Subgroups — C₂²×C₃×C₄×H(3,𝔽₃) #13048972

Order 1296.

1512 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
2	C₂×C₃×C₄×H(3,𝔽₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃³×C₄	C₃	1	4	yes	maximal, complemented
4	C₂²×C₄×H(3,𝔽₃)	C₃	1	9	yes	maximal, direct product factor
5	C₂²×C₃×H(3,𝔽₃)	C₄	1	4	yes	direct product factor
6	C₂²×C₃×H(3,𝔽₃)	C₂²	1	3	yes	—
7	C₃×C₄×H(3,𝔽₃)	C₂²	1	4	yes	direct product factor
8	C₂³×C₃³	C₂×C₃	1	4	yes	—
9	C₂×C₃³×C₄	C₂×C₃	1	24	yes	complemented
10	C₂³×H(3,𝔽₃)	C₂×C₃	1	9	yes	—
11	C₂×C₄×H(3,𝔽₃)	C₂×C₃	1	54	yes	direct product factor
12	C₂×C₃×H(3,𝔽₃)	C₂×C₄	1	6	yes	direct product factor
13	C₂×C₃×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
14	C₂²×C₃²×C₄	—	3	12	no	—
15	C₂²×C₃²×C₄	C₃²	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₃²	1	12	yes	complemented
17	C₂²×C₃³	C₃×C₄	1	16	yes	complemented
18	C₂²×C₃³	C₂²×C₃	1	12	yes	—
19	C₃³×C₄	C₂²×C₃	1	16	yes	complemented
20	C₂²×H(3,𝔽₃)	C₃×C₄	1	36	yes	direct product factor
21	C₂²×H(3,𝔽₃)	C₂²×C₃	1	27	yes	—
22	C₄×H(3,𝔽₃)	C₂²×C₃	1	36	yes	direct product factor
23	C₃×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³×C₃²	—	3	12	no	—
25	C₂³×C₃²	C₂×C₃²	1	12	yes	—
26	C₂³×C₃²	C₂×C₃²	1	1	yes	characteristic
27	C₂×C₃²×C₄	—	3	72	no	—
28	C₂×C₃²×C₄	C₂×C₃²	1	6	yes	—
29	C₂×C₃²×C₄	C₂×C₃²	1	72	yes	complemented
30	C₂×C₃³	C₂×C₃×C₄	1	24	yes	complemented
31	C₂×C₃³	C₂³×C₃	1	4	yes	—
32	C₂×H(3,𝔽₃)	C₂×C₃×C₄	1	54	yes	direct product factor
33	C₂×H(3,𝔽₃)	C₂³×C₃	1	9	yes	—
34	C₂²×C₃×C₄	—	3	12	no	—
35	C₂²×C₃×C₄	H(3,𝔽₃)	1	3	yes	direct product factor
36	C₂²×C₃×C₄	C₃³	1	1	yes	characteristic
37	C₂²×C₃²	—	3	48	no	—
38	C₂²×C₃²	—	3	36	no	—
39	C₂²×C₃²	C₃²×C₄	1	4	yes	—
40	C₂²×C₃²	C₃²×C₄	1	48	yes	complemented
41	C₂²×C₃²	C₂²×C₃²	1	36	yes	—
42	C₂²×C₃²	C₂²×C₃²	1	3	yes	—
43	C₃²×C₄	—	3	48	no	—
44	C₃²×C₄	C₂²×C₃²	1	4	yes	—
45	C₃²×C₄	C₂²×C₃²	1	48	yes	complemented
46	C₃³	C₂²×C₃×C₄	1	4	yes	complemented
47	H(3,𝔽₃)	C₂²×C₃×C₄	1	9	yes	direct product factor
48	C₂³×C₃	—	3	12	no	—
49	C₂³×C₃	C₂×H(3,𝔽₃)	1	3	yes	—
50	C₂³×C₃	C₂×C₃³	1	1	yes	characteristic
51	C₂×C₃×C₄	—	3	72	no	—
52	C₂×C₃×C₄	C₂×H(3,𝔽₃)	1	18	yes	direct product factor
53	C₂×C₃×C₄	C₂×C₃³	1	6	yes	—
54	C₂×C₃²	—	3	72	no	—
55	C₂×C₃²	—	3	12	no	—
56	C₂×C₃²	C₂×C₃²×C₄	1	6	yes	—
57	C₂×C₃²	C₂×C₃²×C₄	1	72	yes	complemented
58	C₂×C₃²	C₂³×C₃²	1	12	yes	—
59	C₂×C₃²	C₂³×C₃²	1	1	yes	characteristic
60	C₂²×C₄	C₃×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₃	—	3	48	no	—
62	C₂²×C₃	—	3	36	no	—
63	C₂²×C₃	C₄×H(3,𝔽₃)	1	12	yes	direct product factor
