Maxillary Artery - The maxillary artery is divided into three divisions each division having five branches.  Not all branches branches will be listed here:
  Mandibular Division - each branch leads to a bony foramen
    Inferior Alveolar Artery - runs to the mandibular foramen for the mandibular teeth.
    Middle Meningeal Artery - to foramen spinosum runs deep to pterion
    Accessory Meningeal - runs through foramen ovale with V3
  Pterygoid Division - each branch serves a muscle
    Masseteric Branch
    Medial and Lateral Pterygoid
    Buccal
    Temporals - two deep temporal arteries
  Pterygopalatine Division - each branch is a terminal branch of the maxillary artery
    Infraorbital Artery - runs in infraorbital foramen with infraorbital branch of V2
    Posterior Superior Alveolar - to maxillary teeth
    (Long) Sphenopalatine Artery - to nasal cavity
    Pharyngeal Branch
    Descending Palatine - becomes the greater and lesser palatine arteries to the hard and soft palate respectively