Subgroups — C₂³×(C₃⋊D₃)⋊C₃ #88911

Order 432.

501 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×(C₃⋊D₃)⋊C₃	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₃×D₃	—	3	1	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	7	yes	complemented
6	C₂×(C₃⋊D₃)⋊C₃	C₂²	1	28	yes	direct product factor
7	C₂³×C₃²	—	2	1	no	—
8	C₂³×C₃²	D₃	1	1	yes	characteristic, complemented
9	C₂³×C₃²	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂²×C₃⋊D₃	C₂×C₃	1	14	yes	complemented
11	C₂²×C₃×D₃	—	3	14	no	—
12	C₂×H(3,𝔽₃)	C₂³	1	7	yes	complemented
13	(C₃⋊D₃)⋊C₃	C₂³	1	8	yes	direct product factor
14	C₂⁴×C₃	—	9	1	no	—
15	C₂³×D₃	—	3	1	no	—
16	C₂³×D₃	—	9	1	no	—
17	C₂²×C₃²	—	2	7	no	—
18	C₂²×C₃²	C₂×D₃	1	7	yes	complemented
19	C₂²×C₃²	C₂²×C₃	1	7	yes	complemented
20	C₂×C₃⋊D₃	C₂²×C₃	1	28	yes	complemented
21	C₂×C₃×D₃	—	3	28	no	—
22	H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented, Sylow
23	C₂³×C₃	—	3	1	no	—
24	C₂³×C₃	—	3	1	no	—
25	C₂³×C₃	—	6	1	no	—
26	C₂³×C₃	—	9	14	no	—
27	C₂³×C₃	C₃×D₃	1	1	yes	characteristic
28	C₂²×D₃	—	3	14	no	—
29	C₂²×D₃	—	9	14	no	—
30	C₂×C₃²	—	2	7	no	—
31	C₂×C₃²	C₂²×D₃	1	7	yes	complemented
32	C₂×C₃²	C₂³×C₃	1	7	yes	complemented
33	C₃⋊D₃	C₂³×C₃	1	8	yes	complemented
34	C₃×D₃	—	3	8	no	—
35	C₂⁴	—	9	1	no	Sylow
36	C₂²×C₃	—	3	7	no	—
37	C₂²×C₃	—	3	7	no	—
38	C₂²×C₃	—	6	7	no	—
39	C₂²×C₃	—	9	28	no	—
40	C₂²×C₃	C₂×C₃×D₃	1	7	yes	—
41	C₂×D₃	—	3	28	no	—
42	C₂×D₃	—	9	28	no	—
43	C₃²	—	2	1	no	—
44	C₃²	C₂³×D₃	1	1	yes	characteristic, complemented
45	C₃²	C₂⁴×C₃	1	1	yes	characteristic, complemented
46	C₂³	—	9	14	no	—
47	C₂³	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, direct product factor
48	C₂×C₃	—	3	7	no	—
49	C₂×C₃	—	3	7	no	—
50	C₂×C₃	—	6	7	no	—
51	C₂×C₃	—	9	8	no	—
52	C₂×C₃	C₂²×C₃×D₃	1	7	yes	—
53	D₃	—	3	8	no	—
54	D₃	—	9	8	no	—
55	C₂²	—	9	28	no	—
56	C₂²	C₂×(C₃⋊D₃)⋊C₃	1	7	yes	direct product factor
57	C₃	—	3	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	6	1	no	—
60	C₃	C₂³×C₃×D₃	1	1	yes	characteristic
61	C₂	—	9	8	no	—
62	C₂	C₂²×(C₃⋊D₃)⋊C₃	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃²	C₂⁴×C₃
Frattini	C₃	C₂³×C₃×D₃
Fitting	C₂³×H(3,𝔽₃)	C₂
socle	C₂³×C₃	C₃×D₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×H(3,𝔽₃)	C₂	semidirect
2. C₂²×(C₃⋊D₃)⋊C₃	C₂	direct (N × H)
3. C₂³×C₃⋊D₃	C₃	semidirect
5. C₂²×H(3,𝔽₃)	C₂²	semidirect
6. C₂×(C₃⋊D₃)⋊C₃	C₂²	direct (N × H)
8. C₂³×C₃²	D₃	semidirect
9. C₂³×C₃²	C₂×C₃	semidirect
10. C₂²×C₃⋊D₃	C₂×C₃	semidirect
12. C₂×H(3,𝔽₃)	C₂³	semidirect
13. (C₃⋊D₃)⋊C₃	C₂³	direct (N × H)
18. C₂²×C₃²	C₂×D₃	semidirect
19. C₂²×C₃²	C₂²×C₃	semidirect
20. C₂×C₃⋊D₃	C₂²×C₃	semidirect
22. H(3,𝔽₃)	C₂⁴	semidirect
31. C₂×C₃²	C₂²×D₃	semidirect
32. C₂×C₃²	C₂³×C₃	semidirect
33. C₃⋊D₃	C₂³×C₃	semidirect
44. C₃²	C₂³×D₃	semidirect
45. C₃²	C₂⁴×C₃	semidirect
47. C₂³	(C₃⋊D₃)⋊C₃	direct (N × H)
56. C₂²	C₂×(C₃⋊D₃)⋊C₃	direct (N × H)
62. C₂	C₂²×(C₃⋊D₃)⋊C₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×H(3,𝔽₃)	2
C₂²×(C₃⋊D₃)⋊C₃	2
C₂³×C₃⋊D₃	3
C₂³×C₃×D₃	3

Subgroups — C23×(C3⋊D3)⋊C3 #88911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×(C₃⋊D₃)⋊C₃ #88911