Subgroups — C₃²×#6180 #14517869

Order 1944.

1040 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₃⋊C₈	—	3	4	no	maximal
3	C₃×#6180	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₃⁴×C₄	D₃	1	4	yes	—
6	C₃×C₄×H(3,𝔽₃)	—	2	16	no	—
7	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	4	yes	—
8	C₃²×H(3,𝔽₃)	C₈	1	1	yes	characteristic, complemented, Sylow
9	C₃³×C₈	—	9	1	no	—
10	C₃²×C₃⋊C₈	—	3	36	no	—
11	C₃²×C₃⋊C₈	—	3	16	no	—
12	#6180	C₃²	1	1	yes	characteristic, direct product factor
13	C₂×C₃⁴	Dic₃	1	4	yes	—
14	C₂×C₃×H(3,𝔽₃)	—	2	16	no	—
15	C₂×C₃×H(3,𝔽₃)	C₃×C₄	1	4	yes	—
16	C₃³×C₄	—	2	16	no	—
17	C₃³×C₄	—	3	36	no	—
18	C₃³×C₄	C₃×D₃	1	16	yes	—
19	C₃³×C₄	C₃⋊D₃	1	1	yes	characteristic
20	C₄×H(3,𝔽₃)	—	2	24	no	—
21	C₄×H(3,𝔽₃)	—	2	16	no	—
22	C₄×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
23	C₃⁴	C₃⋊C₈	1	4	yes	complemented
24	C₃×H(3,𝔽₃)	—	2	16	no	—
25	C₃×H(3,𝔽₃)	C₃×C₈	1	4	yes	complemented
26	C₃²×C₈	—	9	9	no	—
27	C₃²×C₈	—	9	4	no	—
28	C₃×C₃⋊C₈	—	3	48	no	—
29	C₃×C₃⋊C₈	—	3	4	no	—
30	C₂×C₃³	—	2	16	no	—
31	C₂×C₃³	—	3	36	no	—
32	C₂×C₃³	C₃×Dic₃	1	16	yes	—
33	C₂×C₃³	C₃⋊Dic₃	1	1	yes	characteristic
34	C₂×H(3,𝔽₃)	—	2	24	no	—
35	C₂×H(3,𝔽₃)	—	2	16	no	—
36	C₂×H(3,𝔽₃)	C₃²×C₄	1	1	yes	characteristic
37	C₃²×C₄	—	2	16	no	—
38	C₃²×C₄	—	3	48	no	—
39	C₃²×C₄	—	6	48	no	—
40	C₃²×C₄	#272	1	9	yes	—
41	C₃²×C₄	C₃²×D₃	1	4	yes	—
42	C₃²×C₄	C₃×C₃⋊D₃	1	4	yes	—
43	C₃³	—	2	16	no	—
44	C₃³	—	3	36	no	—
45	C₃³	C₃×C₃⋊C₈	1	16	yes	complemented
46	C₃³	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
47	H(3,𝔽₃)	—	2	24	no	—
48	H(3,𝔽₃)	—	2	16	no	—
49	H(3,𝔽₃)	C₃²×C₈	1	1	yes	characteristic, complemented
50	C₃×C₈	—	9	12	no	—
51	C₃×C₈	—	9	1	no	—
52	C₃⋊C₈	—	3	4	no	—
53	C₂×C₃²	—	2	16	no	—
54	C₂×C₃²	—	3	48	no	—
55	C₂×C₃²	—	6	48	no	—
56	C₂×C₃²	#1084	1	9	yes	—
57	C₂×C₃²	C₃²×Dic₃	1	4	yes	—
58	C₂×C₃²	C₃×C₃⋊Dic₃	1	4	yes	—
59	C₃×C₄	—	3	4	no	—
60	C₃×C₄	—	6	16	no	—
61	C₃×C₄	C₃×#272	1	12	yes	—
62	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
63	C₃²	—	2	16	no	—
64	C₃²	—	3	48	no	—
65	C₃²	—	6	48	no	—
66	C₃²	#6180	1	9	yes	direct product factor
67	C₃²	C₃²×C₃⋊C₈	1	4	yes	complemented
68	C₃²	C₃×C₃⋊(C₃⋊C₈)	1	4	yes	—
69	C₈	—	9	1	no	Sylow
70	C₂×C₃	—	3	4	no	—
71	C₂×C₃	—	6	16	no	—
72	C₂×C₃	C₃×#1084	1	12	yes	—
73	C₂×C₃	C₃²×C₃⋊Dic₃	1	1	yes	characteristic
74	C₄	C₃²×#272	1	1	yes	characteristic
75	C₃	—	3	4	no	—
76	C₃	—	6	16	no	—
77	C₃	C₃×#6180	1	12	yes	direct product factor
78	C₃	C₃²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
79	C₂	C₃²×#1084	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃³×C₄	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₃²×C₈
Frattini	C₃×C₄	C₃²×C₃⋊D₃
Fitting	C₃²×C₄×H(3,𝔽₃)	C₂
socle	C₂×C₃³	C₃⋊Dic₃
prime radical	C₂×C₃²×H(3,𝔽₃)	C₄
cosocle	C₄×H(3,𝔽₃)	C₂×C₃²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃³×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×#6180	C₃	direct (N × H)
8. C₃²×H(3,𝔽₃)	C₈	semidirect
12. #6180	C₃²	direct (N × H)
23. C₃⁴	C₃⋊C₈	semidirect
25. C₃×H(3,𝔽₃)	C₃×C₈	semidirect
45. C₃³	C₃×C₃⋊C₈	semidirect
49. H(3,𝔽₃)	C₃²×C₈	semidirect
66. C₃²	#6180	direct (N × H)
67. C₃²	C₃²×C₃⋊C₈	semidirect
77. C₃	C₃×#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
C₃²×C₄×H(3,𝔽₃)	2
C₃³×C₃⋊C₈	3
C₃×#6180	3

Subgroups — C32×#6180 #14517869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6180 #14517869