The recommended height of the Menorah from the ground:[1]

The candles are to be placed between three to ten Tefachim above the ground [24-80cm.[2]].[3] It is not to be placed lower than three[4] Tefach and is a Mitzvah not to be placed higher than ten Tefach.[5] This applies even if one lights the candles inside his house.[6] If one placed the Menorah above ten Tefachim he has nevertheless fulfilled his obligation.[7] Likewise, if one placed the Menorah below three Tefachim, he nevertheless fulfills his obligation.[8] [Some Poskim[9] rule the Menorah is initially to be placed higher than seven Tefach from the ground. Other Poskim[10], however, rule it is initially to be placed exactly three Tefach from the ground. Practically, it is not necessary to follow either of these opinions so long as the Menorah is between 3-10 Tefach from the ground[11], and the Chabad custom is not to be particular in this matter.[12]]

Must the actual flame be below ten Tefachim from the ground?[13]

Initially, both the Menorah and its flame is to be within 10 Tefachim from the ground.

Must the actual Menorah reach above three Tefachim from the ground, or only the flame?

Some Poskim[14] rule it is not necessary for the body of the Menorah to be above three Tefachim from the ground, so long as the actual flames are above three Tefachim. Other Poskim[15], however, rule that also the body of the Menorah is to be above three Tefachim from the ground, and doing so is of greater importance than having the flame be within ten Tefachim. According to this latter opinion, the Menorah is to be placed on a stool which reaches three Tefachim from the ground.

May the actual Menorah be placed on the ground, if its height reaches three Tefachim above the ground?[16]

In all cases, it is proper not to place the Menorah directly on the floor even if it reaches a height above three Tefachim.

[1] 671:6; Tur in name of Maharam Rothenberg

Other opinions: Some Poskim rule the Menorah is to be between 10-18 Tefachim from the ground. [Shaareiy Teshuvah 671:8 in name of Seder Hayom]

[2] So is according to Shiureiy Torah of Grach Na’ah; According to Chazon Ish, the measurement is between 28.8-96 cm; See Piskeiy Teshuvos 671:7 that one may be lenient like whatever opinion he chooses regarding Rabbinical matters, such as Chanukah

[3] Michaber ibid “It is placed higher than three Tefach from the ground, and is a Mitzvah to be placed lower than ten Tefach from the ground.”

One who is lighting on stairs: Stairs that are used for walking, are to have their 3-10 Tefach measured from the floor. [See Aruch Hashulchan 671:22; Piskeiy Teshuvos 671:9]

[4] The reason for higher than three: As if the candles are placed lower than three Tefachim, it appears as if it was placed on the ground, and is not recognizable that the person lit it there. [Bach brought in M”B 671:26; Levush; Peri Chadash 671; Beis Yosef; Kaf Hachaim 671:45]

[5] The reason for lower than ten: As there is greater publication of the miracle when the Menorah is placed lower than ten Tefachim, as a candle that is lit for its light is not lit so low. [Bach brought in M”B 671:27; Levush; Kaf Hachaim 671:47] For the reason according to Kaballah-See Peri Eitz Chaim 19:4; Kaf Hachaim 671:46

[6] Tur; Beis Yosef “So is custom of all the meticulous”; M”B 671:27; Kaf Hachaim 671:48

Other opinions: Some Poskim rule that those who light inside the house, may position the Menorah higher than ten Tefachim. [Elya Raba in name of Mordechai that so is custom, brought in Machatzis Hashekel 671:6 and M”B ibid] Practically, one is not to rely on this ruling, as it opposes the ruling of the Shulchan Aruch and Arizal. [Kaf Hachaim 671:48]

[7] Peri Chadash 671; P”M 671 A”A 6; M”B 671:26; Kaf Hachaim 671:50

[8] Peri Chadash 671:6; P”M A”A 671:6; Misgeres Hashulchan on Kitzur SHU”A 139:7; M”B 671:26; Kaf Hachaim 671:50

Other opinions: Some Poskim rule that it is invalid even Bedieved. [Kitzur SHU”A 139:13; See Nitei Gavriel 16 footnote 11]

[9] Tur Barekes brought in Shaarei Teshuvah 671:8; Ben Ish Chaiy Vayeishev 5 based on Arizal; Shomer Emunim 5; Machazik Bracha 671:3 brought in Kaf Hachaim 671:46

[10] Shomer Emunim 65

[11] Kaf Hachaim 671:46 that based on Arizal there is no purpose of placing it higher than seven Tefachim, and so long as it is between 3-10 Tefachim it is initially valid.

[12] Sefer Haminhagim p. [English] “It is not necessarily higher than seven tefachim [seven handbreadths, viz., 56cm.] nor necessarily approximating a height of three tefachim [viz., 24cm.].”

[13] Aruch Hashulchan 671:22; Kaf Hachaim 671:52 based on P”M 671 M”Z 5; Shaar Hatziyon 671:33; Divrei Yatziv 2:284; Minchas Yitzchak 6:65; Az Nidbaru 5:37; Lehoros Nassan 4:63; Teshuvos Vehanhagos 2:343; Piskeiy Teshuvos 671:9

Other opinions: Some Poskim rule that so long as the bottom part of the Menorah is within 10 Tefachim, it is valid, even if the upper part of the Menorah reaches above ten Tefachim. [Leket Yosher in name of Terumos Hadeshen, brought in Piskeiy Teshuvos 671 footnote 36] There were Chassidic masters who would have Menorah’s that reach above ten Tefach from the ground. [See Nitei Gavriel Chanukah 5 footnote 24]

[14] Lehoros Nassan 4:63; See Piskeiy Teshuvos 671:9

[15] Kinyan Torah 1:131

[16] Piskeiy Teshuvos 671:9