  In this paper, we describe a complete computer classification of the
hemisystems in the two known flock generalized quadrangles of order $(5^2,5)$
and give numerous further examples of hemisystems in all the known flock
generalized quadrangles of order $(s^2,s)$ for $s \le 11$. By analysing the
computational data, we identify two possible new infinite families of
hemisystems in the classical generalized quadrangle $H(3,q^2)$.
