Subgroups — C₂×C₄⋊#272 #88739

Order 432.

297 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
hypercenter	C₂×C₃×C₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#272	C₂	semidirect
2. C₂×C₄×H(3,𝔽₃)	C₂	semidirect
3. C₄⋊#272	C₂	direct (N × H)
7. C₂×#272	C₂²	semidirect
8. C₄×H(3,𝔽₃)	C₂²	semidirect
10. C₂×C₃²×C₄	D₃	semidirect
12. C₂×H(3,𝔽₃)	D₄	semidirect
20. C₃²×C₄	C₂×D₃	semidirect
21. H(3,𝔽₃)	C₂×D₄	semidirect
28. C₂×C₃²	D₁₂	semidirect
40. C₃²	C₂×D₁₂	semidirect
43. C₂×C₄	#272	semidirect
53. C₄	C₂×#272	semidirect
57. C₂	C₄⋊#272	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×C₄×H(3,𝔽₃)	2
C₄⋊#272	2
C₂×C₃×D₁₂	3

Subgroups — C2×C4⋊#272 #88739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#272 #88739