Subgroups — D₃²⋊D₃ #6321

Order 216.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1113	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#157	C₂	1	1	yes	maximal, characteristic, complemented
4	GO⁺(4,𝔽₂)	—	3	1	no	maximal
5	C₃×C₃⋊D₃	—	2	1	no	—
6	C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃²×D₃	—	2	1	no	—
8	C₂×C₃×D₃	—	6	1	no	—
9	D₃²	—	3	1	no	—
10	D₃²	—	6	1	no	—
11	D₃²	D₃	1	1	yes	characteristic, complemented
12	#157	—	3	1	no	—
13	C₃³	D₄	1	1	yes	characteristic, complemented, Sylow
14	#67	—	9	1	no	maximal
15	C₂×C₃²	—	6	1	no	—
16	C₃⋊D₃	—	2	1	no	—
17	C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
18	C₃×D₃	—	2	1	no	—
19	C₃×D₃	—	2	1	no	—
20	C₃×D₃	—	4	1	no	—
21	C₃×D₃	—	6	1	no	—
22	C₃×D₃	—	6	1	no	—
23	C₃×D₃	—	6	1	no	—
24	C₃×D₃	—	6	1	no	—
25	C₃×D₃	—	12	1	no	—
26	C₂²×C₃	—	9	1	no	—
27	C₂×D₃	—	6	1	no	—
28	C₂×D₃	—	9	1	no	—
29	C₂×D₃	—	18	1	no	—
30	Dic₃	—	9	1	no	—
31	C₃²	—	2	1	no	—
32	C₃²	—	2	1	no	—
33	C₃²	—	4	1	no	—
34	C₃²	—	4	1	no	—
35	C₃²	#67	1	1	yes	characteristic, complemented
36	D₄	—	27	1	no	Sylow
37	C₂×C₃	—	6	1	no	—
38	C₂×C₃	—	6	1	no	—
39	C₂×C₃	—	9	1	no	—
40	C₂×C₃	—	12	1	no	—
41	C₂×C₃	—	18	1	no	—
42	D₃	—	2	1	no	—
43	D₃	—	6	1	no	—
44	D₃	—	6	1	no	—
45	D₃	—	6	1	no	—
46	D₃	—	6	1	no	—
47	D₃	—	12	1	no	—
48	C₂²	—	9	1	no	—
49	C₂²	—	27	1	no	—
50	C₄	—	27	1	no	—
51	C₃	—	2	1	no	—
52	C₃	—	2	1	no	—
53	C₃	—	4	1	no	—
54	C₃	—	4	1	no	—
55	C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic, complemented
56	C₂	—	6	1	no	—
57	C₂	—	9	1	no	—
58	C₂	—	18	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃×C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃³	D₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁

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.