Finite Group DatabasebetaFinite Group DatabasebetaOrder 216.
30 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#157 | C2 | 1 | 3 | yes | maximal |
| 2 | PSU(3,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C3×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 4 | #157 | C2×C3 | 1 | 3 | yes | — |
| 5 | C33 | Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C3×Q8 | — | 9 | 1 | no | maximal |
| 7 | C3⋊D3 | C22×C3 | 1 | 1 | yes | characteristic |
| 8 | C3×D3 | — | 12 | 1 | no | — |
| 9 | C3×C4 | — | 9 | 3 | no | — |
| 10 | C32 | — | 4 | 1 | no | — |
| 11 | C32 | — | 8 | 1 | no | — |
| 12 | C32 | C3×Q8 | 1 | 1 | yes | characteristic, complemented |
| 13 | Q8 | — | 9 | 1 | no | Sylow |
| 14 | C2×C3 | — | 9 | 1 | no | — |
| 15 | D3 | — | 12 | 1 | no | — |
| 16 | C4 | — | 9 | 3 | no | — |
| 17 | C3 | — | 4 | 1 | no | — |
| 18 | C3 | — | 8 | 1 | no | — |
| 19 | C3 | PSU(3,𝔽2) | 1 | 1 | yes | characteristic, direct product factor |
| 20 | C2 | — | 9 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | PSU(3,𝔽2) |
| commutator, cosocle | C3⋊D3 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C33 | Q8 |
| prime radical | C3×C3⋊D3 | C22 |
| hypocenter | C32 | C3×Q8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. PSU(3,𝔽2) | C3 | direct (N × H) |
| 5. C33 | Q8 | semidirect |
| 12. C32 | C3×Q8 | semidirect |
| 19. C3 | PSU(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#157 | 2 |
| PSU(3,𝔽2) | 3 |
| C3×Q8 | 9 |