Iron oxide nanoflowers (IONFs) that display singular magnetic properties can be synthesized through a polyol route first introduced almost 2 decades ago by Caruntu et al., presenting a multi-core morphology in which several grains (around 10 nm) are attached together and sintered. These outstanding properties are of great interest for magnetic field hyperthermia, which is considered as a promising therapy against cancer. Although of significantly smaller diameter, the specific adsorption rate (SAR) of IONFs reach values on the order of 1 kW g, as large as "magnetosomes" that are natural magnetic nanoparticles typically ~40 nm found in certain bacteria, which can be grown artificially but with much lower yield compared to chemical synthesis such as the polyol route. This work aims at better understanding the structure-property relationships, linking the internal IONF nanostructure as observed by high resolution transmission electron microscopy (HR-TEM) to their magnetic properties. A library of mono- and multicore IONFs is presented, with diameters ranging from 11 to 30 nm in a narrow size distribution. More particularly, by relating their structural features (diameter, morphology, defects…) to their magnetic properties investigated by utilizing AC magnetometry over a wide range of alternating magnetic field (AMF) conditions, we showed that the SAR values of all synthesized batches vary with overall diameter and number of constituting cores. These variations are in qualitative agreement with theoretical predictions either by the Linear Response Theory (LRT) at low fields or with the Stoner-Wohlfarth (SW) model at larger amplitudes, and with numerical simulations reported previously. More precisely, our results show a continuous (almost quadratic) increase of SAR with IONF diameter for AMF amplitudes of 20 kA m and above, whatever the frequency between 146 and 344 kHz, and a pronounced maximum at an IONF diameter of 22 nm for amplitudes of 16 kA m and below. Thank to this understanding of the impact of size and core multiplicity, stable colloidal solutions of IONPs can be synthesized with diameters targeting a SAR value adapted to the theragnostic approach envisioned.