Subgroups — C₂²×C₃×D₃ #651

Order 72.

69 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₃	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₃	—	3	1	no	maximal
4	C₂²×D₃	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²	C₂²	1	3	yes	complemented
6	C₃×D₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₃	—	2	1	no	—
8	C₂²×C₃	—	3	6	no	—
9	C₂²×C₃	D₃	1	1	yes	characteristic, direct product factor
10	C₂²×C₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₃	C₂×C₃	1	6	yes	direct product factor
12	C₃²	C₂³	1	1	yes	characteristic, complemented, Sylow
13	C₂³	—	3	1	no	Sylow
14	C₂×C₃	—	2	3	no	—
15	C₂×C₃	—	3	4	no	—
16	C₂×C₃	C₂×D₃	1	3	yes	direct product factor
17	C₂×C₃	C₂²×C₃	1	3	yes	complemented
18	D₃	C₂²×C₃	1	4	yes	direct product factor
19	C₂²	—	3	6	no	—
20	C₂²	C₃×D₃	1	1	yes	characteristic, direct product factor
21	C₃	—	2	1	no	—
22	C₃	C₂²×D₃	1	1	yes	characteristic, direct product factor
23	C₃	C₂³×C₃	1	1	yes	characteristic, complemented
24	C₂	—	3	4	no	—
25	C₂	C₂×C₃×D₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃

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.