Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
2760 conjugacy classes of subgroups; only the 756 normal subgroups are shown.
This group has 2760 conjugacy classes of subgroups — too many to draw. Only its 756 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C33 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C33 |
| Frattini | C2 | C22×C33×D3 |
| Fitting | C23×C34 | C2 |
| socle | C22×C34 | C22 |
| hypercenter | C23×C33 | D3 |
| hypocenter | C3 | C2×C33×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C33 ◁ C23×C33 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C34 | C2 | semidirect |
| 2. C22×C33×D3 | C2 | semidirect |
| 3. C33×#67 | C2 | direct (N × H) |
| 4. C2×C33×Dic3 | C2 | semidirect |
| 6. C2×C32×#67 | C3 | direct (N × H) |
| 9. C22×C34 | C22 | semidirect |
| 11. C2×C33×D3 | C22 | semidirect |
| 12. C33×Dic3 | C22 | semidirect |
| 15. C23×C33 | D3 | semidirect |
| 16. C23×C33 | C2×C3 | semidirect |
| 17. C22×C32×D3 | C2×C3 | semidirect |
| 20. C32×#67 | C2×C3 | direct (N × H) |
| 21. C2×C32×Dic3 | C2×C3 | semidirect |
| 23. C2×C34 | D4 | semidirect |
| 27. C2×C3×#67 | C32 | direct (N × H) |
| 36. C22×C33 | C2×D3 | semidirect |
| 38. C22×C33 | C22×C3 | semidirect |
| 40. C2×C32×D3 | C22×C3 | semidirect |
| 42. C32×Dic3 | C22×C3 | semidirect |
| 43. C34 | C2×D4 | semidirect |
| 46. C23×C32 | C3×D3 | semidirect |
| 47. C23×C32 | C2×C32 | semidirect |
| 48. C22×C3×D3 | C2×C32 | semidirect |
| 51. C3×#67 | C2×C32 | direct (N × H) |
| 52. C2×C3×Dic3 | C2×C32 | semidirect |
| 59. C2×C33 | #67 | direct (N × H) |
| 60. C2×C33 | C3×D4 | semidirect |
| 65. C2×#67 | C33 | direct (N × H) |
| 74. C22×C32 | C2×C3×D3 | semidirect |
| 76. C22×C32 | C22×C32 | semidirect |
| 78. C2×C3×D3 | C22×C32 | semidirect |
| 80. C3×Dic3 | C22×C32 | semidirect |
| 82. C33 | C2×#67 | direct (N × H) |
| 83. C33 | C2×C3×D4 | semidirect |
| 86. C23×C3 | C32×D3 | semidirect |
| 87. C23×C3 | C2×C33 | semidirect |
| 88. C22×D3 | C2×C33 | semidirect |
| 91. #67 | C2×C33 | direct (N × H) |
| 92. C2×Dic3 | C2×C33 | semidirect |
| 99. C2×C32 | C3×#67 | direct (N × H) |
| 100. C2×C32 | C32×D4 | semidirect |
| 113. C22×C3 | C2×C32×D3 | semidirect |
| 115. C22×C3 | C22×C33 | semidirect |
| 117. C2×D3 | C22×C33 | semidirect |
| 119. Dic3 | C22×C33 | semidirect |
| 121. C32 | C2×C3×#67 | direct (N × H) |
| 122. C32 | C2×C32×D4 | semidirect |
| 124. C23 | C33×D3 | semidirect |
| 133. C2×C3 | C32×#67 | direct (N × H) |
| 134. C2×C3 | C33×D4 | semidirect |
| 142. C22 | C2×C33×D3 | semidirect |
| 145. C3 | C2×C32×#67 | direct (N × H) |
| 146. C3 | C2×C33×D4 | semidirect |
| 149. C2 | C33×#67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C34 | 2 |
| C22×C33×D3 | 2 |
| C33×#67 | 2 |
| C2×C33×Dic3 | 2 |
| C2×C33×D4 | 3 |
| C2×C32×#67 | 3 |