Subgroups — C₂³×D₃² #64392

Order 288.

1563 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×D₃	C₂	1	2	yes	maximal, complemented
3	C₂²×D₃²	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×D₃	—	3	2	no	maximal
5	C₂³×C₃²	C₂²	1	1	yes	characteristic, complemented
6	C₂²×C₃⋊D₃	C₂²	1	14	yes	complemented
7	C₂²×C₃×D₃	C₂²	1	28	yes	complemented
8	C₂×D₃²	C₂²	1	112	yes	direct product factor
9	C₂⁴×C₃	—	3	2	no	—
10	C₂³×D₃	—	3	2	no	—
11	C₂³×D₃	—	3	56	no	—
12	C₂³×D₃	—	6	1	no	—
13	C₂³×D₃	D₃	1	2	yes	direct product factor
14	C₂²×C₃²	C₂³	1	7	yes	complemented
15	C₂×C₃⋊D₃	C₂³	1	28	yes	complemented
16	C₂×C₃×D₃	C₂³	1	56	yes	complemented
17	D₃²	C₂³	1	64	yes	direct product factor
18	C₂⁵	—	9	1	no	Sylow
19	C₂³×C₃	—	2	1	no	—
20	C₂³×C₃	—	3	28	no	—
21	C₂³×C₃	C₂×D₃	1	2	yes	complemented
22	C₂²×D₃	—	3	28	no	—
23	C₂²×D₃	—	3	224	no	—
24	C₂²×D₃	—	6	14	no	—
25	C₂²×D₃	C₂×D₃	1	28	yes	direct product factor
26	C₂×C₃²	C₂⁴	1	7	yes	complemented
27	C₃⋊D₃	C₂⁴	1	8	yes	complemented
28	C₃×D₃	C₂⁴	1	16	yes	complemented
29	C₂⁴	—	3	2	no	—
30	C₂⁴	—	9	28	no	—
31	C₂⁴	—	9	1	no	—
32	C₂²×C₃	—	2	7	no	—
33	C₂²×C₃	—	3	56	no	—
34	C₂²×C₃	C₂²×D₃	1	14	yes	complemented
35	C₂×D₃	—	3	56	no	—
36	C₂×D₃	—	3	128	no	—
37	C₂×D₃	—	6	28	no	—
38	C₂×D₃	C₂²×D₃	1	56	yes	direct product factor
39	C₃²	C₂⁵	1	1	yes	characteristic, complemented, Sylow
40	C₂³	—	3	28	no	—
41	C₂³	—	9	112	no	—
42	C₂³	—	9	14	no	—
43	C₂³	D₃²	1	1	yes	characteristic, direct product factor
44	C₂×C₃	—	2	7	no	—
45	C₂×C₃	—	3	16	no	—
46	C₂×C₃	C₂³×D₃	1	14	yes	complemented
47	D₃	—	3	16	no	—
48	D₃	—	6	8	no	—
49	D₃	C₂³×D₃	1	16	yes	direct product factor
50	C₂²	—	3	56	no	—
51	C₂²	—	9	64	no	—
52	C₂²	—	9	28	no	—
53	C₂²	C₂×D₃²	1	7	yes	direct product factor
54	C₃	—	2	1	no	—
55	C₃	C₂⁴×D₃	1	2	yes	complemented
56	C₂	—	3	16	no	—
57	C₂	—	9	8	no	—
58	C₂	C₂²×D₃²	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, cosocle, hypocenter	C₃²	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃⋊D₃	C₂	semidirect
2. C₂³×C₃×D₃	C₂	semidirect
3. C₂²×D₃²	C₂	direct (N × H)
5. C₂³×C₃²	C₂²	semidirect
6. C₂²×C₃⋊D₃	C₂²	semidirect
7. C₂²×C₃×D₃	C₂²	semidirect
8. C₂×D₃²	C₂²	direct (N × H)
13. C₂³×D₃	D₃	direct (N × H)
14. C₂²×C₃²	C₂³	semidirect
15. C₂×C₃⋊D₃	C₂³	semidirect
16. C₂×C₃×D₃	C₂³	semidirect
17. D₃²	C₂³	direct (N × H)
21. C₂³×C₃	C₂×D₃	semidirect
25. C₂²×D₃	C₂×D₃	direct (N × H)
26. C₂×C₃²	C₂⁴	semidirect
27. C₃⋊D₃	C₂⁴	semidirect
28. C₃×D₃	C₂⁴	semidirect
34. C₂²×C₃	C₂²×D₃	semidirect
38. C₂×D₃	C₂²×D₃	direct (N × H)
39. C₃²	C₂⁵	semidirect
43. C₂³	D₃²	direct (N × H)
46. C₂×C₃	C₂³×D₃	semidirect
49. D₃	C₂³×D₃	direct (N × H)
53. C₂²	C₂×D₃²	direct (N × H)
55. C₃	C₂⁴×D₃	semidirect
58. C₂	C₂²×D₃²	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×C₃⋊D₃	2
C₂³×C₃×D₃	2
C₂²×D₃²	2
C₂⁴×D₃	3

Subgroups — C23×D32 #64392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₃² #64392