Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
550 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 102 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. C32×C4×SL(2,𝔽3) | C2 | semidirect |
| 3. C32×#859 | C2 | semidirect |
| 4. C3×#5394 | C3 | direct (N × H) |
| 7. C32×GL(2,𝔽3) | C4 | semidirect |
| 11. C32×C4×Q8 | D3 | semidirect |
| 14. C3×C4×SL(2,𝔽3) | C2×C3 | semidirect |
| 15. C3×#859 | C2×C3 | semidirect |
| 23. C32×SL(2,𝔽3) | C2×C4 | semidirect |
| 24. #5394 | C32 | direct (N × H) |
| 31. C3×GL(2,𝔽3) | C3×C4 | semidirect |
| 49. C3×C4×Q8 | C3×D3 | semidirect |
| 52. C4×SL(2,𝔽3) | C2×C32 | semidirect |
| 53. #859 | C2×C32 | semidirect |
| 73. C3×SL(2,𝔽3) | C2×C3×C4 | semidirect |
| 86. GL(2,𝔽3) | C32×C4 | semidirect |
| 114. C4×Q8 | C32×D3 | semidirect |
| 136. SL(2,𝔽3) | C2×C32×C4 | semidirect |
| 180. C32 | #5394 | direct (N × H) |
| 215. C3 | C3×#5394 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×GL(2,𝔽3) | 2 |
| C32×C4×SL(2,𝔽3) | 2 |
| C32×#859 | 2 |
| C3×#5394 | 3 |
| C32×#440 | 3 |
| C32×(C2×D3)⋊C4 | 4 |