From the Luminous Red Galaxy (LRG) sample of the Sloan Digital Sky Survey Data Release 6 (SDSS DR6), we construct a high-concentration LRG sample and a low-concentration LRG sample and investigate the clustering properties of LRGs with different concentration index using cluster analysis. Because the two LRG samples with different concentration index have different number densities, we randomly extracted a subsample from the high-concentration LRG sample which has the same galaxy number and number density as the low-concentration LRG sample, and again performed comparative studies of clustering properties between the high-concentration LRGs and low-concentration LRGs. Our result preferentially shows that the clustering properties of LRGs are not significantly correlated with concentration index, which is not consistent with the result for Main galaxies.