The original question has been answered. As background information, I point out that S. Amitsur classified finite groups which can occur as subgroups of multiplicative groups of divison algebras. These include finite groups which have a faithful "one-dimensional" representation over the quaternions. As noted in Ben's answer, any such finite group (in the general division ring case) has cyclic Sylow $p$-subgroups for each odd prime $p$ and cyclic or generalized quaternion Sylow $2$-subgroups. It also satisfies other group-theoretic properties, such as elements of order $p$ and $q$ commuting when $p$ and $q$ are distinct primes.
In fact, finite groups which occur as subgroups of multiplicative groups of division algebras are Frobenius complements. We note that any generalized quaternion group already occurs - this is not in conflict with Ben's answer: if that group has order greater than $8,$ then it has a non-Abelian dihedral group as a homomorphic image, so not all its irreducible representations over the quaternions are $1$-dimensional. The only non-solvable group which occurs is ${\rm SL}(2,5),$ which does already occur inside the multiplicative group of the real quaternions ( as does each generalized quaternion $2$-group)- again, the fact that ${\rm SL}(2,5)$ has $A_{5}$ as a homomorphic image means that not all its irreducible representations over the quaternions are $1$-dimensional.