  We study the existence of subharmonic solutions in the system $\ddot
{u}(t)=f(t,u(t))$, where $u(t)\in\mathbb{R}^{k}$ and $f$ is an even and
$p$-periodic function in time. Under some additional symmetry conditions on the
function $f$, the problem of finding $mp$-periodic solutions can be
reformulated in a functional space as a $\Gamma\times\mathbb{Z}_{2}\times
D_{m}% $-equivariant equation, where the group $\Gamma\times\mathbb{Z}_{2}$
acts on the space $\mathbb{R}^{k}$ and $D_{m}$ acts on $u(t)$ by time-shifts
and reflection. We apply Brouwer equivariant degree to prove the existence of
an infinite number of subharmonic solutions for the function $f$ that satisfies
additional hypothesis on linear behavior near zero and the Nagumo condition at
infinity. We also discuss the bifurcation of subharmonic solutions when the
system depends on an extra parameter.
