Subgroups — C₂²×#1084 #88751

Order 432.

297 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#1084	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×Dic₃	—	3	4	no	maximal
4	C₂²×H(3,𝔽₃)	C₄	1	4	yes	complemented
5	C₂²×H(3,𝔽₃)	C₂²	1	3	yes	—
6	#1084	C₂²	1	4	yes	direct product factor
7	C₂³×C₃²	D₃	1	4	yes	—
8	C₂×C₃×Dic₃	—	3	24	no	—
9	C₂×H(3,𝔽₃)	C₂×C₄	1	6	yes	complemented
10	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
11	C₂²×C₃×C₄	—	9	1	no	—
12	C₂²×Dic₃	—	3	4	no	—
13	C₂²×C₃²	Dic₃	1	16	yes	complemented
14	C₂²×C₃²	C₂×D₃	1	12	yes	—
15	C₃×Dic₃	—	3	16	no	—
16	H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂³×C₃	—	3	4	no	—
18	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
19	C₂×C₃×C₄	—	9	6	no	—
20	C₂×Dic₃	—	3	24	no	—
21	C₂×C₃²	C₂×Dic₃	1	24	yes	complemented
22	C₂×C₃²	C₂²×D₃	1	4	yes	—
23	C₂²×C₄	—	9	1	no	Sylow
24	C₂²×C₃	—	3	16	no	—
25	C₂²×C₃	—	3	12	no	—
26	C₂²×C₃	C₃⋊Dic₃	1	4	yes	—
27	C₂²×C₃	C₂×C₃⋊D₃	1	3	yes	—
28	C₃×C₄	—	9	4	no	—
29	Dic₃	—	3	16	no	—
30	C₃²	C₂²×Dic₃	1	4	yes	complemented
31	C₂³	#272	1	1	yes	characteristic
32	C₂×C₄	—	9	6	no	—
33	C₂×C₃	—	3	24	no	—
34	C₂×C₃	—	3	4	no	—
35	C₂×C₃	C₂×C₃⋊Dic₃	1	6	yes	—
36	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
37	C₂²	#1084	1	4	yes	direct product factor
38	C₂²	C₂×#272	1	3	yes	—
39	C₄	—	9	4	no	—
40	C₃	—	3	4	no	—
41	C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
42	C₂	C₂×#1084	1	6	yes	direct product factor
43	C₂	C₂²×#272	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#1084	C₂	direct (N × H)
4. C₂²×H(3,𝔽₃)	C₄	semidirect
6. #1084	C₂²	direct (N × H)
9. C₂×H(3,𝔽₃)	C₂×C₄	semidirect
13. C₂²×C₃²	Dic₃	semidirect
16. H(3,𝔽₃)	C₂²×C₄	semidirect
21. C₂×C₃²	C₂×Dic₃	semidirect
30. C₃²	C₂²×Dic₃	semidirect
37. C₂²	#1084	direct (N × H)
42. C₂	C₂×#1084	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×H(3,𝔽₃)	2
C₂×#1084	2
C₂²×C₃×Dic₃	3

Subgroups — C22×#1084 #88751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1084 #88751