  The architecture of infinite structures with non-crystallographic symmetries
can be modeled via aperiodic tilings, but a systematic construction method for
finite structures with non-crystallographic symmetry at different radial levels
is still lacking. We present here a group theoretical method for the
construction of finite nested point set with non-crystallographic symmetry.
Akin to the construction of quasicrystals, we embed a non-crystallographic
group $G$ into the point group $\mathcal{P}$ of a higher dimensional lattice
and construct the chains of all $G$-containing subgroups. We determine the
orbits of lattice points under such subgroups, and show that their projection
into a lower dimensional $G$-invariant subspace consists of nested point sets
with $G$-symmetry at each radial level. The number of different radial levels
is bounded by the index of $G$ in the subgroup of $\mathcal{P}$. In the case of
icosahedral symmetry, we determine all subgroup chains explicitly and
illustrate that these point sets in projection provide blueprints that
approximate the organisation of simple viral capsids, encoding information on
the structural organisation of capsid proteins and the genomic material
collectively, based on two case studies. Contrary to the affine extensions
previously introduced, these orbits endow virus architecture with an underlying
finite group structure, which lends itself better for the modelling of its
dynamic properties than its infinite dimensional counterpart.
