You're going to have to draw the heatmap (just on a pdf(NULL) while saving the drawn heatmap to an object), then use something like get_row_order (or a similar dendrogram getter) on that object to get the portion you're interested then and then extract the names from that. If you only wish to add those as labels (not sure if that'll look any good), you'll need to use the row_annotation as row_labels cannot be selectively hidden (as far as I am aware at least).

EDIT: Like I said, if you read the manual properly, you'll find the solution more often than not: https://jokergoo.github.io/ComplexHeatmap-reference/book/a-single-heatmap.html#get-orders-and-dendrograms-from-heatmap