Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
731 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 731 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.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×C3⋊S4 |
| commutator, cosocle | C32×A4 | C2×C32 |
| Fitting | C22×C3×H(3,𝔽3) | D3 |
| socle | C22×C32 | C32×D3 |
| hypercenter | H(3,𝔽3) | C3⋊S4 |
| hypocenter | C3×A4 | C2×H(3,𝔽3) |
| LCS | G ▷ C32×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×A4×H(3,𝔽3) | C2 | semidirect |
| 2. C32×C3⋊S4 | C3 | semidirect |
| 6. C33×A4 | C2×C3 | semidirect |
| 7. C22×C3×H(3,𝔽3) | D3 | semidirect |
| 11. A4×H(3,𝔽3) | D3 | semidirect |
| 23. C22×C33 | C3×D3 | semidirect |
| 30. C32×A4 | C3×D3 | semidirect |
| 35. C22×H(3,𝔽3) | C3⋊D3 | semidirect |
| 43. C3×H(3,𝔽3) | S4 | semidirect |
| 44. C3⋊S4 | H(3,𝔽3) | direct (N × H) |
| 63. C22×C32 | C3×C3⋊D3 | semidirect |
| 75. C3×A4 | C2×H(3,𝔽3) | semidirect |
| 87. C33 | C3×S4 | semidirect |
| 91. H(3,𝔽3) | C3⋊S4 | direct (N × H) |
| 112. C22×C3 | D3×H(3,𝔽3) | semidirect |
| 117. A4 | D3×H(3,𝔽3) | semidirect |
| 136. C32 | C3×C3⋊S4 | semidirect |
| 148. C22 | C3⋊D3×H(3,𝔽3) | semidirect |
| 156. C3 | S4×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×A4×H(3,𝔽3) | 2 |
| C32×C3⋊S4 | 3 |
| S4×H(3,𝔽3) | 3 |
| #67×H(3,𝔽3) | 3 |
| C3⋊D3×H(3,𝔽3) | 4 |