Subgroups — C₂⁴×C₃×D₈ #10681684

Order 768.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×D₄	C₂	semidirect
2. C₂³×C₃×D₈	C₂	direct (N × H)
3. C₂⁴×C₃×C₈	C₂	semidirect
4. C₂⁴×D₈	C₃	direct (N × H)
7. C₂³×C₃×D₄	C₂²	semidirect
9. C₂²×C₃×D₈	C₂²	direct (N × H)
10. C₂³×C₃×C₈	C₂²	semidirect
11. C₂⁴×D₄	C₂×C₃	semidirect
12. C₂³×D₈	C₂×C₃	direct (N × H)
13. C₂⁴×C₈	C₂×C₃	semidirect
18. C₂²×C₃×D₄	C₂³	semidirect
21. C₂×C₃×D₈	C₂³	direct (N × H)
22. C₂²×C₃×C₈	C₂³	semidirect
25. C₂³×D₄	C₂²×C₃	semidirect
27. C₂²×D₈	C₂²×C₃	direct (N × H)
28. C₂³×C₈	C₂²×C₃	semidirect
31. C₂⁴×C₃	D₈	direct (N × H)
34. C₂×C₃×D₄	C₂⁴	semidirect
37. C₃×D₈	C₂⁴	direct (N × H)
38. C₂×C₃×C₈	C₂⁴	semidirect
43. C₂²×D₄	C₂³×C₃	semidirect
46. C₂×D₈	C₂³×C₃	direct (N × H)
47. C₂²×C₈	C₂³×C₃	semidirect
50. C₂³×C₃	C₂×D₈	direct (N × H)
53. C₃×D₄	C₂⁵	semidirect
56. C₃×C₈	C₂⁵	semidirect
59. C₂⁴	C₃×D₈	direct (N × H)
62. C₂×D₄	C₂⁴×C₃	semidirect
65. D₈	C₂⁴×C₃	direct (N × H)
66. C₂×C₈	C₂⁴×C₃	semidirect
69. C₂²×C₃	C₂²×D₈	direct (N × H)
75. C₂³	C₂×C₃×D₈	direct (N × H)
78. D₄	C₂⁵×C₃	semidirect
81. C₈	C₂⁵×C₃	semidirect
83. C₂×C₃	C₂³×D₈	direct (N × H)
87. C₂²	C₂²×C₃×D₈	direct (N × H)
91. C₃	C₂⁴×D₈	direct (N × H)
93. C₂	C₂³×C₃×D₈	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×C₃×D₄	2
C₂³×C₃×D₈	2
C₂⁴×C₃×C₈	2
C₂⁴×D₈	3

Subgroups — C24×C3×D8 #10681684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×D₈ #10681684