Normal Subgroups and Quotient Groups · Hard problem

In $S_4$, let $H = \langle(1\,2\,3\,4), (1\,3)\rangle$, a subgroup of order $8$, and let $N = \{e,(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3)\}$, the normal subgroup of order $4$. It turns out $N \subseteq H$. Using the Second Isomorphism Theorem relation $H/(H\cap N) \cong HN/N$, what is $|H/(H\cap N)|$?