We report on a comprehensive de Haas-van Alphen (dHvA) study of the iron pnictide LaFe2P2. Our extensive density-functional band-structure calculations can well explain the measured angular-dependent dHvA frequencies. As salient feature, we observe only one quasi-two-dimensional Fermi-surface sheet; i.e., a hole-like Fermi-surface cylinder around Γ, essential for s± pairing, is missing. In spite of considerable mass enhancements due to many-body effects, LaFe2P2 shows no superconductivity. This is likely caused by the absence of any nesting between electron and hole bands. © 2014 American Physical Society.