Subgroups — C₂⁴×D₂₄ #11708962

Order 768.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×D₂₄	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×D₈	—	3	1	no	maximal, Sylow
5	C₂⁵×D₃	—	2	2	no	—
6	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂³×D₁₂	—	2	30	no	—
8	C₂³×D₁₂	C₂²	1	60	yes	complemented
9	C₂³×C₃×C₈	C₂²	1	30	yes	complemented
10	C₂²×D₂₄	C₂²	1	560	yes	direct product factor
11	C₂⁴×D₄	—	3	2	no	—
12	C₂³×D₈	—	3	60	no	—
13	C₂⁴×C₈	D₃	1	1	yes	characteristic, complemented
14	C₂⁵×C₃	D₄	1	1	yes	characteristic
15	C₂⁴×D₃	—	2	60	no	—
16	C₂⁴×D₃	—	4	32	no	—
17	C₂³×C₃×C₄	D₄	1	15	yes	—
18	C₂³×C₃×C₄	C₂³	1	15	yes	—
19	C₂²×D₁₂	—	2	420	no	—
20	C₂²×D₁₂	C₂³	1	280	yes	complemented
21	C₂²×C₃×C₈	C₂³	1	140	yes	complemented
22	C₂×D₂₄	C₂³	1	960	yes	direct product factor
23	C₂⁶	—	6	2	no	—
24	C₂³×D₄	—	3	60	no	—
25	C₂³×D₄	—	6	30	no	—
26	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
27	C₂²×D₈	—	3	560	no	—
28	C₂³×C₈	C₂×D₃	1	30	yes	complemented
29	C₂⁴×C₃	D₈	1	16	yes	complemented
30	C₂⁴×C₃	C₂×D₄	1	15	yes	—
31	C₂³×D₃	—	2	280	no	—
32	C₂³×D₃	—	4	480	no	—
33	C₂²×C₃×C₄	C₂×D₄	1	105	yes	—
34	C₂²×C₃×C₄	C₂⁴	1	35	yes	—
35	C₂×D₁₂	—	2	840	no	—
36	C₂×D₁₂	C₂⁴	1	240	yes	complemented
37	C₂×C₃×C₈	C₂⁴	1	120	yes	complemented
38	D₂₄	C₂⁴	1	256	yes	direct product factor
39	C₂⁵	—	6	60	no	—
40	C₂⁵	—	12	32	no	—
41	C₂⁵	D₁₂	1	1	yes	characteristic
42	C₂²×D₄	—	3	280	no	—
43	C₂²×D₄	—	6	420	no	—
44	C₂³×C₄	D₁₂	1	15	yes	—
45	C₂³×C₄	C₂²×D₃	1	15	yes	—
46	C₂×D₈	—	3	960	no	—
47	C₂²×C₈	C₂²×D₃	1	140	yes	complemented
48	C₂³×C₃	C₂×D₈	1	120	yes	complemented
49	C₂³×C₃	C₂²×D₄	1	35	yes	—
50	C₂²×D₃	—	2	240	no	—
51	C₂²×D₃	—	4	1120	no	—
52	C₂×C₃×C₄	C₂²×D₄	1	105	yes	—
53	C₂×C₃×C₄	C₂⁵	1	15	yes	—
54	D₁₂	—	2	240	no	—
55	D₁₂	C₂⁵	1	32	yes	complemented
56	C₃×C₈	C₂⁵	1	16	yes	complemented
57	C₂⁴	—	6	280	no	—
58	C₂⁴	—	12	480	no	—
59	C₂⁴	D₂₄	1	16	yes	direct product factor
60	C₂⁴	C₂×D₁₂	1	15	yes	—
61	C₂×D₄	—	3	240	no	—
62	C₂×D₄	—	6	840	no	—
63	C₂²×C₄	C₂×D₁₂	1	105	yes	—
64	C₂²×C₄	C₂³×D₃	1	35	yes	—
65	D₈	—	3	256	no	—
66	C₂×C₈	C₂³×D₃	1	120	yes	complemented
