Sub Hopf-algebra of the convolution algebra $C(G)$ necessarily of the form $C(H)$?

Question

Let $G$ be a finite group and $C(G)$ the convolution Hopf algebra of functions $G \to \mathbb{C}$ with comultiplication given by precomposing with the group multiplication and using the isomorphism $C(G\times G)\cong C(G)\otimes C(G)$. Is a sub-Hopf algebra of $C(G)$ necessarily of the form $C(H)$ for some subgroup $H\leq G$? Is this a 1:1-correspondence? I know this for the group ring $kG$, but this is not always self-dual.