We prove upper bounds for the cardinality of constant dimension codes (CDC) which contain a lifted maximum rank distance (LMRD) code as a subset. Thereby we cover all parameters fulfilling <inline-formula> <tex-math notation="LaTeX">k\lt 3d/2 </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> is the codeword dimension and <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> is the minimum subspace distance. The proofs of the bounds additionally show that an LMRD code <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula> can be unioned with a CDC <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula> (of fitting parameters) without violating the subspace distance condition iff each codeword of <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula> intersects the special subspace of <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula> in at least dimension <inline-formula> <tex-math notation="LaTeX">d/2 </tex-math></inline-formula>. This connection is used to find the new largest and sometimes bound achieving CDCs for small parameters.