Does the iliohypogastric nerve have some connections to the gallbladder?the word gastric is making me ask.

No connection: I understand your point. Iliohypogastric nerve innervates areas below the belly button and has no connection with the gallbladder which receives stimulation from the vagus and phrenic nerve (diaphragmatic).

No: Hypogastric is a descriptive term to describe the lower abdominal region. The iliohypogastric nerve supplies muscles and skin sensation to the flank and eventually to the groin area.