Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
900 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 240 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×C32 | D32 |
| commutator, cosocle | C2×C32 | C23×C32 |
| Frattini | C2 | C2×C32×D32 |
| Fitting, socle, prime radical | C22×C34 | C22 |
| hypocenter | C32 | C2×C32×Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C32×#627 | C2 | direct (N × H) |
| 5. C2×C3×#627 | C3 | direct (N × H) |
| 17. C3×#627 | C2×C3 | direct (N × H) |
| 18. C2×C34 | Q8 | semidirect |
| 22. C2×#627 | C32 | direct (N × H) |
| 35. C34 | C2×Q8 | semidirect |
| 48. #627 | C2×C32 | direct (N × H) |
| 55. C2×C33 | Dic6 | semidirect |
| 56. C2×C33 | C3×Q8 | semidirect |
| 83. C33 | C2×Dic6 | semidirect |
| 84. C33 | C2×C3×Q8 | semidirect |
| 106. C2×C32 | #627 | direct (N × H) |
| 107. C2×C32 | C3×Dic6 | semidirect |
| 108. C2×C32 | C32×Q8 | semidirect |
| 131. C32 | C2×#627 | direct (N × H) |
| 132. C32 | C2×C3×Dic6 | semidirect |
| 133. C32 | C2×C32×Q8 | semidirect |
| 143. C2×C3 | C3×#627 | direct (N × H) |
| 144. C2×C3 | C32×Dic6 | semidirect |
| 153. C3 | C2×C3×#627 | direct (N × H) |
| 154. C3 | C2×C32×Dic6 | semidirect |
| 155. C2 | C32×#627 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×C3⋊Dic3 | 2 |
| C2×C33×Dic3 | 2 |
| C32×#627 | 2 |
| C2×C32×Dic6 | 3 |
| C2×C3×#627 | 3 |