Subgroups — C₂×#6180 #88537

Order 432.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#6180	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₃⋊C₈	—	3	4	no	maximal
4	C₂²×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄	D₃	1	4	yes	—
8	C₃×C₃⋊C₈	—	3	8	no	—
9	C₂×H(3,𝔽₃)	C₈	1	2	yes	complemented
10	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₈	—	9	1	no	—
12	C₂×C₃⋊C₈	—	3	4	no	—
13	C₂²×C₃²	Dic₃	1	4	yes	—
14	C₃²×C₄	Dic₃	1	4	yes	—
15	C₃²×C₄	C₂×D₃	1	4	yes	—
16	H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃×C₄	—	3	4	no	—
18	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
19	C₃×C₈	—	9	2	no	—
20	C₃⋊C₈	—	3	8	no	—
21	C₂×C₃²	C₃⋊C₈	1	8	yes	complemented
22	C₂×C₃²	C₂×Dic₃	1	4	yes	—
23	C₂×C₈	—	9	1	no	Sylow
24	C₂²×C₃	—	3	4	no	—
25	C₂²×C₃	C₃⋊Dic₃	1	1	yes	characteristic
26	C₃×C₄	—	3	4	no	—
27	C₃×C₄	—	3	4	no	—
28	C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
29	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
30	C₃²	C₂×C₃⋊C₈	1	4	yes	complemented
31	C₂×C₄	#272	1	1	yes	characteristic
32	C₈	—	9	2	no	—
33	C₂×C₃	—	3	8	no	—
34	C₂×C₃	—	3	4	no	—
35	C₂×C₃	C₃⋊(C₃⋊C₈)	1	2	yes	—
36	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
37	C₂²	#1084	1	1	yes	characteristic
38	C₄	#1084	1	1	yes	characteristic
39	C₄	C₂×#272	1	1	yes	characteristic
40	C₃	—	3	4	no	—
41	C₃	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
42	C₂	#6180	1	2	yes	direct product factor
43	C₂	C₂×#1084	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₂×C₈
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₃⋊Dic₃
prime radical	C₂²×H(3,𝔽₃)	C₄
cosocle	C₄×H(3,𝔽₃)	C₂²
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. #6180	C₂	direct (N × H)
9. C₂×H(3,𝔽₃)	C₈	semidirect
16. H(3,𝔽₃)	C₂×C₈	semidirect
21. C₂×C₃²	C₃⋊C₈	semidirect
30. C₃²	C₂×C₃⋊C₈	semidirect
42. C₂	#6180	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×C₄×H(3,𝔽₃)	2
#6180	2
C₂×C₃×C₃⋊C₈	3

Subgroups — C2×#6180 #88537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6180 #88537