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Order	Exponent	Nilpotency class	Nilpotent

Center	Commutator	Central quotient	Abelianization

Abelian	Cyclic	Solvable	Simple

Perfect	Direct product	Semidirect product	Wreath product

Automorphism group	Automorphisms	Min. transitive degree	Min. permutation degree

Number of subgroups	Num. of normal subs	Num. of conj. classes

Order statistics	Family	Order factorization	Commutator length

Min. degree of $\C$-irrep	Min. degree of $\Q$-irrep	Min. degree of $\C$-rep	Min. degree of $\Q$-rep

Outer aut. group	Outer automorphisms	Metabelian	Metacyclic

Almost simple	Quasisimple	A-group	Z-group

Frattini subgroup	Derived length	Rank	Schur multiplier

Supersolvable	Monomial	Rational

Num. of char. subgroups	Num. of aut. classes	Number of divisions

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Label	Name	Order	Exponent	Nilp. class	Der. length	Comp. length	Rank	$\card{\mathrm{conj}(G)}$	Subgroups	Subgroup classes	Normal subgroups	Center	Central quotient	Commutator	Abelianization	$\card{\mathrm{Aut}(G)}$	$\card{\mathrm{Out}(G)}$	Tr. deg	Perm. deg	$\C$-irrep deg	$\R$-irrep deg	$\Q$-irrep deg	$\C$-rep deg	$\R$-rep deg	$\Q$-rep deg	Type - length
216.1	$C_{27}:C_8$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{3}$	$-1$	$2$	$6$	$2$	60	$52$	$16$	$13$	$C_4$	$D_{27}$	$C_{27}$	$C_{8}$	$2^{3} \cdot 3^{5}$	$2^{2} \cdot 3^{2}$	$216$	$35$	$2$	$4$	$36$	$2$	$4$	$22$	Solvable - 2
216.2	$C_{216}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{3}$	$1$	$1$	$6$	$1$	216	$16$	$16$	$16$	$C_{216}$	$C_1$	$C_1$	$C_{216}$	$2^{3} \cdot 3^{2}$	$2^{3} \cdot 3^{2}$	$216$	$35$	$1$	$2$	$72$	$1$	$2$	$22$	Cyclic
216.3	$Q_8:C_{27}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$3$	$6$	$2$	63	$27$	$15$	$10$	$C_{18}$	$A_4$	$Q_8$	$C_{27}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3^{2}$	$216$	$35$	$2$	$4$	$36$	$2$	$4$	$22$	Solvable - 3
216.4	$C_{27}:Q_8$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$2$	$6$	$2$	57	$132$	$24$	$15$	$C_2$	$D_{54}$	$C_{54}$	$C_{2}^{2}$	$2^{4} \cdot 3^{5}$	$2^{2} \cdot 3^{2}$	$216$	$35$	$2$	$4$	$72$	$2$	$4$	$22$	Solvable - 2
216.5	$C_4\times D_{27}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$2$	$6$	$2$	60	$212$	$32$	$17$	$C_4$	$D_{27}$	$C_{27}$	$C_{2} \times C_{4}$	$2^{3} \cdot 3^{5}$	$2^{2} \cdot 3^{2}$	$108$	$31$	$2$	$4$	$36$	$2$	$4$	$20$	Solvable - 2
216.6	$D_{108}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$2$	$6$	$2$	57	$292$	$32$	$15$	$C_2$	$D_{54}$	$C_{54}$	$C_{2}^{2}$	$2^{4} \cdot 3^{5}$	$2^{2} \cdot 3^{2}$	$108$	$31$	$2$	$2$	$36$	$2$	$2$	$20$	Solvable - 2
216.7	$C_{54}:C_4$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$2$	$6$	$2$	60	$140$	$32$	$23$	$C_2^2$	$D_{27}$	$C_{27}$	$C_{2} \times C_{4}$	$2^{4} \cdot 3^{5}$	$2^{3} \cdot 3^{2}$	$216$	$33$	$-$	$-$	$-$	$3$	$4$	$20$	Solvable - 2
216.8	$C_{27}:D_4$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$2$	$6$	$2$	57	$220$	$32$	$15$	$C_2$	$D_{54}$	$C_{54}$	$C_{2}^{2}$	$2^{3} \cdot 3^{5}$	$2 \cdot 3^{2}$	$108$	$31$	$2$	$4$	$36$	$2$	$4$	$20$	Solvable - 2
216.9	$C_2\times C_{108}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$1$	$1$	$6$	$2$	216	$32$	$32$	$32$	$C_2\times C_{108}$	$C_1$	$C_1$	$C_{2} \times C_{108}$	$2^{4} \cdot 3^{2}$	$2^{4} \cdot 3^{2}$	$216$	$33$	$-$	$-$	$-$	$2$	$3$	$20$	Abelian - 2
