The thermal conductivity ͑T͒ of the crystalline and glassy phases of the two isomers of propyl alcohol has been measured. The two isomers differ by a minor chemical detail involving the position of the hydroxyl group with respect to the carbon backbone. Such a difference in molecular structure leads, however, to disparate behaviors for the temperature dependence of ͑T͒, for both glass and crystal states. The ͑T͒ for the glass shows for 1-propanol an anomalously large plateau region comprising temperatures within 6 -90 K, while data for isomeric 2-propanol show only a small plateau up to 10 K which is comparable to data on lower alcohols. The results emphasize the role played by internal molecular degrees of freedom as sources of strong resonant phonon scattering.