Subgroups — C₂³×C₃²×C₈ #10596366

Order 576.

1014 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₈	C₃	1	4	yes	maximal, direct product factor
4	C₂⁴×C₃²	C₄	1	1	yes	characteristic
5	C₂²×C₃²×C₄	C₄	1	7	yes	—
6	C₂²×C₃²×C₄	C₂²	1	7	yes	—
7	C₂×C₃²×C₈	C₂²	1	28	yes	direct product factor
8	C₂³×C₃×C₄	C₂×C₃	1	4	yes	—
9	C₂²×C₃×C₈	C₂×C₃	1	56	yes	direct product factor
10	C₂³×C₃²	C₈	1	8	yes	direct product factor
11	C₂³×C₃²	C₂×C₄	1	7	yes	—
12	C₂×C₃²×C₄	C₂×C₄	1	21	yes	—
13	C₂×C₃²×C₄	C₂³	1	7	yes	—
14	C₃²×C₈	C₂³	1	8	yes	direct product factor
15	C₂³×C₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
16	C₂⁴×C₃	C₃×C₄	1	4	yes	—
17	C₂²×C₃×C₄	C₃×C₄	1	28	yes	—
18	C₂²×C₃×C₄	C₂²×C₃	1	28	yes	—
19	C₂×C₃×C₈	C₂²×C₃	1	112	yes	direct product factor
20	C₂²×C₃²	C₂×C₈	1	28	yes	direct product factor
21	C₂²×C₃²	C₂²×C₄	1	7	yes	—
22	C₃²×C₄	C₂²×C₄	1	7	yes	—
23	C₃²×C₄	C₂⁴	1	1	yes	characteristic
24	C₂³×C₄	C₂×C₃²	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₃²	1	14	yes	direct product factor
26	C₂³×C₃	C₃×C₈	1	32	yes	direct product factor
27	C₂³×C₃	C₂×C₃×C₄	1	28	yes	—
28	C₂×C₃×C₄	C₂×C₃×C₄	1	84	yes	—
29	C₂×C₃×C₄	C₂³×C₃	1	28	yes	—
30	C₃×C₈	C₂³×C₃	1	32	yes	direct product factor
31	C₂×C₃²	C₂²×C₈	1	14	yes	direct product factor
32	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
33	C₂⁴	C₃²×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₃²×C₄	1	7	yes	—
35	C₂²×C₄	C₂²×C₃²	1	7	yes	—
36	C₂×C₈	C₂²×C₃²	1	28	yes	direct product factor
37	C₂²×C₃	C₂×C₃×C₈	1	112	yes	direct product factor
38	C₂²×C₃	C₂²×C₃×C₄	1	28	yes	—
39	C₃×C₄	C₂²×C₃×C₄	1	28	yes	—
40	C₃×C₄	C₂⁴×C₃	1	4	yes	—
41	C₃²	C₂³×C₈	1	1	yes	characteristic, direct product factor, Sylow
42	C₂³	C₃²×C₈	1	8	yes	direct product factor
43	C₂³	C₂×C₃²×C₄	1	7	yes	—
44	C₂×C₄	C₂×C₃²×C₄	1	21	yes	—
45	C₂×C₄	C₂³×C₃²	1	7	yes	—
46	C₈	C₂³×C₃²	1	8	yes	direct product factor
47	C₂×C₃	C₂²×C₃×C₈	1	56	yes	direct product factor
48	C₂×C₃	C₂³×C₃×C₄	1	4	yes	—
49	C₂²	C₂×C₃²×C₈	1	28	yes	direct product factor
50	C₂²	C₂²×C₃²×C₄	1	7	yes	—
51	C₄	C₂²×C₃²×C₄	1	7	yes	—
52	C₄	C₂⁴×C₃²	1	1	yes	characteristic
53	C₃	C₂³×C₃×C₈	1	4	yes	direct product factor
54	C₂	C₂²×C₃²×C₈	1	14	yes	direct product factor
55	C₂	C₂³×C₃²×C₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₈	C₂	direct (N × H)
3. C₂³×C₃×C₈	C₃	direct (N × H)
7. C₂×C₃²×C₈	C₂²	direct (N × H)
9. C₂²×C₃×C₈	C₂×C₃	direct (N × H)
10. C₂³×C₃²	C₈	direct (N × H)
14. C₃²×C₈	C₂³	direct (N × H)
15. C₂³×C₈	C₃²	direct (N × H)
19. C₂×C₃×C₈	C₂²×C₃	direct (N × H)
20. C₂²×C₃²	C₂×C₈	direct (N × H)
25. C₂²×C₈	C₂×C₃²	direct (N × H)
26. C₂³×C₃	C₃×C₈	direct (N × H)
30. C₃×C₈	C₂³×C₃	direct (N × H)
31. C₂×C₃²	C₂²×C₈	direct (N × H)
36. C₂×C₈	C₂²×C₃²	direct (N × H)
37. C₂²×C₃	C₂×C₃×C₈	direct (N × H)
41. C₃²	C₂³×C₈	direct (N × H)
42. C₂³	C₃²×C₈	direct (N × H)
46. C₈	C₂³×C₃²	direct (N × H)
47. C₂×C₃	C₂²×C₃×C₈	direct (N × H)
49. C₂²	C₂×C₃²×C₈	direct (N × H)
53. C₃	C₂³×C₃×C₈	direct (N × H)
54. C₂	C₂²×C₃²×C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₄	2
C₂²×C₃²×C₈	2
C₂³×C₃×C₈	3

Subgroups — C23×C32×C8 #10596366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×C₈ #10596366