A brute force solution that probably won't work for you is to specify a page break before every occurrence of the sub summary part. That starts every sub heading at the top of the next page, but won't deal with cases where the group of records under still exceeds one page and can leave a lot of wasted space in your report if a lot of your groups are just a few records long.
If there isn't any sliding enabled to produce fields of variable height, it is possible to calculate the number of records and sub summary parts that will fit on your page. It's not simple, but with that data it may be possible to add an additional sub summary part to appear after the page break. The trick is to manage successfully so that you don't get the sub heading twice if the page break and the end of the next record group naturally coincide.