Subgroups — C₂×#6312 #89082

Order 432.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#6312	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#638	—	3	13	no	maximal
6	C₂²×C₃³	—	2	2	no	—
7	C₂²×C₃³	C₂²	1	1	yes	characteristic
8	C₂²×C₃³	C₂²	1	2	yes	complemented
9	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
10	C₂×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
11	C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	complemented
12	C₂³×C₃²	D₃	1	13	yes	complemented
13	C₂²×C₃⋊D₃	—	3	13	no	—
14	#638	—	3	52	no	—
15	C₂×C₃⋊Dic₃	—	3	13	no	—
16	C₂×C₃³	—	2	2	no	—
17	C₂×C₃³	D₄	1	2	yes	complemented
18	C₂×C₃³	C₂³	1	1	yes	characteristic
19	C₃⋊(C₃⋊D₃)	—	2	2	no	—
20	C₂×#67	—	9	13	no	—
21	C₂²×C₃²	—	2	26	no	—
22	C₂²×C₃²	C₂×D₃	1	13	yes	—
23	C₂²×C₃²	C₂×D₃	1	26	yes	complemented
24	C₂×C₃⋊D₃	—	3	26	no	—
25	C₂×C₃⋊D₃	—	6	26	no	—
26	C₃⋊Dic₃	—	3	26	no	—
27	C₃³	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
28	C₂³×C₃	C₃⋊D₃	1	13	yes	complemented
29	C₂²×D₃	—	9	13	no	—
30	#67	—	9	52	no	—
31	C₂×Dic₃	—	9	13	no	—
32	C₂×C₃²	—	2	26	no	—
33	C₂×C₃²	#67	1	26	yes	complemented
34	C₂×C₃²	C₂²×D₃	1	13	yes	—
35	C₃⋊D₃	—	6	26	no	—
36	C₂×D₄	—	27	1	no	Sylow
37	C₂²×C₃	—	2	26	no	—
38	C₂²×C₃	C₂×C₃⋊D₃	1	13	yes	—
39	C₂²×C₃	C₂×C₃⋊D₃	1	26	yes	complemented
40	C₂×D₃	—	9	26	no	—
41	C₂×D₃	—	18	26	no	—
42	Dic₃	—	9	26	no	—
43	C₃²	C₂×#67	1	13	yes	complemented
44	C₂³	—	27	1	no	—
45	C₂³	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
46	D₄	—	27	4	no	—
47	C₂×C₄	—	27	1	no	—
48	C₂×C₃	—	2	26	no	—
49	C₂×C₃	#638	1	26	yes	complemented
50	C₂×C₃	C₂²×C₃⋊D₃	1	13	yes	—
51	D₃	—	18	26	no	—
52	C₂²	—	2	2	no	—
53	C₂²	—	27	2	no	—
54	C₂²	—	54	2	no	—
55	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
56	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
57	C₄	—	27	2	no	—
58	C₃	C₂×#638	1	13	yes	complemented
59	C₂	—	2	2	no	—
60	C₂	—	54	2	no	—
61	C₂	#6312	1	2	yes	direct product factor
62	C₂	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³	C₂³
Frattini	C₂	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂³×C₃³	C₂
socle	C₂²×C₃³	C₂²
hypercenter	C₂³	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×D₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃³	C₂	semidirect
2. C₂²×C₃⋊(C₃⋊D₃)	C₂	semidirect
3. #6312	C₂	direct (N × H)
4. C₂×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
8. C₂²×C₃³	C₂²	semidirect
10. C₂×C₃⋊(C₃⋊D₃)	C₂²	semidirect
11. C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
12. C₂³×C₃²	D₃	semidirect
17. C₂×C₃³	D₄	semidirect
23. C₂²×C₃²	C₂×D₃	semidirect
27. C₃³	C₂×D₄	semidirect
28. C₂³×C₃	C₃⋊D₃	semidirect
33. C₂×C₃²	#67	semidirect
39. C₂²×C₃	C₂×C₃⋊D₃	semidirect
43. C₃²	C₂×#67	semidirect
45. C₂³	C₃⋊(C₃⋊D₃)	semidirect
49. C₂×C₃	#638	semidirect
56. C₂²	C₂×C₃⋊(C₃⋊D₃)	semidirect
58. C₃	C₂×#638	semidirect
61. C₂	#6312	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×C₃³	2
C₂²×C₃⋊(C₃⋊D₃)	2
#6312	2
C₂×C₃⋊(C₃⋊Dic₃)	2
C₂×#638	3

Subgroups — C2×#6312 #89082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6312 #89082