Subgroups — #13163780

Order 1728.

1176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#131	C₂	1	1	yes	maximal, characteristic
2	#11722182	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	3	yes	maximal, complemented
4	#10592839	—	3	13	no	maximal
5	C₃³×#41	C₄	1	1	yes	characteristic
6	C₃³×M₁₆	C₄	1	3	yes	—
7	C₂×C₃³×C₈	C₂²	1	3	yes	—
8	C₃⋊(C₃⋊(C₃⋊C₁₆))	C₂²	1	1	yes	characteristic
9	C₃⋊(C₃⋊(C₃⋊C₁₆))	C₂²	1	3	yes	—
10	C₃²×#131	D₃	1	13	yes	—
11	#63639	—	3	39	no	—
12	C₂×C₃⋊(C₃⋊C₁₆)	—	3	39	no	—
13	C₃³×Q₈	C₈	1	1	yes	characteristic
14	C₃³×D₄	C₈	1	3	yes	—
15	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
16	C₃³×C₈	C₂×C₄	1	3	yes	—
17	C₃³×C₈	C₂³	1	1	yes	characteristic
18	#5140	—	9	13	no	—
19	C₃²×#41	Dic₃	1	13	yes	—
20	C₃²×M₁₆	Dic₃	1	39	yes	—
21	C₂×C₃²×C₈	C₂×D₃	1	39	yes	—
22	C₃⋊(C₃⋊C₁₆)	—	3	13	no	—
23	C₃⋊(C₃⋊C₁₆)	—	3	39	no	—
24	C₂²×C₃³	C₂×C₈	1	3	yes	—
25	C₃³×C₄	C₂×C₈	1	3	yes	—
26	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
27	C₃×#131	C₃⋊D₃	1	13	yes	—
28	#812	—	9	39	no	—
29	C₂×C₃⋊C₁₆	—	9	39	no	—
30	C₃²×Q₈	C₃⋊C₈	1	13	yes	—
31	C₃²×D₄	C₃⋊C₈	1	39	yes	—
32	C₂×C₃²×C₄	C₂×Dic₃	1	39	yes	—
33	C₃²×C₈	C₂×Dic₃	1	39	yes	—
34	C₃²×C₈	C₂²×D₃	1	13	yes	—
35	#504	—	27	1	no	Sylow
36	C₂×C₃³	—	2	3	no	—
37	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
38	C₃×#41	C₃⋊Dic₃	1	13	yes	—
39	C₃×M₁₆	C₃⋊Dic₃	1	39	yes	—
40	C₂×C₃×C₈	C₂×C₃⋊D₃	1	39	yes	—
41	C₃⋊C₁₆	—	9	13	no	—
42	C₃⋊C₁₆	—	9	39	no	—
43	C₂²×C₃²	C₂×C₃⋊C₈	1	39	yes	—
44	C₃²×C₄	C₂×C₃⋊C₈	1	39	yes	—
45	C₃²×C₄	C₂²×Dic₃	1	13	yes	—
46	#131	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
47	M₃₂	—	27	3	no	—
48	C₂×C₁₆	—	27	3	no	—
49	C₃³	#504	1	1	yes	characteristic, complemented, Sylow
50	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	13	yes	—
51	C₃×D₄	C₃⋊(C₃⋊C₈)	1	39	yes	—
52	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	39	yes	—
53	C₃×C₈	C₂×C₃⋊Dic₃	1	39	yes	—
54	C₃×C₈	C₂²×C₃⋊D₃	1	13	yes	—
55	C₂×C₃²	—	2	39	no	—
56	C₂×C₃²	C₂²×C₃⋊C₈	1	13	yes	—
57	#41	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
58	M₁₆	C₃⋊(C₃⋊Dic₃)	1	3	yes	—
59	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
60	C₁₆	—	27	1	no	—
61	C₁₆	—	27	3	no	—
62	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	39	yes	—
63	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	39	yes	—
64	C₃×C₄	C₂²×C₃⋊Dic₃	1	13	yes	—
65	C₃²	#5140	1	13	yes	complemented
66	Q₈	C₃⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
67	D₄	C₃⋊(C₃⋊(C₃⋊C₈))	1	3	yes	—
68	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	3	yes	—
69	C₈	C₂×C₃⋊(C₃⋊Dic₃)	1	3	yes	—
70	C₈	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
71	C₂×C₃	—	2	39	no	—
72	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	13	yes	—
73	C₂²	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	1	3	yes	—
74	C₄	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	1	3	yes	—
75	C₄	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
76	C₃	#10592839	1	13	yes	complemented
77	C₂	—	2	3	no	—
78	C₂	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle	C₂×C₃³	C₂²×C₈
Fitting	C₃³×#131	C₂
prime radical	C₃³×#41	C₄
cosocle	C₃³×C₈	C₂³
hypercenter	#131	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#504
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #11722182	C₂	semidirect
3. C₂×C₃⋊(C₃⋊(C₃⋊C₁₆))	C₂	semidirect
49. C₃³	#504	semidirect
65. C₃²	#5140	semidirect
76. C₃	#10592839	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#131	2
#11722182	2
C₂×C₃⋊(C₃⋊(C₃⋊C₁₆))	2
#10592839	3