Subgroups — #13135843

Order 1728.

253 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11720551	C₂	1	2	yes	maximal
3	C₃×#4504	—	3	4	no	maximal
4	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₄⋊#1084	—	2	2	no	—
6	C₁₆×H(3,𝔽₃)	C₄	1	2	yes	complemented
7	C₂×C₃²×C₁₆	D₃	1	4	yes	—
8	C₃×#818	—	3	8	no	—
9	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
10	C₂×#1084	—	4	2	no	—
11	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₈×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
13	C₃×#366	—	9	1	no	—
14	#4504	—	3	4	no	—
15	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
16	C₃×C₄⋊Dic₃	—	6	8	no	—
17	C₃²×C₁₆	Dic₃	1	8	yes	complemented
18	C₂²×H(3,𝔽₃)	D₈	1	1	yes	characteristic
19	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
20	C₄×H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic
21	#1084	—	8	2	no	—
22	C₂×C₃×C₁₆	—	3	4	no	—
23	C₂×C₃×C₁₆	C₃⋊D₃	1	1	yes	characteristic
24	C₃×#107	—	9	2	no	—
25	#818	—	3	8	no	—
26	C₂×C₃²×C₄	D₁₂	1	4	yes	—
27	C₂×C₃×Dic₃	—	12	8	no	—
28	C₃²×C₈	Dic₆	1	4	yes	—
29	C₃²×C₈	C₂×Dic₃	1	4	yes	—
30	#366	—	9	1	no	Sylow
31	C₂×H(3,𝔽₃)	#107	1	1	yes	characteristic
32	C₂×H(3,𝔽₃)	D₁₆	1	1	yes	characteristic
33	C₂×H(3,𝔽₃)	Q₃₂	1	1	yes	characteristic
34	C₂×C₃×C₈	—	3	4	no	—
35	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
36	C₃×C₄⋊C₄	—	18	2	no	—
37	C₄⋊Dic₃	—	6	8	no	—
38	C₃×C₁₆	—	3	8	no	—
39	C₃×C₁₆	C₃⋊Dic₃	1	2	yes	—
40	C₂²×C₃²	D₂₄	1	4	yes	—
41	C₃²×C₄	Dic₁₂	1	4	yes	—
42	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
43	C₃×Dic₃	—	24	8	no	—
44	C₂×C₁₆	#272	1	1	yes	characteristic
45	#107	—	9	2	no	—
46	H(3,𝔽₃)	#366	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃×C₄	—	3	4	no	—
48	C₂×C₃×C₄	—	36	2	no	—
49	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
50	C₂×Dic₃	—	12	8	no	—
51	C₃×C₈	—	3	4	no	—
52	C₃×C₈	—	3	4	no	—
53	C₃×C₈	#634	1	1	yes	characteristic
54	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
55	C₂×C₃²	D₄₈	1	4	yes	—
56	C₂×C₃²	Dic₂₄	1	4	yes	—
57	C₂×C₃²	#818	1	4	yes	—
58	C₂×C₈	C₂×#272	1	1	yes	characteristic
59	C₄⋊C₄	—	18	2	no	—
60	C₁₆	#1084	1	2	yes	complemented
61	C₂²×C₃	—	3	4	no	—
62	C₂²×C₃	#3745	1	1	yes	characteristic
63	C₃×C₄	—	3	4	no	—
64	C₃×C₄	—	3	4	no	—
65	C₃×C₄	—	72	2	no	—
66	C₃×C₄	#3746	1	1	yes	characteristic
67	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
68	Dic₃	—	24	8	no	—
69	C₃²	#4504	1	4	yes	complemented
70	C₂×C₄	—	36	2	no	—
71	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
72	C₈	#6229	1	1	yes	characteristic
73	C₈	C₂×#1084	1	1	yes	characteristic
74	C₂×C₃	—	3	4	no	—
75	C₂×C₃	—	3	4	no	—
76	C₂×C₃	—	3	4	no	—
77	C₂×C₃	#63626	1	1	yes	characteristic
78	C₂×C₃	#63628	1	1	yes	characteristic
79	C₂×C₃	#63645	1	1	yes	characteristic
80	C₂²	#88529	1	1	yes	characteristic
81	C₄	—	72	2	no	—
82	C₄	#88530	1	1	yes	characteristic
83	C₄	C₄⋊#1084	1	1	yes	characteristic
84	C₃	—	3	4	no	—
85	C₃	#10589370	1	1	yes	characteristic
86	C₂	#11720512	1	1	yes	characteristic
87	C₂	#11720514	1	1	yes	characteristic
88	C₂	#11720551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#3745
commutator	C₈×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₁₆×H(3,𝔽₃)	C₂
prime radical	C₂²×H(3,𝔽₃)	D₈
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#366
LCS	G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

Complements

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

N	H ≅ G/N	type
6. C₁₆×H(3,𝔽₃)	C₄	semidirect
17. C₃²×C₁₆	Dic₃	semidirect
46. H(3,𝔽₃)	#366	semidirect
60. C₁₆	#1084	semidirect
69. C₃²	#4504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×H(3,𝔽₃)	2
#11720551	2
C₃×#4504	3