Non-semisimple Hopf Algebras of Dimension p²

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Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic 0, where p <= q are odd primes. Suppose that S is the antipode of H. If H is not semisimple, then S⁴ᵖ=idₕ and Tr(S²ᵖ) is an integer divisible by p². In particular, if dim H = p², we prove that H is isomorphic to a Taft algebra. We then complete the classification for the Hopf algebras of dimension p².