An F_q-subline of PG(2,q^3) corresponds to a 2-regulus and an F_q-subplane to a Segre variety S_{2;2} in the Bose representation in PG(8,q), with extensions to PG(8,q^3) and PG(8,q^6) used to describe F_q-conics.
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The Bose representation of PG(2,q^3) in PG(8,q)
An F_q-subline of PG(2,q^3) corresponds to a 2-regulus and an F_q-subplane to a Segre variety S_{2;2} in the Bose representation in PG(8,q), with extensions to PG(8,q^3) and PG(8,q^6) used to describe F_q-conics.