Subgroups — C₂×#6198 #88673

Order 432.

205 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×(C₃⋊D₃)⋊C₃	C₂	1	2	yes	maximal, complemented
2	C₂×#1084	C₂	1	1	yes	maximal, characteristic, complemented
3	#6198	C₂	1	4	yes	maximal, direct product factor
4	C₂×#626	—	3	2	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×(C₃⋊D₃)⋊C₃	—	2	4	no	—
7	C₂×(C₃⋊D₃)⋊C₃	C₂²	1	4	yes	complemented
8	#1084	C₂²	1	2	yes	complemented
9	C₂²×C₃⋊D₃	D₃	1	2	yes	complemented
10	C₂²×C₃×D₃	—	3	2	no	—
11	C₂×C₃×Dic₃	—	3	2	no	—
12	C₂×C₃×Dic₃	—	6	1	no	—
13	#626	—	3	8	no	—
14	C₂×H(3,𝔽₃)	D₄	1	2	yes	complemented
15	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	(C₃⋊D₃)⋊C₃	—	2	4	no	—
17	C₂×#67	—	9	2	no	—
18	C₂×D₁₂	—	9	1	no	—
19	C₂²×C₃²	—	2	1	no	—
20	C₂²×C₃²	C₂×D₃	1	2	yes	—
21	C₂×C₃⋊D₃	—	2	4	no	—
22	C₂×C₃⋊D₃	C₂×D₃	1	4	yes	complemented
23	C₂×C₃×D₃	—	3	4	no	—
24	C₂×C₃×D₃	—	6	4	no	—
25	C₃×Dic₃	—	3	4	no	—
26	C₃×Dic₃	—	6	2	no	—
27	H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
28	C₂³×C₃	—	9	2	no	—
29	C₂²×D₃	—	3	2	no	—
30	C₂²×D₃	—	9	2	no	—
31	C₂×C₃×C₄	—	9	1	no	—
32	#67	—	9	8	no	—
33	C₂×Dic₃	—	3	2	no	—
34	C₂×Dic₃	—	6	1	no	—
35	D₁₂	—	9	4	no	—
36	C₂×C₃²	—	2	2	no	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	#67	1	4	yes	complemented
39	C₂×C₃²	C₂²×D₃	1	2	yes	—
40	C₃⋊D₃	—	2	4	no	—
41	C₃×D₃	—	6	4	no	—
42	C₂×D₄	—	27	1	no	Sylow
43	C₂²×C₃	—	3	2	no	—
44	C₂²×C₃	—	6	1	no	—
45	C₂²×C₃	—	9	4	no	—
46	C₂²×C₃	—	18	4	no	—
47	C₂²×C₃	D₃²	1	1	yes	characteristic
48	C₂×D₃	—	3	4	no	—
49	C₂×D₃	—	6	4	no	—
50	C₂×D₃	—	9	4	no	—
51	C₂×D₃	—	18	4	no	—
52	C₃×C₄	—	9	2	no	—
53	Dic₃	—	3	4	no	—
54	Dic₃	—	6	2	no	—
55	C₃²	—	2	1	no	—
56	C₃²	C₂×#67	1	2	yes	complemented
57	C₂³	—	9	2	no	—
58	D₄	—	27	4	no	—
59	C₂×C₄	—	9	1	no	—
60	C₂×C₃	—	3	4	no	—
61	C₂×C₃	—	3	2	no	—
62	C₂×C₃	—	6	2	no	—
63	C₂×C₃	—	6	1	no	—
64	C₂×C₃	—	18	4	no	—
65	C₂×C₃	#625	1	2	yes	—
66	C₂×C₃	C₂×D₃²	1	1	yes	characteristic
67	D₃	—	6	4	no	—
68	D₃	—	18	4	no	—
69	C₂²	—	9	4	no	—
70	C₂²	—	18	4	no	—
71	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
72	C₄	—	9	2	no	—
73	C₃	—	3	2	no	—
74	C₃	—	6	1	no	—
75	C₃	C₂×#625	1	1	yes	characteristic
76	C₂	—	18	4	no	—
77	C₂	#6198	1	2	yes	direct product factor
78	C₂	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃	C₂×D₃²
Fitting	C₂²×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	D₃²
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×(C₃⋊D₃)⋊C₃	C₂	semidirect
2. C₂×#1084	C₂	semidirect
3. #6198	C₂	direct (N × H)
7. C₂×(C₃⋊D₃)⋊C₃	C₂²	semidirect
8. #1084	C₂²	semidirect
9. C₂²×C₃⋊D₃	D₃	semidirect
14. C₂×H(3,𝔽₃)	D₄	semidirect
22. C₂×C₃⋊D₃	C₂×D₃	semidirect
27. H(3,𝔽₃)	C₂×D₄	semidirect
38. C₂×C₃²	#67	semidirect
56. C₃²	C₂×#67	semidirect
77. C₂	#6198	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⋊D₃)⋊C₃	2
C₂×#1084	2
#6198	2
C₂×#626	3

Subgroups — C2×#6198 #88673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6198 #88673