Subgroups — C₂³×C₃×Dic₃ #64353

Order 288.

539 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₃	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₄	—	3	1	no	maximal
4	C₂³×Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂³×C₃²	C₄	1	8	yes	complemented
6	C₂³×C₃²	C₂²	1	7	yes	—
7	C₂×C₃×Dic₃	C₂²	1	28	yes	direct product factor
8	C₂⁴×C₃	—	2	1	no	—
9	C₂⁴×C₃	D₃	1	1	yes	characteristic
10	C₂⁴×C₃	C₂×C₃	1	1	yes	characteristic
11	C₂²×C₃×C₄	—	3	14	no	—
12	C₂²×Dic₃	C₂×C₃	1	14	yes	direct product factor
13	C₂²×C₃²	C₂×C₄	1	28	yes	complemented
14	C₂²×C₃²	C₂³	1	7	yes	—
15	C₃×Dic₃	C₂³	1	8	yes	direct product factor
16	C₂³×C₄	—	3	1	no	Sylow
17	C₂³×C₃	—	2	8	no	—
18	C₂³×C₃	—	2	7	no	—
19	C₂³×C₃	Dic₃	1	8	yes	direct product factor
20	C₂³×C₃	C₃×C₄	1	8	yes	complemented
21	C₂³×C₃	C₂×D₃	1	7	yes	—
22	C₂³×C₃	C₂²×C₃	1	7	yes	—
23	C₂×C₃×C₄	—	3	28	no	—
24	C₂×Dic₃	C₂²×C₃	1	28	yes	direct product factor
25	C₂×C₃²	C₂²×C₄	1	14	yes	complemented
26	C₂×C₃²	C₂⁴	1	1	yes	characteristic
27	C₂⁴	C₃×D₃	1	1	yes	characteristic
28	C₂²×C₄	—	3	14	no	—
29	C₂²×C₃	—	2	28	no	—
30	C₂²×C₃	—	2	7	no	—
31	C₂²×C₃	C₂×Dic₃	1	28	yes	direct product factor
32	C₂²×C₃	C₂×C₃×C₄	1	28	yes	complemented
33	C₂²×C₃	C₂²×D₃	1	7	yes	—
34	C₂²×C₃	C₂³×C₃	1	7	yes	—
35	C₃×C₄	—	3	8	no	—
36	Dic₃	C₂³×C₃	1	8	yes	direct product factor
37	C₃²	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
38	C₂³	C₃×Dic₃	1	8	yes	direct product factor
39	C₂³	C₂×C₃×D₃	1	7	yes	—
40	C₂×C₄	—	3	28	no	—
41	C₂×C₃	—	2	1	no	—
42	C₂×C₃	—	2	14	no	—
43	C₂×C₃	C₂²×Dic₃	1	14	yes	direct product factor
44	C₂×C₃	C₂²×C₃×C₄	1	14	yes	complemented
45	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
46	C₂×C₃	C₂⁴×C₃	1	1	yes	characteristic
47	C₂²	C₂×C₃×Dic₃	1	28	yes	direct product factor
48	C₂²	C₂²×C₃×D₃	1	7	yes	—
49	C₄	—	3	8	no	—
50	C₃	—	2	1	no	—
51	C₃	C₂³×Dic₃	1	1	yes	characteristic, direct product factor
52	C₃	C₂³×C₃×C₄	1	1	yes	characteristic, complemented
53	C₂	C₂²×C₃×Dic₃	1	14	yes	direct product factor
54	C₂	C₂³×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₃	D₃
commutator, hypocenter	C₃	C₂³×C₃×C₄
Frattini	C₂	C₂³×C₃×D₃
Fitting, socle, prime radical	C₂⁴×C₃²	C₂
cosocle	C₂×C₃	C₂⁴×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₃	C₂	direct (N × H)
4. C₂³×Dic₃	C₃	direct (N × H)
5. C₂³×C₃²	C₄	semidirect
7. C₂×C₃×Dic₃	C₂²	direct (N × H)
12. C₂²×Dic₃	C₂×C₃	direct (N × H)
13. C₂²×C₃²	C₂×C₄	semidirect
15. C₃×Dic₃	C₂³	direct (N × H)
19. C₂³×C₃	Dic₃	direct (N × H)
20. C₂³×C₃	C₃×C₄	semidirect
24. C₂×Dic₃	C₂²×C₃	direct (N × H)
25. C₂×C₃²	C₂²×C₄	semidirect
31. C₂²×C₃	C₂×Dic₃	direct (N × H)
32. C₂²×C₃	C₂×C₃×C₄	semidirect
36. Dic₃	C₂³×C₃	direct (N × H)
37. C₃²	C₂³×C₄	semidirect
38. C₂³	C₃×Dic₃	direct (N × H)
43. C₂×C₃	C₂²×Dic₃	direct (N × H)
44. C₂×C₃	C₂²×C₃×C₄	semidirect
47. C₂²	C₂×C₃×Dic₃	direct (N × H)
51. C₃	C₂³×Dic₃	direct (N × H)
52. C₃	C₂³×C₃×C₄	semidirect
53. C₂	C₂²×C₃×Dic₃	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×C₃²	2
C₂²×C₃×Dic₃	2
C₂³×C₃×C₄	3
C₂³×Dic₃	3

Subgroups — C23×C3×Dic3 #64353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×Dic₃ #64353