There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their structure: [Structure and Representations of $\mathbb{Q}$-Groups][1] by F. Kletzing.

  [1]: https://link.springer.com/book/10.1007%2FBFb0103426