  Compactifications of the heterotic string on special T^d/Z_2 orbifolds
realize a landscape of string models with 16 supercharges and a gauge group on
the left-moving sector of reduced rank d+8. The momenta of untwisted and
twisted states span a lattice known as the Mikhailov lattice II_{(d)}, which is
not self-dual for d > 1. By using computer algorithms which exploit the
properties of lattice embeddings, we perform a systematic exploration of the
moduli space for d=1 and 2, and give a list of maximally enhanced points where
the U(1)^{d+8} enhances to a rank d+8 non-Abelian gauge group. For d = 1, these
groups are simply-laced and simply-connected, and in fact can be obtained from
the Dynkin diagram of E_{10}. For d = 2 there are also symplectic and
doubly-connected groups. For the latter we find the precise form of their
fundamental groups from embeddings ofof lattices into the dual of II_{(2)}.
  Our results easily generalize to d > 2.
