Let $G$ be a finite group. Then the rational group algebra $\mathbb{Q}[G]$ has a wedderburn decomposition of the form $\prod_i M_{n_i}(D_i)$ where each $D_i$ is a division algebra.

My question is: for which $G$ do we have $n_i=1$ for all $i$? In other words, for which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras?

Clearly, this is true when $G$ is abelian. Another example is $G=Q_8$, the quaternion group of order $8$. Are there other examples? Is there a complete classification of such groups? If so, can you provide a reference?