Finite Group DatabasebetaFinite Group DatabasebetaOrder 2916.
2016 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 2016 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×D3×C3⋊D3 |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C22×C32 |
| Fitting | C3×H(5,𝔽3) | C22 |
| socle | C32 | C2×C32×C3⋊D3 |
| hypercenter | H(3,𝔽3) | D3×C3⋊D3 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D3×H(5,𝔽3) | C2 | semidirect |
| 2. C3⋊#92392 | C2 | semidirect |
| 3. C3×#92392 | C2 | semidirect |
| 5. C3×D3×#272 | C3 | semidirect |
| 7. C3×H(5,𝔽3) | C22 | semidirect |
| 9. C3×C3⋊#272 | C2×C3 | semidirect |
| 13. #92392 | D3 | direct (N × H) |
| 15. C32×#272 | C2×C3 | semidirect |
| 18. C3×D3×H(3,𝔽3) | D3 | semidirect |
| 19. C3×D3×H(3,𝔽3) | C2×C3 | semidirect |
| 24. D3×#272 | C32 | semidirect |
| 28. H(5,𝔽3) | C2×D3 | semidirect |
| 30. C32×H(3,𝔽3) | C2×D3 | semidirect |
| 31. C32×H(3,𝔽3) | C22×C3 | semidirect |
| 35. C33×D3 | C3×D3 | semidirect |
| 40. C3⋊#272 | C2×C32 | semidirect |
| 44. C3×#272 | C3×D3 | semidirect |
| 45. C3×#272 | C2×C32 | semidirect |
| 63. C34 | C2×C3×D3 | semidirect |
| 79. C3×H(3,𝔽3) | D32 | semidirect |
| 80. C3×H(3,𝔽3) | C2×C3×D3 | semidirect |
| 111. #272 | C32×D3 | semidirect |
| 128. C33 | C3×D32 | semidirect |
| 208. D3 | #92392 | direct (N × H) |
| 217. C3 | C2×#92392 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×H(5,𝔽3) | 2 |
| C3⋊#92392 | 2 |
| C3×#92392 | 2 |
| C2×#92392 | 3 |
| C3×D3×#272 | 3 |
| D32×H(3,𝔽3) | 3 |