67	C₂²×C₃	C₂²×D₈	1	140	yes	complemented
68	C₂²×C₃	C₂³×D₄	1	15	yes	—
69	C₂×D₃	—	2	32	no	—
70	C₂×D₃	—	4	480	no	—
71	C₃×C₄	C₂³×D₄	1	15	yes	—
72	C₃×C₄	C₂⁶	1	1	yes	characteristic
73	C₂³	—	6	240	no	—
74	C₂³	—	12	1120	no	—
75	C₂³	C₂×D₂₄	1	120	yes	direct product factor
76	C₂³	C₂²×D₁₂	1	35	yes	—
77	D₄	—	3	32	no	—
78	D₄	—	6	240	no	—
79	C₂×C₄	C₂²×D₁₂	1	105	yes	—
80	C₂×C₄	C₂⁴×D₃	1	15	yes	—
81	C₈	C₂⁴×D₃	1	16	yes	complemented
82	C₂×C₃	C₂³×D₈	1	30	yes	complemented
83	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
84	D₃	—	4	32	no	—
85	C₂²	—	6	32	no	—
86	C₂²	—	12	480	no	—
87	C₂²	C₂²×D₂₄	1	140	yes	direct product factor
88	C₂²	C₂³×D₁₂	1	15	yes	—
89	C₄	C₂³×D₁₂	1	15	yes	—
90	C₄	C₂⁵×D₃	1	1	yes	characteristic
91	C₃	C₂⁴×D₈	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	12	32	no	—
93	C₂	C₂³×D₂₄	1	30	yes	direct product factor
94	C₂	C₂⁴×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₁₂
commutator, cosocle	C₃×C₄	C₂⁶
Frattini	C₄	C₂⁵×D₃
Fitting	C₂⁴×C₃×C₈	C₂
socle	C₂⁵×C₃	D₄
hypercenter	C₂⁴×C₈	D₃
hypocenter	C₃	C₂⁴×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₁₂	C₂	semidirect
2. C₂⁴×C₃×C₈	C₂	semidirect
3. C₂³×D₂₄	C₂	direct (N × H)
8. C₂³×D₁₂	C₂²	semidirect
9. C₂³×C₃×C₈	C₂²	semidirect
10. C₂²×D₂₄	C₂²	direct (N × H)
13. C₂⁴×C₈	D₃	semidirect
20. C₂²×D₁₂	C₂³	semidirect
21. C₂²×C₃×C₈	C₂³	semidirect
22. C₂×D₂₄	C₂³	direct (N × H)
28. C₂³×C₈	C₂×D₃	semidirect
29. C₂⁴×C₃	D₈	semidirect
36. C₂×D₁₂	C₂⁴	semidirect
37. C₂×C₃×C₈	C₂⁴	semidirect
38. D₂₄	C₂⁴	direct (N × H)
47. C₂²×C₈	C₂²×D₃	semidirect
48. C₂³×C₃	C₂×D₈	semidirect
55. D₁₂	C₂⁵	semidirect
56. C₃×C₈	C₂⁵	semidirect
59. C₂⁴	D₂₄	direct (N × H)
66. C₂×C₈	C₂³×D₃	semidirect
67. C₂²×C₃	C₂²×D₈	semidirect
75. C₂³	C₂×D₂₄	direct (N × H)
81. C₈	C₂⁴×D₃	semidirect
82. C₂×C₃	C₂³×D₈	semidirect
87. C₂²	C₂²×D₂₄	direct (N × H)
91. C₃	C₂⁴×D₈	semidirect
93. C₂	C₂³×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
C₂⁴×D₁₂	2
C₂⁴×C₃×C₈	2
C₂³×D₂₄	2
C₂⁴×D₈	3

Subgroups — C24×D24 #11708962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₂₄ #11708962