No, she was the goddess of wilderness and childbirth.
She was also known as a goddess over magic, witchcraft, and crossroads.
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The Underworld.
No, Hades was married to Persephone, daughter of the Olympian Goddess Demeter. Hecate is a Goddess with an uncertain mythology, though often associated with Persephone and the Underworld.
Hecate was an underworld or moon goddess in Greek mythology. Her parents have been listed as either Perses and Asterie or Zeus and Asterie among others.
Kerberos, the Hound of Hades which guards the Gates of the Underworld is male. Hecate's dog is Trojan Queen Hekabe, who leapt into the sea after the fall of Troy and was transformed by the goddess into her familiar.
There is no goddess of death. There are several female spirits associated with war and suffering, and some goddesses associated with the underworld, like Persephone, queen of the Underworld, and Hecate, goddess of magic, but no "goddess of death." Thanatos is the personification of death, what we today think of as the Grim Reaper, and Hades was lord of the underworld and therefore the dead.