Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
736 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C3⋊#157) |
| commutator, hypocenter | C34 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C4 |
| prime radical | C22×C32×C3⋊D3 | C2 |
| cosocle | C32×C3⋊D3 | C23 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊(C3⋊#157) | C2 | direct (N × H) |
| 4. C22×C34 | C4 | semidirect |
| 5. C2×C32×C3⋊D3 | C4 | semidirect |
| 7. C3⋊(C3⋊#157) | C22 | direct (N × H) |
| 11. C2×C34 | C2×C4 | semidirect |
| 12. C32×C3⋊D3 | C2×C4 | semidirect |
| 18. C22×C33 | Dic3 | semidirect |
| 19. C2×C3×C3⋊D3 | Dic3 | semidirect |
| 24. C34 | C22×C4 | semidirect |
| 32. C2×C33 | C2×Dic3 | semidirect |
| 33. C3×C3⋊D3 | C2×Dic3 | semidirect |
| 44. C22×C32 | C3⋊Dic3 | semidirect |
| 45. C22×C32 | #157 | semidirect |
| 46. C2×C3⋊D3 | C3⋊Dic3 | semidirect |
| 54. C33 | C22×Dic3 | semidirect |
| 64. C2×C32 | C2×C3⋊Dic3 | semidirect |
| 65. C2×C32 | C2×#157 | semidirect |
| 66. C3⋊D3 | C2×C3⋊Dic3 | semidirect |
| 75. C22×C3 | C3⋊#157 | semidirect |
| 83. C32 | C22×C3⋊Dic3 | semidirect |
| 84. C32 | C22×#157 | semidirect |
| 91. C2×C3 | C2×C3⋊#157 | semidirect |
| 96. C22 | C3⋊(C3⋊#157) | direct (N × H) |
| 100. C3 | C22×C3⋊#157 | semidirect |
| 103. C2 | C2×C3⋊(C3⋊#157) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×C3⋊D3 | 2 |
| C2×C3⋊(C3⋊#157) | 2 |
| C22×C3⋊#157 | 3 |
| C22×C3⋊Dic3 | 9 |