Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1212 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 164 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×C3⋊D32 |
| commutator, socle, cosocle | C2×C34 | C23 |
| Fitting | C34×C4 | C22 |
| prime radical | C2×C32×C3⋊D3 | C22 |
| hypercenter | C4 | C3⋊D32 |
| 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 |
| 3. C32×#634 | C2 | semidirect |
| 4. #10620153 | C2 | semidirect |
| 17. C3×#634 | D3 | semidirect |
| 25. C32×C3⋊D3 | Q8 | semidirect |
| 41. C34 | C2×Q8 | semidirect |
| 55. #634 | C3⋊D3 | direct (N × H) |
| 63. C3×C3⋊D3 | Dic6 | semidirect |
| 89. C33 | C2×Dic6 | semidirect |
| 90. C33 | D3×Q8 | semidirect |
| 109. C3⋊D3 | #634 | direct (N × H) |
| 127. C32 | D3×Dic6 | semidirect |
| 128. C32 | C2×#634 | semidirect |
| 129. C32 | Q8×C3⋊D3 | semidirect |
| 146. C3 | D3×#634 | semidirect |
| 147. C3 | C3⋊D3×Dic6 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10620175 | 2 |
| C32×C4×C3⋊D3 | 2 |
| C32×#634 | 2 |
| #10620153 | 2 |
| C3⋊D3×C3⋊Dic3 | 2 |
| C3⋊D3×Dic6 | 3 |
| D3×#634 | 3 |