  Let $H_4$ and $H_8$ be the Sweedler's and Kac-Paljutkin Hopf algebras,
respectively. In this paper we prove that any Hopf algebra which factorizes
through $H_8$ and $H_4$ (equivalently, any bicrossed product between the Hopf
algebras $H_8$ and $H_4$) must be isomorphic to one of the following four Hopf
algebras: $H_8 \otimes H_4, H_{32,1}, H_{32,2}, H_{32,3}$. The set of all
matched pair $(H_8, H_4, \triangleright, \triangleleft)$ is explicitly
described, and then the associated bicrossed products is given by generators
and relations.
