Subspace codes have important applications in random network coding. It is interesting to construct subspace codes with both sizes, and the minimum distances are as large as possible. In particular, cyclic constant dimension subspaces codes have additional properties which can be used to make encoding and decoding more efficient. In this paper, we construct large cyclic constant dimension subspace codes with minimum distances $2k-2$ and $2k$. These codes are contained in $\mathcal{G}_q(n, k)$, where $\mathcal{G}_q(n, k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{F}_{q^n}$. Consequently, some results in \cite{FW}, \cite{NXG}, and \cite{ZT} are extended.