In this paper, we study partial words in relation with pcodes, compatibility, and containment. First, we introduce C⊂(L), the set of all partial words contained by elements of L, and C⊃(L), the set of all partial words containing elements of L, for a set L of partial words. We discuss the relation between C(L), the set of all partial words compatible with elements of the set L, C⊂(L), and C⊃(L). Next, we consider the condition for C(L), C⊂(L), and C⊃(L) to be a pcode when L is a pcode. Furthermore, we introduce some classes of pcodes. An infix pcode and a comma-free pcode are defined, and the inclusion relation among these classes is established.