Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
431 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 431 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 |
|---|---|---|
| 1. C2×A4×H(3,𝔽3) | C2 | semidirect |
| 2. S4×H(3,𝔽3) | C2 | direct (N × H) |
| 3. C2×C32×S4 | C3 | semidirect |
| 6. A4×H(3,𝔽3) | C22 | semidirect |
| 7. C2×C32×A4 | C2×C3 | semidirect |
| 8. C32×S4 | C2×C3 | semidirect |
| 10. C23×H(3,𝔽3) | D3 | semidirect |
| 21. C32×A4 | C22×C3 | semidirect |
| 27. C22×H(3,𝔽3) | C2×D3 | semidirect |
| 32. C23×C32 | C3×D3 | semidirect |
| 48. C2×H(3,𝔽3) | S4 | direct (N × H) |
| 49. C2×S4 | H(3,𝔽3) | direct (N × H) |
| 57. C22×C32 | C2×C3×D3 | semidirect |
| 67. H(3,𝔽3) | C2×S4 | direct (N × H) |
| 74. C2×A4 | C2×H(3,𝔽3) | semidirect |
| 75. S4 | C2×H(3,𝔽3) | direct (N × H) |
| 89. C2×C32 | C3×S4 | semidirect |
| 108. A4 | C22×H(3,𝔽3) | semidirect |
| 115. C32 | C2×C3×S4 | semidirect |
| 117. C23 | D3×H(3,𝔽3) | semidirect |
| 138. C22 | C2×D3×H(3,𝔽3) | semidirect |
| 148. C2 | S4×H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×A4×H(3,𝔽3) | 2 |
| S4×H(3,𝔽3) | 2 |
| C2×C32×S4 | 3 |
| C2×D4×H(3,𝔽3) | 3 |
| C2×D3×H(3,𝔽3) | 4 |