Subgroups — #157⋊D₃ #6322

Order 216.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1113	C₂	1	2	yes	maximal, complemented
2	C₃×#157	C₂	1	1	yes	maximal, characteristic, complemented
3	GO⁺(4,𝔽₂)	—	3	1	no	maximal
4	C₃×C₃⋊D₃	—	2	2	no	—
5	C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	D₃²	—	3	2	no	—
7	D₃²	—	6	2	no	—
8	#157	D₃	1	1	yes	characteristic, complemented
9	C₃³	D₄	1	1	yes	characteristic, complemented, Sylow
10	D₁₂	—	9	1	no	maximal
11	C₃⋊D₃	—	2	2	no	—
12	C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
13	C₃×D₃	—	6	2	no	—
14	C₃×D₃	—	6	2	no	—
15	C₃×D₃	—	6	2	no	—
16	C₃×D₃	—	12	2	no	—
17	C₂×D₃	—	9	2	no	—
18	C₂×D₃	—	18	2	no	—
19	C₃×C₄	—	9	1	no	—
20	C₃²	—	2	2	no	—
21	C₃²	—	4	2	no	—
22	C₃²	D₁₂	1	1	yes	characteristic, complemented
23	D₄	—	27	1	no	Sylow
24	C₂×C₃	—	9	1	no	—
25	C₂×C₃	—	18	2	no	—
26	D₃	—	6	2	no	—
27	D₃	—	6	2	no	—
28	D₃	—	6	2	no	—
29	D₃	—	12	2	no	—
30	C₂²	—	27	2	no	—
31	C₄	—	9	1	no	—
32	C₃	—	2	2	no	—
33	C₃	—	4	2	no	—
34	C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic, complemented
35	C₂	—	9	1	no	—
36	C₂	—	18	2	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.