Remote sensing observations by recent successful missions to small bodies have revealed the difficulty in classifying the materials which cover their surfaces into a conventional classification of meteorites. Although reflectance spectroscopy is a powerful tool for this purpose, it is influenced by many factors, such as space weathering, lighting conditions, and surface physical conditions (e.g., particle size and style of mixing). Thus, complementary information, such as elemental compositions, which can be obtained by X-ray fluorescence (XRF) and gamma-ray spectrometers (GRS), have been considered very important. However, classifying planetary materials solely based on elemental compositions has not been investigated extensively. In this study, we perform principal component and cluster analyses on 12 major and minor elements of the bulk compositions of 500 meteorites reported in the National Institute of Polar Research (NIPR), Japan database. Our unique approach, which includes using hierarchical cluster analysis, indicates that meteorites can be classified into about 10 groups purely by their bulk elemental compositions. We suggest that Si, Fe, Mg, Ca, and Na are the optimal set of elements, as this set has been used successfully to classify meteorites of the NIPR database with more than 94% accuracy. Principal components analysis indicates that elemental compositions of meteorites form eight clusters in the three-dimensional space of the components. The three major principal components (PC1, PC2, and PC3) are interpreted as (1) degree of differentiations of the source body (i.e., primitive versus differentiated), (2) degree of thermal effects, and (3) degree of chemical fractionation, respectively.