Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
280 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 37 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. C22×PSU(3,𝔽2)⋊C3 | C2 | semidirect |
| 2. C4×PSU(3,𝔽2)⋊C3 | C2 | semidirect |
| 3. D4×PSU(3,𝔽2) | C3 | semidirect |
| 7. C22×PSU(3,𝔽2) | C2×C3 | semidirect |
| 11. C4×PSU(3,𝔽2) | C2×C3 | semidirect |
| 12. PSU(3,𝔽2)⋊C3 | D4 | direct (N × H) |
| 50. PSU(3,𝔽2) | C3×D4 | semidirect |
| 53. C32×D4 | SL(2,𝔽3) | semidirect |
| 54. #638 | SL(2,𝔽3) | semidirect |
| 55. C4⋊(C3⋊D3) | SL(2,𝔽3) | semidirect |
| 72. C22×C32 | C2×SL(2,𝔽3) | semidirect |
| 75. C2×C3⋊D3 | C2×SL(2,𝔽3) | semidirect |
| 87. C32×C4 | C2×SL(2,𝔽3) | semidirect |
| 88. C3⋊Dic3 | C2×SL(2,𝔽3) | semidirect |
| 149. C32 | D4×SL(2,𝔽3) | semidirect |
| 157. D4 | PSU(3,𝔽2)⋊C3 | direct (N × H) |
| 180. C22 | C2×PSU(3,𝔽2)⋊C3 | semidirect |
| 186. C4 | C2×PSU(3,𝔽2)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×PSU(3,𝔽2)⋊C3 | 2 |
| C4×PSU(3,𝔽2)⋊C3 | 2 |
| D4×PSU(3,𝔽2) | 3 |
| D4×(C3⋊D3)⋊C3 | 4 |
| D4×SL(2,𝔽3) | 9 |