We study the morphological and structural properties of the host galaxies associated with 57 optically-selected luminous type 2 AGN at z \sim 0.3-0.4 : 16 high-luminosity Seyfert 2 ( HLSy2 , 8.0 \leq log ( L _ { [ OIII ] } / L _ { \odot } ) < 8.3 ) and 41 obscured quasars ( QSO2 , log ( L _ { [ OIII ] } / L _ { \odot } ) \geq 8.3 ) .
With this work , the total number of QSO2 at z < 1 with parametrized galaxies increases from \sim 35 to 76 .
Our analysis is based on HST WFPC2 and ACS images that we fit with GALFIT .
HLSy2 and QSO2 show a wide diversity of galaxy hosts .
The main difference lies in the higher incidence of highly-disturbed systems among QSO2 .
This is consistent with a scenario in which galaxy interactions are the dominant mechanism triggering nuclear activity at the highest AGN power .
There is a strong dependence of galaxy properties with AGN power ( assuming L _ { [ OIII ] } is an adequate proxy ) .
The relative contribution of the spheroidal component to the total galaxy light ( B/T ) increases with L _ { [ OIII ] } .
While systems dominated by the spheoridal component spread across the total range of L _ { [ OIII ] } , most disk-dominated galaxies concentrate at log ( L _ { [ OIII ] } / L _ { \odot } ) < 8.6 .
This is expected if more powerful AGN are powered by more massive black holes which are hosted by more massive bulges or spheroids .
The average galaxy sizes ( \langle r _ { e } \rangle ) are 5.0 \pm 1.5 kpc for HLSy2 and 3.9 \pm 0.6 kpc for HLSy2 and QSO2 respectively .
These are significantly smaller than those found for QSO1 and narrow line radio galaxies at similar z .
We put the results of our work in context of related studies of AGN with quasar-like luminosities .