Cryptographic schemes based on supersingular isogenies have become an active area of research in the field of post-quantum cryptography. We investigate the resistance of these cryptosystems to fault injection attacks. It appears that the iterative structure of the secret isogeny computation renders these schemes vulnerable to loop-abort attacks. Loop-abort faults allow to perform a full key recovery, bypassing all the previously introduced validation methods. Therefore implementing additional countermeasures seems unavoidable for applications where physical attacks are relevant.