Subgroups — C₃²×#338⋊C₃ #13145747

Order 1728.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#338	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#338⋊C₃	C₃	1	12	yes	maximal, direct product factor
3	C₃²×C₄×A₄	—	4	1	no	maximal
4	C₂×C₃²×M₁₆	—	3	1	no	—
5	C₂×C₃²×A₄	—	4	1	no	—
6	C₃×#338	C₃²	1	4	yes	complemented
7	#338⋊C₃	C₃²	1	9	yes	direct product factor
8	C₂²×C₃²×C₄	A₄	1	1	yes	characteristic
9	C₃×C₄×A₄	—	4	12	no	—
10	C₃²×M₁₆	—	6	2	no	—
11	C₂×C₃²×C₈	—	3	2	no	—
12	C₃²×A₄	—	4	1	no	—
13	C₃³×C₄	—	16	1	no	—
14	C₂×C₃×M₁₆	—	3	4	no	—
15	C₂³×C₃²	SL(2,𝔽₃)	1	1	yes	characteristic
16	C₂×C₃×A₄	—	4	12	no	—
17	C₂×C₃²×C₄	—	3	1	no	—
18	C₂×C₃²×C₄	—	3	1	no	—
19	C₃²×C₈	—	6	2	no	—
20	#338	C₃³	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃³	—	16	1	no	—
22	C₂²×C₃×C₄	C₃×A₄	1	4	yes	—
23	C₄×A₄	—	4	9	no	—
24	C₃×M₁₆	—	6	8	no	—
25	C₂×C₃×C₈	—	3	8	no	—
26	C₂²×C₃²	—	3	1	no	—
27	C₂²×C₃²	—	4	1	no	—
28	C₃×A₄	—	4	12	no	—
29	C₃²×C₄	—	3	1	no	—
30	C₃²×C₄	—	16	12	no	—
31	C₃²×C₄	C₄²⋊C₃	1	1	yes	characteristic
32	C₂×M₁₆	—	3	1	no	—
33	C₃³	—	16	1	no	Sylow
34	C₂³×C₃	C₃×SL(2,𝔽₃)	1	4	yes	—
35	C₂×A₄	—	4	9	no	—
36	C₂×C₃×C₄	—	3	4	no	—
37	C₂×C₃×C₄	—	3	4	no	—
38	C₃×C₈	—	6	8	no	—
39	C₂×C₃²	—	6	1	no	—
40	C₂×C₃²	—	16	12	no	—
41	C₂×C₃²	#95⋊C₃	1	1	yes	characteristic
42	C₂²×C₄	C₃²×A₄	1	1	yes	characteristic
43	M₁₆	—	6	2	no	—
44	C₂×C₈	—	3	2	no	—
45	C₂²×C₃	—	3	4	no	—
46	C₂²×C₃	—	4	4	no	—
47	A₄	—	4	9	no	—
48	C₃×C₄	—	3	4	no	—
49	C₃×C₄	—	16	9	no	—
50	C₃×C₄	C₃×C₄²⋊C₃	1	4	yes	—
51	C₃²	—	16	12	no	—
52	C₃²	#338⋊C₃	1	1	yes	characteristic, direct product factor
53	C₂³	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
54	C₂×C₄	—	3	1	no	—
55	C₂×C₄	—	3	1	no	—
56	C₈	—	6	2	no	—
57	C₂×C₃	—	6	4	no	—
58	C₂×C₃	—	16	9	no	—
59	C₂×C₃	C₃×#95⋊C₃	1	4	yes	—
60	C₂²	—	3	1	no	—
61	C₂²	—	4	1	no	—
62	C₄	—	3	1	no	—
63	C₄	C₃²×C₄²⋊C₃	1	1	yes	characteristic
64	C₃	—	16	9	no	—
65	C₃	C₃×#338⋊C₃	1	4	yes	direct product factor
66	C₂	—	6	1	no	—
67	C₂	C₃²×#95⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₄	C₄²⋊C₃
commutator, cosocle, hypocenter	#338	C₃³
Frattini	C₂²×C₄	C₃²×A₄
Fitting	C₃²×#338	C₃
socle	C₂×C₃²	#95⋊C₃
LCS	G ▷ #338
UCS	C₁ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
1. C₃²×#338	C₃	semidirect
2. C₃×#338⋊C₃	C₃	direct (N × H)
6. C₃×#338	C₃²	semidirect
7. #338⋊C₃	C₃²	direct (N × H)
20. #338	C₃³	semidirect
52. C₃²	#338⋊C₃	direct (N × H)
65. C₃	C₃×#338⋊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
C₃²×#338	3
C₃×#338⋊C₃	3
C₃²×C₄×A₄	4

Subgroups — C32×#338⋊C3 #13145747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#338⋊C₃ #13145747