Subgroups — C₃×D₄×A₄ #64332

Order 288.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×A₄	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×A₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×D₄	C₃	1	1	yes	maximal, characteristic, complemented
4	D₄×A₄	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃×A₄	—	2	2	no	—
6	C₂×C₃×A₄	C₂²	1	1	yes	characteristic
7	C₃²×D₄	—	4	1	no	maximal
8	C₂⁴×C₃	C₂×C₃	1	2	yes	complemented
9	C₂²×A₄	C₂×C₃	1	6	yes	complemented
10	C₂×C₃×D₄	—	3	1	no	—
11	C₂×C₃×D₄	—	3	1	no	—
12	C₂×C₃×D₄	—	3	2	no	—
13	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic, complemented
14	C₄×A₄	C₂×C₃	1	3	yes	complemented
15	C₂²×C₃²	—	4	2	no	—
16	C₃×A₄	D₄	1	1	yes	characteristic, direct product factor
17	C₃²×C₄	—	4	1	no	—
18	C₂²×D₄	C₃²	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₃	—	2	2	no	—
20	C₂³×C₃	—	3	2	no	—
21	C₂³×C₃	—	3	2	no	—
22	C₂³×C₃	—	6	2	no	—
23	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
24	C₂×A₄	—	2	6	no	—
25	C₂×A₄	C₂²×C₃	1	3	yes	—
26	C₃×D₄	—	3	2	no	—
27	C₃×D₄	—	3	2	no	—
28	C₃×D₄	—	3	1	no	—
29	C₃×D₄	—	4	3	no	—
30	C₃×D₄	A₄	1	1	yes	characteristic, direct product factor
31	C₂×C₃×C₄	—	3	1	no	—
32	C₂×C₃×C₄	—	3	1	no	—
33	C₂×C₃²	—	4	1	no	—
34	C₂×C₃²	—	8	2	no	—
35	C₂⁴	C₂×C₃²	1	2	yes	complemented
36	C₂×D₄	—	3	1	no	—
37	C₂×D₄	—	3	1	no	—
38	C₂×D₄	—	3	2	no	—
39	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
40	C₂²×C₃	—	3	1	no	—
41	C₂²×C₃	—	3	1	no	—
42	C₂²×C₃	—	3	2	no	—
43	C₂²×C₃	—	4	6	no	—
44	C₂²×C₃	—	6	2	no	—
45	C₂²×C₃	—	6	2	no	—
46	C₂²×C₃	—	6	2	no	—
47	C₂²×C₃	—	6	2	no	—
48	C₂²×C₃	C₃×D₄	1	1	yes	characteristic, complemented
49	C₂²×C₃	C₂×A₄	1	2	yes	complemented
50	A₄	C₃×D₄	1	3	yes	direct product factor
51	C₃×C₄	—	3	1	no	—
52	C₃×C₄	—	4	3	no	—
53	C₃×C₄	C₂×A₄	1	1	yes	characteristic, complemented
54	C₃²	—	4	1	no	Sylow
55	C₂³	—	2	2	no	—
56	C₂³	—	3	2	no	—
57	C₂³	—	3	2	no	—
58	C₂³	—	6	2	no	—
59	C₂³	C₂²×C₃²	1	1	yes	characteristic
60	D₄	—	3	2	no	—
61	D₄	—	3	2	no	—
62	D₄	—	3	1	no	—
63	D₄	C₃×A₄	1	1	yes	characteristic, direct product factor
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	4	3	no	—
70	C₂×C₃	—	6	2	no	—
71	C₂×C₃	—	8	6	no	—
72	C₂×C₃	C₂²×A₄	1	1	yes	characteristic
73	C₂²	—	3	1	no	—
74	C₂²	—	3	1	no	—
75	C₂²	—	3	2	no	—
76	C₂²	—	6	2	no	—
77	C₂²	—	6	2	no	—
78	C₂²	—	6	2	no	—
79	C₂²	—	6	2	no	—
80	C₂²	C₃²×D₄	1	1	yes	characteristic, complemented
81	C₂²	C₂×C₃×A₄	1	2	yes	complemented
82	C₄	—	3	1	no	—
83	C₄	C₂×C₃×A₄	1	1	yes	characteristic, complemented
84	C₃	—	4	3	no	—
85	C₃	D₄×A₄	1	1	yes	characteristic, direct product factor
86	C₂	—	2	2	no	—
87	C₂	—	3	1	no	—
88	C₂	—	3	1	no	—
89	C₂	—	6	2	no	—
90	C₂	C₂²×C₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×A₄
commutator, cosocle	C₂³	C₂²×C₃²
Frattini	C₂	C₂²×C₃×A₄
Fitting	C₂²×C₃×D₄	C₃
socle	C₂³×C₃	C₂²×C₃
hypercenter	C₃×D₄	A₄
hypocenter	C₂²	C₃²×D₄
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×A₄	C₂	semidirect
2. C₃×C₄×A₄	C₂	semidirect
3. C₂²×C₃×D₄	C₃	semidirect
4. D₄×A₄	C₃	direct (N × H)
8. C₂⁴×C₃	C₂×C₃	semidirect
9. C₂²×A₄	C₂×C₃	semidirect
13. C₂²×C₃×C₄	C₂×C₃	semidirect
14. C₄×A₄	C₂×C₃	semidirect
16. C₃×A₄	D₄	direct (N × H)
18. C₂²×D₄	C₃²	semidirect
30. C₃×D₄	A₄	direct (N × H)
35. C₂⁴	C₂×C₃²	semidirect
39. C₂²×C₄	C₂×C₃²	semidirect
48. C₂²×C₃	C₃×D₄	semidirect
49. C₂²×C₃	C₂×A₄	semidirect
50. A₄	C₃×D₄	direct (N × H)
53. C₃×C₄	C₂×A₄	semidirect
63. D₄	C₃×A₄	direct (N × H)
80. C₂²	C₃²×D₄	semidirect
81. C₂²	C₂×C₃×A₄	semidirect
83. C₄	C₂×C₃×A₄	semidirect
85. C₃	D₄×A₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×A₄	2
C₃×C₄×A₄	2
C₂²×C₃×D₄	3
D₄×A₄	3
C₃²×D₄	4

Subgroups — C3×D4×A4 #64332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×A₄ #64332