Each god of Ancient Greece and later Rome held their own symbols of power. Whether major or minor, the gods were identified by these symbols. However, the gods had established rules about each symbol, much like each god's domain. A god is not allowed to steal another god's symbol of power, however demigods are not affected by this rule (which allowed Luke to steal the Master Bolt and the Helm of Darkness). Also Hades, Zeus and Poseidon cannot create another Symbol of Power without the other gods agreeing to it beforehand as a new symbol could upset the balance of power. Below is a list of the gods and some of their attributes.
Olympians
God : Symbols
Aphrodite/Venus: Dove, Sceptre, Myrtle, Dolphin, Rose, Scallop Shell, Sparrow, Girdle, Mirror, Swan
Apollo / Apollo(Phoebus): Lyre, Laurel Wreath, Python, Raven, Bow and Arrows, Sun
Ares /Mars: Boar, Spear, Vulture, Dog, Flaming Torch, Helmet, Chariot
Artemis/Diana: Bow and Arrows, Stag, Hunting dog, Crescent Moon
Athena/Minerva: Owl, Olive tree, Snake, Aegis, Armor, Helmet, Spear
Demeter/Ceres: Torch, Cornucopia, Sheaves of Wheat, Lion
Dionysus/Bacchus: Thyrsus, Grapevine, Leopard Skin, Panther, Tiger
Hades/Pluto: Helm of Darkness, Keys of Hades, Sword, Cerberus, Two-Progned Staff
Hephaestus/Vulcan: Anvil, Flaming Hammer, Tongs, Axe, Quail
Hera/Juno: Cow, Peacock, Diadem, Goatskin Cloak.(Perhaps one should be a female dog XD)
Hermes/Mercury: Winged Shoes, Caduceus, Tortoise, Lyre, Rooster
Hestia/Vesta: Fire, Hearth, Donkey
Poseidon/Neptune: Horse, Trident, Bull, Seafoam, Dolphin
Zeus/Jupiter: Eagle, Shield, Thunderbolt, Oaktree
Minor Gods
God : Symbol
Harmonia/Concordia: Patera, Cornucopia, Caduceus
Eris/Discordia: Golden Apple Of Discord
Tyche/Fortuna: Four-Leaf Clover, Mural Crown, Wheel Of Fortune, Cornucopia, Ship's Rudder
Hecate/Trivia: Torch, Key, Dark Moon, Crossroad, Three-headed Dog, Mask, Candle
Hypnos/Somus: Poppy
Iris/Arcus: Rainbow, Fleur-de-lis
Janus: Door, Key
Morpheus/Somnia: Poppy, Closed Eye, Dark Crown
Nemesis/Invidia: Balancing Scales, Goose, Wheel Of Fortune
Pan/Faunus(Inuus): Woods, Pastures, Syrinx, Reed Pipes
Persephone/Proserpina: Grain, Pomegranate
Pomona: Apple, Platter of Fruit, Cornucopia
Thanatos/Letus: Theta, Poppy, Butterfly, Sword, Inverted Torch
Enyo/Bellona: Crossed Sword, Swan
Eros/Cupid: Bow, arrow, candle, heart, Cupid, wing
Nyx/Nox: Black star and Crescent
Olympians
God : Symbols
Aphrodite/Venus: Dove, Sceptre, Myrtle, Dolphin, Rose, Scallop Shell, Sparrow, Girdle, Mirror, Swan
Apollo / Apollo(Phoebus): Lyre, Laurel Wreath, Python, Raven, Bow and Arrows, Sun
Ares /Mars: Boar, Spear, Vulture, Dog, Flaming Torch, Helmet, Chariot
Artemis/Diana: Bow and Arrows, Stag, Hunting dog, Crescent Moon
Athena/Minerva: Owl, Olive tree, Snake, Aegis, Armor, Helmet, Spear
Demeter/Ceres: Torch, Cornucopia, Sheaves of Wheat, Lion
Dionysus/Bacchus: Thyrsus, Grapevine, Leopard Skin, Panther, Tiger
Hades/Pluto: Helm of Darkness, Keys of Hades, Sword, Cerberus, Two-Progned Staff
Hephaestus/Vulcan: Anvil, Flaming Hammer, Tongs, Axe, Quail
Hera/Juno: Cow, Peacock, Diadem, Goatskin Cloak.(Perhaps one should be a female dog XD)
Hermes/Mercury: Winged Shoes, Caduceus, Tortoise, Lyre, Rooster
Hestia/Vesta: Fire, Hearth, Donkey
Poseidon/Neptune: Horse, Trident, Bull, Seafoam, Dolphin
Zeus/Jupiter: Eagle, Shield, Thunderbolt, Oaktree
Minor Gods
God : Symbol
Harmonia/Concordia: Patera, Cornucopia, Caduceus
Eris/Discordia: Golden Apple Of Discord
Tyche/Fortuna: Four-Leaf Clover, Mural Crown, Wheel Of Fortune, Cornucopia, Ship's Rudder
Hecate/Trivia: Torch, Key, Dark Moon, Crossroad, Three-headed Dog, Mask, Candle
Hypnos/Somus: Poppy
Iris/Arcus: Rainbow, Fleur-de-lis
Janus: Door, Key
Morpheus/Somnia: Poppy, Closed Eye, Dark Crown
Nemesis/Invidia: Balancing Scales, Goose, Wheel Of Fortune
Pan/Faunus(Inuus): Woods, Pastures, Syrinx, Reed Pipes
Persephone/Proserpina: Grain, Pomegranate
Pomona: Apple, Platter of Fruit, Cornucopia
Thanatos/Letus: Theta, Poppy, Butterfly, Sword, Inverted Torch
Enyo/Bellona: Crossed Sword, Swan
Eros/Cupid: Bow, arrow, candle, heart, Cupid, wing
Nyx/Nox: Black star and Crescent