Subgroups — C₂⁴×C₃²×D₄ #11770637

Order 1152.

31656 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₃²	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃²×D₄	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₃×D₄	C₃	1	4	yes	maximal, direct product factor
5	C₂⁵×C₃²	—	2	32	no	—
6	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
7	C₂⁵×C₃²	C₂²	1	60	yes	complemented
8	C₂²×C₃²×D₄	C₂²	1	560	yes	direct product factor
9	C₂³×C₃²×C₄	C₂²	1	30	yes	complemented
10	C₂⁶×C₃	C₂×C₃	1	8	yes	complemented
11	C₂³×C₃×D₄	C₂×C₃	1	240	yes	direct product factor
12	C₂⁴×C₃×C₄	C₂×C₃	1	4	yes	complemented
13	C₂⁴×C₃²	—	2	480	no	—
14	C₂⁴×C₃²	D₄	1	16	yes	direct product factor
15	C₂⁴×C₃²	C₂³	1	15	yes	—
16	C₂⁴×C₃²	C₂³	1	280	yes	complemented
17	C₂×C₃²×D₄	C₂³	1	960	yes	direct product factor
18	C₂²×C₃²×C₄	C₂³	1	140	yes	complemented
19	C₂⁴×D₄	C₃²	1	1	yes	characteristic, direct product factor, Sylow
20	C₂⁵×C₃	—	2	128	no	—
21	C₂⁵×C₃	C₂²×C₃	1	4	yes	—
22	C₂⁵×C₃	C₂²×C₃	1	240	yes	complemented
23	C₂²×C₃×D₄	C₂²×C₃	1	2240	yes	direct product factor
24	C₂³×C₃×C₄	C₂²×C₃	1	120	yes	complemented
25	C₂³×C₃²	—	2	1120	no	—
26	C₂³×C₃²	C₂×D₄	1	120	yes	direct product factor
27	C₂³×C₃²	C₂⁴	1	35	yes	—
28	C₂³×C₃²	C₂⁴	1	240	yes	complemented
29	C₃²×D₄	C₂⁴	1	256	yes	direct product factor
30	C₂×C₃²×C₄	C₂⁴	1	120	yes	complemented
31	C₂⁶	C₂×C₃²	1	2	yes	complemented
32	C₂³×D₄	C₂×C₃²	1	60	yes	direct product factor
33	C₂⁴×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
34	C₂⁴×C₃	—	2	1920	no	—
35	C₂⁴×C₃	C₃×D₄	1	64	yes	direct product factor
36	C₂⁴×C₃	C₂³×C₃	1	60	yes	—
37	C₂⁴×C₃	C₂³×C₃	1	1120	yes	complemented
38	C₂×C₃×D₄	C₂³×C₃	1	3840	yes	direct product factor
39	C₂²×C₃×C₄	C₂³×C₃	1	560	yes	complemented
40	C₂²×C₃²	—	2	480	no	—
41	C₂²×C₃²	C₂²×D₄	1	140	yes	direct product factor
42	C₂²×C₃²	C₂⁵	1	15	yes	—
43	C₂²×C₃²	C₂⁵	1	32	yes	complemented
44	C₃²×C₄	C₂⁵	1	16	yes	complemented
45	C₂⁵	—	2	32	no	—
46	C₂⁵	C₂²×C₃²	1	1	yes	characteristic
47	C₂⁵	C₂²×C₃²	1	60	yes	complemented
48	C₂²×D₄	C₂²×C₃²	1	560	yes	direct product factor
49	C₂³×C₄	C₂²×C₃²	1	30	yes	complemented
50	C₂³×C₃	—	2	4480	no	—
51	C₂³×C₃	C₂×C₃×D₄	1	480	yes	direct product factor
52	C₂³×C₃	C₂⁴×C₃	1	140	yes	—
53	C₂³×C₃	C₂⁴×C₃	1	960	yes	complemented
54	C₃×D₄	C₂⁴×C₃	1	1024	yes	direct product factor
55	C₂×C₃×C₄	C₂⁴×C₃	1	480	yes	complemented
56	C₂×C₃²	—	2	32	no	—
57	C₂×C₃²	C₂³×D₄	1	30	yes	direct product factor
58	C₂×C₃²	C₂⁶	1	1	yes	characteristic
59	C₂⁴	—	2	480	no	—
60	C₂⁴	C₃²×D₄	1	16	yes	direct product factor
61	C₂⁴	C₂³×C₃²	1	15	yes	—
62	C₂⁴	C₂³×C₃²	1	280	yes	complemented
63	C₂×D₄	C₂³×C₃²	1	960	yes	direct product factor
64	C₂²×C₄	C₂³×C₃²	1	140	yes	complemented
65	C₂²×C₃	—	2	1920	no	—
66	C₂²×C₃	C₂²×C₃×D₄	1	560	yes	direct product factor
67	C₂²×C₃	C₂⁵×C₃	1	60	yes	—
68	C₂²×C₃	C₂⁵×C₃	1	128	yes	complemented
