Nonomino

When rotations and reflections are not considered to be distinct shapes, there are 1,285 different free nonominoes.

[2] The 1,285 free nonominoes can be classified according to their symmetry groups:[2]

[3][4] Therefore a complete set cannot be packed into a rectangle and not all nonominoes have tilings.

Two additional nonominoes admit tilings, but satisfy neither of the previous criteria.

[5] This is the lowest order of polyomino for which such exceptions exist.

A nonomino or Jigsaw Sudoku puzzle, as seen in The Sunday Telegraph
The two nonominoes which can tile the plane, but cannot form a patch which satisfies the Conway criterion.