  In this paper, we give a refinement of a generalized Dedekind's theorem. In
addition, we show that all possible values of integer group determinants of any
group are also possible values of integer group determinants of its any abelian
subgroup. By applying the refinement of a generalized Dedekind's theorem, we
determine all possible values of integer group determinants of the direct
product group of the cyclic group of order $8$ and the cyclic group of order
$2$.
