(n, q) denote the group of n by n matrices of determinant 1 over the field GF(c7) of q elements ; let PSL(n, q) be equal to SL(n, q) modulo its center. The subgroups of PSL(2, q) were determined by Dickson [12]. Those of PSL(3, q) were determined for odd q by Mitchell [19], using geometric methods. (The results for even q are given by Hartley [18].) In this paper we show that more modern group-theoretic methods can be used for a new determination of the subgroups of PSL(3, q), at least when q is odd. (For a result relevant to of the case of even q, see Suzuki [28].) Our major result is Theorem 1.1. Let q=pa be a power of an odd prime p, and let & be a subgroup ofPSL(3,q) of order > 1. Assume © has no normal elementary-abelian subgroup of order > 1. Then © is isomorphic to one of the following: (1) PSL(3,//);ß|«. (2) PU(3,/>«); 2ß|a. (3) in the case where 3 | (pe-1) and 3ß|a, a group containing the subgroup of type (1) with index 3. (4) in the case where 3 \ (pß + 1) and 6ß\a,a group containing the subgroup of type (2) with index 3. (5) PSLÍ2,//) or PGL(2,/A with ß\a andp^3. (6) PSLÍ2, 5), with q=±l (mod 10). (7) PSL(2, 7), with q3=l (mod 7). (8) A6, A7, or a group containing Ae with index 2, with p = 5 and a even. (9) A6, with q= 1 or 19 (mod 30). Moreover, PSL(3, q) has exactly one subgroup © of each type mentioned (for each indicated value of q, ß), up to conjugacy in GL(3, c7)/Z(SL(3, q)). Here GL(n, q) denotes the group of nonsingular matrices of degree n over the field GF(<7); U(n,q) is the subgroup of SL(n, q2), and U*(n,q) the subgroup of GL(n, q2), consisting of matrices A such that A "1 is the transpose of the matrix obtained from A via the automorphism c-» c" of GF(<72). For any group ®, P@ denotes ®/Z(@) where Z(@) is the center of ©. An, Sn denote the alternating and symmetric groups on n letters. We will give explicit representatives (in matrix form) of all conjugacy classes
