Given two non-abelian finite p-groups $P_1$ and $P_2$ of the same order that are not isomorphic.
Can $P_1$ and $P_2$ have isomorphic subgroup lattices?
(I'm not experienced with group theory, sorry in case this is not appropriate for MO).
The answer is yes. You can find references at page 277 of the book
Roland Schmidt: Subgroup lattices of groups, De Gruyter Expositions in Mathematics 14, Berlin: Walter de Gruyter. xv, 572 p. (1994). ZBL0843.20003.