Subgroups — C₂³×C₃×#272 #13049353

Order 1296.

2336 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×C₃²×D₃	—	3	4	no	maximal
4	C₂³×#272	C₃	1	1	yes	maximal, characteristic, direct product factor
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×C₃²×D₃	—	3	56	no	—
9	C₂³×H(3,𝔽₃)	—	2	4	no	—
10	C₂³×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂²×#272	C₂×C₃	1	14	yes	direct product factor
12	C₂×C₃×H(3,𝔽₃)	C₂³	1	7	yes	complemented
13	C₃×#272	C₂³	1	8	yes	direct product factor
14	C₂⁴×C₃²	—	9	1	no	—
15	C₂³×C₃×D₃	—	3	12	no	—
16	C₂³×C₃×D₃	—	3	4	no	—
17	C₂²×C₃³	C₂×D₃	1	28	yes	complemented
18	C₂×C₃²×D₃	—	3	112	no	—
19	C₂²×H(3,𝔽₃)	—	2	28	no	—
20	C₂²×H(3,𝔽₃)	C₂²×C₃	1	7	yes	complemented
21	C₂×#272	C₂²×C₃	1	28	yes	direct product factor
22	C₃×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented, Sylow
23	C₂³×C₃²	—	2	4	no	—
24	C₂³×C₃²	—	3	12	no	—
25	C₂³×C₃²	—	9	14	no	—
26	C₂³×C₃²	C₃×D₃	1	4	yes	complemented
27	C₂³×C₃²	C₃⋊D₃	1	1	yes	characteristic
28	C₂²×C₃×D₃	—	3	168	no	—
29	C₂²×C₃×D₃	—	3	56	no	—
30	C₂×C₃³	C₂²×D₃	1	28	yes	complemented
31	C₃²×D₃	—	3	32	no	—
32	C₂×H(3,𝔽₃)	—	2	28	no	—
33	C₂×H(3,𝔽₃)	C₂³×C₃	1	7	yes	complemented
34	#272	C₂³×C₃	1	8	yes	direct product factor
35	C₂⁴×C₃	—	9	3	no	—
36	C₂⁴×C₃	—	9	1	no	—
37	C₂³×D₃	—	3	4	no	—
38	C₂²×C₃²	—	2	28	no	—
39	C₂²×C₃²	—	3	84	no	—
40	C₂²×C₃²	—	9	28	no	—
41	C₂²×C₃²	C₂×C₃×D₃	1	28	yes	complemented
42	C₂²×C₃²	C₂×C₃⋊D₃	1	7	yes	—
43	C₂×C₃×D₃	—	3	336	no	—
44	C₂×C₃×D₃	—	3	112	no	—
45	C₃³	C₂³×D₃	1	4	yes	complemented
46	H(3,𝔽₃)	—	2	4	no	—
47	H(3,𝔽₃)	C₂⁴×C₃	1	1	yes	characteristic, complemented
48	C₂³×C₃	—	3	4	no	—
49	C₂³×C₃	—	6	4	no	—
50	C₂³×C₃	—	9	42	no	—
51	C₂³×C₃	—	9	14	no	—
52	C₂³×C₃	#272	1	3	yes	direct product factor
53	C₂³×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
54	C₂²×D₃	—	3	56	no	—
55	C₂×C₃²	—	2	28	no	—
56	C₂×C₃²	—	3	84	no	—
57	C₂×C₃²	—	9	8	no	—
58	C₂×C₃²	C₂²×C₃×D₃	1	28	yes	complemented
59	C₂×C₃²	C₂²×C₃⋊D₃	1	7	yes	—
60	C₃×D₃	—	3	96	no	—
61	C₃×D₃	—	3	32	no	—
62	C₂⁴	—	9	1	no	Sylow
63	C₂²×C₃	—	3	28	no	—
64	C₂²×C₃	—	6	28	no	—
65	C₂²×C₃	—	9	84	no	—
66	C₂²×C₃	—	9	28	no	—
67	C₂²×C₃	C₂×#272	1	21	yes	direct product factor
68	C₂²×C₃	C₂×C₃×C₃⋊D₃	1	7	yes	—
69	C₂×D₃	—	3	112	no	—
70	C₃²	—	2	4	no	—
71	C₃²	—	3	12	no	—
72	C₃²	C₂³×C₃×D₃	1	4	yes	complemented
73	C₃²	C₂³×C₃⋊D₃	1	1	yes	characteristic
74	C₂³	—	9	14	no	—
75	C₂³	C₃×#272	1	1	yes	characteristic, direct product factor
76	C₂×C₃	—	3	28	no	—
77	C₂×C₃	—	6	28	no	—
78	C₂×C₃	—	9	8	no	—
79	C₂×C₃	—	9	24	no	—
80	C₂×C₃	C₂²×#272	1	21	yes	direct product factor
81	C₂×C₃	C₂²×C₃×C₃⋊D₃	1	7	yes	—
82	D₃	—	3	32	no	—
83	C₂²	—	9	28	no	—
84	C₂²	C₂×C₃×#272	1	7	yes	direct product factor
85	C₃	—	3	4	no	—
86	C₃	—	6	4	no	—
87	C₃	C₂³×#272	1	3	yes	direct product factor
88	C₃	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
89	C₂	—	9	8	no	—
90	C₂	C₂²×C₃×#272	1	7	yes	direct product factor

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

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)
4. C₂³×#272	C₃	direct (N × H)
5. C₂²×C₃×H(3,𝔽₃)	C₂²	semidirect
6. C₂×C₃×#272	C₂²	direct (N × H)
7. C₂³×C₃³	D₃	semidirect
10. C₂³×H(3,𝔽₃)	C₂×C₃	semidirect
11. C₂²×#272	C₂×C₃	direct (N × H)
12. C₂×C₃×H(3,𝔽₃)	C₂³	semidirect
13. C₃×#272	C₂³	direct (N × H)
17. C₂²×C₃³	C₂×D₃	semidirect
20. C₂²×H(3,𝔽₃)	C₂²×C₃	semidirect
21. C₂×#272	C₂²×C₃	direct (N × H)
22. C₃×H(3,𝔽₃)	C₂⁴	semidirect
26. C₂³×C₃²	C₃×D₃	semidirect
30. C₂×C₃³	C₂²×D₃	semidirect
33. C₂×H(3,𝔽₃)	C₂³×C₃	semidirect
34. #272	C₂³×C₃	direct (N × H)
41. C₂²×C₃²	C₂×C₃×D₃	semidirect
45. C₃³	C₂³×D₃	semidirect
47. H(3,𝔽₃)	C₂⁴×C₃	semidirect
52. C₂³×C₃	#272	direct (N × H)
58. C₂×C₃²	C₂²×C₃×D₃	semidirect
67. C₂²×C₃	C₂×#272	direct (N × H)
72. C₃²	C₂³×C₃×D₃	semidirect
75. C₂³	C₃×#272	direct (N × H)
80. C₂×C₃	C₂²×#272	direct (N × H)
84. C₂²	C₂×C₃×#272	direct (N × H)
87. C₃	C₂³×#272	direct (N × H)
90. 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×C₃²×D₃	3
C₂³×#272	3

Subgroups — C23×C3×#272 #13049353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#272 #13049353