Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1532 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 322 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 |
| prime radical | C2×C33×D3 | C22 |
| hypercenter | C4 | D3×C3⋊(C3⋊D3) |
| hypocenter | C34 | C2×Q8 |
| 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. #10620175 | C2 | semidirect |
| 2. C3×#6308 | C2 | semidirect |
| 4. #10620148 | C2 | semidirect |
| 19. #6308 | D3 | direct (N × H) |
| 25. C33×D3 | Q8 | semidirect |
| 40. C34 | C2×Q8 | semidirect |
| 56. C32×D3 | Dic6 | semidirect |
| 72. C33 | C2×Dic6 | semidirect |
| 73. C33 | D3×Q8 | semidirect |
| 89. C3×D3 | #634 | semidirect |
| 104. C32 | D3×Dic6 | semidirect |
| 105. C32 | C2×#634 | semidirect |
| 115. D3 | #6308 | direct (N × H) |
| 122. C3 | D3×#634 | semidirect |
| 123. C3 | C2×#6308 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10620175 | 2 |
| C3×#6308 | 2 |
| C33×C4×D3 | 2 |
| #10620148 | 2 |
| D3×C3⋊(C3⋊Dic3) | 2 |
| C2×#6308 | 3 |
| D3×#634 | 3 |