Subgroups — C₃×#88907 #13049342

Order 1296.

826 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃³	C₃	1	1	yes	maximal, characteristic
2	#88907	C₃	1	9	yes	maximal, direct product factor
3	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	3	yes	maximal, complemented
4	C₃×#1094	—	4	5	no	maximal
5	C₂³×C₃³	—	3	5	no	—
6	C₂⁴×C₃²	—	3	3	no	—
7	C₂⁴×C₃²	C₃²	1	3	yes	—
8	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
9	C₂²⋊(C₂²⋊C₉)	C₃²	1	9	yes	complemented
10	C₂²×C₃³	—	3	10	no	—
11	C₂²×C₃³	A₄	1	5	yes	—
12	#1094	—	4	45	no	—
13	C₃×C₂²⋊C₉	—	4	15	no	—
14	C₃×3₋¹⁺²	—	16	1	no	Sylow
15	C₂³×C₃²	—	3	45	no	—
16	C₂³×C₃²	—	3	15	no	—
17	C₂³×C₃²	—	3	5	no	—
18	C₂×C₃³	—	3	5	no	—
19	C₂⁴×C₃	—	3	3	no	—
20	C₂⁴×C₃	3₋¹⁺²	1	3	yes	complemented
21	C₂⁴×C₃	C₃³	1	1	yes	characteristic
22	C₂²×C₃²	—	3	90	no	—
23	C₂²×C₃²	—	3	30	no	—
24	C₂²×C₃²	—	3	10	no	—
25	C₂²×C₃²	—	3	15	no	—
26	C₂²×C₃²	C₃×A₄	1	15	yes	—
27	C₂²×C₃²	C₃×A₄	1	5	yes	—
28	C₂²⋊C₉	—	4	45	no	—
29	C₃³	C₂²⋊A₄	1	1	yes	characteristic
30	3₋¹⁺²	—	16	9	no	—
31	C₃×C₉	—	16	3	no	—
32	C₂³×C₃	—	3	45	no	—
33	C₂³×C₃	—	3	15	no	—
34	C₂³×C₃	—	3	5	no	—
35	C₂×C₃²	—	3	45	no	—
36	C₂×C₃²	—	3	15	no	—
37	C₂×C₃²	—	3	5	no	—
38	C₂⁴	C₃×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₃	—	3	90	no	—
40	C₂²×C₃	—	3	10	no	—
41	C₂²×C₃	—	3	30	no	—
42	C₂²×C₃	—	3	15	no	—
43	C₂²×C₃	#1094	1	15	yes	complemented
44	C₂²×C₃	C₃²×A₄	1	5	yes	—
45	C₃²	—	3	3	no	—
46	C₃²	C₃×C₂²⋊A₄	1	3	yes	—
47	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
48	C₉	—	16	9	no	—
49	C₂³	—	3	5	no	—
50	C₂×C₃	—	3	45	no	—
51	C₂×C₃	—	3	5	no	—
52	C₂×C₃	—	3	15	no	—
53	C₂²	—	3	10	no	—
54	C₂²	C₃×#1094	1	5	yes	complemented
55	C₃	—	3	3	no	—
56	C₃	#88907	1	3	yes	direct product factor
57	C₃	C₃²×C₂²⋊A₄	1	1	yes	characteristic
58	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃²	C₃×C₂²⋊A₄
commutator, cosocle	C₂⁴×C₃	C₃³
Frattini	C₃	C₃²×C₂²⋊A₄
Fitting, prime radical	C₂⁴×C₃³	C₃
socle	C₂⁴×C₃²	C₃²
hypercenter	C₃³	C₂²⋊A₄
hypocenter	C₂⁴	C₃×3₋¹⁺²
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃² ◁ C₃³

Complements

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

N	H ≅ G/N	type
2. #88907	C₃	direct (N × H)
3. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
9. C₂²⋊(C₂²⋊C₉)	C₃²	semidirect
20. C₂⁴×C₃	3₋¹⁺²	semidirect
38. C₂⁴	C₃×3₋¹⁺²	semidirect
43. C₂²×C₃	#1094	semidirect
54. C₂²	C₃×#1094	semidirect
56. C₃	#88907	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×C₃³	3
#88907	3
C₃×C₂²⋊(C₂²⋊C₉)	3
C₃×#1094	4

Subgroups — C3×#88907 #13049342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#88907 #13049342