Subgroups — C₂⁴×Dic₁₂ #11708969

Order 768.

11166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₆	C₂	1	2	yes	maximal
2	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂³×Dic₁₂	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×Q₁₆	—	3	1	no	maximal, Sylow
5	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₃	—	2	2	no	—
7	C₂³×Dic₆	—	2	30	no	—
8	C₂³×Dic₆	C₂²	1	60	yes	—
9	C₂³×C₃×C₈	C₂²	1	30	yes	—
10	C₂²×Dic₁₂	C₂²	1	560	yes	direct product factor
11	C₂⁴×Q₈	—	3	2	no	—
12	C₂³×Q₁₆	—	3	60	no	—
13	C₂⁴×C₈	D₃	1	1	yes	characteristic
14	C₂⁵×C₃	D₄	1	1	yes	characteristic
15	C₂³×C₃×C₄	D₄	1	15	yes	—
16	C₂³×C₃×C₄	C₂³	1	15	yes	—
17	C₂³×Dic₃	—	2	60	no	—
18	C₂²×Dic₆	—	2	420	no	—
19	C₂²×Dic₆	C₂³	1	280	yes	—
20	C₂²×C₃×C₈	C₂³	1	140	yes	—
21	C₂×Dic₁₂	C₂³	1	960	yes	direct product factor
22	C₂³×Q₈	—	3	60	no	—
23	C₂³×Q₈	—	6	30	no	—
24	C₂⁴×C₄	—	6	2	no	—
25	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
26	C₂²×Q₁₆	—	3	560	no	—
27	C₂³×C₈	C₂×D₃	1	30	yes	—
28	C₂⁴×C₃	Q₁₆	1	16	yes	complemented
29	C₂⁴×C₃	C₂×D₄	1	15	yes	—
30	C₂²×C₃×C₄	C₂×D₄	1	105	yes	—
31	C₂²×C₃×C₄	C₂⁴	1	35	yes	—
32	C₂²×Dic₃	—	2	280	no	—
33	C₂×Dic₆	—	2	840	no	—
34	C₂×Dic₆	C₂⁴	1	240	yes	—
35	C₂×C₃×C₈	C₂⁴	1	120	yes	—
36	Dic₁₂	C₂⁴	1	256	yes	direct product factor
37	C₂⁵	D₁₂	1	1	yes	characteristic
38	C₂²×Q₈	—	3	280	no	—
39	C₂²×Q₈	—	6	420	no	—
40	C₂³×C₄	—	6	60	no	—
41	C₂³×C₄	D₁₂	1	15	yes	—
42	C₂³×C₄	C₂²×D₃	1	15	yes	—
43	C₂×Q₁₆	—	3	960	no	—
44	C₂²×C₈	C₂²×D₃	1	140	yes	—
45	C₂³×C₃	C₂×Q₁₆	1	120	yes	complemented
46	C₂³×C₃	C₂²×D₄	1	35	yes	—
47	C₂×C₃×C₄	C₂²×D₄	1	105	yes	—
48	C₂×C₃×C₄	C₂⁵	1	15	yes	—
49	C₂×Dic₃	—	2	240	no	—
50	Dic₆	—	2	240	no	—
51	Dic₆	C₂⁵	1	32	yes	—
52	C₃×C₈	C₂⁵	1	16	yes	—
53	C₂⁴	Dic₁₂	1	16	yes	direct product factor
54	C₂⁴	C₂×D₁₂	1	15	yes	—
55	C₂×Q₈	—	3	240	no	—
56	C₂×Q₈	—	6	840	no	—
57	C₂²×C₄	—	6	280	no	—
58	C₂²×C₄	C₂×D₁₂	1	105	yes	—
59	C₂²×C₄	C₂³×D₃	1	35	yes	—
60	Q₁₆	—	3	256	no	—
61	C₂×C₈	C₂³×D₃	1	120	yes	—
62	C₂²×C₃	C₂²×Q₁₆	1	140	yes	complemented
63	C₂²×C₃	C₂³×D₄	1	15	yes	—
64	C₃×C₄	C₂³×D₄	1	15	yes	—
65	C₃×C₄	C₂⁶	1	1	yes	characteristic
66	Dic₃	—	2	32	no	—
67	C₂³	C₂×Dic₁₂	1	120	yes	direct product factor
68	C₂³	C₂²×D₁₂	1	35	yes	—
69	Q₈	—	3	32	no	—
70	Q₈	—	6	240	no	—
71	C₂×C₄	—	6	240	no	—
72	C₂×C₄	C₂²×D₁₂	1	105	yes	—
73	C₂×C₄	C₂⁴×D₃	1	15	yes	—
74	C₈	C₂⁴×D₃	1	16	yes	—
75	C₂×C₃	C₂³×Q₁₆	1	30	yes	complemented
76	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
77	C₂²	C₂²×Dic₁₂	1	140	yes	direct product factor
78	C₂²	C₂³×D₁₂	1	15	yes	—
79	C₄	—	6	32	no	—
80	C₄	C₂³×D₁₂	1	15	yes	—
81	C₄	C₂⁵×D₃	1	1	yes	characteristic
82	C₃	C₂⁴×Q₁₆	1	1	yes	characteristic, complemented, Sylow
83	C₂	C₂³×Dic₁₂	1	30	yes	direct product factor
84	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, prime radical	C₂⁵×C₃	D₄
hypercenter	C₂⁴×C₈	D₃
hypocenter	C₃	C₂⁴×Q₁₆
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
3. C₂³×Dic₁₂	C₂	direct (N × H)
10. C₂²×Dic₁₂	C₂²	direct (N × H)
21. C₂×Dic₁₂	C₂³	direct (N × H)
28. C₂⁴×C₃	Q₁₆	semidirect
36. Dic₁₂	C₂⁴	direct (N × H)
45. C₂³×C₃	C₂×Q₁₆	semidirect
53. C₂⁴	Dic₁₂	direct (N × H)
62. C₂²×C₃	C₂²×Q₁₆	semidirect
67. C₂³	C₂×Dic₁₂	direct (N × H)
75. C₂×C₃	C₂³×Q₁₆	semidirect
77. C₂²	C₂²×Dic₁₂	direct (N × H)
82. C₃	C₂⁴×Q₁₆	semidirect
83. C₂	C₂³×Dic₁₂	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×Dic₆	2
C₂⁴×C₃×C₈	2
C₂³×Dic₁₂	2
C₂⁴×Q₁₆	3

Subgroups — C24×Dic12 #11708969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Dic₁₂ #11708969