  The free streaming motion of dark matter particles imprints a cutoff in the
matter power spectrum and set the scale of the smallest dark matter halo.
Recent cosmological $N$-body simulations have shown that the central density
cusp is much steeper in haloes near the free streaming scale than in more
massive haloes. Here, we study the abundance and structure of subhaloes near
the free streaming scale at very high redshift using a suite of unprecedentedly
large cosmological $N$-body simulations, over a wide range of the host halo
mass. The subhalo abundance is suppressed strongly below the free streaming
scale, but the ratio between the subhalo mass function in the cutoff and no
cutoff simulations is well fitted by a single correction function regardless of
the host halo mass and the redshift. In subhaloes, the central slopes are
considerably shallower than in field haloes, however, are still steeper than
that of the NFW profile. Contrary, the concentrations are significantly larger
in subhaloes than haloes and depend on the subhalo mass. We compare two methods
to extrapolate the mass-concentration relation of haloes and subhaloes to z=0
and provide a new simple fitting function for subhaloes, based on a suite of
large cosmological $N$-body simulations. Finally, we estimate the annihilation
boost factor of a Milky-Way sized halo to be between 1.8 and 6.2.
