Subgroups — C₂×#1088⋊C₂ #88883

Order 432.

145 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	C₂	1	2	yes	maximal, complemented
2	C₂×#1088	C₂	1	1	yes	maximal, characteristic, complemented
3	#1088⋊C₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×#272	C₂²	1	1	yes	characteristic
5	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
6	((C₃⋊D₃)⋊C₃)⋊C₂	—	2	2	no	—
7	((C₃⋊D₃)⋊C₃)⋊C₂	C₂²	1	4	yes	complemented
8	#1088	C₂²	1	2	yes	complemented
9	C₂×D₃²	—	6	2	no	—
10	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic, complemented
11	#272	D₄	1	1	yes	characteristic, complemented
12	#272	C₂³	1	1	yes	characteristic
13	(C₃⋊D₃)⋊C₃	—	2	4	no	—
14	C₂×D₁₂	—	9	1	no	maximal
15	C₂×C₃⋊D₃	—	2	2	no	—
16	C₂×C₃×D₃	—	6	2	no	—
17	C₂×C₃×D₃	—	6	2	no	—
18	D₃²	—	6	4	no	—
19	D₃²	—	6	4	no	—
20	H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×D₃	—	9	2	no	—
22	C₂²×D₃	—	18	2	no	—
23	C₂×C₃×C₄	—	9	1	no	—
24	D₁₂	—	9	4	no	—
25	C₂×C₃²	—	2	2	no	—
26	C₃⋊D₃	—	2	4	no	—
27	C₃×D₃	—	6	2	no	—
28	C₃×D₃	—	6	2	no	—
29	C₃×D₃	—	6	4	no	—
30	C₂×D₄	—	27	1	no	Sylow
31	C₂²×C₃	—	9	1	no	—
32	C₂²×C₃	—	18	2	no	—
33	C₂×D₃	—	6	2	no	—
34	C₂×D₃	—	6	2	no	—
35	C₂×D₃	—	9	4	no	—
36	C₂×D₃	—	18	2	no	—
37	C₂×D₃	—	18	4	no	—
38	C₂×D₃	—	18	2	no	—
39	C₂×D₃	—	18	4	no	—
40	C₃×C₄	—	9	2	no	—
41	C₃²	—	2	2	no	—
42	C₂³	—	27	2	no	—
43	D₄	—	27	4	no	—
44	C₂×C₄	—	9	1	no	—
45	C₂×C₃	—	6	2	no	—
46	C₂×C₃	—	9	1	no	—
47	C₂×C₃	—	9	1	no	—
48	C₂×C₃	—	18	4	no	—
49	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
50	D₃	—	6	2	no	—
51	D₃	—	6	2	no	—
52	D₃	—	6	4	no	—
53	D₃	—	18	4	no	—
54	C₂²	—	9	1	no	—
55	C₂²	—	18	2	no	—
56	C₂²	—	27	4	no	—
57	C₂²	—	54	2	no	—
58	C₄	—	9	2	no	—
59	C₃	—	6	2	no	—
60	C₃	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
61	C₂	—	9	1	no	—
62	C₂	—	9	1	no	—
63	C₂	—	18	4	no	—
64	C₂	#1088⋊C₂	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#1088⋊C₂
commutator, cosocle	#272	C₂³
Frattini	C₃	C₂×GO⁺(4,𝔽₂)
Fitting	C₂×H(3,𝔽₃)	D₄
socle	C₂×C₃	GO⁺(4,𝔽₂)
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×((C₃⋊D₃)⋊C₃)⋊C₂	C₂	semidirect
2. C₂×#1088	C₂	semidirect
3. #1088⋊C₂	C₂	direct (N × H)
7. ((C₃⋊D₃)⋊C₃)⋊C₂	C₂²	semidirect
8. #1088	C₂²	semidirect
10. C₂×H(3,𝔽₃)	D₄	semidirect
11. #272	D₄	semidirect
20. H(3,𝔽₃)	C₂×D₄	semidirect
64. C₂	#1088⋊C₂	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⋊D₃)⋊C₃)⋊C₂	2
C₂×#1088	2
#1088⋊C₂	2
C₂×D₁₂	9

Subgroups — C2×#1088⋊C2 #88883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1088⋊C₂ #88883