Subgroups — C₂⁴×C₃×D₉ #11724077

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×C₃×D₉	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃×D₃	—	3	1	no	maximal
4	C₂⁴×D₉	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂³×C₃×C₉	C₂²	1	15	yes	complemented
6	C₂²×C₃×D₉	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₃²	D₃	1	1	yes	characteristic
8	C₂³×C₃×D₃	—	3	30	no	—
9	C₂⁴×C₉	—	2	1	no	—
10	C₂⁴×C₉	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂³×D₉	C₂×C₃	1	30	yes	direct product factor
12	C₂²×C₃×C₉	C₂³	1	35	yes	complemented
13	C₂×C₃×D₉	C₂³	1	120	yes	direct product factor
14	C₂⁵×C₃	—	9	1	no	—
15	C₂⁴×D₃	—	3	1	no	—
16	C₂³×C₃²	C₂×D₃	1	15	yes	—
17	C₂²×C₃×D₃	—	3	140	no	—
18	C₂³×C₉	—	2	15	no	—
19	C₂³×C₉	C₂²×C₃	1	15	yes	complemented
20	C₂²×D₉	C₂²×C₃	1	140	yes	direct product factor
21	C₂×C₃×C₉	C₂⁴	1	15	yes	complemented
22	C₃×D₉	C₂⁴	1	16	yes	direct product factor
23	C₂⁴×C₃	—	2	1	no	—
24	C₂⁴×C₃	—	9	30	no	—
25	C₂⁴×C₃	D₉	1	1	yes	characteristic, direct product factor
26	C₂⁴×C₃	C₃×D₃	1	1	yes	characteristic
27	C₂³×D₃	—	3	30	no	—
28	C₂²×C₃²	C₂²×D₃	1	35	yes	—
29	C₂×C₃×D₃	—	3	120	no	—
30	C₂²×C₉	—	2	35	no	—
31	C₂²×C₉	C₂³×C₃	1	35	yes	complemented
32	C₂×D₉	C₂³×C₃	1	120	yes	direct product factor
33	C₂⁵	—	9	1	no	Sylow
34	C₃×C₉	C₂⁵	1	1	yes	characteristic, complemented, Sylow
35	C₂³×C₃	—	2	15	no	—
36	C₂³×C₃	—	9	140	no	—
37	C₂³×C₃	C₂×D₉	1	15	yes	direct product factor
38	C₂³×C₃	C₂×C₃×D₃	1	15	yes	—
39	C₂²×D₃	—	3	140	no	—
40	C₂×C₃²	C₂³×D₃	1	15	yes	—
41	C₃×D₃	—	3	16	no	—
42	C₂×C₉	—	2	15	no	—
43	C₂×C₉	C₂⁴×C₃	1	15	yes	complemented
44	D₉	C₂⁴×C₃	1	16	yes	direct product factor
45	C₂⁴	—	9	30	no	—
46	C₂⁴	C₃×D₉	1	1	yes	characteristic, direct product factor
47	C₂²×C₃	—	2	35	no	—
48	C₂²×C₃	—	9	120	no	—
49	C₂²×C₃	C₂²×D₉	1	35	yes	direct product factor
50	C₂²×C₃	C₂²×C₃×D₃	1	35	yes	—
51	C₂×D₃	—	3	120	no	—
52	C₃²	C₂⁴×D₃	1	1	yes	characteristic
53	C₉	—	2	1	no	—
54	C₉	C₂⁵×C₃	1	1	yes	characteristic, complemented
55	C₂³	—	9	140	no	—
56	C₂³	C₂×C₃×D₉	1	15	yes	direct product factor
57	C₂×C₃	—	2	15	no	—
58	C₂×C₃	—	9	16	no	—
59	C₂×C₃	C₂³×D₉	1	15	yes	direct product factor
60	C₂×C₃	C₂³×C₃×D₃	1	15	yes	—
61	D₃	—	3	16	no	—
62	C₂²	—	9	120	no	—
63	C₂²	C₂²×C₃×D₉	1	35	yes	direct product factor
64	C₃	—	2	1	no	—
65	C₃	C₂⁴×D₉	1	1	yes	characteristic, direct product factor
66	C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
67	C₂	—	9	16	no	—
68	C₂	C₂³×C₃×D₉	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴×C₃	D₉
commutator, cosocle, hypocenter	C₉	C₂⁵×C₃
Frattini	C₃	C₂⁴×C₃×D₃
Fitting	C₂⁴×C₃×C₉	C₂
socle	C₂⁴×C₃²	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₉	C₂	semidirect
2. C₂³×C₃×D₉	C₂	direct (N × H)
4. C₂⁴×D₉	C₃	direct (N × H)
5. C₂³×C₃×C₉	C₂²	semidirect
6. C₂²×C₃×D₉	C₂²	direct (N × H)
10. C₂⁴×C₉	C₂×C₃	semidirect
11. C₂³×D₉	C₂×C₃	direct (N × H)
12. C₂²×C₃×C₉	C₂³	semidirect
13. C₂×C₃×D₉	C₂³	direct (N × H)
19. C₂³×C₉	C₂²×C₃	semidirect
20. C₂²×D₉	C₂²×C₃	direct (N × H)
21. C₂×C₃×C₉	C₂⁴	semidirect
22. C₃×D₉	C₂⁴	direct (N × H)
25. C₂⁴×C₃	D₉	direct (N × H)
31. C₂²×C₉	C₂³×C₃	semidirect
32. C₂×D₉	C₂³×C₃	direct (N × H)
34. C₃×C₉	C₂⁵	semidirect
37. C₂³×C₃	C₂×D₉	direct (N × H)
43. C₂×C₉	C₂⁴×C₃	semidirect
44. D₉	C₂⁴×C₃	direct (N × H)
46. C₂⁴	C₃×D₉	direct (N × H)
49. C₂²×C₃	C₂²×D₉	direct (N × H)
54. C₉	C₂⁵×C₃	semidirect
56. C₂³	C₂×C₃×D₉	direct (N × H)
59. C₂×C₃	C₂³×D₉	direct (N × H)
63. C₂²	C₂²×C₃×D₉	direct (N × H)
65. C₃	C₂⁴×D₉	direct (N × H)
68. C₂	C₂³×C₃×D₉	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×C₃×D₉	2
C₂⁴×C₃×D₃	3
C₂⁴×D₉	3

Subgroups — C24×C3×D9 #11724077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×D₉ #11724077