Nonsingular $H$-matrices play important roles in the theory and application of numerical linear algebra, therefore, it is very important for us to study the determinate conditions for a nonsingular $H$-matrix. In this paper, we present some sufficient conditions for nonsingular $H$-matrices according to the properties of generalized strictly $\alpha$-chain diagonally dominant matrices, generalized strictly $\alpha$-diagonally dominant matrices and by introducing the iteration factors. The proposed conditions improve some related results and the theory of nonsingular $H$-matrices is also enriched. Finally, the effectiveness of these sufficient conditions is illustrated with numerical examples.