Subgroups — C₂×C₃⋊#272 #66458

Order 324.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊#272	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₃⋊D₃	—	3	4	no	maximal
4	C₂×#272	—	3	9	no	maximal
5	C₃×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	D₃	1	4	yes	complemented
7	C₃×C₃⋊D₃	—	3	8	no	—
8	C₂×H(3,𝔽₃)	D₃	1	9	yes	complemented
9	#272	—	3	18	no	—
10	C₂×C₃⋊D₃	—	3	4	no	—
11	C₂×C₃×D₃	—	9	1	no	—
12	C₂×C₃×D₃	—	9	12	no	—
13	C₃³	C₂×D₃	1	4	yes	complemented
14	H(3,𝔽₃)	C₂×D₃	1	9	yes	complemented
15	C₂×C₃²	—	3	4	no	—
16	C₂×C₃²	—	6	4	no	—
17	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic
18	C₂×C₃²	C₃⋊D₃	1	12	yes	complemented
19	C₃⋊D₃	—	3	8	no	—
20	C₃×D₃	—	9	2	no	—
21	C₃×D₃	—	9	24	no	—
22	C₂²×C₃	—	27	1	no	—
23	C₂×D₃	—	9	12	no	—
24	C₂×D₃	—	9	1	no	—
25	C₃²	—	3	4	no	—
26	C₃²	—	6	4	no	—
27	C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic
28	C₃²	C₂×C₃⋊D₃	1	12	yes	complemented
29	C₂×C₃	—	2	1	no	—
30	C₂×C₃	—	3	12	no	—
31	C₂×C₃	—	27	2	no	—
32	C₂×C₃	#272	1	1	yes	characteristic, complemented
33	C₂×C₃	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
34	D₃	—	9	24	no	—
35	D₃	—	9	2	no	—
36	C₂²	—	27	1	no	Sylow
37	C₃	—	2	1	no	—
38	C₃	—	3	12	no	—
39	C₃	C₂×#272	1	1	yes	characteristic, complemented
40	C₃	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
41	C₂	—	27	2	no	—
42	C₂	C₃⋊#272	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂²
Frattini	C₃	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃×H(3,𝔽₃)	C₂
socle	C₂×C₃²	C₃⋊D₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃

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.