Subgroups — C₃⋊(C₃⋊#4628) #13157022

Order 1728.

318 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#4628	—	3	12	no	maximal
3	#10589427	—	3	1	no	maximal
4	C₃²×#41⋊C₃	C₄	1	1	yes	characteristic
5	C₃⋊(C₃⋊(C₃⋊C₁₆))	—	4	1	no	maximal
6	C₃²×#131	D₃	1	1	yes	characteristic
7	C₃×#131⋊C₃	D₃	1	12	yes	—
8	#63639	—	3	1	no	—
9	C₂×C₃⋊(C₃⋊C₁₆)	—	3	1	no	—
10	C₃²×SL(2,𝔽₃)	C₈	1	1	yes	characteristic
11	C₃³×C₈	—	4	1	no	—
12	#4628	—	9	9	no	—
13	#4561	—	9	4	no	—
14	C₃²×#41	Dic₃	1	1	yes	characteristic
15	C₃×#41⋊C₃	Dic₃	1	12	yes	—
16	C₃²×M₁₆	—	3	1	no	—
17	C₂×C₃²×C₈	—	3	1	no	—
18	C₃⋊(C₃⋊C₁₆)	—	6	1	no	—
19	C₃⋊(C₃⋊C₁₆)	—	6	1	no	—
20	C₃⋊(C₃⋊C₁₆)	—	12	12	no	—
21	C₃³×C₄	—	4	1	no	—
22	C₃×#131	C₃⋊D₃	1	4	yes	—
23	#131⋊C₃	C₃⋊D₃	1	9	yes	—
24	#812	—	9	4	no	—
25	C₂×C₃⋊C₁₆	—	9	4	no	—
26	C₃²×Q₈	C₃⋊C₈	1	1	yes	characteristic
27	C₃²×D₄	—	3	1	no	—
28	C₂×C₃²×C₄	—	3	1	no	—
29	C₃×SL(2,𝔽₃)	C₃⋊C₈	1	12	yes	—
30	C₃²×C₈	—	3	1	no	—
31	C₃²×C₈	—	4	12	no	—
32	C₃²×C₈	S₄	1	1	yes	characteristic
33	#350	—	27	1	no	Sylow
34	C₂×C₃³	—	4	1	no	—
35	C₃×#41	C₃⋊Dic₃	1	4	yes	—
36	#41⋊C₃	C₃⋊Dic₃	1	9	yes	—
37	C₃×M₁₆	—	3	4	no	—
38	C₂×C₃×C₈	—	3	4	no	—
39	C₃⋊C₁₆	—	18	4	no	—
40	C₃⋊C₁₆	—	18	4	no	—
41	C₃⋊C₁₆	—	36	9	no	—
42	C₂²×C₃²	—	3	1	no	—
43	C₃²×C₄	—	3	1	no	—
44	C₃²×C₄	—	4	12	no	—
45	C₃²×C₄	SL(2,ℤ/4)	1	1	yes	characteristic
46	#131	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
47	M₃₂	—	27	1	no	—
48	C₂×C₁₆	—	27	1	no	—
49	C₃³	—	4	1	no	Sylow
50	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	4	yes	—
51	C₃×D₄	—	3	4	no	—
52	C₂×C₃×C₄	—	3	4	no	—
53	SL(2,𝔽₃)	C₃⋊(C₃⋊C₈)	1	9	yes	—
54	C₃×C₈	—	3	4	no	—
55	C₃×C₈	—	4	9	no	—
56	C₃×C₈	C₃⋊S₄	1	4	yes	—
57	C₂×C₃²	—	4	12	no	—
58	C₂×C₃²	—	6	1	no	—
59	C₂×C₃²	A₄⋊C₈	1	1	yes	characteristic
60	#41	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
61	M₁₆	—	3	1	no	—
62	C₂×C₈	—	3	1	no	—
63	C₁₆	—	54	1	no	—
64	C₁₆	—	54	1	no	—
65	C₂²×C₃	—	3	4	no	—
66	C₃×C₄	—	3	4	no	—
67	C₃×C₄	—	4	9	no	—
68	C₃×C₄	C₃⋊SL(2,ℤ/4)	1	4	yes	—
69	C₃²	—	4	12	no	—
70	C₃²	#4628	1	1	yes	characteristic, complemented
71	Q₈	C₃⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
72	D₄	—	3	1	no	—
73	C₂×C₄	—	3	1	no	—
74	C₈	—	3	1	no	—
75	C₈	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic
76	C₂×C₃	—	4	9	no	—
77	C₂×C₃	—	6	4	no	—
78	C₂×C₃	C₃⋊(A₄⋊C₈)	1	4	yes	—
79	C₂²	—	3	1	no	—
80	C₄	—	3	1	no	—
81	C₄	C₃⋊(C₃⋊SL(2,ℤ/4))	1	1	yes	characteristic
82	C₃	—	4	9	no	—
83	C₃	C₃⋊#4628	1	4	yes	complemented
84	C₂	—	6	1	no	—
85	C₂	C₃⋊(C₃⋊(A₄⋊C₈))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₈	C₃⋊(C₃⋊S₄)
commutator, hypocenter	C₃²×SL(2,𝔽₃)	C₈
Fitting	C₃²×#131	D₃
socle	C₂×C₃²	A₄⋊C₈
prime radical	C₃²×#41⋊C₃	C₄
cosocle	C₃²×#131⋊C₃	C₂
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
70. C₃²	#4628	semidirect
83. C₃	C₃⋊#4628	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#131⋊C₃	2
C₃⋊#4628	3
#10589427	3
C₃⋊(C₃⋊(C₃⋊C₁₆))	4

Subgroups — C3⋊(C3⋊#4628) #13157022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊#4628) #13157022