  We study properties of a $p-$type subcritical branching process in random
environment initiated at moment zero by a vector $\mathbf{z}=\left(
z_{1},..,z_{p}\right) $\ of particles of different types. Assuming that the
process belongs to the class of the so-called strongly subcritical processes we
show that its survival probability to moment $n$\ behaves for large $n$\ as
$C(\mathbf{z})\lambda ^{n}$\ where $\lambda $\ is the upper Lyapunov exponent
for the product of mean matrices of the process and $C(\mathbf{z})$% \ is an
explicitly given constant. We also demonstrate that the limiting conditional
distribution of the number of particles given the survival of the process for a
long time does not depend on the vector $\mathbf{z}$ of the number of particles
initiated the process.
