Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
7491 conjugacy classes of subgroups; only the 1133 normal subgroups are shown.
This group has 7491 conjugacy classes of subgroups — too many to draw. Only its 1133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×#272 | C2 | semidirect |
| 2. C23×C4×H(3,𝔽3) | C2 | semidirect |
| 3. C22×C4⋊#272 | C2 | direct (N × H) |
| 7. C23×#272 | C22 | semidirect |
| 8. C22×C4×H(3,𝔽3) | C22 | semidirect |
| 9. C2×C4⋊#272 | C22 | direct (N × H) |
| 11. C23×C32×C4 | D3 | semidirect |
| 13. C23×H(3,𝔽3) | D4 | semidirect |
| 16. C22×#272 | C23 | semidirect |
| 17. C2×C4×H(3,𝔽3) | C23 | semidirect |
| 18. C4⋊#272 | C23 | direct (N × H) |
| 24. C22×C32×C4 | C2×D3 | semidirect |
| 26. C22×H(3,𝔽3) | C2×D4 | semidirect |
| 29. C2×#272 | C24 | semidirect |
| 30. C4×H(3,𝔽3) | C24 | semidirect |
| 37. C23×C32 | D12 | semidirect |
| 41. C2×C32×C4 | C22×D3 | semidirect |
| 44. C2×H(3,𝔽3) | C22×D4 | semidirect |
| 57. C22×C32 | C2×D12 | semidirect |
| 61. C32×C4 | C23×D3 | semidirect |
| 64. C23×C4 | #272 | semidirect |
| 65. H(3,𝔽3) | C23×D4 | semidirect |
| 78. C2×C32 | C22×D12 | semidirect |
| 85. C22×C4 | C2×#272 | semidirect |
| 96. C32 | C23×D12 | semidirect |
| 99. C23 | C4⋊#272 | direct (N × H) |
| 102. C2×C4 | C22×#272 | semidirect |
| 111. C22 | C2×C4⋊#272 | direct (N × H) |
| 113. C4 | C23×#272 | semidirect |
| 117. C2 | C22×C4⋊#272 | 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 |
| C23×C4×H(3,𝔽3) | 2 |
| C22×C4⋊#272 | 2 |
| C23×C3×D12 | 3 |