In this dissertation, we study the classification of Hopf algebras of dimension 24, and more generally, 8p where p is an odd prime. In particular, we show if a 24-dimensional Hopf algebra has a nontrivial grouplike element, then it must also have a nontrivial skew primitive element. Further, a nonsemisimple p-dimensional Hopf algebra cannot contain a semisimple Hopf subalgebra of dimension 8. Finally, we classify the nonsemisimple Hopf algebras of dimension 8p with the Chevalley property. In particular, we find such a Hopf algebra must either be pointed, or have a coradical of dimension 4p. Further the 4p-dimensional coradical can only be isomorphic to the dual of the dihedral group algebra, the dual of the dicyclic group algebra, or the self-dual, noncommutative semisimple Hopf algebra A+ of dimension 4p.