We study mutually unbiased bases (MUBs) in which all the bases are unextendible maximally entangled ones. We first present a necessary and sufficient condition of constructing a pair of MUBs in \(C^2 \otimes C^4\). Based on this condition, an analytical and necessary condition for constructing MUBs is given. Moreover we illustrate our approach by some detailed examples in \(C^2 \otimes C^4\). The results are generalized to \(C^2 \otimes C^d\) \((d\geq 3)\) and a concrete example in \(C^2 \otimes C^8\) is given.