The sextic field \(K=Q(\vartheta)\) with \(\vartheta\) satisfying \(x^6+ax^4+bx^2+e=0\) is said to be the lift of the cubic field \(C=Q(\alpha)\) with \(\alpha\) satisfying \(x^3+ax^2+bx+e=0\). For \(e=\pm 1\) the authors investigate monogenic cubic fields whose lift is also monogenic. There are eight Galois groups of sextic fields when this is possible. For five of these Galois groups infinitely many monogenic sextic fields are obtained this way, while for the remaining three Galois groups only finitely many. This result is an important contribution to the theory of monogenic fields.