Soldiers & Sailors (http://www.itd.nps.gov/cwss/soldiers.htm)
shows a Franklin Cox in both "Douthat's Company, Virginia Light Artillery (Botetourt Artillery)" and "28th Regiment, Virginia Infantry".

A service summary for the Botetourt Art'y may explain why he's listed also in the 28th Infantry:

"The origin of the unit dates back to 1859 when they were formed as a militia unit. This unit was among the first to answer the call when the war began. They first saw service at the First Battle of Manassas as members of Company I, 28th Virginia Infantry Regiment. In early 1862 they transferred to the artillery branch and were assigned to Kirby Smith's small army in Eastern Tennessee. Later that same year they participated in the invasion of Kentucky. In December of 1862 they received orders to move south to Vicksburg, Mississippi."

"The Botetourt Artillery saw service through the entire 1863 Vicksburg Campaign and suffered grievous losses. They were actively engaged in the battles of Port Gibson and Champions Hill and were in the trenches during the 47 day seige of Vicksburg. Following the Vicksburg ordeal they returned to Virginia in the fall of 1863 and were stationed in Southwest Virginia."

"During the 1864 Valley Campaign they participated in the Battle of Lynchburg and were with General Early's command until December 1864."

"In April 1865 the war ended for the Botetourt Artillery in Christiansburg, Virginia where the unit was disbanded."

"Captains included Joseph W. Anderson (to major April 4, 1863), John W. Johnston (to major Sept. 2, 1863), and Henry Clay Douthat."

(from Markham and posted online here: http://www.geocities.com/Heartland/Hills/1850/botetourtarty.html)

Compiled service records for soldiers in this unit are at the National Archives (how-to: http://www.archives.gov/veterans/research/genealogy.html)

Someplace to start ...