Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
457 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 63 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. A4×PSU(3,𝔽2) | C2 | direct (N × H) |
| 3. C23×PSU(3,𝔽2) | C3 | semidirect |
| 11. C22×PSU(3,𝔽2) | C2×C3 | semidirect |
| 12. C2×C32×A4 | Q8 | semidirect |
| 14. A4×C3⋊D3 | Q8 | semidirect |
| 30. C2×PSU(3,𝔽2) | A4 | direct (N × H) |
| 32. C32×A4 | C2×Q8 | semidirect |
| 36. C23×C32 | C3×Q8 | semidirect |
| 41. C22×C3⋊D3 | C3×Q8 | semidirect |
| 55. PSU(3,𝔽2) | C2×A4 | direct (N × H) |
| 66. C22×C32 | C2×C3×Q8 | semidirect |
| 97. C2×A4 | PSU(3,𝔽2) | direct (N × H) |
| 104. C2×C32 | Q8×A4 | semidirect |
| 107. C3⋊D3 | Q8×A4 | semidirect |
| 137. A4 | C2×PSU(3,𝔽2) | direct (N × H) |
| 141. C32 | C2×Q8×A4 | semidirect |
| 148. C23 | C3×PSU(3,𝔽2) | semidirect |
| 180. C22 | C2×C3×PSU(3,𝔽2) | semidirect |
| 192. C2 | A4×PSU(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×A4×#157 | 2 |
| A4×PSU(3,𝔽2) | 2 |
| C23×PSU(3,𝔽2) | 3 |
| C2×C3×PSU(3,𝔽2) | 4 |
| C2×Q8×A4 | 9 |