Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
494 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 61 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 | D3×((C3⋊D3)⋊C3)⋊C2 |
| commutator, Fitting, cosocle | C2×C3×H(3,𝔽3) | C23 |
| Frattini | C2×C3 | D33 |
| socle | C2×C32 | C2×D32 |
| hypocenter | C3×H(3,𝔽3) | C2×D4 |
| LCS | G ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D3×(C3⋊D3)⋊C3 | C2 | semidirect |
| 2. #10619902 | C2 | semidirect |
| 3. D3×#1084 | C2 | semidirect |
| 4. #10619884 | C2 | semidirect |
| 5. C3×#6198 | C2 | semidirect |
| 8. C2×C3⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 10. C2×C3×(C3⋊D3)⋊C3 | C22 | semidirect |
| 12. C3⋊#1084 | C22 | semidirect |
| 13. C3×#1084 | C22 | semidirect |
| 14. C2×D3×C3⋊D3 | D3 | semidirect |
| 27. #6198 | D3 | direct (N × H) |
| 30. D3×H(3,𝔽3) | D4 | semidirect |
| 35. C2×C3⋊(C3⋊D3) | C2×D3 | semidirect |
| 37. C2×C3×C3⋊D3 | C2×D3 | semidirect |
| 52. C2×(C3⋊D3)⋊C3 | C2×D3 | semidirect |
| 56. #1084 | C2×D3 | semidirect |
| 57. C3×H(3,𝔽3) | C2×D4 | semidirect |
| 88. C32×D3 | #67 | semidirect |
| 109. C2×C3⋊D3 | D32 | semidirect |
| 139. C33 | C2×#67 | semidirect |
| 142. H(3,𝔽3) | D3×D4 | semidirect |
| 230. C32 | D3×#67 | semidirect |
| 257. D3 | #6198 | direct (N × H) |
| 269. C3 | C2×#6198 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×(C3⋊D3)⋊C3 | 2 |
| #10619902 | 2 |
| D3×#1084 | 2 |
| #10619884 | 2 |
| C3×#6198 | 2 |
| D3×#626 | 3 |
| C2×#6198 | 3 |