The Picard group \mathbf{P} is a discrete subgroup of PSL(2,\Bbb{C}) with Gaussian integer coefficients. Here it is shown that the total number of conjugacy classes of elliptic elements of order 2 and 3 in \mathbf{P}, which is given as seven by B. Fine \left[ 3\right] , can actually be reduced to four and using this, the conditions for the maximal Fuchsian subgroups of \mathbf{P} to have elliptic elements of orders 2 and 3 are found.

The Picard group \mathbf{P} is a discrete subgroup of PSL(2,\Bbb{C}) with Gaussian integer coefficients. Here it is shown that the total number of conjugacy classes of elliptic elements of order 2 and 3 in \mathbf{P}, which is given as seven by B. Fine \left[ 3\right] , can actually be reduced to four and using this, the conditions for the maximal Fuchsian subgroups of \mathbf{P} to have elliptic elements of orders 2 and 3 are found.