  By utilizing an irreducible inclusion of type III$_{q^{2}} $ factors coming
from a free-product type action of the quantum group $ SU_{q}(2) $, we show
that the free group factor $ L(\mathbb {F}_{\infty}) $ possesses irreducible
subfactors of arbitrary index $ >4 $. Combined with earlier results of
Radulescu, this shows that $ L(\mathbb {F}_{\infty}) $ has irreducible
subfactors with any index value in $ \{4\cos ^{2}(\pi /n):n\geq 3\}\cup
[4,+\infty) $.
