Existence, Uniqueness and Counterexamples · Superchallenge problem

For how many subsets $A$ of $\{1, 2, 3, 4, 5, 6\}$ is this statement true: "for every $x$ in $A$ there is a $y$ in $A$ with $y \ne x$ such that $x \mid y$ or $y \mid x$"? (Count the empty set if the statement is true for it.)