Your problem is closely related to order statistics for normal random variables, so you may find this paper useful: [Percentage Points and Modes of Order Statistics from the Normal Distribution][1] by Shanti S. Gupta 


  [1]: https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-32/issue-3/Percentage-Points-and-Modes-of-Order-Statistics-from-the-Normal/10.1214/aoms/1177704982.full