Let $m$ be greater than or equal to 2 and $n$ be a multiple of $m$. We will call a spanning subgraph whose components are $K_m$ of the complete graph $K_{n}$ a $K_m$-spanning subgraph of $K_{n}$. The Dihedral group $D_{n}$ acts on the complete graph $K_{n}$ naturally. This action of $D_{n}$ induces the action on the set of the $K_m$-spanning subgraphs of the complete graph $K_{n}$ . In [3], we calculated the number of the equivalence classes of the 1-regular spanning subgraphs of the complete graph $K_n$ of even order $n$ by this action by using Burnside's Lemma. This is in the case $m=2$. In this paper, we generalize this results and calculate the number of the non-equivalent $K_m$-spanning subgraphs of $K_{n}$ for all $m$ and $n$.