In decentralized control systems with linear dynamics, quadratic cost, and Gaussian disturbance (also called decentralized LQG systems) linear control strategies are not always optimal. Nonetheless, linear control strategies are appealing due to their analytic and implementation simplicity. In this paper, we investigate decentralized LQG systems with partial history sharing information structure and identify finite-dimensional sufficient statistics for such systems. Unlike prior work on decentralized LQG systems, we do not assume partially nestedness or quadratic invariance. Our approach is based on the common information approach of Nayyar et al., 2013 and exploits the linearity of the system dynamics and control strategies. To illustrate our methodology, we identify sufficient statistics for linear strategies in decentralized systems where controllers communicate over a strongly connected graph with finite delays, and for decentralized systems consisting of coupled subsystems with control sharing or one-sided one step delay sharing information structures.