216.10	$D_4\times C_{27}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$2$	$2$	$6$	$2$	135	$40$	$32$	$24$	$C_{54}$	$C_2^2$	$C_2$	$C_{2} \times C_{54}$	$2^{4} \cdot 3^{2}$	$2^{2} \cdot 3^{2}$	$108$	$31$	$2$	$4$	$36$	$2$	$4$	$20$	Nilpotent - 2
216.11	$Q_8\times C_{27}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$2$	$2$	$6$	$2$	135	$24$	$24$	$24$	$C_{54}$	$C_2^2$	$C_2$	$C_{2} \times C_{54}$	$2^{4} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$216$	$35$	$2$	$4$	$36$	$2$	$4$	$22$	Nilpotent - 2
216.12	$C_9:C_{24}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	72	$56$	$30$	$20$	$C_{12}$	$D_9$	$C_9$	$C_{24}$	$2^{4} \cdot 3^{3}$	$2^{3} \cdot 3$	$72$	$20$	$2$	$4$	$24$	$2$	$4$	$12$	Solvable - 2
216.13	$C_3:C_{72}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	108	$42$	$30$	$21$	$C_{36}$	$S_3$	$C_3$	$C_{72}$	$2^{4} \cdot 3^{2}$	$2^{3} \cdot 3$	$72$	$20$	$2$	$4$	$24$	$2$	$4$	$10$	Solvable - 2
216.14	$C_3^2:C_{24}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3$	$-1$	$2$	$6$	$2$	40	$92$	$34$	$17$	$C_4$	$C_3^2:C_6$	$C_3^2$	$C_{24}$	$2^{4} \cdot 3^{3}$	$2^{3}$	$72$	$17$	$6$	$12$	$12$	$6$	$8$	$10$	Solvable - 2
216.15	$C_9:C_{24}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	40	$56$	$27$	$17$	$C_4$	$C_9:C_6$	$C_9$	$C_{24}$	$2^{3} \cdot 3^{3}$	$2^{2}$	$72$	$17$	$6$	$12$	$12$	$6$	$8$	$10$	Solvable - 2
216.16	$C_{12}.D_9$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{2}$	$-1$	$2$	$6$	$3$	60	$106$	$40$	$31$	$C_4$	$C_3:D_9$	$C_3\times C_9$	$C_{8}$	$2^{4} \cdot 3^{6}$	$2^{3} \cdot 3^{3}$	$216$	$20$	$-$	$-$	$-$	$4$	$6$	$10$	Solvable - 2
216.17	$\He_3:C_8$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3$	$-1$	$3$	$6$	$3$	40	$100$	$44$	$22$	$C_{12}$	$C_3:S_3$	$\He_3$	$C_{8}$	$2^{6} \cdot 3^{3}$	$2^{5} \cdot 3$	$72$	$17$	$3$	$6$	$24$	$3$	$6$	$10$	Solvable - 3
216.18	$C_3\times C_{72}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{2}$	$1$	$1$	$6$	$2$	216	$40$	$40$	$40$	$C_3\times C_{72}$	$C_1$	$C_1$	$C_{3} \times C_{72}$	$2^{4} \cdot 3^{3}$	$2^{4} \cdot 3^{3}$	$216$	$20$	$-$	$-$	$-$	$2$	$4$	$12$	Abelian - 2
216.19	$C_8\times \He_3$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3$	$2$	$2$	$6$	$2$	88	$76$	$44$	$28$	$C_{24}$	$C_3^2$	$C_3$	$C_{3} \times C_{24}$	$2^{6} \cdot 3^{3}$	$2^{6} \cdot 3$	$72$	$17$	$3$	$6$	$24$	$3$	$6$	$10$	Nilpotent - 2
216.20	$C_9:C_{24}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{2}$	$2$	$2$	$6$	$2$	88	$40$	$32$	$28$	$C_{24}$	$C_3^2$	$C_3$	$C_{3} \times C_{24}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3$	$72$	$17$	$3$	$6$	$24$	$3$	$6$	$10$	Nilpotent - 2
216.21	$C_2^2:D_{27}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$-1$	$3$	$6$	$2$	21	$220$	$25$	$8$	$C_1$	$C_2^2:D_{27}$	$C_2^2:C_{27}$	$C_{2}$	$2^{3} \cdot 3^{5}$	$3^{2}$	$54$	$31$	$6$	$6$	$18$	$5$	$5$	$18$	Solvable - 3
216.22	$C_2^2:C_{54}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3^{3}$	$-1$	$2$	$6$	$2$	72	$58$	$28$	$14$	$C_{18}$	$A_4$	$C_2^2$	$C_{54}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3^{2}$	$54$	$33$	$3$	$6$	$18$	$3$	$5$	$18$	Solvable - 2
216.23	$C_2\times D_{54}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3^{3}$	$-1$	$2$	$6$	$3$	60	$460$	$64$	$31$	$C_2^2$	$D_{27}$	$C_{27}$	$C_{2}^{3}$	$2^{4} \cdot 3^{6}$	$2^{3} \cdot 3^{3}$	$108$	$31$	$-$	$-$	$-$	$3$	$3$	$19$	Solvable - 2
