Subgroups — #88552

Order 432.

187 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	2	yes	maximal, complemented
3	C₃×#217	—	3	4	no	maximal
4	C₂²×H(3,𝔽₃)	—	2	2	no	—
5	C₂²×H(3,𝔽₃)	C₄	1	2	yes	complemented
6	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	#1084	—	2	2	no	—
8	C₂³×C₃²	D₃	1	4	yes	—
9	C₂×C₃×Dic₃	—	3	8	no	—
10	C₂×H(3,𝔽₃)	—	2	2	no	—
11	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₂×H(3,𝔽₃)	D₄	1	2	yes	—
13	C₃×C₂²⋊C₄	—	9	1	no	—
14	#217	—	3	4	no	—
15	C₂²×C₃²	—	2	8	no	—
16	C₂²×C₃²	Dic₃	1	8	yes	complemented
17	C₂²×C₃²	C₂×D₃	1	4	yes	—
18	C₃×Dic₃	—	6	8	no	—
19	H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
20	C₂³×C₃	—	3	4	no	—
21	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
22	C₂×C₃×C₄	—	9	2	no	—
23	C₂×Dic₃	—	3	8	no	—
24	C₂×C₃²	—	2	8	no	—
25	C₂×C₃²	C₂×Dic₃	1	4	yes	—
26	C₂×C₃²	#67	1	8	yes	—
27	C₂²⋊C₄	—	9	1	no	Sylow
28	C₂²×C₃	—	2	2	no	—
29	C₂²×C₃	—	3	8	no	—
30	C₂²×C₃	—	3	4	no	—
31	C₂²×C₃	—	6	8	no	—
32	C₂²×C₃	C₃⋊Dic₃	1	2	yes	—
33	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
34	C₃×C₄	—	18	2	no	—
35	Dic₃	—	6	8	no	—
36	C₃²	#217	1	4	yes	complemented
37	C₂³	#272	1	1	yes	characteristic
38	C₂×C₄	—	9	2	no	—
39	C₂×C₃	—	2	2	no	—
40	C₂×C₃	—	3	4	no	—
41	C₂×C₃	—	3	8	no	—
42	C₂×C₃	—	6	8	no	—
43	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
44	C₂×C₃	#638	1	2	yes	—
45	C₂²	—	2	2	no	—
46	C₂²	#1084	1	2	yes	complemented
47	C₂²	C₂×#272	1	1	yes	characteristic
48	C₄	—	18	2	no	—
49	C₃	—	3	4	no	—
50	C₃	#3757	1	1	yes	characteristic
51	C₂	—	2	2	no	—
52	C₂	C₂×#1084	1	1	yes	characteristic
53	C₂	#6235	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#1084	C₂	semidirect
5. C₂²×H(3,𝔽₃)	C₄	semidirect
16. C₂²×C₃²	Dic₃	semidirect
19. H(3,𝔽₃)	C₂²⋊C₄	semidirect
36. C₃²	#217	semidirect
46. C₂²	#1084	semidirect

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
C₃×#217	3