Subgroups — #11721869

Order 864.

341 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#88538	C₂	1	3	yes	maximal, complemented
3	C₂×#6180	C₂	1	3	yes	maximal, complemented
4	C₃×#948	—	3	4	no	maximal
5	Q₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	D₄×H(3,𝔽₃)	C₄	1	3	yes	—
7	C₂×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	#6180	C₂²	1	1	yes	characteristic
9	#6180	C₂²	1	3	yes	—
10	C₃²×#41	D₃	1	4	yes	—
11	C₃×#208	—	3	12	no	—
12	C₂×C₃×C₃⋊C₈	—	3	12	no	—
13	C₂²×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
14	C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
15	C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₃×#131	—	9	1	no	—
17	#948	—	3	4	no	—
18	C₃²×Q₈	Dic₃	1	4	yes	—
19	C₃²×D₄	Dic₃	1	12	yes	—
20	C₂×C₃²×C₄	C₂×D₃	1	12	yes	—
21	C₃×C₃⋊C₈	—	3	4	no	—
22	C₃×C₃⋊C₈	—	3	12	no	—
23	C₂×H(3,𝔽₃)	—	2	3	no	—
24	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
25	C₃×#41	—	3	4	no	—
26	C₃×#41	C₃⋊D₃	1	1	yes	characteristic
27	C₃×M₁₆	—	9	3	no	—
28	C₂×C₃×C₈	—	9	3	no	—
29	#208	—	3	12	no	—
30	C₂×C₃⋊C₈	—	3	12	no	—
31	C₂²×C₃²	C₂×Dic₃	1	12	yes	—
32	C₃²×C₄	C₂×Dic₃	1	12	yes	—
33	C₃²×C₄	C₂²×D₃	1	4	yes	—
34	#131	—	9	1	no	Sylow
35	H(3,𝔽₃)	#131	1	1	yes	characteristic, complemented, Sylow
36	C₃×Q₈	—	3	4	no	—
37	C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
38	C₃×D₄	—	3	12	no	—
39	C₃×D₄	C₃⋊Dic₃	1	3	yes	—
40	C₂×C₃×C₄	—	3	12	no	—
41	C₂×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
42	C₃×C₈	—	9	1	no	—
43	C₃×C₈	—	9	3	no	—
44	C₃⋊C₈	—	3	4	no	—
45	C₃⋊C₈	—	3	12	no	—
46	C₂×C₃²	—	2	12	no	—
47	C₂×C₃²	C₂²×Dic₃	1	4	yes	—
48	#41	#272	1	1	yes	characteristic
49	M₁₆	—	9	3	no	—
50	C₂×C₈	—	9	3	no	—
51	C₂²×C₃	—	3	12	no	—
52	C₂²×C₃	C₂×C₃⋊Dic₃	1	3	yes	—
53	C₃×C₄	—	3	12	no	—
54	C₃×C₄	—	3	4	no	—
55	C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
56	C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
57	C₃²	#948	1	4	yes	complemented
58	Q₈	#1084	1	1	yes	characteristic
59	D₄	#1084	1	3	yes	—
60	C₂×C₄	C₂×#272	1	3	yes	—
61	C₈	—	9	1	no	—
62	C₈	—	9	3	no	—
63	C₂×C₃	—	2	3	no	—
64	C₂×C₃	—	3	4	no	—
65	C₂×C₃	—	6	12	no	—
66	C₂×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
67	C₂²	C₂×#1084	1	3	yes	—
68	C₄	C₂×#1084	1	3	yes	—
69	C₄	C₂²×#272	1	1	yes	characteristic
70	C₃	—	3	4	no	—
71	C₃	#64157	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	C₂²×#1084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂²×C₄
Fitting, prime radical	#41×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₂²×C₃⋊Dic₃
cosocle	C₄×H(3,𝔽₃)	C₂³
hypercenter	C₃×#41	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#131
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#41

Complements

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

N	H ≅ G/N	type
2. #88538	C₂	semidirect
3. C₂×#6180	C₂	semidirect
35. H(3,𝔽₃)	#131	semidirect
57. C₃²	#948	semidirect

Maximal subgroups

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

subgroup	index
#41×H(3,𝔽₃)	2
#88538	2
C₂×#6180	2
C₃×#948	3