Subgroups — C₃²×#585 #10596769

Order 576.

2430 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×2₋¹⁺⁴	C₂	1	6	yes	maximal, complemented
2	C₃²×H(5,𝔽₂)	C₂	1	10	yes	maximal, complemented
3	C₂×C₃²×#41	C₂	1	15	yes	maximal, complemented
4	C₃×#585	C₃	1	4	yes	maximal, direct product factor
5	C₃²×#41	C₂²	1	20	yes	complemented
6	C₃²×#41	C₂²	1	60	yes	complemented
7	C₂×C₃²×Q₈	C₂²	1	15	yes	complemented
8	C₂×C₃²×D₄	C₂²	1	45	yes	complemented
9	C₂²×C₃²×C₄	C₂²	1	15	yes	complemented
10	C₃×2₋¹⁺⁴	C₂×C₃	1	24	yes	complemented
11	C₃×H(5,𝔽₂)	C₂×C₃	1	40	yes	complemented
12	C₂×C₃×#41	C₂×C₃	1	60	yes	complemented
13	C₂³×C₃²	C₂³	1	15	yes	—
14	C₃²×Q₈	C₂³	1	20	yes	—
15	C₃²×D₄	C₂³	1	60	yes	—
16	C₂×C₃²×C₄	C₂³	1	45	yes	—
17	C₂×C₃²×C₄	C₂³	1	15	yes	—
18	#585	C₃²	1	1	yes	characteristic, direct product factor, Sylow
19	C₃×#41	C₂²×C₃	1	80	yes	complemented
20	C₃×#41	C₂²×C₃	1	240	yes	complemented
21	C₂×C₃×Q₈	C₂²×C₃	1	60	yes	complemented
22	C₂×C₃×D₄	C₂²×C₃	1	180	yes	complemented
23	C₂²×C₃×C₄	C₂²×C₃	1	60	yes	complemented
24	C₂²×C₃²	—	4	15	no	—
25	C₂²×C₃²	C₂⁴	1	15	yes	—
26	C₃²×C₄	C₂⁴	1	15	yes	—
27	C₃²×C₄	C₂⁴	1	1	yes	characteristic
28	2₋¹⁺⁴	C₂×C₃²	1	6	yes	complemented
29	H(5,𝔽₂)	C₂×C₃²	1	10	yes	complemented
30	C₂×#41	C₂×C₃²	1	15	yes	complemented
31	C₂³×C₃	C₂³×C₃	1	60	yes	—
32	C₃×Q₈	C₂³×C₃	1	80	yes	—
33	C₃×D₄	C₂³×C₃	1	240	yes	—
34	C₂×C₃×C₄	C₂³×C₃	1	60	yes	—
35	C₂×C₃×C₄	C₂³×C₃	1	180	yes	—
36	C₂×C₃²	—	2	15	no	—
37	C₂×C₃²	C₂⁵	1	1	yes	characteristic
38	#41	C₂²×C₃²	1	20	yes	complemented
39	#41	C₂²×C₃²	1	60	yes	complemented
40	C₂×Q₈	C₂²×C₃²	1	15	yes	complemented
41	C₂×D₄	C₂²×C₃²	1	45	yes	complemented
42	C₂²×C₄	C₂²×C₃²	1	15	yes	complemented
43	C₂²×C₃	—	4	60	no	—
44	C₂²×C₃	C₂⁴×C₃	1	60	yes	—
45	C₃×C₄	C₂⁴×C₃	1	4	yes	—
46	C₃×C₄	C₂⁴×C₃	1	60	yes	—
47	C₃²	#585	1	1	yes	characteristic, direct product factor, Sylow
48	C₂³	C₂³×C₃²	1	15	yes	—
49	Q₈	C₂³×C₃²	1	20	yes	—
50	D₄	C₂³×C₃²	1	60	yes	—
51	C₂×C₄	C₂³×C₃²	1	15	yes	—
52	C₂×C₄	C₂³×C₃²	1	45	yes	—
53	C₂×C₃	—	2	60	no	—
54	C₂×C₃	C₂⁵×C₃	1	4	yes	—
55	C₂²	—	4	15	no	—
56	C₂²	C₂⁴×C₃²	1	15	yes	—
57	C₄	C₂⁴×C₃²	1	1	yes	characteristic
58	C₄	C₂⁴×C₃²	1	15	yes	—
59	C₃	C₃×#585	1	4	yes	direct product factor
60	C₂	—	2	15	no	—
61	C₂	C₂⁵×C₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₃²
socle	C₂×C₃²	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×2₋¹⁺⁴	C₂	semidirect
2. C₃²×H(5,𝔽₂)	C₂	semidirect
3. C₂×C₃²×#41	C₂	semidirect
4. C₃×#585	C₃	direct (N × H)
5. C₃²×#41	C₂²	semidirect
6. C₃²×#41	C₂²	semidirect
7. C₂×C₃²×Q₈	C₂²	semidirect
8. C₂×C₃²×D₄	C₂²	semidirect
9. C₂²×C₃²×C₄	C₂²	semidirect
10. C₃×2₋¹⁺⁴	C₂×C₃	semidirect
11. C₃×H(5,𝔽₂)	C₂×C₃	semidirect
12. C₂×C₃×#41	C₂×C₃	semidirect
18. #585	C₃²	direct (N × H)
19. C₃×#41	C₂²×C₃	semidirect
20. C₃×#41	C₂²×C₃	semidirect
21. C₂×C₃×Q₈	C₂²×C₃	semidirect
22. C₂×C₃×D₄	C₂²×C₃	semidirect
23. C₂²×C₃×C₄	C₂²×C₃	semidirect
28. 2₋¹⁺⁴	C₂×C₃²	semidirect
29. H(5,𝔽₂)	C₂×C₃²	semidirect
30. C₂×#41	C₂×C₃²	semidirect
38. #41	C₂²×C₃²	semidirect
39. #41	C₂²×C₃²	semidirect
40. C₂×Q₈	C₂²×C₃²	semidirect
41. C₂×D₄	C₂²×C₃²	semidirect
42. C₂²×C₄	C₂²×C₃²	semidirect
47. C₃²	#585	direct (N × H)
59. C₃	C₃×#585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×2₋¹⁺⁴	2
C₃²×H(5,𝔽₂)	2
C₂×C₃²×#41	2
C₃×#585	3

Subgroups — C32×#585 #10596769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#585 #10596769