Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
577 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A4×C3⋊D3 |
| commutator, cosocle, hypocenter | C22×H(3,𝔽3) | C22×C3 |
| Frattini | C3 | C2×A4×C3⋊D3 |
| Fitting | C23×H(3,𝔽3) | C2×C3 |
| socle | C23×C3 | C3×C3⋊D3 |
| LCS | G ▷ C22×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×A4×H(3,𝔽3) | C2 | semidirect |
| 2. A4×#272 | C2 | direct (N × H) |
| 4. C23×#272 | C3 | semidirect |
| 6. A4×H(3,𝔽3) | C22 | semidirect |
| 7. C2×C32×A4 | D3 | semidirect |
| 9. C23×H(3,𝔽3) | C2×C3 | semidirect |
| 13. C22×#272 | C2×C3 | semidirect |
| 21. C32×A4 | C2×D3 | semidirect |
| 24. C22×H(3,𝔽3) | C22×C3 | semidirect |
| 30. C2×#272 | A4 | direct (N × H) |
| 33. C23×C32 | C3×D3 | semidirect |
| 48. C2×H(3,𝔽3) | C2×A4 | semidirect |
| 50. #272 | C2×A4 | direct (N × H) |
| 57. C22×C32 | C2×C3×D3 | semidirect |
| 70. H(3,𝔽3) | C22×A4 | semidirect |
| 83. C2×A4 | #272 | direct (N × H) |
| 90. C2×C32 | D3×A4 | semidirect |
| 115. A4 | C2×#272 | direct (N × H) |
| 119. C32 | C2×D3×A4 | semidirect |
| 124. C23 | C3×#272 | semidirect |
| 146. C22 | C2×C3×#272 | semidirect |
| 155. C2 | A4×#272 | 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 |
| A4×#272 | 2 |
| C2×C3×D3×A4 | 3 |
| C23×#272 | 3 |
| C2×C3×#272 | 4 |