Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3539 conjugacy classes of subgroups; only the 457 normal subgroups are shown.
This group has 3539 conjugacy classes of subgroups — too many to draw. Only its 457 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #1088⋊C2 |
| commutator, cosocle | #272 | C25 |
| Frattini | C3 | C23×GO+(4,𝔽2) |
| Fitting | C23×H(3,𝔽3) | D4 |
| socle | C23×C3 | GO+(4,𝔽2) |
| hypocenter | H(3,𝔽3) | C23×D4 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×((C3⋊D3)⋊C3)⋊C2 | C2 | semidirect |
| 2. C23×#1088 | C2 | semidirect |
| 3. C22×#1088⋊C2 | C2 | direct (N × H) |
| 7. C22×((C3⋊D3)⋊C3)⋊C2 | C22 | semidirect |
| 8. C22×#1088 | C22 | semidirect |
| 9. C2×#1088⋊C2 | C22 | direct (N × H) |
| 11. C23×H(3,𝔽3) | D4 | semidirect |
| 12. C22×#272 | D4 | semidirect |
| 16. C2×((C3⋊D3)⋊C3)⋊C2 | C23 | semidirect |
| 17. C2×#1088 | C23 | semidirect |
| 18. #1088⋊C2 | C23 | direct (N × H) |
| 25. C22×H(3,𝔽3) | C2×D4 | semidirect |
| 26. C2×#272 | C2×D4 | semidirect |
| 30. ((C3⋊D3)⋊C3)⋊C2 | C24 | semidirect |
| 31. #1088 | C24 | semidirect |
| 44. C2×H(3,𝔽3) | C22×D4 | semidirect |
| 45. #272 | C22×D4 | semidirect |
| 69. H(3,𝔽3) | C23×D4 | semidirect |
| 116. C23 | #1088⋊C2 | direct (N × H) |
| 133. C22 | C2×#1088⋊C2 | direct (N × H) |
| 140. C2 | C22×#1088⋊C2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×((C3⋊D3)⋊C3)⋊C2 | 2 |
| C23×#1088 | 2 |
| C22×#1088⋊C2 | 2 |
| C23×D12 | 9 |