Let \(K\) be a number field. An elliptic curve \(E/K\) is said to have a non-abelian entanglement if there are relatively prime positive integers, \(m_1\) and \(m_2\), such that \(K(E[m_1])\cap K(E[m_2])\) is a non-abelian Galois extension of \(K\). In this talk, we will discuss our (joint with Nathan Jones) ongoing efforts to classify, using explicit methods, all infinite families of elliptic curves \(E/K\), for a fixed \(K\), that have a non-abelian entanglement.