Subgroups — #11720777

Order 864.

89 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88627	C₂	1	2	yes	maximal
2	C₈×#272	C₂	1	1	yes	maximal, characteristic
3	C₄×#272	C₂²	1	1	yes	characteristic
4	#6188	—	2	2	no	—
5	C₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	#6180	C₄	1	1	yes	characteristic
7	C₃×D₃×C₈	—	6	2	no	—
8	C₂×#272	Q₈	1	1	yes	characteristic
9	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
10	#1084	D₄	1	1	yes	characteristic
11	C₃×#108	—	9	1	no	maximal
12	C₃×C₄×D₃	—	6	2	no	—
13	C₃²×C₈	—	2	2	no	—
14	C₃×C₃⋊C₈	—	6	2	no	—
15	C₂×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
16	#272	—	2	1	no	—
17	C₃×M₁₆	—	9	2	no	—
18	C₂×C₃×C₈	—	9	1	no	—
19	D₃×C₈	—	6	2	no	—
20	C₂×C₃×D₃	—	6	2	no	—
21	C₃²×C₄	—	2	2	no	—
22	C₃×Dic₃	—	6	2	no	—
23	#108	—	9	1	no	Sylow
24	H(3,𝔽₃)	#108	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃×C₄	—	9	1	no	—
26	C₄×D₃	—	6	2	no	—
27	C₃×C₈	—	6	2	no	—
28	C₃×C₈	—	9	1	no	—
29	C₃×C₈	—	18	2	no	—
30	C₃×C₈	#157	1	1	yes	characteristic
31	C₃⋊C₈	—	6	2	no	—
32	C₂×C₃²	—	2	2	no	—
33	C₃×D₃	—	6	4	no	—
34	M₁₆	—	9	2	no	—
35	C₂×C₈	—	9	1	no	—
36	C₂²×C₃	—	9	1	no	—
37	C₂×D₃	—	6	2	no	—
38	C₃×C₄	—	6	2	no	—
39	C₃×C₄	—	9	1	no	—
40	C₃×C₄	C₂×#157	1	1	yes	characteristic
41	Dic₃	—	6	2	no	—
42	C₃²	—	2	2	no	—
43	C₂×C₄	—	9	1	no	—
44	C₈	—	9	1	no	—
45	C₈	—	18	2	no	—
46	C₈	#1088	1	1	yes	characteristic
47	C₂×C₃	—	6	2	no	—
48	C₂×C₃	—	18	1	no	—
49	C₂×C₃	C₄⋊#157	1	1	yes	characteristic
50	D₃	—	6	4	no	—
51	C₂²	—	9	1	no	—
52	C₄	—	9	1	no	—
53	C₄	C₂×#1088	1	1	yes	characteristic
54	C₃	—	6	2	no	—
55	C₃	#63771	1	1	yes	characteristic
56	C₂	—	18	1	no	—
57	C₂	C₄⋊#1088	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₄⋊#157
commutator	C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₃×C₄	C₂×#157
Fitting	C₈×H(3,𝔽₃)	C₄
prime radical	C₂×#272	Q₈
cosocle	C₄×#272	C₂²
hypercenter	C₃×C₈	#157
hypocenter	H(3,𝔽₃)	#108
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈

Complements

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

N	H ≅ G/N	type
24. H(3,𝔽₃)	#108	semidirect

Maximal subgroups

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

subgroup	index
#88627	2
C₈×#272	2
C₃×#108	9