Subgroups — C₂×#6235 #88752

Order 432.

297 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	1	yes	maximal, characteristic, complemented
3	#6235	C₂	1	4	yes	maximal, direct product factor
4	C₂×#1084	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#67	—	3	4	no	maximal
6	C₂²×H(3,𝔽₃)	—	2	2	no	—
7	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₂²×H(3,𝔽₃)	C₂²	1	2	yes	complemented
9	C₂×#272	—	2	2	no	—
10	C₂×#272	C₂²	1	2	yes	complemented
11	#1084	C₂²	1	2	yes	complemented
12	C₂³×C₃²	D₃	1	4	yes	complemented
13	C₂²×C₃×D₃	—	3	4	no	—
14	C₃×#67	—	3	16	no	—
15	C₂×C₃×Dic₃	—	3	4	no	—
16	C₂×H(3,𝔽₃)	—	2	2	no	—
17	C₂×H(3,𝔽₃)	D₄	1	2	yes	complemented
18	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
19	#272	—	2	2	no	—
20	C₂×C₃×D₄	—	9	1	no	—
21	C₂×#67	—	3	4	no	—
22	C₂²×C₃²	—	2	8	no	—
23	C₂²×C₃²	C₂×D₃	1	4	yes	—
24	C₂²×C₃²	C₂×D₃	1	8	yes	complemented
25	C₂×C₃×D₃	—	3	8	no	—
26	C₂×C₃×D₃	—	6	8	no	—
27	C₃×Dic₃	—	3	8	no	—
28	H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
29	C₂³×C₃	—	3	4	no	—
30	C₂³×C₃	—	9	1	no	—
31	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
32	C₂²×D₃	—	3	4	no	—
33	C₃×D₄	—	9	4	no	—
34	C₂×C₃×C₄	—	9	1	no	—
35	#67	—	3	16	no	—
36	C₂×Dic₃	—	3	4	no	—
37	C₂×C₃²	—	2	8	no	—
38	C₂×C₃²	#67	1	8	yes	complemented
39	C₂×C₃²	C₂²×D₃	1	4	yes	—
40	C₃×D₃	—	6	8	no	—
41	C₂×D₄	—	9	1	no	Sylow
42	C₂²×C₃	—	2	2	no	—
43	C₂²×C₃	—	3	4	no	—
44	C₂²×C₃	—	3	8	no	—
45	C₂²×C₃	—	6	8	no	—
46	C₂²×C₃	—	9	2	no	—
47	C₂²×C₃	—	18	2	no	—
48	C₂²×C₃	C₂×C₃⋊D₃	1	2	yes	—
49	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₂×D₃	—	3	8	no	—
51	C₂×D₃	—	6	8	no	—
52	C₃×C₄	—	9	2	no	—
53	Dic₃	—	3	8	no	—
54	C₃²	C₂×#67	1	4	yes	complemented
55	C₂³	—	9	1	no	—
56	C₂³	#272	1	1	yes	characteristic, complemented
57	D₄	—	9	4	no	—
58	C₂×C₄	—	9	1	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	3	8	no	—
61	C₂×C₃	—	3	4	no	—
62	C₂×C₃	—	6	8	no	—
63	C₂×C₃	—	18	2	no	—
64	C₂×C₃	#638	1	2	yes	—
65	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
66	D₃	—	6	8	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	9	2	no	—
69	C₂²	—	18	2	no	—
70	C₂²	C₂×#272	1	1	yes	characteristic
71	C₂²	C₂×#272	1	2	yes	complemented
72	C₄	—	9	2	no	—
73	C₃	—	3	4	no	—
74	C₃	C₂×#638	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	—	18	2	no	—
77	C₂	#6235	1	2	yes	direct product factor
78	C₂	C₂²×#272	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃	C₂²×C₃⋊D₃
Fitting	C₂³×H(3,𝔽₃)	C₂
hypercenter	C₂³×C₃	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

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×#272	C₂	semidirect
3. #6235	C₂	direct (N × H)
4. C₂×#1084	C₂	semidirect
8. C₂²×H(3,𝔽₃)	C₂²	semidirect
10. C₂×#272	C₂²	semidirect
11. #1084	C₂²	semidirect
12. C₂³×C₃²	D₃	semidirect
17. C₂×H(3,𝔽₃)	D₄	semidirect
24. C₂²×C₃²	C₂×D₃	semidirect
28. H(3,𝔽₃)	C₂×D₄	semidirect
38. C₂×C₃²	#67	semidirect
54. C₃²	C₂×#67	semidirect
56. C₂³	#272	semidirect
71. C₂²	C₂×#272	semidirect
77. C₂	#6235	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×#272	2
#6235	2
C₂×#1084	2
C₂×C₃×#67	3

Subgroups — C2×#6235 #88752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6235 #88752