Subgroups — D₃×#157 #6319

Order 216.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#157	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#157	—	3	1	no	maximal
5	C₃⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic, complemented
6	C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃²×D₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃⋊D₃	—	3	1	no	—
9	D₃²	—	6	2	no	—
10	#157	—	3	1	no	—
11	#157	D₃	1	1	yes	characteristic, direct product factor
12	C₃³	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
13	C₄×D₃	—	9	1	no	maximal
14	C₂×C₃²	—	3	1	no	—
15	C₃⋊D₃	—	3	1	no	—
16	C₃⋊D₃	—	6	2	no	—
17	C₃⋊D₃	—	12	2	no	—
18	C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
19	C₃×D₃	—	2	2	no	—
20	C₃×D₃	—	6	2	no	—
21	C₂×D₃	—	9	1	no	—
22	C₂×D₃	—	18	2	no	—
23	C₃×C₄	—	9	1	no	—
24	Dic₃	—	9	1	no	—
25	C₃²	—	2	2	no	—
26	C₃²	—	4	2	no	—
27	C₃²	C₄×D₃	1	1	yes	characteristic, complemented
28	C₂×C₄	—	27	1	no	Sylow
29	C₂×C₃	—	6	2	no	—
30	C₂×C₃	—	9	1	no	—
31	D₃	—	6	2	no	—
32	D₃	—	9	1	no	—
33	D₃	—	18	2	no	—
34	D₃	—	36	2	no	—
35	D₃	#157	1	1	yes	characteristic, direct product factor
36	C₂²	—	27	1	no	—
37	C₄	—	9	1	no	—
38	C₄	—	27	1	no	—
39	C₃	—	2	2	no	—
40	C₃	—	4	2	no	—
41	C₃	C₂×#157	1	1	yes	characteristic, complemented
42	C₂	—	3	1	no	—
43	C₂	—	9	1	no	—
44	C₂	—	27	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³	C₂×C₄
prime radical	D₃×C₃⋊D₃	C₂
cosocle	C₃×C₃⋊D₃	C₂²
LCS	G ▷ 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.