Subgroups — C₂×C₄×#272 #88738

Order 432.

297 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×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1084	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#272	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₄×D₃	—	3	4	no	maximal
6	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×#272	C₄	1	4	yes	direct product factor
8	C₂×#272	C₂²	1	2	yes	—
9	C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
10	#1084	C₂²	1	2	yes	complemented
11	C₂²×C₃×D₃	—	3	4	no	—
12	C₂×C₃²×C₄	D₃	1	4	yes	complemented
13	C₂×C₃×Dic₃	—	3	4	no	—
14	C₃×C₄×D₃	—	3	16	no	—
15	C₂×H(3,𝔽₃)	C₂×C₄	1	2	yes	complemented
16	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
17	#272	C₂×C₄	1	4	yes	direct product factor
18	C₂²×C₃×C₄	—	9	1	no	—
19	C₂×C₄×D₃	—	3	4	no	—
20	C₂²×C₃²	C₂×D₃	1	4	yes	—
21	C₂×C₃×D₃	—	3	16	no	—
22	C₂×C₃×D₃	—	3	8	no	—
23	C₃²×C₄	C₂×D₃	1	8	yes	complemented
24	C₃×Dic₃	—	3	8	no	—
25	H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂³×C₃	—	9	1	no	—
27	C₂²×D₃	—	3	4	no	—
28	C₂×C₃×C₄	—	3	4	no	—
29	C₂×C₃×C₄	—	9	1	no	—
30	C₂×C₃×C₄	—	9	4	no	—
31	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
32	C₂×Dic₃	—	3	4	no	—
33	C₄×D₃	—	3	16	no	—
34	C₂×C₃²	C₄×D₃	1	8	yes	complemented
35	C₂×C₃²	C₂²×D₃	1	4	yes	—
36	C₃×D₃	—	3	16	no	—
37	C₂²×C₄	—	9	1	no	Sylow
38	C₂²×C₃	—	3	4	no	—
39	C₂²×C₃	—	9	4	no	—
40	C₂²×C₃	—	9	2	no	—
41	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
42	C₂×D₃	—	3	8	no	—
43	C₂×D₃	—	3	16	no	—
44	C₃×C₄	—	3	8	no	—
45	C₃×C₄	—	9	2	no	—
46	C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
47	Dic₃	—	3	8	no	—
48	C₃²	C₂×C₄×D₃	1	4	yes	complemented
49	C₂³	—	9	1	no	—
50	C₂×C₄	—	9	1	no	—
51	C₂×C₄	—	9	4	no	—
52	C₂×C₄	#272	1	1	yes	characteristic, direct product factor
53	C₂×C₃	—	3	8	no	—
54	C₂×C₃	—	3	4	no	—
55	C₂×C₃	—	9	4	no	—
56	C₂×C₃	C₄×C₃⋊D₃	1	2	yes	—
57	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
58	D₃	—	3	16	no	—
59	C₂²	—	9	4	no	—
60	C₂²	—	9	2	no	—
61	C₂²	C₂×#272	1	1	yes	characteristic
62	C₄	—	9	2	no	—
63	C₄	C₂×#272	1	2	yes	direct product factor
64	C₃	—	3	4	no	—
65	C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
66	C₂	—	9	4	no	—
67	C₂	C₄×#272	1	2	yes	direct product factor
68	C₂	C₂²×#272	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₂²×C₄
Frattini	C₂×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₂×C₃⋊D₃
prime radical	C₂²×#272	C₂
cosocle	C₂×H(3,𝔽₃)	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×H(3,𝔽₃)	C₂	semidirect
3. C₂×#1084	C₂	semidirect
4. C₄×#272	C₂	direct (N × H)
7. C₂×#272	C₄	direct (N × H)
9. C₄×H(3,𝔽₃)	C₂²	semidirect
10. #1084	C₂²	semidirect
12. C₂×C₃²×C₄	D₃	semidirect
15. C₂×H(3,𝔽₃)	C₂×C₄	semidirect
17. #272	C₂×C₄	direct (N × H)
23. C₃²×C₄	C₂×D₃	semidirect
25. H(3,𝔽₃)	C₂²×C₄	semidirect
34. C₂×C₃²	C₄×D₃	semidirect
48. C₃²	C₂×C₄×D₃	semidirect
52. C₂×C₄	#272	direct (N × H)
63. C₄	C₂×#272	direct (N × H)
67. 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
C₂×#1084	2
C₄×#272	2
C₂×C₃×C₄×D₃	3

Subgroups — C2×C4×#272 #88738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#272 #88738