  Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order
$p \ge 5$, then there exists an involution $y$ such that $<x,y>$ is not
solvable. Also, if $x$ is an involution then there exist three conjugates of
$x$ that generate a nonsolvable group, unless $x$ belongs to a short list of
exceptions, which are described explicitly. We also prove that if $x$ has order
$6$ or $9$, then there exists two conjugates that generate a nonsolvable group.
