Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5515 conjugacy classes of subgroups; only the 795 normal subgroups are shown.
This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 795 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×H(3,𝔽3) | C22 |
| socle | C24×C3 | D32 |
| hypocenter | H(3,𝔽3) | C23×D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×#272 | C2 | semidirect |
| 2. C24×(C3⋊D3)⋊C3 | C2 | semidirect |
| 3. C23×(C3⋊Dic3)⋊C3 | C2 | semidirect |
| 4. C22×#6200 | C2 | direct (N × H) |
| 9. C23×#272 | C22 | semidirect |
| 11. C23×(C3⋊D3)⋊C3 | C22 | semidirect |
| 12. C22×(C3⋊Dic3)⋊C3 | C22 | semidirect |
| 13. C2×#6200 | C22 | direct (N × H) |
| 14. C24×C3⋊D3 | D3 | semidirect |
| 19. C23×C3⋊Dic3 | D3 | semidirect |
| 23. C23×H(3,𝔽3) | D4 | semidirect |
| 26. C22×#272 | C23 | semidirect |
| 28. C22×(C3⋊D3)⋊C3 | C23 | semidirect |
| 29. C2×(C3⋊Dic3)⋊C3 | C23 | semidirect |
| 30. #6200 | C23 | direct (N × H) |
| 38. C23×C3⋊D3 | C2×D3 | semidirect |
| 47. C22×C3⋊Dic3 | C2×D3 | semidirect |
| 51. C22×H(3,𝔽3) | C2×D4 | semidirect |
| 54. C2×#272 | C24 | semidirect |
| 56. C2×(C3⋊D3)⋊C3 | C24 | semidirect |
| 57. (C3⋊Dic3)⋊C3 | C24 | semidirect |
| 73. C23×C32 | D12 | semidirect |
| 74. C23×C32 | #67 | semidirect |
| 78. C22×C3⋊D3 | C22×D3 | semidirect |
| 87. C2×C3⋊Dic3 | C22×D3 | semidirect |
| 92. C2×H(3,𝔽3) | C22×D4 | semidirect |
| 122. C22×C32 | C2×D12 | semidirect |
| 123. C22×C32 | C2×#67 | semidirect |
| 127. C2×C3⋊D3 | C23×D3 | semidirect |
| 136. C3⋊Dic3 | C23×D3 | semidirect |
| 142. H(3,𝔽3) | C23×D4 | semidirect |
| 173. C2×C32 | C22×D12 | semidirect |
| 174. C2×C32 | C22×#67 | semidirect |
| 215. C32 | C23×D12 | semidirect |
| 216. C32 | C23×#67 | semidirect |
| 221. C23 | #6200 | direct (N × H) |
| 244. C22 | C2×#6200 | direct (N × H) |
| 253. C2 | C22×#6200 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#272 | 2 |
| C24×(C3⋊D3)⋊C3 | 2 |
| C23×(C3⋊Dic3)⋊C3 | 2 |
| C22×#6200 | 2 |
| C23×#626 | 3 |
| C23×#625 | 3 |