This paper illustrates the Mobius groups $M$ and $M'$ on $Q(\sqrt{m})$, where $M'=\langle xy,yx\rangle$ is a subgroup of $M$. The system of linear congruence is used to discover classes $[a,b,c]\pmod{12}$ of elements of $Q*(\sqrt{n})$ and then by means of these classes, we explored several $M'$-subsets of $Q'''(\sqrt{n})$ which assist in finding more $M$-subsets of $Q(\sqrt{m})$.