Subgroups — C₂×C₃×D₂₄ #64026

Order 288.

163 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×D₈	—	3	1	no	maximal
5	C₂×D₂₄	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×D₃	—	2	2	no	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₃×D₁₂	—	2	2	no	—
9	C₃×D₁₂	C₂²	1	4	yes	complemented
10	C₃²×C₈	C₂²	1	2	yes	complemented
11	C₂×C₃×D₄	—	3	2	no	—
12	C₂×D₁₂	C₂×C₃	1	2	yes	complemented
13	C₃×D₈	—	3	4	no	—
14	C₂×C₃×C₈	—	2	1	no	—
15	C₂×C₃×C₈	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
17	D₂₄	C₂×C₃	1	4	yes	direct product factor
18	C₂²×C₃²	D₄	1	1	yes	characteristic
19	C₂×C₃×D₃	—	2	4	no	—
20	C₂×C₃×D₃	—	4	4	no	—
21	C₃²×C₄	D₄	1	1	yes	characteristic
22	C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₂×D₈	—	3	1	no	Sylow
24	C₂³×C₃	—	6	2	no	—
25	C₂²×D₃	—	2	2	no	—
26	C₃×D₄	—	3	4	no	—
27	C₃×D₄	—	6	2	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
31	D₁₂	—	2	2	no	—
32	D₁₂	C₂²×C₃	1	4	yes	complemented
33	C₃×C₈	—	2	2	no	—
34	C₃×C₈	C₂×D₃	1	2	yes	complemented
35	C₃×C₈	C₂²×C₃	1	2	yes	complemented
36	C₂×C₃²	D₈	1	2	yes	complemented
37	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
38	C₃×D₃	—	4	4	no	—
39	C₂×D₄	—	3	2	no	—
40	D₈	—	3	4	no	—
41	C₂×C₈	C₃×D₃	1	1	yes	characteristic, complemented
42	C₂²×C₃	—	2	1	no	—
43	C₂²×C₃	—	6	4	no	—
44	C₂²×C₃	—	12	4	no	—
45	C₂²×C₃	D₁₂	1	1	yes	characteristic
46	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
47	C₂×D₃	—	2	4	no	—
48	C₂×D₃	—	4	4	no	—
49	C₃×C₄	—	2	1	no	—
50	C₃×C₄	—	2	1	no	—
51	C₃×C₄	D₁₂	1	1	yes	characteristic
52	C₃×C₄	C₃×D₄	1	1	yes	characteristic
53	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
54	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
55	C₃²	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
56	C₂³	—	6	2	no	—
57	D₄	—	3	4	no	—
58	D₄	—	6	2	no	—
59	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
60	C₈	C₂×C₃×D₃	1	2	yes	complemented
61	C₂×C₃	—	2	2	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	—	12	4	no	—
64	C₂×C₃	D₂₄	1	2	yes	direct product factor
65	C₂×C₃	C₃×D₈	1	2	yes	complemented
66	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
67	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
68	D₃	—	4	4	no	—
69	C₂²	—	6	4	no	—
70	C₂²	—	12	4	no	—
71	C₂²	C₃×D₁₂	1	1	yes	characteristic
72	C₄	C₃×D₁₂	1	1	yes	characteristic
73	C₄	C₂²×C₃×D₃	1	1	yes	characteristic
74	C₃	—	2	1	no	—
75	C₃	C₂×D₂₄	1	1	yes	characteristic, direct product factor
76	C₃	C₂×C₃×D₈	1	1	yes	characteristic, complemented
77	C₂	—	12	4	no	—
78	C₂	C₃×D₂₄	1	2	yes	direct product factor
79	C₂	C₂×C₃×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	D₄
hypercenter	C₂×C₃×C₈	D₃
hypocenter	C₃	C₂×C₃×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₁₂	C₂	semidirect
2. C₂×C₃²×C₈	C₂	semidirect
3. C₃×D₂₄	C₂	direct (N × H)
5. C₂×D₂₄	C₃	direct (N × H)
9. C₃×D₁₂	C₂²	semidirect
10. C₃²×C₈	C₂²	semidirect
12. C₂×D₁₂	C₂×C₃	semidirect
15. C₂×C₃×C₈	D₃	semidirect
16. C₂×C₃×C₈	C₂×C₃	semidirect
17. D₂₄	C₂×C₃	direct (N × H)
32. D₁₂	C₂²×C₃	semidirect
34. C₃×C₈	C₂×D₃	semidirect
35. C₃×C₈	C₂²×C₃	semidirect
36. C₂×C₃²	D₈	semidirect
41. C₂×C₈	C₃×D₃	semidirect
55. C₃²	C₂×D₈	semidirect
60. C₈	C₂×C₃×D₃	semidirect
64. C₂×C₃	D₂₄	direct (N × H)
65. C₂×C₃	C₃×D₈	semidirect
75. C₃	C₂×D₂₄	direct (N × H)
76. C₃	C₂×C₃×D₈	semidirect
78. C₂	C₃×D₂₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₂	2
C₂×C₃²×C₈	2
C₃×D₂₄	2
C₂×C₃×D₈	3
C₂×D₂₄	3

Subgroups — C2×C3×D24 #64026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₂₄ #64026