Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3211 conjugacy classes of subgroups; only the 1204 normal subgroups are shown.
This group has 3211 conjugacy classes of subgroups — too many to draw. Only its 1204 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 |
|---|---|---|
| 3. C22×#634⋊C3 | C2 | direct (N × H) |
| 4. C23×#634 | C3 | semidirect |
| 9. C2×#634⋊C3 | C22 | direct (N × H) |
| 14. C22×#634 | C2×C3 | semidirect |
| 17. C23×H(3,𝔽3) | Q8 | semidirect |
| 21. #634⋊C3 | C23 | direct (N × H) |
| 32. C2×#634 | C22×C3 | semidirect |
| 35. C22×H(3,𝔽3) | C2×Q8 | semidirect |
| 51. C23×C32 | Dic6 | semidirect |
| 52. C23×C32 | C3×Q8 | semidirect |
| 59. #634 | C23×C3 | semidirect |
| 63. C2×H(3,𝔽3) | C22×Q8 | semidirect |
| 81. C22×C32 | C2×Dic6 | semidirect |
| 82. C22×C32 | C2×C3×Q8 | semidirect |
| 93. H(3,𝔽3) | C23×Q8 | semidirect |
| 114. C2×C32 | C22×Dic6 | semidirect |
| 115. C2×C32 | C22×C3×Q8 | semidirect |
| 138. C32 | C23×Dic6 | semidirect |
| 139. C32 | C23×C3×Q8 | semidirect |
| 140. C23 | #634⋊C3 | direct (N × H) |
| 153. C22 | C2×#634⋊C3 | direct (N × H) |
| 161. C2 | C22×#634⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×H(3,𝔽3) | 2 |
| C23×(C3⋊Dic3)⋊C3 | 2 |
| C22×#634⋊C3 | 2 |
| C23×#634 | 3 |
| C23×C3×Dic6 | 3 |