  Coset diagrams have been used to study quotients, orbits, subgroups and
structure of the finitely generated groups. In this paper we use coset diagrams
and modular arithmetic to determine the $G$-orbits of $\QQ^*(\sqrt{p^k})$,
$\QQ^*(\sqrt{2p^k})$, $\QQ^*(\sqrt{2^2p^k})$, and in general
$\QQ^*(\sqrt{2^lp^k})$, for each $l\geq3$ and $k=2h+1\geq3$, for each odd prime
$p$.
