Competing individually, Strnadová reached two major outdoor finals in the 800 m, placing seventh at the 1983 World Championships and fifth at the 1986 European Athletics Championships. Her greatest individual achievement was at the 1984 European Athletics Indoor Championships, where she won a gold medal in the 800 m with a championship record time of 1:59.52 minutes. This remained as the record for 18 years before being bettered by Slovenia's Jolanda Ceplak in 2002. She was runner-up to her world record-breaking compatriot Kratochvílová at the 1985 IAAF Grand Prix Final, and won full points for her country at the European Cup B Final in 1991. Her last international appearances were as semi-finalist at the 1991 World Championships in Athletics and 1992 European Athletics Indoor Championships.