For dihedral and D_n groups, the paper gives a partial order that exactly controls which Specht ideals contain which, and proves type D cannot be described by orbit types alone.
Darstellungstheorie der Hyperoktaedergruppe
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Posets for Specht ideals of essential real reflection groups
For dihedral and D_n groups, the paper gives a partial order that exactly controls which Specht ideals contain which, and proves type D cannot be described by orbit types alone.