Doctrinally, a CSG serves corps units, not division units, but in practice CSGs typically have a close working relationship with a particular division especially in garrison.
A CSG's subordinate units also may be parceled out to different commands in theater, thus their wartime organization will normally significantly differ from their garrison organization.
Typically, a CSG and its subordinate units will only deploy as far forward as the Division Support Area.
A typical CSG will be organized with a Headquarters Company and a varying number of Corps Support Battalions (CSB's) which may or may not be from the same Fort as the headquarters.
The US Army has been transforming many CSG's into Sustainment Brigades while Corps Support Battalions are becoming Combat Sustainment Support Battalions (CSSBs).