Class
The social hierarchy is an unavoidable reality in Britain, and it is interesting to watch it play out in the work of a socialist playwright. Shaw includes members of all social classes from the lowest (Liza) to the servant class (Mrs. Pearce) to the middle class (Doolittle after his inheritance) to the genteel poor (the Eynsford Hills) to the upper class (Pickering and the Higginses). The general sense is that class structures are rigid and should not be tampered with, so the example of Liza's class mobility is most shocking. The issue of language is tied up in class quite closely; the fact that Higgins is able to identify where people were born by their accents is telling. British class and identity are very much tied up in their land and their birthplace, so it becomes hard to be socially mobile if your accent marks you as coming from a certain location.
Gentility and Manners
Good manners (or any manners at all) were mostly associated with the upper class at this time. Shaw's position on manners is somewhat unclear; as a socialist, one would think that he would have no time for them because they are a marker of class divisions. Yet, Higgins's pattern of treating everyone like dirt--while just as democratic as Pickering's of treating everyone like a duke or duchess--is less satisfactory than Pickering's. It is a poignant moment at the end of Pygmalion when Liza thanks Pickering for teaching her manners and pointedly comments that otherwise she would have had no way of learning them.
Marriage and Prostitution
These institutions are very much related in Shaw's plays, especially in Mrs. Warren's profession. From his unusual standpoint of being committed to a celibate marriage, Shaw apparently feels free to denounce marriage as an exchange of sexuality for money similar to prostitution (even though this was not happening in his own marriage). Ironically, while her father expresses no regrets when he is led to believe that Liza will take up this profession, it is she who denounces it. She declares that she was less degraded as a flower-seller than as a "genteel" lady trying to make an appropriate marriage--because as a flower-seller, at least, she wasn't selling her body.
Myths of Creation
Of all Shaw's plays, Pygmalion has the most references to Greek and Roman mythology. Higgins represents Pygmalion, a Greek sculptor who lived alone because he hated women. Pygmalion created a sculpture of a perfect woman and fell in love with it; after he prayed, Aphrodite brought it to life for him. This statue is named Galatea, and it is represented in Shaw's play by Liza. Unlike the myth, Shaw's play does not end in a marriage between the pair, and Liza is infuriated with Higgins's suggestion that her success is his success and that he has made her what she is. She has worked to recreate her identity as well.
Language
In this play and in British society at large, language is closely tied with class. From a person's accent, one can determine where the person comes from and usually what the person's socioeconomic background is. Because accents are not very malleable, poor people are marked as poor for life. Higgins's teachings are somewhat radical in that they disrupt this social marker, allowing for greater social mobility.
Professionalism
At the time that this play was written, the idea of female professionals was somewhat new. Aside from the profession of prostitution, women were generally housewives before this period, and there is some residual resistance to the idea of normally male professions being entered by females in the play.
Moreover, Pickering is initially horrified by the idea of Eliza opening a flower shop, since being involved in a trade was a mark of belonging to the lower class. Pickering is shaken similarly after his experience of watching Eliza fool everyone at a garden and dinner party, saying that she played her part almost too well. The idea of a professional female socialite is somehow threatening to him.
Gender Solidarity or Antagonism
Although British society is supposed to break down along class lines, Shaw makes a point of highlighting gender loyalties in this play. Although Mrs. Higgins initially is horrified by the idea that her son might bring a flower-girl into her home, she quickly grows sympathetic to Liza. As a woman, she is the first to express a concern for what will be done with the girl after the experiment--the idea that her training makes her highly unmarriageable by anyone anywhere on the social scale. When Liza runs away from Wimpole St., she instinctively knows that Mrs. Higgins will take good care of her. Higgins's mother sides with Liza before even her son, not revealing that Liza is in the house while Higgins is dialing the police.
In contrast, relations between people of opposite genders are generally portrayed by Shaw as antagonistic. Higgins and his mother have a troubled relationship, as do the professor and Mrs. Pearce. Freddy and Liza get along better perhaps only due to his more passive, feminine demeanor.