216.24	$C_2^2\times C_{54}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3^{3}$	$1$	$1$	$6$	$3$	216	$64$	$64$	$64$	$C_2^2\times C_{54}$	$C_1$	$C_1$	$C_{2}^{2} \times C_{54}$	$2^{4} \cdot 3^{3} \cdot 7$	$2^{4} \cdot 3^{3} \cdot 7$	$216$	$33$	$-$	$-$	$-$	$3$	$4$	$20$	Abelian - 3
216.25	$\He_3:C_8$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3$	$-1$	$3$	$6$	$2$	28	$100$	$24$	$8$	$C_6$	$C_3^2:C_4$	$\He_3$	$C_{8}$	$2^{5} \cdot 3^{2}$	$2^{3}$	$72$	$26$	$3$	$6$	$24$	$3$	$6$	$10$	Solvable - 3
216.26	$C_6.D_{18}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	33	$186$	$42$	$19$	$C_2$	$S_3\times D_9$	$C_3\times C_{18}$	$C_{2}^{2}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$72$	$20$	$4$	$8$	$24$	$4$	$6$	$12$	Solvable - 2
216.27	$C_6.D_{18}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	36	$238$	$54$	$23$	$C_2$	$S_3\times D_9$	$C_3\times C_9$	$C_{2} \times C_{4}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$72$	$16$	$4$	$8$	$24$	$4$	$6$	$10$	Solvable - 2
216.28	$C_6.D_{18}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	36	$338$	$58$	$21$	$C_2$	$S_3\times D_9$	$C_3\times C_9$	$C_{2} \times C_{4}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$36$	$16$	$4$	$4$	$12$	$4$	$4$	$10$	Solvable - 2
216.29	$C_3:D_{36}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	33	$390$	$58$	$19$	$C_2$	$S_3\times D_9$	$C_3\times C_{18}$	$C_{2}^{2}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$36$	$16$	$4$	$4$	$12$	$4$	$4$	$10$	Solvable - 2
216.30	$C_6.D_{18}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	36	$210$	$54$	$25$	$C_2$	$S_3\times D_9$	$C_3\times C_9$	$C_{2} \times C_{4}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$72$	$16$	$4$	$8$	$24$	$4$	$6$	$10$	Solvable - 2
216.31	$D_6:D_9$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	33	$262$	$54$	$19$	$C_2$	$S_3\times D_9$	$C_3\times C_{18}$	$C_{2}^{2}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$72$	$16$	$4$	$8$	$24$	$4$	$6$	$10$	Solvable - 2
216.32	$C_9:D_{12}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	33	$362$	$58$	$19$	$C_2$	$S_3\times D_9$	$C_3\times C_{18}$	$C_{2}^{2}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3$	$36$	$16$	$4$	$4$	$12$	$4$	$4$	$10$	Solvable - 2
216.33	$C_6.S_3^2$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	19	$212$	$48$	$16$	$C_2$	$C_3^2:D_6$	$C_2\times \He_3$	$C_{2}^{2}$	$2^{5} \cdot 3^{3}$	$2^{3}$	$72$	$17$	$6$	$12$	$24$	$6$	$10$	$10$	Solvable - 3
216.34	$C_6.S_3^2$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	22	$282$	$62$	$20$	$C_2$	$C_3^2:D_6$	$\He_3$	$C_{2} \times C_{4}$	$2^{4} \cdot 3^{3}$	$2^{2}$	$36$	$13$	$6$	$12$	$12$	$6$	$8$	$8$	Solvable - 3
216.35	$\He_3:D_4$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	19	$352$	$62$	$16$	$C_2$	$C_3^2:D_6$	$C_2\times \He_3$	$C_{2}^{2}$	$2^{5} \cdot 3^{3}$	$2^{3}$	$36$	$13$	$6$	$6$	$12$	$6$	$6$	$8$	Solvable - 3
216.36	$C_6.S_3^2$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	22	$298$	$66$	$18$	$C_2$	$C_3^2:D_6$	$\He_3$	$C_{2} \times C_{4}$	$2^{5} \cdot 3^{3}$	$2^{3}$	$36$	$13$	$6$	$12$	$12$	$6$	$8$	$8$	Solvable - 3
