Subgroups — #11720771

Order 864.

83 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#272	C₂	1	1	yes	maximal, characteristic
2	#88410	C₂	1	2	yes	maximal, complemented
3	C₄×#272	C₄	1	1	yes	characteristic
4	C₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	#6180	C₂²	1	1	yes	characteristic
6	C₃×D₃×C₈	—	6	2	no	—
7	C₂×#272	C₈	1	1	yes	characteristic
8	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
9	#1084	C₈	1	1	yes	characteristic
10	C₃×M₃₂	—	9	1	no	maximal
11	C₃×C₄×D₃	—	6	2	no	—
12	C₃²×C₈	—	2	2	no	—
13	C₃×C₃⋊C₈	—	6	2	no	—
14	C₂×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
15	#272	—	2	1	no	—
16	C₂×C₃×C₈	—	9	1	no	—
17	D₃×C₈	—	6	2	no	—
18	C₃×C₁₆	—	9	2	no	—
19	C₂×C₃×D₃	—	6	2	no	—
20	C₃²×C₄	—	2	2	no	—
21	C₃×Dic₃	—	6	2	no	—
22	M₃₂	—	9	1	no	Sylow
23	H(3,𝔽₃)	M₃₂	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃×C₄	—	9	1	no	—
25	C₄×D₃	—	6	2	no	—
26	C₃×C₈	—	6	2	no	—
27	C₃×C₈	—	9	1	no	—
28	C₃×C₈	#157	1	1	yes	characteristic
29	C₃⋊C₈	—	6	2	no	—
30	C₂×C₃²	—	2	2	no	—
31	C₃×D₃	—	6	4	no	—
32	C₂×C₈	—	9	1	no	—
33	C₁₆	—	9	2	no	—
34	C₂²×C₃	—	9	1	no	—
35	C₂×D₃	—	6	2	no	—
36	C₃×C₄	—	6	2	no	—
37	C₃×C₄	—	9	1	no	—
38	C₃×C₄	C₂×#157	1	1	yes	characteristic
39	Dic₃	—	6	2	no	—
40	C₃²	—	2	2	no	—
41	C₂×C₄	—	9	1	no	—
42	C₈	—	9	1	no	—
43	C₈	#1088	1	1	yes	characteristic
44	C₂×C₃	—	6	2	no	—
45	C₂×C₃	—	18	1	no	—
46	C₂×C₃	#3787	1	1	yes	characteristic
47	D₃	—	6	4	no	—
48	C₂²	—	9	1	no	—
49	C₄	—	9	1	no	—
50	C₄	C₂×#1088	1	1	yes	characteristic
51	C₃	—	6	2	no	—
52	C₃	#63765	1	1	yes	characteristic
53	C₂	—	18	1	no	—
54	C₂	#88626	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₄	C₂×#157
commutator	C₂×H(3,𝔽₃)	C₂×C₈
Fitting	C₈×H(3,𝔽₃)	C₄
socle	C₂×C₃	#3787
prime radical	C₂×#272	C₈
cosocle	#6180	C₂²
hypercenter	C₃×C₈	#157
hypocenter	H(3,𝔽₃)	M₃₂
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₃×C₈

Complements

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

N	H ≅ G/N	type
2. #88410	C₂	semidirect
23. H(3,𝔽₃)	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×#272	2
#88410	2
C₃×M₃₂	9