In this paper we consider a reduced complexity discrete bit loading for Multicarrier systems based on the greedy power allocation (GPA) under the constraints of transmit power budget, target BER, and maximum permissible QAM modulation order. Compared to the standard GPA, which is optimal in terms of maximising the data throughput, three suboptimal schemes are proposed, which perform GPA on subsets of subcarriers only. These subsets are created by considering the minimum SNR boundaries of QAM levels for a given BER. We demonstrate how these schemes can reduce complexity. Two of the proposed algorithms can achieve near optimal performance by including a transfer of residual power between groups at the expense of a very small extra cost. It is shown that the two near optimal schemes,while greatly reducing complexity, perform best in two separate and distinct SNR regions.