Subgroups — C₂³×3₋¹⁺² #6279

Order 216.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×3₋¹⁺²	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃²	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₉	C₃	1	3	yes	maximal, complemented
4	C₂×3₋¹⁺²	C₂²	1	7	yes	direct product factor
5	C₂²×C₃²	C₂×C₃	1	7	yes	—
6	C₂²×C₉	C₂×C₃	1	21	yes	complemented
7	3₋¹⁺²	C₂³	1	1	yes	characteristic, direct product factor, Sylow
8	C₂³×C₃	—	3	1	no	—
9	C₂³×C₃	C₃²	1	1	yes	characteristic
10	C₂×C₃²	C₂²×C₃	1	7	yes	—
11	C₂×C₉	C₂²×C₃	1	21	yes	complemented
12	C₂²×C₃	—	3	7	no	—
13	C₂²×C₃	C₂×C₃²	1	7	yes	—
14	C₃²	C₂³×C₃	1	1	yes	characteristic
15	C₉	C₂³×C₃	1	3	yes	complemented
16	C₂³	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
17	C₂×C₃	—	3	7	no	—
18	C₂×C₃	C₂²×C₃²	1	7	yes	—
19	C₂²	C₂×3₋¹⁺²	1	7	yes	direct product factor
20	C₃	—	3	1	no	—
21	C₃	C₂³×C₃²	1	1	yes	characteristic
22	C₂	C₂²×3₋¹⁺²	1	7	yes	direct product factor

	subgroup	G/H
center, socle	C₂³×C₃	C₃²
commutator, Frattini, cosocle	C₃	C₂³×C₃²
prime radical	C₂³×C₃²	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.