Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | PSU(3,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | (C3⋊D3)⋊C3 | — | 4 | 1 | no | maximal |
| 3 | #157 | — | 3 | 1 | no | — |
| 4 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 5 | SL(2,𝔽3) | — | 9 | 1 | no | maximal |
| 6 | C3⋊D3 | A4 | 1 | 1 | yes | characteristic |
| 7 | C3×D3 | — | 12 | 1 | no | — |
| 8 | C32 | — | 4 | 1 | no | — |
| 9 | C32 | — | 8 | 1 | no | — |
| 10 | C32 | SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 11 | Q8 | — | 9 | 1 | no | Sylow |
| 12 | C2×C3 | — | 36 | 1 | no | — |
| 13 | D3 | — | 12 | 1 | no | — |
| 14 | C4 | — | 27 | 1 | no | — |
| 15 | C3 | — | 4 | 1 | no | — |
| 16 | C3 | — | 12 | 1 | no | — |
| 17 | C3 | — | 24 | 1 | no | — |
| 18 | C2 | — | 9 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | PSU(3,𝔽2) | C3 |
| Fitting, socle | C32 | SL(2,𝔽3) |
| LCS | G ▷ PSU(3,𝔽2) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. PSU(3,𝔽2) | C3 | semidirect |
| 10. C32 | SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| PSU(3,𝔽2) | 3 |
| (C3⋊D3)⋊C3 | 4 |
| SL(2,𝔽3) | 9 |