The designs of the Iowa Statehood Centennial Half Dollar feature a portion of the Iowa State Seal on the obverse, and the Old Stone Capitol in Iowa City (now a part of the University of Iowa) on the reverse. Though fitting designs that are representative of Iowa's history, neither is part of the official logo of the Iowa Centennial. (You can read more about the designs via the link below.)
Iowa Statehood Centennial Half Dollar

Shown here is a pin-back button manufactured by the St. Louis Button Company for the Iowa Centennial Committee; it presents a close adaptation of the official logo. The button is monochromatic (using red as its print color) and was often seen being worn by attendees of Centennial celebration events.
The Centennial Logo has an outline map of the State at its center with "IOWA" printed across it. Above the map is a covered wagon being drawn by a pair of oxen, meant to symbolize a typical mode of transportation circa 1846, while below the map is seen a train, an automobile and an airplane, all representing circa 1946 transportation methods. "IOWA CENTENNIAL CELEBRATION" encircles at the outer rim above and the anniversary dates "1846-1946" appear below; the dates are flanked by ears of corn (one on each side); corn was/is a major Iowa cash crop.
Iowa Centennial Pin-back Button

I picked this example of the button up via auction at a very reasonable price - it was not a popular item!
For other of my posts about commemorative coins and medals, including more on the Iowa half dollar and its design, see:
Commems Collection.