  We analyse the properties of substructures within dark matter halos
(subhalos) using a set of high-resolution numerical simulations of the
formation of structure in a Lambda-CDM Universe. Our simulation set includes 11
high-resolution simulations of massive clusters as well as a region of mean
density, allowing us to study the spatial and mass distribution of
substructures down to a mass resolution limit of 10^9 h^(-1)Mo. We also
investigate how the properties of substructures vary as a function of the mass
of the `parent' halo in which they are located. We find that the substructure
mass function depends at most weakly on the mass of the parent halo and is well
described by a power-law. The radial number density profiles of substructures
are steeper in low mass halos than in high mass halos. More massive
substructures tend to avoid the centres of halos and are preferentially located
in the external regions of their parent halos. We also study the mass accretion
and merging histories of substructures, which we find to be largely independent
of environment. We find that a significant fraction of the substructures
residing in clusters at the present day were accreted at redshifts z < 1. This
implies that a significant fraction of present-day `passive' cluster galaxies
should have been still outside the cluster progenitor and more active at z~1.
