Subgroups — #13144261

Order 1728.

759 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11720574	C₂	1	6	yes	maximal
3	C₃×#4937	—	3	4	no	maximal
4	C₂²×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
5	C₂²×#1084	C₂²	1	4	yes	—
6	C₄²×H(3,𝔽₃)	C₄	1	4	yes	complemented
7	C₂×C₃²×C₄²	D₃	1	4	yes	—
8	C₃×C₃⋊#95	—	3	24	no	—
9	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
10	C₂×C₄×H(3,𝔽₃)	—	2	6	no	—
11	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	6	yes	—
12	C₂×#1084	—	2	12	no	—
13	C₃×#383	—	9	1	no	—
14	#4937	—	3	4	no	—
15	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
16	C₂²×C₃×Dic₃	—	3	16	no	—
17	C₃²×C₄²	Dic₃	1	16	yes	complemented
18	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
19	C₂²×H(3,𝔽₃)	#41	1	6	yes	—
20	C₄×H(3,𝔽₃)	—	2	6	no	—
21	#1084	—	4	4	no	—
22	C₂×C₃×C₄²	—	3	4	no	—
23	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
24	C₃×#95	—	9	6	no	—
25	C₃⋊#95	—	3	24	no	—
26	C₂³×C₃²	C₂²×D₃	1	4	yes	—
27	C₂×C₃²×C₄	—	2	24	no	—
28	C₂×C₃²×C₄	C₂×Dic₃	1	24	yes	—
29	C₂×C₃×Dic₃	—	6	48	no	—
30	#383	—	9	1	no	Sylow
31	C₂×H(3,𝔽₃)	#117	1	3	yes	—
32	C₂×H(3,𝔽₃)	#126	1	4	yes	—
33	C₂²×C₃×C₄	—	3	12	no	—
34	C₂²×C₃×C₄	—	9	4	no	—
35	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
36	C₂²×Dic₃	—	3	16	no	—
37	C₃×C₄²	—	3	16	no	—
38	C₃×C₄²	C₃⋊Dic₃	1	4	yes	—
39	C₂²×C₃²	#235	1	24	yes	—
40	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
41	C₃²×C₄	—	2	24	no	—
42	C₃×Dic₃	—	12	16	no	—
43	C₂×C₄²	#272	1	1	yes	characteristic
44	#95	—	9	6	no	—
45	H(3,𝔽₃)	#383	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₃	—	3	4	no	—
47	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	6	no	—
49	C₂×C₃×C₄	—	3	24	no	—
50	C₂×C₃×C₄	—	6	24	no	—
51	C₂×C₃×C₄	—	18	12	no	—
52	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
53	C₂×Dic₃	—	6	48	no	—
54	C₂×C₃²	#876	1	16	yes	—
55	C₂×C₃²	#926	1	12	yes	—
56	C₂²×C₄	—	9	4	no	—
57	C₂²×C₄	C₂×#272	1	3	yes	—
58	C₄²	#1084	1	4	yes	complemented
59	C₂²×C₃	—	3	24	no	—
60	C₂²×C₃	—	3	4	no	—
61	C₂²×C₃	#3828	1	6	yes	—
62	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
63	C₃×C₄	—	2	6	no	—
64	C₃×C₄	—	6	24	no	—
65	C₃×C₄	—	36	4	no	—
66	Dic₃	—	12	16	no	—
67	C₃²	#4937	1	4	yes	complemented
68	C₂³	C₂²×#272	1	1	yes	characteristic
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	18	12	no	—
71	C₂×C₄	C₂×#1084	1	6	yes	—
72	C₂×C₃	—	3	12	no	—
73	C₂×C₃	—	3	16	no	—
74	C₂×C₃	#64085	1	4	yes	—
75	C₂×C₃	#64135	1	3	yes	—
76	C₂²	#88740	1	6	yes	—
77	C₂²	C₂²×#1084	1	1	yes	characteristic
78	C₄	—	2	6	no	—
79	C₄	—	36	4	no	—
80	C₃	—	3	4	no	—
81	C₃	#10592636	1	1	yes	characteristic
82	C₂	#11721722	1	4	yes	—
83	C₂	#11721826	1	3	yes	—

	subgroup	G/H
center, Frattini, socle	C₂³×C₃	C₂²×C₃⋊D₃
commutator	C₂²×H(3,𝔽₃)	C₂²×C₄
Fitting	C₂×C₄²×H(3,𝔽₃)	C₂
prime radical, cosocle	C₂³×H(3,𝔽₃)	C₂³
hypercenter	C₂×C₃×C₄²	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#383
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×H(3,𝔽₃)	C₄	semidirect
17. C₃²×C₄²	Dic₃	semidirect
45. H(3,𝔽₃)	#383	semidirect
58. C₄²	#1084	semidirect
67. C₃²	#4937	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×H(3,𝔽₃)	2
#11720574	2
C₃×#4937	3