Strong couplings between localized surface plasmon resonance (LSPR) modes and single quantum emitters have been intensively investigated recently, and meanwhile the coupling between different LSPR modes inside individual metallic nanoparticle is still rarely researched. Herein, the strong coupling is investigated for different LSPR modes inside individual rectangular-grating-shaped gold nanostructure composed of one main-cuboid and two side-attached subcuboids. Original uncoupled LSPR modes are revealed to be longitudinal and U-shaped LSPRs. For the nanostructure with increased main-cuboid length, the dispersion curves of simulated dual-original LSPR scattering wavelengths and dual-coupled LSPR scattering wavelengths show typical strong coupling patterns featuring anti-crossing and large Rabi splitting of 598.1 meV. The strong coupling is considered to be caused by the overlap of longitudinal and U-shaped LSPR oscillation modes in the gold nanostructure. The extracted coupling strength is found in order of 10¹³ Hz and it increases with the LSPR overlap length. The spatial mode and time evolution of the coupled LSPR modes are also numerically investigated. The simulated results are well comprehended with the classical strong coupling model of oscillators, further confirming the coupling between the longitudinal and the U-shaped LSPRs. The experimental dark-field scattering spectrum shows the existence of U-shaped LSPR mode in the gold nanostructure.