Subgroups — C₂×C₃×GO⁺(4,𝔽₂) #89107

Order 432.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃²	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×GO⁺(4,𝔽₂)	C₂	1	4	yes	maximal, direct product factor
4	C₂×GO⁺(4,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₂×C₃²×D₃	—	2	2	no	—
7	C₃×D₃²	—	2	2	no	—
8	C₃×D₃²	C₂²	1	4	yes	complemented
9	C₃×#157	C₂²	1	2	yes	complemented
10	C₂²×C₃×D₃	—	6	2	no	—
11	C₂×D₃²	C₂×C₃	1	2	yes	complemented
12	C₂×#157	C₂×C₃	1	1	yes	characteristic, complemented
13	GO⁺(4,𝔽₂)	C₂×C₃	1	4	yes	direct product factor
14	C₂×C₃³	D₄	1	1	yes	characteristic, complemented
15	C₃×C₃⋊D₃	D₄	1	1	yes	characteristic, complemented
16	C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
17	C₃²×D₃	—	2	4	no	—
18	C₂×C₃×D₄	—	9	1	no	maximal
19	C₂²×C₃²	—	6	2	no	—
20	C₂×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
21	C₂×C₃×D₃	—	2	2	no	—
22	C₂×C₃×D₃	—	2	2	no	—
23	C₂×C₃×D₃	—	4	2	no	—
24	C₂×C₃×D₃	—	6	4	no	—
25	C₂×C₃×D₃	—	6	2	no	—
26	C₂×C₃×D₃	—	6	4	no	—
27	D₃²	—	2	2	no	—
28	D₃²	C₂²×C₃	1	4	yes	complemented
29	#157	C₂²×C₃	1	2	yes	complemented
30	C₃³	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
31	C₂³×C₃	—	9	2	no	—
32	C₂²×D₃	—	6	2	no	—
33	C₃×D₄	—	9	4	no	—
34	C₂×C₃×C₄	—	9	1	no	—
35	C₂×C₃²	—	2	2	no	—
36	C₂×C₃²	—	4	2	no	—
37	C₂×C₃²	—	6	4	no	—
38	C₂×C₃²	C₃×D₄	1	1	yes	characteristic, complemented
39	C₃⋊D₃	C₃×D₄	1	1	yes	characteristic, complemented
40	C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
41	C₃×D₃	—	2	4	no	—
42	C₃×D₃	—	2	4	no	—
43	C₃×D₃	—	4	4	no	—
44	C₃×D₃	—	6	2	no	—
45	C₃×D₃	—	6	2	no	—
46	C₂×D₄	—	9	1	no	Sylow
47	C₂²×C₃	—	6	2	no	—
48	C₂²×C₃	—	6	2	no	—
49	C₂²×C₃	—	9	4	no	—
50	C₂²×C₃	—	9	1	no	—
51	C₂²×C₃	—	12	2	no	—
52	C₂²×C₃	—	18	2	no	—
53	C₂×D₃	—	2	2	no	—
54	C₂×D₃	—	6	4	no	—
55	C₂×D₃	—	6	2	no	—
56	C₂×D₃	—	6	4	no	—
57	C₃×C₄	—	9	2	no	—
58	C₃²	—	2	2	no	—
59	C₃²	—	4	2	no	—
60	C₃²	C₂×C₃×D₄	1	1	yes	characteristic, complemented
61	C₂³	—	9	2	no	—
62	D₄	—	9	4	no	—
63	C₂×C₄	—	9	1	no	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	—	4	2	no	—
66	C₂×C₃	—	6	4	no	—
67	C₂×C₃	—	6	4	no	—
68	C₂×C₃	—	9	1	no	—
69	C₂×C₃	—	9	1	no	—
70	C₂×C₃	—	12	4	no	—
71	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic, direct product factor
72	D₃	—	2	4	no	—
73	D₃	—	6	2	no	—
74	D₃	—	6	2	no	—
75	C₂²	—	6	2	no	—
76	C₂²	—	9	4	no	—
77	C₂²	—	9	1	no	—
78	C₂²	—	18	2	no	—
79	C₄	—	9	2	no	—
80	C₃	—	2	2	no	—
81	C₃	—	4	2	no	—
82	C₃	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic, direct product factor
83	C₂	—	6	4	no	—
84	C₂	—	9	1	no	—
85	C₂	—	9	1	no	—
86	C₂	C₃×GO⁺(4,𝔽₂)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊D₃	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃³	D₄
hypocenter	C₃²	C₂×C₃×D₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₃²	C₂	semidirect
2. C₂×C₃×#157	C₂	semidirect
3. C₃×GO⁺(4,𝔽₂)	C₂	direct (N × H)
4. C₂×GO⁺(4,𝔽₂)	C₃	direct (N × H)
8. C₃×D₃²	C₂²	semidirect
9. C₃×#157	C₂²	semidirect
11. C₂×D₃²	C₂×C₃	semidirect
12. C₂×#157	C₂×C₃	semidirect
13. GO⁺(4,𝔽₂)	C₂×C₃	direct (N × H)
14. C₂×C₃³	D₄	semidirect
15. C₃×C₃⋊D₃	D₄	semidirect
28. D₃²	C₂²×C₃	semidirect
29. #157	C₂²×C₃	semidirect
30. C₃³	C₂×D₄	semidirect
38. C₂×C₃²	C₃×D₄	semidirect
39. C₃⋊D₃	C₃×D₄	semidirect
60. C₃²	C₂×C₃×D₄	semidirect
71. C₂×C₃	GO⁺(4,𝔽₂)	direct (N × H)
82. C₃	C₂×GO⁺(4,𝔽₂)	direct (N × H)
86. C₂	C₃×GO⁺(4,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃²	2
C₂×C₃×#157	2
C₃×GO⁺(4,𝔽₂)	2
C₂×GO⁺(4,𝔽₂)	3
C₂×C₃×D₄	9

Subgroups — C2×C3×GO+(4,𝔽2) #89107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×GO⁺(4,𝔽₂) #89107