Seeking Roster for 45th Inf Division WW2

Short version, I know that my father was transferred from the Army Air Corps to the infantry, I don't know to what division, in the winter of 1944, when the Allies were lined up along the Rhine.

His discharge papers list him as part of HQ & HQ Company 290 Infantry, which, if I'm right was part of the 75th Division, but they were farther to the north, involved in the Bulge. His old dress uniform has a 3rd Infantry Division shoulder patch.

Dad told us he was at Dachau. Just that, no other details. I know that the 45th, the 42nd, and the 3rd Divisions swept down through that area. The 3rd Division was not directly involved in the liberation of Dachau, but Dad, if he was with the 3rd Division, could have been sent over to see the camp. Eisenhower wanted as many as possible to witness the camps.

In his effects was a hard cover book, similar to school year books, of the 180 Infantry, 45th Division, published in Munich,1945. Dad's photo or name is not in the book. Circumstance may have prevented that. Possibly because he had enough points to be shipped home. I have to wonder why he would have that particular book.

Primarily I'm looking for a roster of the 45th Division, but also interested in the same for the 3rd Division.

I look forward to any and all replies.

Thank you.