Subgroups — #88447

Order 432.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1084	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal
3	Dic₃²	—	3	2	no	maximal
4	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	#1084	C₄	1	2	yes	complemented
6	(C₃⋊Dic₃)⋊C₃	C₄	1	4	yes	complemented
7	C₂×C₃⋊Dic₃	D₃	1	2	yes	—
8	C₂×C₃×Dic₃	—	3	2	no	—
9	C₂×C₃×Dic₃	—	3	2	no	—
10	C₂×C₃×Dic₃	—	6	1	no	—
11	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₂×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
13	C₄×Dic₃	—	9	1	no	—
14	C₄×Dic₃	—	9	2	no	—
15	C₂²×C₃²	—	2	1	no	—
16	C₂²×C₃²	C₂×D₃	1	2	yes	—
17	C₃⋊Dic₃	Dic₃	1	4	yes	complemented
18	C₃×Dic₃	—	3	4	no	—
19	C₃×Dic₃	—	3	4	no	—
20	C₃×Dic₃	—	6	2	no	—
21	H(3,𝔽₃)	C₄²	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₄	—	9	2	no	—
23	C₂×C₃×C₄	—	9	1	no	—
24	C₂×Dic₃	—	3	2	no	—
25	C₂×Dic₃	—	3	2	no	—
26	C₂×Dic₃	—	6	1	no	—
27	C₂×Dic₃	—	9	2	no	—
28	C₂×C₃²	—	2	2	no	—
29	C₂×C₃²	—	2	1	no	—
30	C₂×C₃²	C₄×D₃	1	2	yes	—
31	C₂×C₃²	C₄×D₃	1	2	yes	—
32	C₂×C₃²	C₂×Dic₃	1	2	yes	—
33	C₄²	—	27	1	no	Sylow
34	C₂²×C₃	—	3	2	no	—
35	C₂²×C₃	—	6	1	no	—
36	C₂²×C₃	D₃²	1	1	yes	characteristic
37	C₃×C₄	—	9	4	no	—
38	C₃×C₄	—	9	2	no	—
39	Dic₃	—	3	4	no	—
40	Dic₃	—	3	4	no	—
41	Dic₃	—	6	2	no	—
42	Dic₃	—	9	4	no	—
43	C₃²	—	2	1	no	—
44	C₃²	C₄×Dic₃	1	2	yes	complemented
45	C₂×C₄	—	9	1	no	—
46	C₂×C₄	—	9	2	no	—
47	C₂×C₃	—	3	2	no	—
48	C₂×C₃	—	3	2	no	—
49	C₂×C₃	—	3	2	no	—
50	C₂×C₃	—	6	2	no	—
51	C₂×C₃	—	6	1	no	—
52	C₂×C₃	D₃×Dic₃	1	2	yes	—
53	C₂×C₃	#624	1	1	yes	characteristic
54	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
55	C₄	—	9	4	no	—
56	C₄	—	9	2	no	—
57	C₃	—	3	2	no	—
58	C₃	—	6	1	no	—
59	C₃	Dic₃²	1	1	yes	characteristic
60	C₂	((C₃⋊D₃)⋊C₃)⋊C₄	1	2	yes	—
61	C₂	#6199	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, hypocenter	H(3,𝔽₃)	C₄²
Frattini, socle	C₂²×C₃	D₃²
Fitting, prime radical, cosocle	C₂²×H(3,𝔽₃)	C₂²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
5. #1084	C₄	semidirect
6. (C₃⋊Dic₃)⋊C₃	C₄	semidirect
17. C₃⋊Dic₃	Dic₃	semidirect
21. H(3,𝔽₃)	C₄²	semidirect
44. C₃²	C₄×Dic₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1084	2
C₂×(C₃⋊Dic₃)⋊C₃	2
Dic₃²	3