 22 | C2×D3 | — | 9 | 3 | no | — |
| 23 | C2×D3 | — | 9 | 1 | no | — |
| 24 | A4 | — | 3 | 1 | no | — |
| 25 | A4 | — | 6 | 1 | no | — |
| 26 | C32 | — | 4 | 1 | no | — |
| 27 | C32 | — | 8 | 1 | no | — |
| 28 | C32 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 29 | C23 | — | 9 | 1 | no | Sylow |
| 30 | C2×C3 | — | 3 | 3 | no | — |
| 31 | C2×C3 | — | 3 | 1 | no | — |
| 32 | C2×C3 | — | 36 | 1 | no | — |
| 33 | D3 | — | 3 | 1 | no | — |
| 34 | D3 | — | 9 | 1 | no | — |
| 35 | D3 | — | 9 | 3 | no | — |
| 36 | D3 | — | 9 | 1 | no | — |
| 37 | C22 | — | 27 | 1 | no | — |
| 38 | C22 | — | 27 | 1 | no | — |
| 39 | C22 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 40 | C3 | — | 3 | 1 | no | — |
| 41 | C3 | — | 12 | 1 | no | — |
| 42 | C3 | — | 24 | 1 | no | — |
| 43 | C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 44 | C2 | — | 3 | 1 | no | — |
| 45 | C2 | — | 9 | 1 | no | — |
| 46 | C2 | — | 27 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C32 | C2×C3 |
| Frattini | C3 | D3×A4 |
| socle | C22×C3 | C3×D3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊H(3,𝔽3) | C2 | semidirect |
| 2. C22×C3⋊D3 | C3 | semidirect |
| 5. C22×C32 | C2×C3 | semidirect |
| 9. C3×A4 | D3 | semidirect |
| 16. C3⋊D3 | A4 | semidirect |
| 28. C32 | C2×A4 | semidirect |
| 39. C22 | (C3⋊D3)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22⋊H(3,𝔽3) | 2 |
| C22×C3⋊D3 | 3 |
| D3×A4 | 3 |
| (C3⋊D3)⋊C3 | 4 |