64	C₂²×C₃	C₂²×H(3,𝔽₃)	1	9	yes	—
65	C₂²×C₃	C₃³×C₄	1	4	yes	—
66	C₂²×C₃	C₂²×C₃³	1	3	yes	—
67	C₃×C₄	—	3	48	no	—
68	C₃×C₄	C₂²×H(3,𝔽₃)	1	12	yes	direct product factor
69	C₃×C₄	C₂²×C₃³	1	4	yes	—
70	C₃²	—	3	12	no	—
71	C₃²	C₂²×C₃²×C₄	1	1	yes	characteristic
72	C₃²	C₂²×C₃²×C₄	1	12	yes	complemented
73	C₂³	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₂×C₄	C₂×C₃×H(3,𝔽₃)	1	6	yes	direct product factor
75	C₂×C₃	—	3	72	no	—
76	C₂×C₃	—	3	12	no	—
77	C₂×C₃	C₂×C₄×H(3,𝔽₃)	1	18	yes	direct product factor
78	C₂×C₃	C₂³×H(3,𝔽₃)	1	3	yes	—
79	C₂×C₃	C₂×C₃³×C₄	1	6	yes	—
80	C₂×C₃	C₂³×C₃³	1	1	yes	characteristic
81	C₂²	C₃×C₄×H(3,𝔽₃)	1	4	yes	direct product factor
82	C₂²	C₂²×C₃×H(3,𝔽₃)	1	3	yes	—
83	C₄	C₂²×C₃×H(3,𝔽₃)	1	4	yes	direct product factor
84	C₃	—	3	12	no	—
85	C₃	C₂²×C₄×H(3,𝔽₃)	1	3	yes	direct product factor
86	C₃	C₂²×C₃³×C₄	1	1	yes	characteristic
87	C₂	C₂×C₃×C₄×H(3,𝔽₃)	1	6	yes	direct product factor
88	C₂	C₂³×C₃×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²×C₄	C₃²
commutator	C₃	C₂²×C₃³×C₄
Frattini, cosocle	C₂×C₃	C₂³×C₃³
socle	C₂³×C₃²	C₂×C₃²
prime radical	C₂³×C₃×H(3,𝔽₃)	C₂
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂²×C₃³×C₄	C₃	semidirect
4. C₂²×C₄×H(3,𝔽₃)	C₃	direct (N × H)
5. C₂²×C₃×H(3,𝔽₃)	C₄	direct (N × H)
7. C₃×C₄×H(3,𝔽₃)	C₂²	direct (N × H)
9. C₂×C₃³×C₄	C₂×C₃	semidirect
11. C₂×C₄×H(3,𝔽₃)	C₂×C₃	direct (N × H)
12. C₂×C₃×H(3,𝔽₃)	C₂×C₄	direct (N × H)
16. C₂²×C₃²×C₄	C₃²	semidirect
17. C₂²×C₃³	C₃×C₄	semidirect
19. C₃³×C₄	C₂²×C₃	semidirect
20. C₂²×H(3,𝔽₃)	C₃×C₄	direct (N × H)
22. C₄×H(3,𝔽₃)	C₂²×C₃	direct (N × H)
23. C₃×H(3,𝔽₃)	C₂²×C₄	direct (N × H)
29. C₂×C₃²×C₄	C₂×C₃²	semidirect
30. C₂×C₃³	C₂×C₃×C₄	semidirect
32. C₂×H(3,𝔽₃)	C₂×C₃×C₄	direct (N × H)
35. C₂²×C₃×C₄	H(3,𝔽₃)	direct (N × H)
40. C₂²×C₃²	C₃²×C₄	semidirect
45. C₃²×C₄	C₂²×C₃²	semidirect
46. C₃³	C₂²×C₃×C₄	semidirect
47. H(3,𝔽₃)	C₂²×C₃×C₄	direct (N × H)
52. C₂×C₃×C₄	C₂×H(3,𝔽₃)	direct (N × H)
57. C₂×C₃²	C₂×C₃²×C₄	semidirect
60. C₂²×C₄	C₃×H(3,𝔽₃)	direct (N × H)
63. C₂²×C₃	C₄×H(3,𝔽₃)	direct (N × H)
68. C₃×C₄	C₂²×H(3,𝔽₃)	direct (N × H)
72. C₃²	C₂²×C₃²×C₄	semidirect
74. C₂×C₄	C₂×C₃×H(3,𝔽₃)	direct (N × H)
77. C₂×C₃	C₂×C₄×H(3,𝔽₃)	direct (N × H)
81. C₂²	C₃×C₄×H(3,𝔽₃)	direct (N × H)
83. C₄	C₂²×C₃×H(3,𝔽₃)	direct (N × H)
85. C₃	C₂²×C₄×H(3,𝔽₃)	direct (N × H)
87. C₂	C₂×C₃×C₄×H(3,𝔽₃)	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
C₂×C₃×C₄×H(3,𝔽₃)	2
C₂²×C₃³×C₄	3
C₂²×C₄×H(3,𝔽₃)	3

Subgroups — C22×C3×C4×H(3,𝔽3) #13048972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×H(3,𝔽₃) #13048972