  In this article, the partial plane spreads in $PG(6,2)$ of maximum possible
size $17$ and of size $16$ are classified. Based on this result, we obtain the
classification of the following closely related combinatorial objects: Vector
space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD
codes of minimum rank distance $3$, and subspace codes with parameters
$(7,17,6)_2$ and $(7,34,5)_2$.
