Subgroups — D₃×Dic₃ #623

Order 72.

35 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃	—	3	1	no	maximal
5	C₄×D₃	—	3	1	no	maximal
6	C₂×C₃²	C₂²	1	1	yes	characteristic
7	C₃×D₃	C₄	1	2	yes	complemented
8	C₂²×C₃	—	3	1	no	—
9	C₂×D₃	D₃	1	1	yes	characteristic
10	C₃×C₄	—	3	1	no	—
11	Dic₃	—	3	1	no	—
12	Dic₃	—	3	1	no	—
13	Dic₃	—	6	1	no	—
14	Dic₃	D₃	1	1	yes	characteristic, direct product factor
15	C₃²	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₄	—	9	1	no	Sylow
17	C₂×C₃	—	2	1	no	—
18	C₂×C₃	—	3	2	no	—
19	C₂×C₃	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃	C₂×D₃	1	1	yes	characteristic
21	D₃	Dic₃	1	2	yes	direct product factor
22	C₂²	—	3	1	no	—
23	C₄	—	3	1	no	—
24	C₄	—	9	1	no	—
25	C₃	—	2	1	no	—
26	C₃	C₄×D₃	1	1	yes	characteristic, complemented
27	C₃	C₂×Dic₃	1	1	yes	characteristic, complemented
28	C₂	—	3	2	no	—
29	C₂	D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²
commutator, hypocenter	C₃²	C₂×C₄
Fitting, socle, cosocle	C₂×C₃²	C₂²
prime radical	C₂×C₃×D₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂

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.