Private company

Yes a private ltd company also shoud keep the register of charges

Section 143 in The Companies Act, 1956

143. Company' s register of charges.

(1) Every company shall keep at its registered office a register of charges and enter therein all charges specifically affecting property of the company and all floating charges on the undertaking or on any property of the company, giving in each case- (i) a short descripttion of the property charged; (ii) the amount of the charge; and (iii) except in the case of securities to bearer, the names of the persons entitled to the charge.

(2) If any officer of the company knowingly omits, or wilfully authorises or permits the omission of, any entry required to be made in pursuance of sub- section (1), he shall be punishable with fine which may extend to five hundred rupees.

Agree with Dhruva ji