Subgroups — Dic₃² #3720

Order 144.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₃	C₂	1	2	yes	maximal
3	C₄×Dic₃	—	3	2	no	maximal
4	C₂²×C₃²	C₂²	1	1	yes	characteristic
5	C₃⋊Dic₃	C₄	1	2	yes	complemented
6	C₃×Dic₃	C₄	1	4	yes	complemented
7	C₂×C₃×C₄	—	3	2	no	—
8	C₂×Dic₃	—	3	2	no	—
9	C₂×Dic₃	—	6	1	no	—
10	C₂×Dic₃	D₃	1	2	yes	—
11	C₂×C₃²	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃²	C₂×C₄	1	2	yes	—
13	C₄²	—	9	1	no	Sylow
14	C₂²×C₃	—	2	1	no	—
15	C₂²×C₃	C₂×D₃	1	2	yes	—
16	C₃×C₄	—	3	4	no	—
17	Dic₃	—	3	4	no	—
18	Dic₃	—	6	2	no	—
19	Dic₃	Dic₃	1	4	yes	direct product factor
20	C₃²	C₄²	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₄	—	3	2	no	—
22	C₂×C₄	—	9	1	no	—
23	C₂×C₃	—	2	2	no	—
24	C₂×C₃	—	2	1	no	—
25	C₂×C₃	C₄×D₃	1	2	yes	—
26	C₂×C₃	C₄×D₃	1	2	yes	—
27	C₂×C₃	C₂×Dic₃	1	2	yes	—
28	C₂²	D₃²	1	1	yes	characteristic
29	C₄	—	3	4	no	—
30	C₄	—	9	2	no	—
31	C₃	—	2	1	no	—
32	C₃	C₄×Dic₃	1	2	yes	complemented
33	C₂	D₃×Dic₃	1	2	yes	—
34	C₂	#624	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃²
commutator, hypocenter	C₃²	C₄²
Fitting, socle, prime radical, cosocle	C₂²×C₃²	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²

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.