Subgroups — C₂³×D₂₄ #87148

Order 384.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₂₄	C₂	1	28	yes	maximal, direct product factor
4	C₂³×D₈	—	3	1	no	maximal, Sylow
5	C₂⁴×D₃	—	2	2	no	—
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×D₁₂	—	2	14	no	—
8	C₂²×D₁₂	C₂²	1	28	yes	complemented
9	C₂²×C₃×C₈	C₂²	1	14	yes	complemented
10	C₂×D₂₄	C₂²	1	112	yes	direct product factor
11	C₂³×D₄	—	3	2	no	—
12	C₂²×D₈	—	3	28	no	—
13	C₂³×C₈	D₃	1	1	yes	characteristic, complemented
14	C₂⁴×C₃	D₄	1	1	yes	characteristic
15	C₂³×D₃	—	2	28	no	—
16	C₂³×D₃	—	4	16	no	—
17	C₂²×C₃×C₄	D₄	1	7	yes	—
18	C₂²×C₃×C₄	C₂³	1	7	yes	—
19	C₂×D₁₂	—	2	84	no	—
20	C₂×D₁₂	C₂³	1	56	yes	complemented
21	C₂×C₃×C₈	C₂³	1	28	yes	complemented
22	D₂₄	C₂³	1	64	yes	direct product factor
23	C₂⁵	—	6	2	no	—
24	C₂²×D₄	—	3	28	no	—
25	C₂²×D₄	—	6	14	no	—
26	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
27	C₂×D₈	—	3	112	no	—
28	C₂²×C₈	C₂×D₃	1	14	yes	complemented
29	C₂³×C₃	D₈	1	8	yes	complemented
30	C₂³×C₃	C₂×D₄	1	7	yes	—
31	C₂²×D₃	—	2	56	no	—
32	C₂²×D₃	—	4	112	no	—
33	C₂×C₃×C₄	C₂×D₄	1	21	yes	—
34	C₂×C₃×C₄	C₂⁴	1	7	yes	—
35	D₁₂	—	2	56	no	—
36	D₁₂	C₂⁴	1	16	yes	complemented
37	C₃×C₈	C₂⁴	1	8	yes	complemented
38	C₂⁴	—	6	28	no	—
39	C₂⁴	—	12	16	no	—
40	C₂⁴	D₁₂	1	1	yes	characteristic
41	C₂×D₄	—	3	56	no	—
42	C₂×D₄	—	6	84	no	—
43	C₂²×C₄	D₁₂	1	7	yes	—
44	C₂²×C₄	C₂²×D₃	1	7	yes	—
45	D₈	—	3	64	no	—
46	C₂×C₈	C₂²×D₃	1	28	yes	complemented
47	C₂²×C₃	C₂×D₈	1	28	yes	complemented
48	C₂²×C₃	C₂²×D₄	1	7	yes	—
49	C₂×D₃	—	2	16	no	—
50	C₂×D₃	—	4	112	no	—
51	C₃×C₄	C₂²×D₄	1	7	yes	—
52	C₃×C₄	C₂⁵	1	1	yes	characteristic
53	C₂³	—	6	56	no	—
54	C₂³	—	12	112	no	—
55	C₂³	D₂₄	1	8	yes	direct product factor
56	C₂³	C₂×D₁₂	1	7	yes	—
57	D₄	—	3	16	no	—
58	D₄	—	6	56	no	—
59	C₂×C₄	C₂×D₁₂	1	21	yes	—
60	C₂×C₄	C₂³×D₃	1	7	yes	—
61	C₈	C₂³×D₃	1	8	yes	complemented
62	C₂×C₃	C₂²×D₈	1	14	yes	complemented
63	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
64	D₃	—	4	16	no	—
65	C₂²	—	6	16	no	—
66	C₂²	—	12	112	no	—
67	C₂²	C₂×D₂₄	1	28	yes	direct product factor
68	C₂²	C₂²×D₁₂	1	7	yes	—
69	C₄	C₂²×D₁₂	1	7	yes	—
70	C₄	C₂⁴×D₃	1	1	yes	characteristic
71	C₃	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	12	16	no	—
73	C₂	C₂²×D₂₄	1	14	yes	direct product factor
74	C₂	C₂³×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₁₂
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂³×C₃×C₈	C₂
socle	C₂⁴×C₃	D₄
hypercenter	C₂³×C₈	D₃
hypocenter	C₃	C₂³×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₁₂	C₂	semidirect
2. C₂³×C₃×C₈	C₂	semidirect
3. C₂²×D₂₄	C₂	direct (N × H)
8. C₂²×D₁₂	C₂²	semidirect
9. C₂²×C₃×C₈	C₂²	semidirect
10. C₂×D₂₄	C₂²	direct (N × H)
13. C₂³×C₈	D₃	semidirect
20. C₂×D₁₂	C₂³	semidirect
21. C₂×C₃×C₈	C₂³	semidirect
22. D₂₄	C₂³	direct (N × H)
28. C₂²×C₈	C₂×D₃	semidirect
29. C₂³×C₃	D₈	semidirect
36. D₁₂	C₂⁴	semidirect
37. C₃×C₈	C₂⁴	semidirect
46. C₂×C₈	C₂²×D₃	semidirect
47. C₂²×C₃	C₂×D₈	semidirect
55. C₂³	D₂₄	direct (N × H)
61. C₈	C₂³×D₃	semidirect
62. C₂×C₃	C₂²×D₈	semidirect
67. C₂²	C₂×D₂₄	direct (N × H)
71. C₃	C₂³×D₈	semidirect
73. C₂	C₂²×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
C₂³×D₁₂	2
C₂³×C₃×C₈	2
C₂²×D₂₄	2
C₂³×D₈	3

Subgroups — C23×D24 #87148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₂₄ #87148