Subgroups — C₃×#576 #5906

Order 192.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₃×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
3	C₃×#135	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×D₈	C₂	1	3	yes	maximal, complemented
5	C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	#576	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#41	—	2	2	no	—
8	C₃×#41	C₂²	1	1	yes	characteristic
9	C₃×#41	C₂²	1	6	yes	complemented
10	C₂×C₃×D₄	—	2	6	no	—
11	C₂×C₃×D₄	C₂²	1	6	yes	complemented
12	C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₃×SD₁₆	C₂²	1	6	yes	complemented
14	C₃×D₈	C₂²	1	3	yes	complemented
15	C₃×D₈	C₂²	1	6	yes	complemented
16	C₃×M₁₆	C₂²	1	3	yes	complemented
17	C₂×C₃×C₈	C₂²	1	3	yes	complemented
18	H(5,𝔽₂)	C₂×C₃	1	2	yes	complemented
19	AGL(1,ℤ/8)	C₂×C₃	1	6	yes	complemented
20	#135	C₂×C₃	1	3	yes	complemented
21	C₂×D₈	C₂×C₃	1	3	yes	complemented
22	#131	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂³×C₃	—	2	6	no	—
24	C₃×Q₈	D₄	1	1	yes	characteristic
25	C₃×Q₈	C₂³	1	2	yes	—
26	C₃×D₄	—	2	6	no	—
27	C₃×D₄	—	2	6	no	—
28	C₃×D₄	D₄	1	3	yes	—
29	C₃×D₄	C₂³	1	6	yes	—
30	C₂×C₃×C₄	—	2	6	no	—
31	C₂×C₃×C₄	C₂³	1	3	yes	—
32	C₃×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₈	C₂³	1	3	yes	—
34	#41	—	2	2	no	—
35	#41	C₂²×C₃	1	1	yes	characteristic
36	#41	C₂²×C₃	1	6	yes	complemented
37	C₂×D₄	—	2	6	no	—
38	C₂×D₄	C₂²×C₃	1	6	yes	complemented
39	Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
40	SD₁₆	C₂²×C₃	1	6	yes	complemented
41	D₈	C₂²×C₃	1	3	yes	complemented
42	D₈	C₂²×C₃	1	6	yes	complemented
43	M₁₆	C₂²×C₃	1	3	yes	complemented
44	C₂×C₈	C₂²×C₃	1	3	yes	complemented
45	C₂²×C₃	—	2	6	no	—
46	C₂²×C₃	—	8	6	no	—
47	C₂²×C₃	C₂×D₄	1	3	yes	—
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	C₂×D₄	1	3	yes	—
50	C₃×C₄	C₂⁴	1	1	yes	characteristic
51	C₂³	—	2	6	no	—
52	Q₈	C₃×D₄	1	1	yes	characteristic
53	Q₈	C₂³×C₃	1	2	yes	—
54	D₄	—	2	6	no	—
55	D₄	—	2	6	no	—
56	D₄	C₃×D₄	1	3	yes	—
57	D₄	C₂³×C₃	1	6	yes	—
58	C₂×C₄	—	2	6	no	—
59	C₂×C₄	C₂³×C₃	1	3	yes	—
60	C₈	C₂³×C₃	1	1	yes	characteristic
61	C₈	C₂³×C₃	1	3	yes	—
62	C₂×C₃	—	2	3	no	—
63	C₂×C₃	—	4	6	no	—
64	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
65	C₂²	—	2	6	no	—
66	C₂²	—	8	6	no	—
67	C₂²	C₂×C₃×D₄	1	3	yes	—
68	C₄	—	2	2	no	—
69	C₄	C₂×C₃×D₄	1	3	yes	—
70	C₄	C₂⁴×C₃	1	1	yes	characteristic
71	C₃	#576	1	1	yes	characteristic, direct product factor, Sylow
72	C₂	—	2	3	no	—
73	C₂	—	4	6	no	—
74	C₂	C₂²×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₂)	C₂	semidirect
2. C₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₃×#135	C₂	semidirect
4. C₂×C₃×D₈	C₂	semidirect
5. C₃×#131	C₂	semidirect
6. #576	C₃	direct (N × H)
9. C₃×#41	C₂²	semidirect
11. C₂×C₃×D₄	C₂²	semidirect
12. C₃×Q₁₆	C₂²	semidirect
13. C₃×SD₁₆	C₂²	semidirect
14. C₃×D₈	C₂²	semidirect
15. C₃×D₈	C₂²	semidirect
16. C₃×M₁₆	C₂²	semidirect
17. C₂×C₃×C₈	C₂²	semidirect
18. H(5,𝔽₂)	C₂×C₃	semidirect
19. AGL(1,ℤ/8)	C₂×C₃	semidirect
20. #135	C₂×C₃	semidirect
21. C₂×D₈	C₂×C₃	semidirect
22. #131	C₂×C₃	semidirect
36. #41	C₂²×C₃	semidirect
38. C₂×D₄	C₂²×C₃	semidirect
39. Q₁₆	C₂²×C₃	semidirect
40. SD₁₆	C₂²×C₃	semidirect
41. D₈	C₂²×C₃	semidirect
42. D₈	C₂²×C₃	semidirect
43. M₁₆	C₂²×C₃	semidirect
44. C₂×C₈	C₂²×C₃	semidirect
71. C₃	#576	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₂)	2
C₃×AGL(1,ℤ/8)	2
C₃×#135	2
C₂×C₃×D₈	2
C₃×#131	2
#576	3

Subgroups — C3×#576 #5906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#576 #5906