Subgroups — #88540

Order 432.

143 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	C₂×#1084	C₂	1	1	yes	maximal, characteristic
3	C₂×#1084	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃⋊C₄	—	3	4	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄×H(3,𝔽₃)	—	2	1	no	—
7	#1084	—	2	1	no	—
8	#1084	C₄	1	2	yes	complemented
9	C₂×C₃²×C₄	D₃	1	4	yes	—
10	C₂×C₃×Dic₃	—	3	4	no	—
11	C₂×C₃×Dic₃	—	3	4	no	—
12	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
14	C₂×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
15	C₃×C₄⋊C₄	—	9	1	no	—
16	Dic₃⋊C₄	—	3	4	no	—
17	C₂²×C₃²	C₂×D₃	1	4	yes	—
18	C₃²×C₄	—	2	4	no	—
19	C₃×Dic₃	—	3	8	no	—
20	C₃×Dic₃	—	6	4	no	—
21	H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₄	—	3	4	no	—
23	C₂×C₃×C₄	—	9	1	no	—
24	C₂×C₃×C₄	—	9	1	no	—
25	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
26	C₂×Dic₃	—	3	4	no	—
27	C₂×Dic₃	—	3	4	no	—
28	C₂×C₃²	Dic₆	1	4	yes	—
29	C₂×C₃²	C₄×D₃	1	4	yes	—
30	C₂×C₃²	#67	1	4	yes	—
31	C₄⋊C₄	—	9	1	no	Sylow
32	C₂²×C₃	—	3	4	no	—
33	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
34	C₃×C₄	—	2	1	no	—
35	C₃×C₄	—	6	4	no	—
36	C₃×C₄	—	9	2	no	—
37	C₃×C₄	—	18	1	no	—
38	Dic₃	—	3	8	no	—
39	Dic₃	—	6	4	no	—
40	C₃²	Dic₃⋊C₄	1	4	yes	complemented
41	C₂×C₄	—	9	1	no	—
42	C₂×C₄	—	9	1	no	—
43	C₂×C₄	#272	1	1	yes	characteristic
44	C₂×C₃	—	3	4	no	—
45	C₂×C₃	—	3	4	no	—
46	C₂×C₃	—	3	4	no	—
47	C₂×C₃	#634	1	1	yes	characteristic
48	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
49	C₂×C₃	#638	1	1	yes	characteristic
50	C₂²	C₂×#272	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	9	2	no	—
53	C₄	—	18	1	no	—
54	C₃	—	3	4	no	—
55	C₃	#3750	1	1	yes	characteristic
56	C₂	#6229	1	1	yes	characteristic
57	C₂	C₄×#272	1	1	yes	characteristic
58	C₂	#6235	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂×C₄
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
prime radical, cosocle	C₂²×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₄⋊C₄
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
8. #1084	C₄	semidirect
21. H(3,𝔽₃)	C₄⋊C₄	semidirect
40. C₃²	Dic₃⋊C₄	semidirect

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
C₂×#1084	2
C₃×Dic₃⋊C₄	3