  Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we study
the distribution of rectangles in a given set in $\mathbb{F}_q^2$. More
precisely, for any $0<\delta\le 1$, we prove that there exists an integer
$q_0=q_0(\delta)$ with the following property: if $q\ge q_0$ and $A$ is a
multiplicative subgroup of $\mathbb{F}^*_q$ with $|A|\ge q^{2/3}$, then any set
$S\subset \mathbb{F}_q^2$ with $|S|\ge \delta q^2$ contains at least $\gg
\frac{|S|^4|A|^2}{q^5}$ rectangles with side-lengths in $A$. We also consider
the case of rectangles with one fixed side-length and the other in a
multiplicative subgroup $A$.
