Can countable dense additive subgroups of the reals be well-ordered up to isomorphism by inclusion?


If so, is $\mathbb{Q}$ the smallest (up to isomorphism) countable dense subgroup of the reals, and what is the second smallest (up to isomorphism)?



Countable Dense Sub-Groups of the Reals...
$\{a2^b:a,b\in\mathbb Z\}$ and $\{a3^b:a,b\in\mathbb Z\}$ are both countable dense additive subgroups of the reals, and they are not embeddable in each other (hence $\mathbb Q$ is embeddable in neither).


Also, let $\{p\_k:k\in\mathbb N\}$ be an enumeration of primes, and let $A\_k$ consist of all fractions $a/b$ of integers such that $b$ is not divisible by $p\_0,\dots,p\_k$. Then $A\_k$ is an additive group, and $A\_0\supset A\_1\supset A\_2\supset\dots$ is an infinite strictly decreasing chain with respect to either inclusion or embeddability. Hence the poset of countable dense subgroups is not even well-founded.


