S4 decompose into D4 and cosets?

Question

Can $S_4$ (symmetry group of $4$) be represented by the union of $D_4$ (dihedral group of $4$) and the cosets (in $S_4$) thereof? If not why not?

Answer 1

Presumably, "dihedral group of 4" means the group of symmetries of a square (this is often denoted $D_8$ since it has 8 elements). If so, then answer is Yes - it's a subgroup of $S_4$ and, therefore, its cosets cover the whole group.

Answer 2

You have effectively answered your own question by talking about "the cosets of $D_4$ in $S_4$". This only makes sense because $D_4$ can be regarded as a subgroup of $S_4$ (namely the subgroup of permutations of the corners of a square that can be obtained by rotations and reflections). For every subgroup of a group, the group is the union of the subgroup and its cosets, so the answer is "yes".