  We study the magnetic phase diagram of the $J_1$--$J_2$ Heisenberg
antiferromagnet on a honeycomb lattice at the strongly frustrated point
$J_2/J_1=1/2$ using large-scale Monte Carlo simulations. At low temperatures we
find three different field regimes, each characterized by different broken
discrete symmetries. In low magnetic fields up to $h_{c1}/J_1\approx 2.9$ the
$Z_3$ rotational lattice symmetry is spontaneously broken while a
1/2-magnetization plateau is stabilized around $h_{c2}/J_1=4$. The collinear
plateau state and the coplanar state in higher fields break the $Z_4$
translational symmetry and correspond to triple-$q$ magnetic structures. The
intermediate phase $h_{c1}<h<h_{c2}$ has an interesting symmetry structure,
breaking simultaneously the $Z_3$ and $Z_4$ symmetries. At much lower
temperatures the spatial broken discrete symmetries coexist with the quasi
long-range order of the transverse spin components.
