Subgroups — C₂⁴×D₃×C₉ #11724078

Order 864.

2378 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₃×C₉	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃×D₃	C₃	1	1	yes	maximal, characteristic
4	C₂⁵×C₉	—	3	1	no	maximal
5	C₂³×C₃×C₉	C₂²	1	15	yes	complemented
6	C₂²×D₃×C₉	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₃²	C₂×C₃	1	1	yes	characteristic
8	C₂³×C₃×D₃	C₂×C₃	1	30	yes	—
9	C₂⁴×C₉	—	2	1	no	—
10	C₂⁴×C₉	—	3	30	no	—
11	C₂⁴×C₉	D₃	1	1	yes	characteristic, direct product factor
12	C₂²×C₃×C₉	C₂³	1	35	yes	complemented
13	C₂×D₃×C₉	C₂³	1	120	yes	direct product factor
14	C₂⁵×C₃	—	3	1	no	—
15	C₂⁴×D₃	C₉	1	1	yes	characteristic, direct product factor
16	C₂³×C₃²	C₂²×C₃	1	15	yes	—
17	C₂²×C₃×D₃	C₂²×C₃	1	140	yes	—
18	C₂³×C₉	—	2	15	no	—
19	C₂³×C₉	—	3	140	no	—
20	C₂³×C₉	C₂×D₃	1	15	yes	direct product factor
21	C₂×C₃×C₉	C₂⁴	1	15	yes	complemented
22	D₃×C₉	C₂⁴	1	16	yes	direct product factor
23	C₂⁴×C₃	—	2	1	no	—
24	C₂⁴×C₃	—	3	30	no	—
25	C₂⁴×C₃	C₂×C₉	1	1	yes	characteristic, complemented
26	C₂⁴×C₃	C₃×D₃	1	1	yes	characteristic
27	C₂³×D₃	C₂×C₉	1	30	yes	direct product factor
28	C₂²×C₃²	C₂³×C₃	1	35	yes	—
29	C₂×C₃×D₃	C₂³×C₃	1	120	yes	—
30	C₂²×C₉	—	2	35	no	—
31	C₂²×C₉	—	3	120	no	—
32	C₂²×C₉	C₂²×D₃	1	35	yes	direct product factor
33	C₂⁵	—	3	1	no	Sylow
34	C₃×C₉	C₂⁵	1	1	yes	characteristic, complemented, Sylow
35	C₂³×C₃	—	2	15	no	—
36	C₂³×C₃	—	3	140	no	—
37	C₂³×C₃	C₂²×C₉	1	15	yes	complemented
38	C₂³×C₃	C₂×C₃×D₃	1	15	yes	—
39	C₂²×D₃	C₂²×C₉	1	140	yes	direct product factor
40	C₂×C₃²	C₂⁴×C₃	1	15	yes	—
41	C₃×D₃	C₂⁴×C₃	1	16	yes	—
42	C₂×C₉	—	2	15	no	—
43	C₂×C₉	—	3	16	no	—
44	C₂×C₉	C₂³×D₃	1	15	yes	direct product factor
45	C₂⁴	—	3	30	no	—
46	C₂⁴	D₃×C₉	1	1	yes	characteristic, direct product factor
47	C₂²×C₃	—	2	35	no	—
48	C₂²×C₃	—	3	120	no	—
49	C₂²×C₃	C₂³×C₉	1	35	yes	complemented
50	C₂²×C₃	C₂²×C₃×D₃	1	35	yes	—
51	C₂×D₃	C₂³×C₉	1	120	yes	direct product factor
52	C₃²	C₂⁵×C₃	1	1	yes	characteristic
53	C₉	—	2	1	no	—
54	C₉	C₂⁴×D₃	1	1	yes	characteristic, direct product factor
55	C₂³	—	3	140	no	—
56	C₂³	C₂×D₃×C₉	1	15	yes	direct product factor
57	C₂×C₃	—	2	15	no	—
58	C₂×C₃	—	3	16	no	—
59	C₂×C₃	C₂⁴×C₉	1	15	yes	complemented
60	C₂×C₃	C₂³×C₃×D₃	1	15	yes	—
61	D₃	C₂⁴×C₉	1	16	yes	direct product factor
62	C₂²	—	3	120	no	—
63	C₂²	C₂²×D₃×C₉	1	35	yes	direct product factor
64	C₃	—	2	1	no	—
65	C₃	C₂⁵×C₉	1	1	yes	characteristic, complemented
66	C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
67	C₂	—	3	16	no	—
68	C₂	C₂³×D₃×C₉	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴×C₉	D₃
commutator, hypocenter	C₃	C₂⁵×C₉
Frattini	C₃	C₂⁴×C₃×D₃
Fitting	C₂⁴×C₃×C₉	C₂
socle	C₂⁴×C₃²	C₂×C₃
prime radical	C₂⁴×C₃×D₃	C₃
cosocle	C₃²	C₂⁵×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁴×C₉

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₉	C₂	semidirect
2. C₂³×D₃×C₉	C₂	direct (N × H)
5. C₂³×C₃×C₉	C₂²	semidirect
6. C₂²×D₃×C₉	C₂²	direct (N × H)
11. C₂⁴×C₉	D₃	direct (N × H)
12. C₂²×C₃×C₉	C₂³	semidirect
13. C₂×D₃×C₉	C₂³	direct (N × H)
15. C₂⁴×D₃	C₉	direct (N × H)
20. C₂³×C₉	C₂×D₃	direct (N × H)
21. C₂×C₃×C₉	C₂⁴	semidirect
22. D₃×C₉	C₂⁴	direct (N × H)
25. C₂⁴×C₃	C₂×C₉	semidirect
27. C₂³×D₃	C₂×C₉	direct (N × H)
32. C₂²×C₉	C₂²×D₃	direct (N × H)
34. C₃×C₉	C₂⁵	semidirect
37. C₂³×C₃	C₂²×C₉	semidirect
39. C₂²×D₃	C₂²×C₉	direct (N × H)
44. C₂×C₉	C₂³×D₃	direct (N × H)
46. C₂⁴	D₃×C₉	direct (N × H)
49. C₂²×C₃	C₂³×C₉	semidirect
51. C₂×D₃	C₂³×C₉	direct (N × H)
54. C₉	C₂⁴×D₃	direct (N × H)
56. C₂³	C₂×D₃×C₉	direct (N × H)
59. C₂×C₃	C₂⁴×C₉	semidirect
61. D₃	C₂⁴×C₉	direct (N × H)
63. C₂²	C₂²×D₃×C₉	direct (N × H)
65. C₃	C₂⁵×C₉	semidirect
68. C₂	C₂³×D₃×C₉	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×C₉	2
C₂³×D₃×C₉	2
C₂⁴×C₃×D₃	3
C₂⁵×C₉	3

Subgroups — C24×D3×C9 #11724078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₃×C₉ #11724078