We present the first chemical analysis of stars on the double subgiant branch (SGB) of the globular cluster NGC 1851. We obtained 48 Magellan IMACS spectra of subgiants and fainter stars covering the spectral region between 3650-6750 Å to derive C and N abundances from the spectral features at 4300 Å (G-band) and at ~3883 Å (CN). We added to our sample ~45 unvevolved stars previously observed with FORS2 at the VLT. These two datasets were homogeneously reduced and analyzed. We derived abundances of C and N for a total of 64 stars and found considerable star-to-star variations in both [C/H] and [N/H] at all luminosities extending to the red giant branch (RGB) base (V ~ 18.9). These abundances appear to be strongly anticorrelated, as would be expected from the CN-cycle enrichment, but we did not detect any bimodality in the C or N content. We used Hubble space telescope (HST) and ground-based photometry to select two groups of faint- and bright-SGB stars from the visual and Strömgren color-magnitude diagrams. Significant variations in the carbon and nitrogen abundances are present among stars of each group, which indicates that each SGB hosts multiple subgenerations of stars. Bright- and faint-SGB stars differ in the total C+N content, where the fainter SGB have about 2.5 times the C+N content of the brighter ones. Coupling our results with literature photometric data and abundance determinations from high-resolution studies, we identify the fainter SGB with the red-RGB population, which also should be richer on average in Ba and other s-process elements, as well as in Na and N, when compared to brighter SGB and the blue-RGB population. Based on observations taken with the 6.5 meter Magellan Telescope at Las Campanas Observatory, Chile and with the Very Large Telescope at ESO La Silla Paranal Observatory, Chile, under programme ID 68.D-0510.