Huh. Maybe this is a US thing, but in the US, "class" is determined by your level of wealth. If you're wealthy, you're not middle class.
Although how much wealth counts as "wealthy" depends a lot on where you live in the US. A level that is wealthy in some areas is a level that's poor in others.
Class is not determined by wealth in every country. But certainly going by wealth in this country, there are middle class people who are rich and some who are not. The upper middle class tends to be better off. Grand bourgeois or haut bourgeois are the fancy terms for the richer end.
Americans class plumbers as middle class while they would be working class in most places. We do have working class people here who are wealthier than middle class — we determine class on social origin and the nature of their work.
Although how much wealth counts as "wealthy" depends a lot on where you live in the US. A level that is wealthy in some areas is a level that's poor in others.