Subgroups — C₂²×C₃×C₄×D₃ #64341

Order 288.

499 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×D₃	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₃×C₄	—	3	1	no	maximal
6	C₂²×C₄×D₃	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃²	C₂²	1	1	yes	characteristic
8	C₂²×C₃×D₃	C₄	1	8	yes	direct product factor
9	C₂²×C₃×D₃	C₂²	1	6	yes	—
10	C₂×C₃²×C₄	C₂²	1	6	yes	complemented
11	C₂×C₃×Dic₃	C₂²	1	6	yes	complemented
12	C₃×C₄×D₃	C₂²	1	16	yes	direct product factor
13	C₂⁴×C₃	—	3	1	no	—
14	C₂³×D₃	C₂×C₃	1	1	yes	characteristic
15	C₂²×C₃×C₄	—	2	1	no	—
16	C₂²×C₃×C₄	—	3	1	no	—
17	C₂²×C₃×C₄	—	3	12	no	—
18	C₂²×C₃×C₄	D₃	1	1	yes	characteristic, direct product factor
19	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂²×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₄×D₃	C₂×C₃	1	12	yes	direct product factor
22	C₂²×C₃²	C₂×C₄	1	4	yes	complemented
23	C₂²×C₃²	C₂³	1	3	yes	—
24	C₂×C₃×D₃	C₂×C₄	1	24	yes	direct product factor
25	C₂×C₃×D₃	C₂³	1	4	yes	—
26	C₃²×C₄	C₂³	1	4	yes	complemented
27	C₃×Dic₃	C₂³	1	4	yes	complemented
28	C₂³×C₄	—	3	1	no	Sylow
29	C₂³×C₃	—	2	1	no	—
30	C₂³×C₃	—	3	8	no	—
31	C₂³×C₃	—	3	6	no	—
32	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
33	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
34	C₂²×D₃	C₃×C₄	1	8	yes	direct product factor
35	C₂²×D₃	C₂²×C₃	1	6	yes	—
36	C₂×C₃×C₄	—	2	6	no	—
37	C₂×C₃×C₄	—	3	6	no	—
38	C₂×C₃×C₄	—	3	16	no	—
39	C₂×C₃×C₄	C₂×D₃	1	6	yes	direct product factor
40	C₂×C₃×C₄	C₂²×C₃	1	6	yes	complemented
41	C₂×Dic₃	C₂²×C₃	1	6	yes	complemented
42	C₄×D₃	C₂²×C₃	1	16	yes	direct product factor
43	C₂×C₃²	C₂²×C₄	1	6	yes	complemented
44	C₂×C₃²	C₂⁴	1	1	yes	characteristic
45	C₃×D₃	C₂²×C₄	1	8	yes	direct product factor
46	C₂⁴	—	3	1	no	—
47	C₂²×C₄	—	3	1	no	—
48	C₂²×C₄	—	3	12	no	—
49	C₂²×C₄	C₃×D₃	1	1	yes	characteristic, direct product factor
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	—	2	3	no	—
52	C₂²×C₃	—	3	24	no	—
53	C₂²×C₃	—	3	4	no	—
54	C₂²×C₃	C₄×D₃	1	4	yes	direct product factor
55	C₂²×C₃	C₂×C₃×C₄	1	4	yes	complemented
56	C₂²×C₃	C₂²×D₃	1	3	yes	—
57	C₂²×C₃	C₂³×C₃	1	3	yes	—
58	C₂×D₃	C₂×C₃×C₄	1	24	yes	direct product factor
59	C₂×D₃	C₂³×C₃	1	4	yes	—
60	C₃×C₄	—	2	4	no	—
61	C₃×C₄	—	3	4	no	—
62	C₃×C₄	C₂²×D₃	1	4	yes	direct product factor
63	C₃×C₄	C₂³×C₃	1	4	yes	complemented
64	Dic₃	C₂³×C₃	1	4	yes	complemented
65	C₃²	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	3	8	no	—
67	C₂³	—	3	6	no	—
68	C₂³	C₂×C₃×D₃	1	1	yes	characteristic
69	C₂×C₄	—	3	6	no	—
70	C₂×C₄	—	3	16	no	—
