Subgroups — C₂⁴×C₃⋊(C₃⋊Dic₃) #13181768

Order 1728.

19068 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃³	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊(C₃⋊Dic₃)	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃⋊Dic₃	—	3	13	no	maximal
4	C₂⁴×C₃³	C₄	1	16	yes	complemented
5	C₂⁴×C₃³	C₂²	1	15	yes	—
6	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	140	yes	direct product factor
7	C₂⁵×C₃²	D₃	1	13	yes	—
8	C₂³×C₃⋊Dic₃	—	3	390	no	—
9	C₂³×C₃³	C₂×C₄	1	120	yes	complemented
10	C₂³×C₃³	C₂³	1	35	yes	—
11	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	120	yes	direct product factor
12	C₂⁴×Dic₃	—	9	13	no	—
13	C₂⁴×C₃²	Dic₃	1	208	yes	complemented
14	C₂⁴×C₃²	C₂×D₃	1	195	yes	—
15	C₂²×C₃⋊Dic₃	—	3	1820	no	—
16	C₂²×C₃³	C₂²×C₄	1	140	yes	complemented
17	C₂²×C₃³	C₂⁴	1	15	yes	—
18	C₃⋊(C₃⋊Dic₃)	C₂⁴	1	16	yes	direct product factor
19	C₂⁵×C₃	C₃⋊D₃	1	13	yes	—
20	C₂³×Dic₃	—	9	390	no	—
21	C₂³×C₃²	C₂×Dic₃	1	1560	yes	complemented
22	C₂³×C₃²	C₂²×D₃	1	455	yes	—
23	C₂×C₃⋊Dic₃	—	3	1560	no	—
24	C₂⁴×C₄	—	27	1	no	Sylow
25	C₂×C₃³	C₂³×C₄	1	30	yes	complemented
26	C₂×C₃³	C₂⁵	1	1	yes	characteristic
27	C₂⁴×C₃	C₃⋊Dic₃	1	208	yes	complemented
28	C₂⁴×C₃	C₂×C₃⋊D₃	1	195	yes	—
29	C₂²×Dic₃	—	9	1820	no	—
30	C₂²×C₃²	C₂²×Dic₃	1	1820	yes	complemented
31	C₂²×C₃²	C₂³×D₃	1	195	yes	—
32	C₃⋊Dic₃	—	3	208	no	—
33	C₂⁵	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
34	C₂³×C₄	—	27	30	no	—
35	C₃³	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₃	C₂×C₃⋊Dic₃	1	1560	yes	complemented
37	C₂³×C₃	C₂²×C₃⋊D₃	1	455	yes	—
38	C₂×Dic₃	—	9	1560	no	—
39	C₂×C₃²	C₂³×Dic₃	1	390	yes	complemented
40	C₂×C₃²	C₂⁴×D₃	1	13	yes	—
41	C₂⁴	C₃⋊(C₃⋊Dic₃)	1	16	yes	direct product factor
42	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	15	yes	—
43	C₂²×C₄	—	27	140	no	—
44	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1820	yes	complemented
45	C₂²×C₃	C₂³×C₃⋊D₃	1	195	yes	—
46	Dic₃	—	9	208	no	—
47	C₃²	C₂⁴×Dic₃	1	13	yes	complemented
48	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	120	yes	direct product factor
49	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	35	yes	—
50	C₂×C₄	—	27	120	no	—
51	C₂×C₃	C₂³×C₃⋊Dic₃	1	390	yes	complemented
52	C₂×C₃	C₂⁴×C₃⋊D₃	1	13	yes	—
53	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	140	yes	direct product factor
54	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	15	yes	—
55	C₄	—	27	16	no	—
56	C₃	C₂⁴×C₃⋊Dic₃	1	13	yes	complemented
57	C₂	C₂³×C₃⋊(C₃⋊Dic₃)	1	30	yes	direct product factor
58	C₂	C₂⁴×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₂⁴×C₄
Frattini	C₂	C₂⁴×C₃⋊(C₃⋊D₃)
Fitting, socle, prime radical	C₂⁵×C₃³	C₂
cosocle	C₂×C₃³	C₂⁵
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊(C₃⋊Dic₃)	C₂	direct (N × H)
4. C₂⁴×C₃³	C₄	semidirect
6. C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	direct (N × H)
9. C₂³×C₃³	C₂×C₄	semidirect
11. C₂×C₃⋊(C₃⋊Dic₃)	C₂³	direct (N × H)
13. C₂⁴×C₃²	Dic₃	semidirect
16. C₂²×C₃³	C₂²×C₄	semidirect
18. C₃⋊(C₃⋊Dic₃)	C₂⁴	direct (N × H)
21. C₂³×C₃²	C₂×Dic₃	semidirect
25. C₂×C₃³	C₂³×C₄	semidirect
27. C₂⁴×C₃	C₃⋊Dic₃	semidirect
30. C₂²×C₃²	C₂²×Dic₃	semidirect
35. C₃³	C₂⁴×C₄	semidirect
36. C₂³×C₃	C₂×C₃⋊Dic₃	semidirect
39. C₂×C₃²	C₂³×Dic₃	semidirect
41. C₂⁴	C₃⋊(C₃⋊Dic₃)	direct (N × H)
44. C₂²×C₃	C₂²×C₃⋊Dic₃	semidirect
47. C₃²	C₂⁴×Dic₃	semidirect
48. C₂³	C₂×C₃⋊(C₃⋊Dic₃)	direct (N × H)
51. C₂×C₃	C₂³×C₃⋊Dic₃	semidirect
53. C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	direct (N × H)
56. C₃	C₂⁴×C₃⋊Dic₃	semidirect
57. C₂	C₂³×C₃⋊(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
C₂⁵×C₃³	2
C₂³×C₃⋊(C₃⋊Dic₃)	2
C₂⁴×C₃⋊Dic₃	3

Subgroups — C24×C3⋊(C3⋊Dic3) #13181768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃⋊(C₃⋊Dic₃) #13181768