The daemons of the grimoire called the Goetia are listed as seventy-two specific entities, and are paired with the seventy-two angels of the ha-Shem. They are also considered divided and apportioned, as are the ha-Shem angels, to the seventy-two zodiacal quinarians. The ruling hierarchy would consist of the archangels of the twelve signs of the Zodiac and the angelic rulers of the thirty-six decans. This creates a tight hierarchy where the angels and daemons form an integral whole. There is a long tradition of pairing the daemons with the angels of the ha-Shem, but this would seem to nullify another hierarchical structure, namely the Infernal Hierarchy of Hell itself.
While such a hierarchy was proposed and became part of the doctrine of the Jewish orthodoxy before the current epoch and was also subsequently mirrored in Christian and Muslim orthodoxy, in our current era such a hierarchy would require a degree of faith and dogmatism that would go against even the most basic philosophical and occult tenants of today's post modern world. It has always been the understanding of most occultists that Satan and the hierarchy of Hell is a medieval throwback to a dualistic perspective of good and evil, where angels and daemons vie for supremacy and seek to manipulate and control the lives, and therefore, the souls of human beings.
However, in Qabbalistic lore, which represents the ground basis of practical ritual magic, the Deity represents a perfect and harmonious unity, which means that everything else is subsumed into it and an integral part of it. Nothing is outside or apart from that unity, which is to say that all things spiritual and ensouled are actually one and indivisible at the level of the Godhead. Thus there is no duality and no conflict between good and evil except in the material world of humanity. We could even say that there is no difference between good and evil amongst the living creatures of the earth - only humanity makes such distinctions. Therefore the Infernal Hierarchy is part of a human perspective of the spiritual world that divides it into good and evil, and of course, that division is illusory and at higher levels of consciousness, even absurd.
We can, however, deal with this infernal hierarchy if it is defined as being the mirror image of the positive angelic hierarchy, and then only if they are both considered integral and in some kind of union. This would make the infernal hierarchy a kind of dark side or unconscious shadow of the angelic reality of light. It may also benefit us to realize that the daemons of the Goetia have an additional hierarchy that further joins them to the greater hierarchy of the Godhead. So at the level of the Qabbalistic world of Atziluth, these differences become nullified, but may have a reality in Briah, Yetzirah and especially Assiah, where good and evil are encountered in the acts and judgments of humanity. As the Qliphoth is said to shadow the Sephiroth and acts as the backside of the Tree of Life, so too would the daemons represent the dark reflections of the angelic beings that they mirror.
The Infernal Hierarchy is not as diverse or complete as the angelic one, so we would expect that it would mirror the most simplistic structure of the angelic hierarchy. The basic numeric structure of this hierarchy of devils is that of the numbers three, four, eight, nine, eleven and seventy-two. Embedded in this structure are the numbers seven and eighty (twice forty). We can see in these numbers a kind of aping of the sacred systems that are represented by the angelic hierarchies and the Trinity, in its various forms - Jewish Qabbalah, Christian and Sufi Islam. The trinity as it is defined in the Infernal Hierarchy are the three great Kings of the Infernal Regions, and these are Lucifer, Beelzebub, and Satan. There are also nine orders of daemons (perhaps three of each being ruled by the three great Kings), and four Demon Princes of the Air, who are ruled by Satan. These four Demon Princes rule over a demonic Bishop or Governor, and these rule over the Goetic daemons. The Hierarchy can be presented in the following tables.
(See "Keys to the Gateway of Magic: Summoning the Solomonic Archangels and Demon Princes", p. 102 - 103 - where these spirits are listed and partially qualified.)
The Four Demon Princes of the Four Directions (Elements)
Direction - Element Demonic Prince Govenor (Bishop)
East (Air) Urieus - Oriens Theltryon
South (Fire) Amaymon Boytheon
West (Water) Paymon Sperion
North (Earth) Egin Mayerion
The nine orders of daemons are as follows: False Gods, Spirits of Lies, Vessels of Iniquities, Evil Avengers, Aerial Powers, Furies, Accusers or Inquisitors, and Tempters and Ensnarers. These orders are quite Christian in their composition, and wouldn't have much to offer a ritual magician unless one was interested in dealing with a daemonic hierarchy apart and distinct from the Godhead. This perspective could be considered a gross fantasy, perversion or twisted delusion, representing more of a Satanic magickal perspective that would certainly place one squarely in the Christian mediaeval spiritual dogma as a diabolic adversary. This is a perspective that does not fit well with the Qabbalah, for reasons already sited above.
To summarize the above considerations, an Infernal Hierarchy is an optional structure that magicians may use if they wish to, as long as it is understood that such a hierarchy does not abrogate the necessary unity of the Godhead, and that daemonic entities are not considered as separate or distinct from their associated spiritual context, which is the dark, shadowy or unconscious aspect of the angelic entities and their hierarchies. I believe that magicians must refrain from the error of dualism and the excesses that it can cause in the actions and practices of magicians. It is my earnest opinion that the Infernal Hierarchy is probably an archaic relic and one that is no longer really relevant to the system of invocation and evocation that I have forged for the Order and its practicing ritual magicians.