Subgroups — C₂³×C₃⋊#272 #13049358

Order 1296.

3162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃⋊#272	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₃⋊D₃	—	3	4	no	maximal
4	C₂³×#272	—	3	9	no	maximal
5	C₂²×C₃×H(3,𝔽₃)	C₂²	1	7	yes	complemented
6	C₂×C₃⋊#272	C₂²	1	28	yes	direct product factor
7	C₂³×C₃³	D₃	1	4	yes	complemented
8	C₂²×C₃×C₃⋊D₃	—	3	56	no	—
9	C₂³×H(3,𝔽₃)	D₃	1	9	yes	complemented
10	C₂²×#272	—	3	126	no	—
11	C₂×C₃×H(3,𝔽₃)	C₂³	1	7	yes	complemented
12	C₃⋊#272	C₂³	1	8	yes	direct product factor
13	C₂³×C₃⋊D₃	—	3	4	no	—
14	C₂³×C₃×D₃	—	9	1	no	—
15	C₂³×C₃×D₃	—	9	12	no	—
16	C₂²×C₃³	C₂×D₃	1	28	yes	complemented
17	C₂×C₃×C₃⋊D₃	—	3	112	no	—
18	C₂²×H(3,𝔽₃)	C₂×D₃	1	63	yes	complemented
19	C₂×#272	—	3	252	no	—
20	C₃×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₃²	—	3	4	no	—
22	C₂³×C₃²	—	6	4	no	—
23	C₂³×C₃²	C₃⋊D₃	1	1	yes	characteristic
24	C₂³×C₃²	C₃⋊D₃	1	12	yes	complemented
25	C₂²×C₃⋊D₃	—	3	56	no	—
26	C₂²×C₃×D₃	—	9	14	no	—
27	C₂²×C₃×D₃	—	9	168	no	—
28	C₂×C₃³	C₂²×D₃	1	28	yes	complemented
29	C₃×C₃⋊D₃	—	3	32	no	—
30	C₂×H(3,𝔽₃)	C₂²×D₃	1	63	yes	complemented
31	#272	—	3	72	no	—
32	C₂⁴×C₃	—	27	1	no	—
33	C₂³×D₃	—	9	12	no	—
34	C₂³×D₃	—	9	1	no	—
35	C₂²×C₃²	—	3	28	no	—
36	C₂²×C₃²	—	6	28	no	—
37	C₂²×C₃²	C₂×C₃⋊D₃	1	7	yes	—
38	C₂²×C₃²	C₂×C₃⋊D₃	1	84	yes	complemented
39	C₂×C₃⋊D₃	—	3	112	no	—
40	C₂×C₃×D₃	—	9	28	no	—
41	C₂×C₃×D₃	—	9	336	no	—
42	C₃³	C₂³×D₃	1	4	yes	complemented
43	H(3,𝔽₃)	C₂³×D₃	1	9	yes	complemented
44	C₂³×C₃	—	2	1	no	—
45	C₂³×C₃	—	3	12	no	—
46	C₂³×C₃	—	27	14	no	—
47	C₂³×C₃	#272	1	1	yes	characteristic, complemented
48	C₂³×C₃	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
49	C₂²×D₃	—	9	168	no	—
50	C₂²×D₃	—	9	14	no	—
51	C₂×C₃²	—	3	28	no	—
52	C₂×C₃²	—	6	28	no	—
53	C₂×C₃²	C₂²×C₃⋊D₃	1	7	yes	—
54	C₂×C₃²	C₂²×C₃⋊D₃	1	84	yes	complemented
55	C₃⋊D₃	—	3	32	no	—
56	C₃×D₃	—	9	8	no	—
57	C₃×D₃	—	9	96	no	—
58	C₂⁴	—	27	1	no	Sylow
59	C₂²×C₃	—	2	7	no	—
60	C₂²×C₃	—	3	84	no	—
61	C₂²×C₃	—	27	28	no	—
62	C₂²×C₃	C₂×#272	1	7	yes	complemented
63	C₂²×C₃	C₂×C₃⋊(C₃⋊D₃)	1	7	yes	—
64	C₂×D₃	—	9	336	no	—
65	C₂×D₃	—	9	28	no	—
66	C₃²	—	3	4	no	—
67	C₃²	—	6	4	no	—
68	C₃²	C₂³×C₃⋊D₃	1	1	yes	characteristic
69	C₃²	C₂³×C₃⋊D₃	1	12	yes	complemented
70	C₂³	—	27	14	no	—
71	C₂³	C₃⋊#272	1	1	yes	characteristic, direct product factor
72	C₂×C₃	—	2	7	no	—
73	C₂×C₃	—	3	84	no	—
74	C₂×C₃	—	27	8	no	—
75	C₂×C₃	C₂²×#272	1	7	yes	complemented
76	C₂×C₃	C₂²×C₃⋊(C₃⋊D₃)	1	7	yes	—
77	D₃	—	9	96	no	—
78	D₃	—	9	8	no	—
79	C₂²	—	27	28	no	—
80	C₂²	C₂×C₃⋊#272	1	7	yes	direct product factor
81	C₃	—	2	1	no	—
82	C₃	—	3	12	no	—
83	C₃	C₂³×#272	1	1	yes	characteristic, complemented
84	C₃	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
85	C₂	—	27	8	no	—
86	C₂	C₂²×C₃⋊#272	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₃	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂⁴
Frattini	C₃	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₂³×C₃×H(3,𝔽₃)	C₂
socle	C₂³×C₃²	C₃⋊D₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×H(3,𝔽₃)	C₂	semidirect
2. C₂²×C₃⋊#272	C₂	direct (N × H)
5. C₂²×C₃×H(3,𝔽₃)	C₂²	semidirect
6. C₂×C₃⋊#272	C₂²	direct (N × H)
7. C₂³×C₃³	D₃	semidirect
9. C₂³×H(3,𝔽₃)	D₃	semidirect
11. C₂×C₃×H(3,𝔽₃)	C₂³	semidirect
12. C₃⋊#272	C₂³	direct (N × H)
16. C₂²×C₃³	C₂×D₃	semidirect
18. C₂²×H(3,𝔽₃)	C₂×D₃	semidirect
20. C₃×H(3,𝔽₃)	C₂⁴	semidirect
24. C₂³×C₃²	C₃⋊D₃	semidirect
28. C₂×C₃³	C₂²×D₃	semidirect
30. C₂×H(3,𝔽₃)	C₂²×D₃	semidirect
38. C₂²×C₃²	C₂×C₃⋊D₃	semidirect
42. C₃³	C₂³×D₃	semidirect
43. H(3,𝔽₃)	C₂³×D₃	semidirect
47. C₂³×C₃	#272	semidirect
54. C₂×C₃²	C₂²×C₃⋊D₃	semidirect
62. C₂²×C₃	C₂×#272	semidirect
69. C₃²	C₂³×C₃⋊D₃	semidirect
71. C₂³	C₃⋊#272	direct (N × H)
75. C₂×C₃	C₂²×#272	semidirect
80. C₂²	C₂×C₃⋊#272	direct (N × H)
83. C₃	C₂³×#272	semidirect
86. C₂	C₂²×C₃⋊#272	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×H(3,𝔽₃)	2
C₂²×C₃⋊#272	2
C₂³×C₃×C₃⋊D₃	3
C₂³×#272	3

Subgroups — C23×C3⋊#272 #13049358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃⋊#272 #13049358