Fix a prime $p$ and let $x$ be a $p$-adic integer that is not an integer.
Now let $M$ be the group of pairs of rational numbers $(r,s)$ so that $rx-s$ is 
a $p$-adic integer. 
Then $M$ maps onto $\mathbb Q$ but it does not contain a copy of $\mathbb Q$,
because that would mean that some $r/s$ would not just be a good approximation of $x$,
but a perfect one.