Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1744 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1744 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C34 | C23 |
| Fitting | C34×C4 | C22 |
| hypercenter | C4 | D3×C3⋊(C3⋊D3) |
| hypocenter | C34 | #41 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊(C3⋊(C3⋊(C3⋊D3))) | C2 | semidirect |
| 2. C3×C4×C3⋊(C3⋊D3) | C2 | semidirect |
| 3. C33×Dic6 | C2 | semidirect |
| 4. #10620147 | C2 | semidirect |
| 5. #10620144 | C2 | semidirect |
| 19. C4×C3⋊(C3⋊D3) | D3 | semidirect |
| 21. C32×Dic6 | D3 | semidirect |
| 41. C34 | #41 | semidirect |
| 50. C3×Dic6 | C3⋊D3 | semidirect |
| 76. C33 | #235 | semidirect |
| 77. C33 | #239 | semidirect |
| 88. Dic6 | C3⋊(C3⋊D3) | semidirect |
| 111. C32 | #3799 | semidirect |
| 112. C32 | #3832 | semidirect |
| 132. C3 | #89018 | semidirect |
| 133. C3 | #89080 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊(C3⋊(C3⋊(C3⋊D3))) | 2 |
| C3×C4×C3⋊(C3⋊D3) | 2 |
| C33×Dic6 | 2 |
| #10620147 | 2 |
| #10620144 | 2 |
| #89080 | 3 |
| #89018 | 3 |