69	C₃×C₄	C₂⁵×C₃	1	64	yes	complemented
70	C₃²	C₂⁴×D₄	1	1	yes	characteristic, direct product factor, Sylow
71	C₂³	—	2	1120	no	—
72	C₂³	C₂×C₃²×D₄	1	120	yes	direct product factor
73	C₂³	C₂⁴×C₃²	1	35	yes	—
74	C₂³	C₂⁴×C₃²	1	240	yes	complemented
75	D₄	C₂⁴×C₃²	1	256	yes	direct product factor
76	C₂×C₄	C₂⁴×C₃²	1	120	yes	complemented
77	C₂×C₃	—	2	128	no	—
78	C₂×C₃	C₂³×C₃×D₄	1	120	yes	direct product factor
79	C₂×C₃	C₂⁶×C₃	1	4	yes	—
80	C₂²	—	2	480	no	—
81	C₂²	C₂²×C₃²×D₄	1	140	yes	direct product factor
82	C₂²	C₂⁵×C₃²	1	15	yes	—
83	C₂²	C₂⁵×C₃²	1	32	yes	complemented
84	C₄	C₂⁵×C₃²	1	16	yes	complemented
85	C₃	C₂⁴×C₃×D₄	1	4	yes	direct product factor
86	C₂	—	2	32	no	—
87	C₂	C₂³×C₃²×D₄	1	30	yes	direct product factor
88	C₂	C₂⁶×C₃²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₃²	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₃²	C₂	semidirect
2. C₂³×C₃²×D₄	C₂	direct (N × H)
3. C₂⁴×C₃²×C₄	C₂	semidirect
4. C₂⁴×C₃×D₄	C₃	direct (N × H)
7. C₂⁵×C₃²	C₂²	semidirect
8. C₂²×C₃²×D₄	C₂²	direct (N × H)
9. C₂³×C₃²×C₄	C₂²	semidirect
10. C₂⁶×C₃	C₂×C₃	semidirect
11. C₂³×C₃×D₄	C₂×C₃	direct (N × H)
12. C₂⁴×C₃×C₄	C₂×C₃	semidirect
14. C₂⁴×C₃²	D₄	direct (N × H)
16. C₂⁴×C₃²	C₂³	semidirect
17. C₂×C₃²×D₄	C₂³	direct (N × H)
18. C₂²×C₃²×C₄	C₂³	semidirect
19. C₂⁴×D₄	C₃²	direct (N × H)
22. C₂⁵×C₃	C₂²×C₃	semidirect
23. C₂²×C₃×D₄	C₂²×C₃	direct (N × H)
24. C₂³×C₃×C₄	C₂²×C₃	semidirect
26. C₂³×C₃²	C₂×D₄	direct (N × H)
28. C₂³×C₃²	C₂⁴	semidirect
29. C₃²×D₄	C₂⁴	direct (N × H)
30. C₂×C₃²×C₄	C₂⁴	semidirect
31. C₂⁶	C₂×C₃²	semidirect
32. C₂³×D₄	C₂×C₃²	direct (N × H)
33. C₂⁴×C₄	C₂×C₃²	semidirect
35. C₂⁴×C₃	C₃×D₄	direct (N × H)
37. C₂⁴×C₃	C₂³×C₃	semidirect
38. C₂×C₃×D₄	C₂³×C₃	direct (N × H)
39. C₂²×C₃×C₄	C₂³×C₃	semidirect
41. C₂²×C₃²	C₂²×D₄	direct (N × H)
43. C₂²×C₃²	C₂⁵	semidirect
44. C₃²×C₄	C₂⁵	semidirect
47. C₂⁵	C₂²×C₃²	semidirect
48. C₂²×D₄	C₂²×C₃²	direct (N × H)
49. C₂³×C₄	C₂²×C₃²	semidirect
51. C₂³×C₃	C₂×C₃×D₄	direct (N × H)
53. C₂³×C₃	C₂⁴×C₃	semidirect
54. C₃×D₄	C₂⁴×C₃	direct (N × H)
55. C₂×C₃×C₄	C₂⁴×C₃	semidirect
57. C₂×C₃²	C₂³×D₄	direct (N × H)
60. C₂⁴	C₃²×D₄	direct (N × H)
62. C₂⁴	C₂³×C₃²	semidirect
63. C₂×D₄	C₂³×C₃²	direct (N × H)
64. C₂²×C₄	C₂³×C₃²	semidirect
66. C₂²×C₃	C₂²×C₃×D₄	direct (N × H)
68. C₂²×C₃	C₂⁵×C₃	semidirect
69. C₃×C₄	C₂⁵×C₃	semidirect
70. C₃²	C₂⁴×D₄	direct (N × H)
72. C₂³	C₂×C₃²×D₄	direct (N × H)
74. C₂³	C₂⁴×C₃²	semidirect
75. D₄	C₂⁴×C₃²	direct (N × H)
76. C₂×C₄	C₂⁴×C₃²	semidirect
78. C₂×C₃	C₂³×C₃×D₄	direct (N × H)
81. C₂²	C₂²×C₃²×D₄	direct (N × H)
83. C₂²	C₂⁵×C₃²	semidirect
84. C₄	C₂⁵×C₃²	semidirect
85. C₃	C₂⁴×C₃×D₄	direct (N × H)
87. C₂	C₂³×C₃²×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₃²	2
C₂³×C₃²×D₄	2
C₂⁴×C₃²×C₄	2
C₂⁴×C₃×D₄	3

Subgroups — C24×C32×D4 #11770637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃²×D₄ #11770637