Latin square from group subgroup

Question

Let $G$ be a group and $H$ be a subgroup of $G$ of index $n$. Then from the cosets $G/H$ we can construct a Latin square of order $n$ using the complete set of representatives of $H$ in $G$. Is it true in the other direction? That is given a Latin square Can we construct $G$ and $H$ such that its Cosets gives the Latin square back? Or is there any counter example?