Subgroups — C₂³×#272 #88914

Order 432.

737 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#272	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×D₃	—	3	4	no	maximal
4	C₂²×H(3,𝔽₃)	C₂²	1	7	yes	complemented
5	C₂×#272	C₂²	1	28	yes	direct product factor
6	C₂³×C₃²	D₃	1	4	yes	complemented
7	C₂²×C₃×D₃	—	3	56	no	—
8	C₂×H(3,𝔽₃)	C₂³	1	7	yes	complemented
9	#272	C₂³	1	8	yes	direct product factor
10	C₂⁴×C₃	—	9	1	no	—
11	C₂³×D₃	—	3	4	no	—
12	C₂²×C₃²	C₂×D₃	1	28	yes	complemented
13	C₂×C₃×D₃	—	3	112	no	—
14	H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented, Sylow
15	C₂³×C₃	—	3	4	no	—
16	C₂³×C₃	—	9	14	no	—
17	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
18	C₂²×D₃	—	3	56	no	—
19	C₂×C₃²	C₂²×D₃	1	28	yes	complemented
20	C₃×D₃	—	3	32	no	—
21	C₂⁴	—	9	1	no	Sylow
22	C₂²×C₃	—	3	28	no	—
23	C₂²×C₃	—	9	28	no	—
24	C₂²×C₃	C₂×C₃⋊D₃	1	7	yes	—
25	C₂×D₃	—	3	112	no	—
26	C₃²	C₂³×D₃	1	4	yes	complemented
27	C₂³	—	9	14	no	—
28	C₂³	#272	1	1	yes	characteristic, direct product factor
29	C₂×C₃	—	3	28	no	—
30	C₂×C₃	—	9	8	no	—
31	C₂×C₃	C₂²×C₃⋊D₃	1	7	yes	—
32	D₃	—	3	32	no	—
33	C₂²	—	9	28	no	—
34	C₂²	C₂×#272	1	7	yes	direct product factor
35	C₃	—	3	4	no	—
36	C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
37	C₂	—	9	8	no	—
38	C₂	C₂²×#272	1	7	yes	direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂³×C₃	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂⁴
Frattini	C₃	C₂³×C₃⋊D₃
Fitting	C₂³×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×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.