yes, if ω is a primitive cube root of unity (a root of x^2 + x + 1 = (x^3 - 1)/(x - 1).).
Q(2^(1/3)) only contains the single real root of x^3 - 2. there are two other complex roots: (2^(1/3))ω, and (2^(1/3))ω^2.
clearly any splitting field K for x^3 - 2 has to contain these three roots, so Q(2^(1/3),ω) is contained in K.
on the other hand, Q(2^(1/3),ω) is the smallest extension of Q containing all three roots. this field is of degree 6 over Q.