  We consider a planar Hamiltonian system of the type $Jz' = \nabla_z H(t,z)$,
where $H: \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}$ is a function periodic
in the time variable, such that $\nabla_z H(t,0) \equiv 0$ and $\nabla_z
H(t,z)$ is asymptotically linear for $\vert z \vert \to +\infty$. After
revisiting the index theory for linear planar Hamiltonian systems, by using the
Poincar\'e-Birkhoff fixed point theorem we prove that the above nonlinear
system has subharmonic solutions of any order $k$ large enough, whenever the
rotation numbers (or, equivalently, the mean Conley-Zehnder indices) of the
linearizations of the system at zero and at infinity are different.
Applications are given to the case of planar Hamiltonian systems coming from
second order scalar ODEs.
