Sorting multiple grouped rows

Is it possible to sort multiple grouped rows of data by a cell within each grouping?

For example...

Row 10 has a bunch info like a start date in "I10"
Rows 11 through 28 are grouped and are data for row 10
(probably best to use the entirerow)
Row 29 has similar info in structure to row 10 like the start date in "I29"
Rows 30 through 47 are grouped and represent row 29.

Can the criteria in rows 10 and 29 be sorted by start date with their corresponding data remaining grouped and beneath each representative row?

I've been looking into the Range.sort method and the groupings are giving some issues.

I've taken a stab at this...

The same region that was grouped had the first cell merged over the same range so I used the 'mergecells' property instead of grouping. I have a sheet which is named "Gantt" and the code uses the bubble-sort method and a copied sheet to pull data from:

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It's really slow maybe because of how 'mechanical' it is.... sort out the start date, shuffle the start and end points according to the start date, copy the data source, make a copy and delete the original range and then paste in the data according to the sorted startdate array.

Maybe this gives a better picture of the tool? Is there a better way to accomplish this do you think?