Subgroups — #88925

Order 432.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#622	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#622	C₂	1	1	yes	maximal, characteristic, complemented
4	#3792	—	3	1	no	maximal
5	C₂×C₃²×D₃	C₄	1	1	yes	characteristic
6	C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
7	C₃×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊Dic₃	—	3	1	no	—
9	D₃×Dic₃	—	6	2	no	—
10	#622	—	3	1	no	—
11	#622	D₃	1	1	yes	characteristic, complemented
12	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
13	C₃²×D₃	—	2	1	no	—
14	ΓL(1,𝔽₂₅)	—	9	1	no	maximal
15	C₂²×C₃²	—	3	1	no	—
16	C₂×C₃×D₃	—	2	2	no	—
17	C₃⋊Dic₃	—	3	1	no	—
18	C₃⋊Dic₃	—	6	2	no	—
19	C₃⋊Dic₃	—	12	2	no	—
20	C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
21	C₃×Dic₃	—	6	2	no	—
22	C₃³	M₁₆	1	1	yes	characteristic, complemented, Sylow
23	C₂×Dic₃	—	18	2	no	—
24	C₄×D₃	—	9	1	no	—
25	C₃×C₈	—	9	1	no	—
26	C₃⋊C₈	—	9	1	no	—
27	C₂×C₃²	—	2	2	no	—
28	C₂×C₃²	—	4	2	no	—
29	C₂×C₃²	—	6	1	no	—
30	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
31	C₃×D₃	—	2	4	no	—
32	M₁₆	—	27	1	no	Sylow
33	C₂²×C₃	—	6	2	no	—
34	C₂×D₃	#157	1	1	yes	characteristic
35	C₃×C₄	—	9	1	no	—
36	Dic₃	—	6	2	no	—
37	Dic₃	—	9	1	no	—
38	Dic₃	—	18	2	no	—
39	Dic₃	—	36	2	no	—
40	C₃²	—	2	2	no	—
41	C₃²	—	4	2	no	—
42	C₃²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
43	C₂×C₄	—	27	1	no	—
44	C₈	—	9	1	no	—
45	C₈	—	27	1	no	—
46	C₂×C₃	—	2	2	no	—
47	C₂×C₃	—	4	2	no	—
48	C₂×C₃	—	6	4	no	—
49	C₂×C₃	C₂×#157	1	1	yes	characteristic
50	D₃	—	2	1	no	—
51	C₂²	—	3	1	no	—
52	C₄	—	9	1	no	—
53	C₄	—	27	1	no	—
54	C₃	—	2	2	no	—
55	C₃	—	4	2	no	—
56	C₃	#3792	1	1	yes	characteristic, complemented
57	C₂	—	6	1	no	—
58	C₂	D₃×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×#157
commutator, Fitting, socle	C₂×C₃³	C₂×C₄
prime radical	C₂×C₃²×D₃	C₄
cosocle	C₃×C₃⋊Dic₃	C₂²
hypocenter	C₃³	M₁₆
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃⋊#622	C₂	semidirect
3. C₃×#622	C₂	semidirect
11. #622	D₃	semidirect
22. C₃³	M₁₆	semidirect
42. C₃²	ΓL(1,𝔽₂₅)	semidirect
56. C₃	#3792	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₃⋊Dic₃	2
C₃⋊#622	2
C₃×#622	2
#3792	3
ΓL(1,𝔽₂₅)	9