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The spatial extension of a gamma-ray source is an essential ingredient to determine its spectral properties as well as its potential multi-wavelength counterpart. The capability to spatially resolve gamma-ray sources is greatly improved by the newly delivered Fermi-Large Area Telescope (LAT) Pass 8 event-level analysis which provides a greater acceptance and an improved point spread function, two crucial factors for the detection of extended sources. Here, we present a complete search for extended sources located within 7 degrees from the Galactic plane, using 6 years of LAT data above 10 GeV. We find 46 extended sources and provide their morphological and spectral characteristics. This constitutes the first catalog of hard LAT extended sources, named the Fermi Galactic Extended Source Catalog, which allows a thorough study of the properties of the Galactic plane in the sub-TeV domain.