71	C₂×C₄	C₂×C₃×D₃	1	6	yes	direct product factor
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	6	no	—
74	C₂×C₃	—	3	8	no	—
75	C₂×C₃	C₂×C₄×D₃	1	6	yes	direct product factor
76	C₂×C₃	C₂²×C₃×C₄	1	6	yes	complemented
77	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
78	C₂×C₃	C₂⁴×C₃	1	1	yes	characteristic
79	D₃	C₂²×C₃×C₄	1	8	yes	direct product factor
80	C₂²	—	3	4	no	—
81	C₂²	—	3	24	no	—
82	C₂²	C₃×C₄×D₃	1	4	yes	direct product factor
83	C₂²	C₂²×C₃×D₃	1	3	yes	—
84	C₄	—	3	4	no	—
85	C₄	C₂²×C₃×D₃	1	4	yes	direct product factor
86	C₃	—	2	1	no	—
87	C₃	C₂²×C₄×D₃	1	1	yes	characteristic, direct product factor
88	C₃	C₂³×C₃×C₄	1	1	yes	characteristic, complemented
89	C₂	—	3	8	no	—
90	C₂	C₂×C₃×C₄×D₃	1	6	yes	direct product factor
91	C₂	C₂³×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₄	D₃
commutator, hypocenter	C₃	C₂³×C₃×C₄
Frattini	C₂	C₂³×C₃×D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂³×C₃²	C₂²
prime radical	C₂³×C₃×D₃	C₂
cosocle	C₂×C₃	C₂⁴×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄	C₂	semidirect
3. C₂²×C₃×Dic₃	C₂	semidirect
4. C₂×C₃×C₄×D₃	C₂	direct (N × H)
6. C₂²×C₄×D₃	C₃	direct (N × H)
8. C₂²×C₃×D₃	C₄	direct (N × H)
10. C₂×C₃²×C₄	C₂²	semidirect
11. C₂×C₃×Dic₃	C₂²	semidirect
12. C₃×C₄×D₃	C₂²	direct (N × H)
18. C₂²×C₃×C₄	D₃	direct (N × H)
19. C₂²×C₃×C₄	C₂×C₃	semidirect
20. C₂²×Dic₃	C₂×C₃	semidirect
21. C₂×C₄×D₃	C₂×C₃	direct (N × H)
22. C₂²×C₃²	C₂×C₄	semidirect
24. C₂×C₃×D₃	C₂×C₄	direct (N × H)
26. C₃²×C₄	C₂³	semidirect
27. C₃×Dic₃	C₂³	semidirect
34. C₂²×D₃	C₃×C₄	direct (N × H)
39. C₂×C₃×C₄	C₂×D₃	direct (N × H)
40. C₂×C₃×C₄	C₂²×C₃	semidirect
41. C₂×Dic₃	C₂²×C₃	semidirect
42. C₄×D₃	C₂²×C₃	direct (N × H)
43. C₂×C₃²	C₂²×C₄	semidirect
45. C₃×D₃	C₂²×C₄	direct (N × H)
49. C₂²×C₄	C₃×D₃	direct (N × H)
54. C₂²×C₃	C₄×D₃	direct (N × H)
55. C₂²×C₃	C₂×C₃×C₄	semidirect
58. C₂×D₃	C₂×C₃×C₄	direct (N × H)
62. C₃×C₄	C₂²×D₃	direct (N × H)
63. C₃×C₄	C₂³×C₃	semidirect
64. Dic₃	C₂³×C₃	semidirect
65. C₃²	C₂³×C₄	semidirect
71. C₂×C₄	C₂×C₃×D₃	direct (N × H)
75. C₂×C₃	C₂×C₄×D₃	direct (N × H)
76. C₂×C₃	C₂²×C₃×C₄	semidirect
79. D₃	C₂²×C₃×C₄	direct (N × H)
82. C₂²	C₃×C₄×D₃	direct (N × H)
85. C₄	C₂²×C₃×D₃	direct (N × H)
87. C₃	C₂²×C₄×D₃	direct (N × H)
88. C₃	C₂³×C₃×C₄	semidirect
90. C₂	C₂×C₃×C₄×D₃	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×D₃	2
C₂²×C₃²×C₄	2
C₂²×C₃×Dic₃	2
C₂×C₃×C₄×D₃	2
C₂³×C₃×C₄	3
C₂²×C₄×D₃	3

Subgroups — C22×C3×C4×D3 #64341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×D₃ #64341