In this paper, the concept of a subspined product of completely regular semigroups are introduced. Firstly, we give a necessary and sufficient condition for a subspined product A[＃45度]B of cryptogroups A and B to be also a cryptogroup. Secondly, it is shown that a subspined product A [＃45度] B is necessarily a cryptogroup if one of A and B is a band and the other is a cryptogroup. It is also shown that any subspined product A ~ B coincides with the spined product A 〓 B if one of A and B is a Clifford semrgroup and the other is a band.
Finally, the concept of a subspined product is extended to the concept of a P-subspined product for the class of completely P-regular semigroups, and some considerations are given for P-subspined products of P-cryptogroups.