Subgroups — C₂×#6199 #88674

Order 432.

221 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#272	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6199	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₃×Dic₃	—	3	2	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×#272	C₄	1	4	yes	complemented
7	C₂×#272	C₂²	1	2	yes	—
8	(C₃⋊Dic₃)⋊C₃	C₂²	1	4	yes	complemented
9	C₂²×C₃×D₃	—	3	2	no	—
10	C₂²×C₃×D₃	—	6	1	no	—
11	C₂×C₃⋊Dic₃	D₃	1	2	yes	complemented
12	C₂×C₃×Dic₃	—	3	2	no	—
13	D₃×Dic₃	—	3	8	no	—
14	C₂×H(3,𝔽₃)	C₂×C₄	1	2	yes	complemented
15	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	#272	C₂×C₄	1	4	yes	complemented
17	C₂²×Dic₃	—	9	1	no	—
18	C₂×C₄×D₃	—	9	2	no	—
19	C₂²×C₃²	—	2	1	no	—
20	C₂²×C₃²	C₂×D₃	1	2	yes	—
21	C₂×C₃×D₃	—	3	8	no	—
22	C₂×C₃×D₃	—	3	4	no	—
23	C₂×C₃×D₃	—	6	4	no	—
24	C₂×C₃×D₃	—	6	2	no	—
25	C₃⋊Dic₃	C₂×D₃	1	4	yes	complemented
26	C₃×Dic₃	—	3	4	no	—
27	H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
28	C₂³×C₃	—	9	1	no	—
29	C₂²×D₃	—	3	2	no	—
30	C₂²×D₃	—	6	1	no	—
31	C₂×C₃×C₄	—	9	2	no	—
32	C₂×Dic₃	—	3	2	no	—
33	C₂×Dic₃	—	9	2	no	—
34	C₂×Dic₃	—	9	4	no	—
35	C₄×D₃	—	9	8	no	—
36	C₂×C₃²	—	2	2	no	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	C₄×D₃	1	4	yes	complemented
39	C₂×C₃²	C₂²×D₃	1	2	yes	—
40	C₃×D₃	—	3	8	no	—
41	C₃×D₃	—	6	4	no	—
42	C₂²×C₄	—	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	—	9	2	no	—
47	C₂²×C₃	D₃²	1	1	yes	characteristic
48	C₂×D₃	—	3	4	no	—
49	C₂×D₃	—	3	8	no	—
50	C₂×D₃	—	6	4	no	—
51	C₂×D₃	—	6	2	no	—
52	C₃×C₄	—	9	4	no	—
53	Dic₃	—	3	4	no	—
54	Dic₃	—	9	4	no	—
55	C₃²	—	2	1	no	—
56	C₃²	C₂×C₄×D₃	1	2	yes	complemented
57	C₂³	—	9	1	no	—
58	C₂×C₄	—	9	2	no	—
59	C₂×C₄	—	27	4	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	—	9	4	no	—
65	C₂×C₃	#624	1	2	yes	—
66	C₂×C₃	C₂×D₃²	1	1	yes	characteristic
67	D₃	—	3	8	no	—
68	D₃	—	6	4	no	—
69	C₂²	—	9	4	no	—
70	C₂²	—	9	2	no	—
71	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
72	C₄	—	9	4	no	—
73	C₃	—	3	2	no	—
74	C₃	—	6	1	no	—
75	C₃	C₂×#624	1	1	yes	characteristic
76	C₂	—	9	4	no	—
77	C₂	#6199	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, hypocenter	H(3,𝔽₃)	C₂²×C₄
Frattini	C₂×C₃	C₂×D₃²
Fitting	C₂²×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	D₃²
prime radical	C₂²×#272	C₂
cosocle	C₂×H(3,𝔽₃)	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×(C₃⋊Dic₃)⋊C₃	C₂	semidirect
3. #6199	C₂	direct (N × H)
6. C₂×#272	C₄	semidirect
8. (C₃⋊Dic₃)⋊C₃	C₂²	semidirect
11. C₂×C₃⋊Dic₃	D₃	semidirect
14. C₂×H(3,𝔽₃)	C₂×C₄	semidirect
16. #272	C₂×C₄	semidirect
25. C₃⋊Dic₃	C₂×D₃	semidirect
27. H(3,𝔽₃)	C₂²×C₄	semidirect
38. C₂×C₃²	C₄×D₃	semidirect
56. C₃²	C₂×C₄×D₃	semidirect
77. C₂	#6199	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×#272	2
C₂×(C₃⋊Dic₃)⋊C₃	2
#6199	2
C₂×D₃×Dic₃	3

Subgroups — C2×#6199 #88674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6199 #88674