Subgroups — #13135844

Order 1728.

220 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	#11720552	C₂	1	2	yes	maximal
3	C₃×#4506	—	3	4	no	maximal
4	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	#88538	—	2	2	no	—
6	C₁₆×H(3,𝔽₃)	C₄	1	2	yes	—
7	C₂×C₃²×C₁₆	D₃	1	4	yes	—
8	C₃×#819	—	3	8	no	—
9	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
10	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
11	C₈×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
12	#6180	—	4	2	no	—
13	C₃×#368	—	9	1	no	—
14	#4506	—	3	4	no	—
15	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
16	C₃×#208	—	6	8	no	—
17	C₃²×C₁₆	Dic₃	1	8	yes	—
18	C₂²×H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic
19	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
20	C₄×H(3,𝔽₃)	D₈	1	1	yes	characteristic
21	C₂×C₃×C₁₆	—	3	4	no	—
22	C₂×C₃×C₁₆	C₃⋊D₃	1	1	yes	characteristic
23	C₃×#108	—	9	2	no	—
24	#819	—	3	8	no	—
25	C₂×C₃²×C₄	D₁₂	1	4	yes	—
26	C₃²×C₈	Dic₆	1	4	yes	—
27	C₃²×C₈	C₂×Dic₃	1	4	yes	—
28	C₃×C₃⋊C₈	—	12	8	no	—
29	#368	—	9	1	no	Sylow
30	C₂×H(3,𝔽₃)	—	2	1	no	—
31	C₂×H(3,𝔽₃)	#107	1	1	yes	characteristic
32	C₃×M₁₆	—	18	2	no	—
33	C₂×C₃×C₈	—	3	4	no	—
34	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
35	#208	—	6	8	no	—
36	C₃×C₁₆	—	3	8	no	—
37	C₃×C₁₆	C₃⋊Dic₃	1	2	yes	—
38	C₂²×C₃²	Dic₁₂	1	4	yes	—
39	C₃²×C₄	D₂₄	1	4	yes	—
40	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
41	C₂×C₁₆	#272	1	1	yes	characteristic
42	#108	—	9	2	no	—
43	H(3,𝔽₃)	#368	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₄	—	3	4	no	—
45	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
46	C₃×C₈	—	3	4	no	—
47	C₃×C₈	—	3	4	no	—
48	C₃×C₈	—	36	2	no	—
49	C₃×C₈	#634	1	1	yes	characteristic
50	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
51	C₃⋊C₈	—	12	8	no	—
52	C₂×C₃²	—	2	4	no	—
53	C₂×C₃²	#818	1	4	yes	—
54	M₁₆	—	18	2	no	—
55	C₂×C₈	C₂×#272	1	1	yes	characteristic
56	C₁₆	#1084	1	2	yes	—
57	C₂²×C₃	—	3	4	no	—
58	C₂²×C₃	#3746	1	1	yes	characteristic
59	C₃×C₄	—	3	4	no	—
60	C₃×C₄	—	3	4	no	—
61	C₃×C₄	#3745	1	1	yes	characteristic
62	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
63	C₃²	#4506	1	4	yes	complemented
64	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
65	C₈	—	36	2	no	—
66	C₈	#6229	1	1	yes	characteristic
67	C₈	C₂×#1084	1	1	yes	characteristic
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	3	4	no	—
70	C₂×C₃	—	6	4	no	—
71	C₂×C₃	#63645	1	1	yes	characteristic
72	C₂²	#88530	1	1	yes	characteristic
73	C₄	#88529	1	1	yes	characteristic
74	C₄	C₄⋊#1084	1	1	yes	characteristic
75	C₃	—	3	4	no	—
76	C₃	#10589372	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#11720551	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#3745
commutator	C₈×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₁₆×H(3,𝔽₃)	C₂
socle	C₂×C₃	#63645
prime radical	C₂²×H(3,𝔽₃)	Q₁₆
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#368
LCS	G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ 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
43. H(3,𝔽₃)	#368	semidirect
63. C₃²	#4506	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
#11720552	2
C₃×#4506	3