Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
7491 conjugacy classes of subgroups; only the 1389 normal subgroups are shown.
This group has 7491 conjugacy classes of subgroups — too many to draw. Only its 1389 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 |
|---|---|---|
| 2. C23×C4×H(3,𝔽3) | C2 | semidirect |
| 3. C23×#1084 | C2 | semidirect |
| 4. C22×C4×#272 | C2 | direct (N × H) |
| 7. C23×#272 | C4 | direct (N × H) |
| 9. C22×C4×H(3,𝔽3) | C22 | semidirect |
| 10. C22×#1084 | C22 | semidirect |
| 11. C2×C4×#272 | C22 | direct (N × H) |
| 13. C23×C32×C4 | D3 | semidirect |
| 16. C23×H(3,𝔽3) | C2×C4 | semidirect |
| 18. C22×#272 | C2×C4 | direct (N × H) |
| 20. C2×C4×H(3,𝔽3) | C23 | semidirect |
| 21. C2×#1084 | C23 | semidirect |
| 22. C4×#272 | C23 | direct (N × H) |
| 28. C22×C32×C4 | C2×D3 | semidirect |
| 31. C22×H(3,𝔽3) | C22×C4 | semidirect |
| 33. C2×#272 | C22×C4 | direct (N × H) |
| 35. C4×H(3,𝔽3) | C24 | semidirect |
| 36. #1084 | C24 | semidirect |
| 45. C23×C32 | C4×D3 | semidirect |
| 49. C2×C32×C4 | C22×D3 | semidirect |
| 53. C2×H(3,𝔽3) | C23×C4 | semidirect |
| 55. #272 | C23×C4 | direct (N × H) |
| 68. C22×C32 | C2×C4×D3 | semidirect |
| 72. C32×C4 | C23×D3 | semidirect |
| 77. C23×C4 | #272 | direct (N × H) |
| 78. H(3,𝔽3) | C24×C4 | semidirect |
| 93. C2×C32 | C22×C4×D3 | semidirect |
| 101. C22×C4 | C2×#272 | direct (N × H) |
| 114. C32 | C23×C4×D3 | semidirect |
| 117. C23 | C4×#272 | direct (N × H) |
| 121. C2×C4 | C22×#272 | direct (N × H) |
| 130. C22 | C2×C4×#272 | direct (N × H) |
| 133. C4 | C23×#272 | direct (N × H) |
| 137. 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 |
| C23×#1084 | 2 |
| C22×C4×#272 | 2 |
| C23×C3×C4×D3 | 3 |