The relation $x\sim y$ iff $S(x)=S(y)$ for $S(x)=\{z\in G:\langle x,z\rangle=G\}$ for a finite, non-cyclic group $G$.

Question

Set $G$ a finite 2-generated group (not cyclic) and if $x\in G$, call $S(x)=\{z\in G: \langle x,z \rangle =G \}$. Define on the elements of $G -\{e\}$ this equivalence relation: $x \sim y$ if and only if $S(x)=S(y)$.

Are there examples of group with $|[x]|>1$ for every $x\in G$, except the identity?

The dihedral group $D_p$ ($p$ prime) is an example, are there many others?