Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
630 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 123 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⋊(C9⋊S4) |
| commutator, cosocle, hypocenter | C3×C9×A4 | C22 |
| Frattini | C3 | C2×C3⋊(C3⋊S4) |
| Fitting | C23×C3×C9 | D3 |
| socle | C23×C32 | C3⋊D3 |
| LCS | G ▷ C3×C9×A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C9×A4 | C2 | semidirect |
| 2. C3⋊(C9⋊S4) | C2 | direct (N × H) |
| 8. C3×C9×A4 | C22 | semidirect |
| 11. C23×C3×C9 | D3 | semidirect |
| 13. C2×C3×C22⋊C9 | D3 | semidirect |
| 14. C2×C9×A4 | D3 | semidirect |
| 31. C22×C3×C9 | C2×D3 | semidirect |
| 37. C3×C22⋊C9 | C2×D3 | semidirect |
| 38. C9×A4 | C2×D3 | semidirect |
| 43. C2×C3×A4 | D9 | semidirect |
| 50. C23×C9 | C3⋊D3 | semidirect |
| 52. C2×C22⋊C9 | C3⋊D3 | semidirect |
| 63. C2×C3×C9 | S4 | semidirect |
| 78. C3×A4 | C2×D9 | semidirect |
| 83. C22×C9 | C2×C3⋊D3 | semidirect |
| 88. C22⋊C9 | C2×C3⋊D3 | semidirect |
| 93. C3×C9 | C2×S4 | semidirect |
| 94. C23×C3 | C3⋊D9 | semidirect |
| 98. C2×A4 | C3⋊D9 | semidirect |
| 117. C2×C9 | C3⋊S4 | semidirect |
| 125. C22×C3 | C2×C3⋊D9 | semidirect |
| 134. A4 | C2×C3⋊D9 | semidirect |
| 141. C9 | C2×C3⋊S4 | semidirect |
| 143. C23 | C3⋊(C3⋊D9) | semidirect |
| 152. C2×C3 | C9⋊S4 | semidirect |
| 162. C22 | C2×C3⋊(C3⋊D9) | semidirect |
| 165. C3 | C2×C9⋊S4 | semidirect |
| 170. C2 | C3⋊(C9⋊S4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C9×A4 | 2 |
| C3⋊(C9⋊S4) | 2 |
| C2×C3⋊(C3⋊S4) | 3 |
| C2×C3⋊((C22⋊C9)⋊C2) | 3 |
| C2×C9⋊S4 | 3 |
| C2×#6233 | 3 |
| C2×C3⋊(C3⋊D9) | 4 |