Subnearrings of $(\mathbb{Z}[x],+,\circ )$

We show that the nearring $(\mathbb{Z}[x],+,\circ )$ of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that are generated by subsets of $\{1,x,x^2,x^3\}$.