216.37	$C_3^2:D_{12}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	19	$368$	$66$	$16$	$C_2$	$C_3^2:D_6$	$C_2\times \He_3$	$C_{2}^{2}$	$2^{4} \cdot 3^{3}$	$2^{2}$	$36$	$13$	$6$	$12$	$12$	$6$	$8$	$8$	Solvable - 3
216.38	$C_9\times \SL(2,3)$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$3$	$6$	$2$	63	$81$	$33$	$16$	$C_{18}$	$A_4$	$Q_8$	$C_{3} \times C_{9}$	$2^{4} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$72$	$17$	$2$	$4$	$12$	$2$	$4$	$10$	Solvable - 3
216.39	$C_9:\SL(2,3)$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$3$	$6$	$2$	31	$81$	$27$	$13$	$C_6$	$C_3\times A_4$	$C_3\times Q_8$	$C_{3}^{2}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3$	$72$	$17$	$6$	$12$	$12$	$5$	$10$	$10$	Solvable - 3
216.40	$C_3^2.\SL(2,3)$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$3$	$6$	$2$	63	$72$	$36$	$22$	$C_3\times C_6$	$A_4$	$Q_8$	$C_{3} \times C_{9}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3^{3}$	$216$	$20$	$-$	$-$	$-$	$3$	$6$	$10$	Solvable - 3
216.41	$C_3^2.\SL(2,3)$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$3$	$6$	$2$	31	$72$	$30$	$13$	$C_6$	$C_3\times A_4$	$C_3\times Q_8$	$C_{3}^{2}$	$2^{3} \cdot 3^{4}$	$2 \cdot 3^{2}$	$72$	$17$	$6$	$12$	$12$	$5$	$10$	$10$	Solvable - 3
216.42	$C_3^2:\SL(2,3)$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	31	$153$	$39$	$13$	$C_6$	$C_3\times A_4$	$C_3\times Q_8$	$C_{3}^{2}$	$2^{4} \cdot 3^{4}$	$2^{2} \cdot 3^{2}$	$72$	$17$	$6$	$12$	$12$	$5$	$10$	$10$	Solvable - 3
216.43	$C_{36}.C_6$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	63	$108$	$42$	$24$	$C_6$	$D_{18}$	$C_{18}$	$C_{2} \times C_{6}$	$2^{5} \cdot 3^{3}$	$2^{3} \cdot 3$	$72$	$20$	$2$	$4$	$24$	$2$	$4$	$10$	Solvable - 2
216.44	$C_{12}.C_{18}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	81	$66$	$42$	$27$	$C_{18}$	$D_6$	$C_6$	$C_{2} \times C_{18}$	$2^{5} \cdot 3^{2}$	$2^{3} \cdot 3$	$72$	$20$	$2$	$4$	$24$	$2$	$4$	$12$	Solvable - 2
216.45	$C_{12}\times D_9$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	72	$160$	$54$	$28$	$C_{12}$	$D_9$	$C_9$	$C_{2} \times C_{12}$	$2^{4} \cdot 3^{3}$	$2^{3} \cdot 3$	$72$	$16$	$2$	$4$	$24$	$2$	$4$	$10$	Solvable - 2
216.46	$C_3\times D_{36}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	63	$212$	$54$	$24$	$C_6$	$D_{18}$	$C_{18}$	$C_{2} \times C_{6}$	$2^{5} \cdot 3^{3}$	$2^{3} \cdot 3$	$72$	$16$	$2$	$4$	$24$	$2$	$4$	$10$	Solvable - 2
216.47	$S_3\times C_{36}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	108	$90$	$54$	$33$	$C_{36}$	$S_3$	$C_3$	$C_{2} \times C_{36}$	$2^{4} \cdot 3^{2}$	$2^{3} \cdot 3$	$72$	$16$	$2$	$4$	$24$	$2$	$4$	$10$	Solvable - 2
216.48	$C_9\times D_{12}$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$6$	$2$	81	$114$	$54$	$27$	$C_{18}$	$D_6$	$C_6$	$C_{2} \times C_{18}$	$2^{5} \cdot 3^{2}$	$2^{3} \cdot 3$	$72$	$16$	$2$	$4$	$24$	$2$	$4$	$10$	Solvable - 2
216.49	$\He_3:Q_8$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	31	$162$	$48$	$21$	$C_2$	$C_3^2:D_6$	$C_3\times C_6$	$C_{2} \times C_{6}$	$2^{5} \cdot 3^{3}$	$2^{3}$	$72$	$17$	$6$	$12$	$24$	$6$	$10$	$10$	Solvable - 2
216.50	$C_4\times C_3^2:C_6$	$2^{3} \cdot 3^{3}$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	40	$232$	$62$	$25$	$C_4$	$C_3^2:C_6$	$C_3^2$	$C_{2} \times C_{12}$	$2^{4} \cdot 3^{3}$	$2^{3}$	$36$	$13$	$6$	$12$	$12$	$6$	$8$	$8$	Solvable - 2

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