  We present the phase diagram in a magnetic field of a 2D isotropic Heisenberg
antiferromagnet on a triangular lattice. We consider spin-$S$ model with
nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) interactions. We
focus on the range of $1/8<J_2/J_1<1$, where the ordered states are different
from those in the model with only nearest neighbor exchange. A classical ground
state in this range has four sublattices and is infinitely degenerate in any
field. The actual order is then determined by quantum fluctuations via "order
from disorder" phenomenon. We argue that the phase diagram is rich due to
competition between competing four-sublattice quantum states which break either
$\mathbb{Z}_3$ orientational symmetry or $\mathbb{Z}_4$ sublattice symmetry. At
small and high fields, the ground state is a $\mathbb{Z}_3$-breaking canted
stripe state, but at intermediate fields the ordered states break
$\mathbb{Z}_4$ sublattice symmetry. The most noticeable of such states is
"three up, one down" state in which spins in three sublattices are directed
along the field and in one sublattice opposite to the field. Such a state
breaks no continuous symmetry and has gapped excitations. As the consequence,
magnetization has a plateau at exactly one half of the saturation value. We
identify gapless states, which border the "three up, one down" state and
discuss the transitions between these states and the canted stripe state.
