The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) notes that the eight Aliyot read on Shabbat (the seven primary Aliyot, and the Maftir) have different levels of importance. The sequence, he writes, is as follows, in descending order of importance: Shishi (the sixth Aliya), Shelishi (the third Aliya), Kohen, Levi, Rebi’i, Hamishi, Shebi’i, Maftir.
It is customary to recite half-Kaddish after the Torah reading is completed, that is, after the seven required Aliyot on Shabbat, or the five required Aliyot on Yom Tob. The Kaddish is recited before the Aliya of Maftir in order to show that the Maftir reading is separate and apart from the seven required Aliyot. On weekdays, the Kaddish is recited after the three required Aliyot, or after the four required Aliyot on Rosh Hodesh. Kaddish is not recited when the Torah is read at Minha, such as on Shabbat and on fast days. The reason, as the Ben Ish Hai explains, is because half-Kaddish is in any event recited before the Amida, soon after the Torah reading. On Shabbat, the Kaddish before the Amida of Musaf is not recited soon after the Torah reading, as "Ashreh" and other prayers are recited in the interim, and thus an additional Kaddish is recited after the Torah reading. At Minha, however, the Kaddish and Amida are recited very soon after the Torah reading, and so there is no need to recite an additional Kaddish after the reading.
