Subgroups — C₂×C₄×C₃⋊(C₃⋊D₃) #89074

Order 432.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃⋊(C₃⋊D₃)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×C₃⋊D₃	—	3	13	no	maximal
6	C₂²×C₃³	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊D₃)	C₄	1	4	yes	direct product factor
8	C₂×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
9	C₃³×C₄	C₂²	1	2	yes	complemented
10	C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	complemented
11	C₂²×C₃⋊D₃	—	3	13	no	—
12	C₂×C₃²×C₄	D₃	1	13	yes	complemented
13	C₂×C₃⋊Dic₃	—	3	13	no	—
14	C₄×C₃⋊D₃	—	3	52	no	—
15	C₂×C₃³	C₂×C₄	1	2	yes	complemented
16	C₂×C₃³	C₂³	1	1	yes	characteristic
17	C₃⋊(C₃⋊D₃)	C₂×C₄	1	4	yes	direct product factor
18	C₂×C₄×D₃	—	9	13	no	—
19	C₂²×C₃²	C₂×D₃	1	13	yes	—
20	C₂×C₃⋊D₃	—	3	52	no	—
21	C₂×C₃⋊D₃	—	3	26	no	—
22	C₃²×C₄	C₂×D₃	1	26	yes	complemented
23	C₃⋊Dic₃	—	3	26	no	—
24	C₃³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
25	C₂²×D₃	—	9	13	no	—
26	C₂×C₃×C₄	C₃⋊D₃	1	13	yes	complemented
27	C₂×Dic₃	—	9	13	no	—
28	C₄×D₃	—	9	52	no	—
29	C₂×C₃²	C₄×D₃	1	26	yes	complemented
30	C₂×C₃²	C₂²×D₃	1	13	yes	—
31	C₃⋊D₃	—	3	52	no	—
32	C₂²×C₄	—	27	1	no	Sylow
33	C₂²×C₃	C₂×C₃⋊D₃	1	13	yes	—
34	C₂×D₃	—	9	52	no	—
35	C₂×D₃	—	9	26	no	—
36	C₃×C₄	C₂×C₃⋊D₃	1	26	yes	complemented
37	Dic₃	—	9	26	no	—
38	C₃²	C₂×C₄×D₃	1	13	yes	complemented
39	C₂³	—	27	1	no	—
40	C₂×C₄	—	27	1	no	—
41	C₂×C₄	—	27	4	no	—
42	C₂×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
43	C₂×C₃	C₄×C₃⋊D₃	1	26	yes	complemented
44	C₂×C₃	C₂²×C₃⋊D₃	1	13	yes	—
45	D₃	—	9	52	no	—
46	C₂²	—	27	4	no	—
47	C₂²	—	27	2	no	—
48	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
49	C₄	—	27	2	no	—
50	C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	direct product factor
51	C₃	C₂×C₄×C₃⋊D₃	1	13	yes	complemented
52	C₂	—	27	4	no	—
53	C₂	C₄×C₃⋊(C₃⋊D₃)	1	2	yes	direct product factor
54	C₂	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₂²×C₄
Frattini	C₂	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₄	C₂
socle	C₂²×C₃³	C₂²
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₂
cosocle	C₂×C₃³	C₂³
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₄	C₂	semidirect
3. C₂×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
4. C₄×C₃⋊(C₃⋊D₃)	C₂	direct (N × H)
7. C₂×C₃⋊(C₃⋊D₃)	C₄	direct (N × H)
9. C₃³×C₄	C₂²	semidirect
10. C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
12. C₂×C₃²×C₄	D₃	semidirect
15. C₂×C₃³	C₂×C₄	semidirect
17. C₃⋊(C₃⋊D₃)	C₂×C₄	direct (N × H)
22. C₃²×C₄	C₂×D₃	semidirect
24. C₃³	C₂²×C₄	semidirect
26. C₂×C₃×C₄	C₃⋊D₃	semidirect
29. C₂×C₃²	C₄×D₃	semidirect
36. C₃×C₄	C₂×C₃⋊D₃	semidirect
38. C₃²	C₂×C₄×D₃	semidirect
42. C₂×C₄	C₃⋊(C₃⋊D₃)	direct (N × H)
43. C₂×C₃	C₄×C₃⋊D₃	semidirect
50. C₄	C₂×C₃⋊(C₃⋊D₃)	direct (N × H)
51. C₃	C₂×C₄×C₃⋊D₃	semidirect
53. C₂	C₄×C₃⋊(C₃⋊D₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊D₃)	2
C₂×C₃³×C₄	2
C₂×C₃⋊(C₃⋊Dic₃)	2
C₄×C₃⋊(C₃⋊D₃)	2
C₂×C₄×C₃⋊D₃	3

Subgroups — C2×C4×C3⋊(C3⋊D3) #89074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃⋊(C₃⋊D₃) #89074