Subgroups — C₂²×GO⁺(4,𝔽₂) #64383

Order 288.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃²	C₂	1	2	yes	maximal, complemented
2	C₂²×#157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×GO⁺(4,𝔽₂)	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂²×C₃×D₃	—	2	2	no	—
6	C₂×D₃²	—	2	6	no	—
7	C₂×D₃²	C₂²	1	12	yes	complemented
8	C₂×#157	C₂²	1	6	yes	complemented
9	GO⁺(4,𝔽₂)	C₂²	1	16	yes	direct product factor
10	C₂³×D₃	—	6	2	no	—
11	C₂²×C₃²	D₄	1	1	yes	characteristic, complemented
12	C₂×C₃⋊D₃	D₄	1	3	yes	complemented
13	C₂×C₃⋊D₃	C₂³	1	3	yes	—
14	C₂×C₃×D₃	—	2	12	no	—
15	D₃²	—	2	12	no	—
16	D₃²	C₂³	1	8	yes	complemented
17	#157	C₂³	1	4	yes	complemented
18	C₂²×D₄	—	9	1	no	maximal, Sylow
19	C₂³×C₃	—	6	2	no	—
20	C₂²×D₃	—	2	2	no	—
21	C₂²×D₃	—	6	12	no	—
22	C₂²×D₃	—	6	2	no	—
23	C₂²×D₃	—	6	12	no	—
24	C₂×C₃²	C₂×D₄	1	3	yes	complemented
25	C₃⋊D₃	C₂×D₄	1	3	yes	complemented
26	C₃⋊D₃	C₂⁴	1	1	yes	characteristic
27	C₃×D₃	—	2	8	no	—
28	C₂⁴	—	9	2	no	—
29	C₂×D₄	—	9	12	no	—
30	C₂²×C₄	—	9	1	no	—
31	C₂²×C₃	—	2	2	no	—
32	C₂²×C₃	—	6	12	no	—
33	C₂×D₃	—	2	12	no	—
34	C₂×D₃	—	6	8	no	—
35	C₂×D₃	—	6	6	no	—
36	C₂×D₃	—	6	6	no	—
37	C₂×D₃	—	6	24	no	—
38	C₃²	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
39	C₂³	—	6	2	no	—
40	C₂³	—	9	12	no	—
41	C₂³	—	9	1	no	—
42	C₂³	—	18	6	no	—
43	D₄	—	9	16	no	—
44	C₂×C₄	—	9	6	no	—
45	C₂×C₃	—	2	6	no	—
46	C₂×C₃	—	6	8	no	—
47	D₃	—	2	8	no	—
48	D₃	—	6	6	no	—
49	D₃	—	6	2	no	—
50	C₂²	—	6	12	no	—
51	C₂²	—	9	8	no	—
52	C₂²	—	9	3	no	—
53	C₂²	—	9	3	no	—
54	C₂²	—	18	12	no	—
55	C₂²	GO⁺(4,𝔽₂)	1	1	yes	characteristic, direct product factor
56	C₄	—	9	4	no	—
57	C₃	—	2	2	no	—
58	C₂	—	6	8	no	—
59	C₂	—	9	1	no	—
60	C₂	—	9	3	no	—
61	C₂	C₂×GO⁺(4,𝔽₂)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊D₃	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	D₄
hypocenter	C₃²	C₂²×D₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₃²	C₂	semidirect
2. C₂²×#157	C₂	semidirect
3. C₂×GO⁺(4,𝔽₂)	C₂	direct (N × H)
7. C₂×D₃²	C₂²	semidirect
8. C₂×#157	C₂²	semidirect
9. GO⁺(4,𝔽₂)	C₂²	direct (N × H)
11. C₂²×C₃²	D₄	semidirect
12. C₂×C₃⋊D₃	D₄	semidirect
16. D₃²	C₂³	semidirect
17. #157	C₂³	semidirect
24. C₂×C₃²	C₂×D₄	semidirect
25. C₃⋊D₃	C₂×D₄	semidirect
38. C₃²	C₂²×D₄	semidirect
55. C₂²	GO⁺(4,𝔽₂)	direct (N × H)
61. C₂	C₂×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
C₂²×D₃²	2
C₂²×#157	2
C₂×GO⁺(4,𝔽₂)	2
C₂²×D₄	9

Subgroups — C22×GO+(4,𝔽2) #64383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×GO⁺(4,𝔽₂) #64383