Subgroups — C₃×C₃²⋊(C₄²⋊C₃) #13047788

Order 1296.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₄²⋊C₃	C₃	1	3	yes	maximal, complemented
3	C₃²⋊(C₄²⋊C₃)	C₃	1	9	yes	maximal, direct product factor
4	C₃×C₂²⋊H(3,𝔽₃)	—	4	1	no	maximal
5	C₂×C₃³×C₄	—	3	1	no	—
6	C₃²×C₄²	—	3	3	no	—
7	C₃²×C₄²	C₃²	1	1	yes	characteristic
8	C₃²×C₄²	C₃²	1	3	yes	complemented
9	C₃×C₄²⋊C₃	—	3	9	no	—
10	C₃×C₄²⋊C₃	C₃²	1	9	yes	complemented
11	C₂²×C₃³	A₄	1	1	yes	characteristic
12	C₃²×A₄	—	4	3	no	—
13	C₃³×C₄	—	3	2	no	—
14	C₂²⋊H(3,𝔽₃)	—	4	9	no	—
15	C₃×H(3,𝔽₃)	—	16	1	no	Sylow
16	C₂×C₃²×C₄	—	3	9	no	—
17	C₂×C₃²×C₄	—	3	1	no	—
18	C₂×C₃²×C₄	—	3	3	no	—
19	C₂×C₃³	—	3	1	no	—
20	C₃×C₄²	—	3	3	no	—
21	C₃×C₄²	H(3,𝔽₃)	1	3	yes	complemented
22	C₃×C₄²	C₃³	1	1	yes	characteristic
23	C₄²⋊C₃	—	3	9	no	—
24	C₂²×C₃²	—	3	3	no	—
25	C₂²×C₃²	C₃×A₄	1	3	yes	—
26	C₂²×C₃²	C₃×A₄	1	1	yes	characteristic
27	C₃×A₄	—	4	9	no	—
28	C₃×A₄	—	12	9	no	—
29	C₃²×C₄	—	3	18	no	—
30	C₃²×C₄	—	3	2	no	—
31	C₃²×C₄	—	3	6	no	—
32	C₃³	—	16	3	no	—
33	C₃³	C₄²⋊C₃	1	1	yes	characteristic, complemented
34	H(3,𝔽₃)	—	16	9	no	—
35	C₂×C₃×C₄	—	3	9	no	—
36	C₂×C₃×C₄	—	3	3	no	—
37	C₂×C₃×C₄	—	3	1	no	—
38	C₂×C₃²	—	3	9	no	—
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	3	3	no	—
41	C₄²	C₃×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₃	—	3	3	no	—
43	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	3	yes	—
44	C₂²×C₃	C₃²×A₄	1	1	yes	characteristic
45	A₄	—	12	9	no	—
46	C₃×C₄	—	3	18	no	—
47	C₃×C₄	—	3	6	no	—
48	C₃×C₄	—	3	2	no	—
49	C₃²	—	3	3	no	—
50	C₃²	—	16	9	no	—
51	C₃²	—	48	9	no	—
52	C₃²	C₃×C₄²⋊C₃	1	1	yes	characteristic
53	C₃²	C₃×C₄²⋊C₃	1	3	yes	complemented
54	C₂×C₄	—	3	1	no	—
55	C₂×C₃	—	3	9	no	—
56	C₂×C₃	—	3	3	no	—
57	C₂×C₃	—	3	1	no	—
58	C₂²	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
59	C₄	—	3	2	no	—
60	C₃	—	3	3	no	—
61	C₃	—	48	9	no	—
62	C₃	C₃²⋊(C₄²⋊C₃)	1	3	yes	direct product factor
63	C₃	C₃²×C₄²⋊C₃	1	1	yes	characteristic
64	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₃×C₄²⋊C₃
commutator, cosocle	C₃×C₄²	C₃³
Frattini	C₂²×C₃	C₃²×A₄
Fitting	C₃³×C₄²	C₃
socle	C₂²×C₃²	C₃×A₄
hypercenter	C₃³	C₄²⋊C₃
hypocenter	C₄²	C₃×H(3,𝔽₃)
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₃² ◁ C₃³

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄²	C₃	semidirect
2. C₃²×C₄²⋊C₃	C₃	semidirect
3. C₃²⋊(C₄²⋊C₃)	C₃	direct (N × H)
8. C₃²×C₄²	C₃²	semidirect
10. C₃×C₄²⋊C₃	C₃²	semidirect
21. C₃×C₄²	H(3,𝔽₃)	semidirect
33. C₃³	C₄²⋊C₃	semidirect
41. C₄²	C₃×H(3,𝔽₃)	semidirect
53. C₃²	C₃×C₄²⋊C₃	semidirect
62. C₃	C₃²⋊(C₄²⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄²	3
C₃²×C₄²⋊C₃	3
C₃²⋊(C₄²⋊C₃)	3
C₃×C₂²⋊H(3,𝔽₃)	4

Subgroups — C3×C32⋊(C42⋊C3) #13047788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃²⋊(C₄²⋊C₃) #13047788