Subgroups — C₂⁴×D₁₂ #87554

Order 384.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₃	C₂	1	2	yes	maximal, complemented
2	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×D₁₂	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×D₄	—	3	1	no	maximal, Sylow
5	C₂⁵×C₃	C₂²	1	1	yes	characteristic
6	C₂⁴×D₃	—	2	32	no	—
7	C₂⁴×D₃	C₂²	1	60	yes	complemented
8	C₂³×C₃×C₄	C₂²	1	30	yes	complemented
9	C₂²×D₁₂	C₂²	1	560	yes	direct product factor
10	C₂⁶	—	3	2	no	—
11	C₂³×D₄	—	3	60	no	—
12	C₂⁴×C₄	D₃	1	1	yes	characteristic, complemented
13	C₂⁴×C₃	D₄	1	16	yes	complemented
14	C₂⁴×C₃	C₂³	1	15	yes	—
15	C₂³×D₃	—	2	480	no	—
16	C₂³×D₃	C₂³	1	280	yes	complemented
17	C₂²×C₃×C₄	C₂³	1	140	yes	complemented
18	C₂×D₁₂	C₂³	1	960	yes	direct product factor
19	C₂⁵	—	3	60	no	—
20	C₂⁵	—	6	32	no	—
21	C₂⁵	C₂×D₃	1	1	yes	characteristic
22	C₂²×D₄	—	3	560	no	—
23	C₂³×C₄	C₂×D₃	1	30	yes	complemented
24	C₂³×C₃	C₂×D₄	1	120	yes	complemented
25	C₂³×C₃	C₂⁴	1	35	yes	—
26	C₂²×D₃	—	2	1120	no	—
27	C₂²×D₃	C₂⁴	1	240	yes	complemented
28	C₂×C₃×C₄	C₂⁴	1	120	yes	complemented
29	D₁₂	C₂⁴	1	256	yes	direct product factor
30	C₂⁴	—	3	280	no	—
31	C₂⁴	—	6	480	no	—
32	C₂⁴	D₁₂	1	16	yes	direct product factor
33	C₂⁴	C₂²×D₃	1	15	yes	—
34	C₂×D₄	—	3	960	no	—
35	C₂²×C₄	C₂²×D₃	1	140	yes	complemented
36	C₂²×C₃	C₂²×D₄	1	140	yes	complemented
37	C₂²×C₃	C₂⁵	1	15	yes	—
38	C₂×D₃	—	2	480	no	—
39	C₂×D₃	C₂⁵	1	32	yes	complemented
40	C₃×C₄	C₂⁵	1	16	yes	complemented
41	C₂³	—	3	240	no	—
42	C₂³	—	6	1120	no	—
43	C₂³	C₂×D₁₂	1	120	yes	direct product factor
44	C₂³	C₂³×D₃	1	35	yes	—
45	D₄	—	3	256	no	—
46	C₂×C₄	C₂³×D₃	1	120	yes	complemented
47	C₂×C₃	C₂³×D₄	1	30	yes	complemented
48	C₂×C₃	C₂⁶	1	1	yes	characteristic
49	D₃	—	2	32	no	—
50	C₂²	—	3	32	no	—
51	C₂²	—	6	480	no	—
52	C₂²	C₂²×D₁₂	1	140	yes	direct product factor
53	C₂²	C₂⁴×D₃	1	15	yes	—
54	C₄	C₂⁴×D₃	1	16	yes	complemented
55	C₃	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	6	32	no	—
57	C₂	C₂³×D₁₂	1	30	yes	direct product factor
58	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂⁴×C₃×C₄	C₂
socle	C₂⁵×C₃	C₂²
hypercenter	C₂⁴×C₄	D₃
hypocenter	C₃	C₂⁴×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₃	C₂	semidirect
2. C₂⁴×C₃×C₄	C₂	semidirect
3. C₂³×D₁₂	C₂	direct (N × H)
7. C₂⁴×D₃	C₂²	semidirect
8. C₂³×C₃×C₄	C₂²	semidirect
9. C₂²×D₁₂	C₂²	direct (N × H)
12. C₂⁴×C₄	D₃	semidirect
13. C₂⁴×C₃	D₄	semidirect
16. C₂³×D₃	C₂³	semidirect
17. C₂²×C₃×C₄	C₂³	semidirect
18. C₂×D₁₂	C₂³	direct (N × H)
23. C₂³×C₄	C₂×D₃	semidirect
24. C₂³×C₃	C₂×D₄	semidirect
27. C₂²×D₃	C₂⁴	semidirect
28. C₂×C₃×C₄	C₂⁴	semidirect
29. D₁₂	C₂⁴	direct (N × H)
32. C₂⁴	D₁₂	direct (N × H)
35. C₂²×C₄	C₂²×D₃	semidirect
36. C₂²×C₃	C₂²×D₄	semidirect
39. C₂×D₃	C₂⁵	semidirect
40. C₃×C₄	C₂⁵	semidirect
43. C₂³	C₂×D₁₂	direct (N × H)
46. C₂×C₄	C₂³×D₃	semidirect
47. C₂×C₃	C₂³×D₄	semidirect
52. C₂²	C₂²×D₁₂	direct (N × H)
54. C₄	C₂⁴×D₃	semidirect
55. C₃	C₂⁴×D₄	semidirect
57. C₂	C₂³×D₁₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₃	2
C₂⁴×C₃×C₄	2
C₂³×D₁₂	2
C₂⁴×D₄	3

Subgroups — C24×D12 #87554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₁₂ #87554