Subgroups — #11720559

Order 864.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#88538	C₂	1	1	yes	maximal, characteristic
3	C₂×#6180	C₂	1	1	yes	maximal, characteristic
4	C₃×#822	—	3	4	no	maximal
5	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₈×H(3,𝔽₃)	—	2	1	no	—
7	#6180	—	2	1	no	—
8	#6180	C₄	1	2	yes	—
9	C₃²×M₁₆	D₃	1	4	yes	—
10	C₃×#208	—	3	4	no	—
11	C₂×C₃×C₃⋊C₈	—	3	4	no	—
12	C₂²×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
15	C₃×#108	—	9	1	no	—
16	#822	—	3	4	no	—
17	C₂×C₃²×C₄	C₂×D₃	1	4	yes	—
18	C₃²×C₈	—	2	4	no	—
19	C₃×C₃⋊C₈	—	3	8	no	—
20	C₃×C₃⋊C₈	—	6	4	no	—
21	C₂×H(3,𝔽₃)	—	2	1	no	—
22	C₂×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
23	C₃×M₁₆	—	6	4	no	—
24	C₃×M₁₆	—	9	1	no	—
25	C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
26	C₂×C₃×C₈	—	9	1	no	—
27	#208	—	3	4	no	—
28	C₂×C₃⋊C₈	—	3	4	no	—
29	C₂²×C₃²	Dic₆	1	4	yes	—
30	C₃²×C₄	C₄×D₃	1	4	yes	—
31	C₃²×C₄	#67	1	4	yes	—
32	#108	—	9	1	no	Sylow
33	H(3,𝔽₃)	#108	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₄	—	6	4	no	—
35	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
36	C₃×C₈	—	2	1	no	—
37	C₃×C₈	—	6	4	no	—
38	C₃×C₈	—	9	2	no	—
39	C₃×C₈	—	18	1	no	—
40	C₃⋊C₈	—	3	8	no	—
41	C₃⋊C₈	—	6	4	no	—
42	C₂×C₃²	—	2	4	no	—
43	C₂×C₃²	Dic₃⋊C₄	1	4	yes	—
44	M₁₆	—	9	1	no	—
45	M₁₆	#272	1	1	yes	characteristic
46	C₂×C₈	—	9	1	no	—
47	C₂²×C₃	—	6	4	no	—
48	C₂²×C₃	#634	1	1	yes	characteristic
49	C₃×C₄	—	6	4	no	—
50	C₃×C₄	—	6	4	no	—
51	C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
52	C₃×C₄	#638	1	1	yes	characteristic
53	C₃²	#822	1	4	yes	complemented
54	C₂×C₄	C₂×#272	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	—	9	2	no	—
57	C₈	—	18	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	6	4	no	—
60	C₂×C₃	—	12	4	no	—
61	C₂×C₃	#3750	1	1	yes	characteristic
62	C₂²	#6229	1	1	yes	characteristic
63	C₄	C₄×#272	1	1	yes	characteristic
64	C₄	#6235	1	1	yes	characteristic
65	C₃	—	6	4	no	—
66	C₃	#63649	1	1	yes	characteristic
67	C₂	—	2	1	no	—
68	C₂	#88540	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#638
commutator	C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×C₃⋊D₃
Fitting	M₁₆×H(3,𝔽₃)	C₂
socle	C₂×C₃	#3750
prime radical	C₂²×H(3,𝔽₃)	Q₈
cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₃×M₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#108
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×M₁₆

Complements

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

N	H ≅ G/N	type
33. H(3,𝔽₃)	#108	semidirect
53. C₃²	#822	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×H(3,𝔽₃)	2
#88538	2
C₂×#6180	2
C₃×#822	3