Subgroups — #11722451

Order 864.

173 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88748	C₂	1	1	yes	maximal, characteristic
2	#88627	C₂	1	3	yes	maximal, complemented
3	#88626	C₂	1	3	yes	maximal, complemented
4	Q₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₄⋊#272	C₄	1	3	yes	—
6	C₄×#272	C₂²	1	3	yes	—
7	#6188	C₂²	1	1	yes	characteristic
8	#6188	C₂²	1	3	yes	—
9	C₃×#239	—	6	2	no	—
10	C₂×#272	C₂×C₄	1	3	yes	—
11	C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
12	#1084	C₂³	1	1	yes	characteristic
13	C₃×#131	—	9	1	no	maximal
14	C₃²×Q₈	—	2	2	no	—
15	C₃×D₁₂	—	6	6	no	—
16	C₃×C₄×D₃	—	6	6	no	—
17	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
18	#272	—	2	3	no	—
19	C₃×#41	—	9	1	no	—
20	#239	—	6	2	no	—
21	C₃×M₁₆	—	9	3	no	—
22	C₂×C₃×C₈	—	9	3	no	—
23	C₂×C₃×D₃	—	6	6	no	—
24	C₃²×C₄	—	2	6	no	—
25	C₃×Dic₃	—	6	2	no	—
26	#131	—	9	1	no	Sylow
27	H(3,𝔽₃)	#131	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	6	2	no	—
29	C₃×Q₈	#157	1	1	yes	characteristic
30	C₃×D₄	—	9	3	no	—
31	C₂×C₃×C₄	—	9	3	no	—
32	D₁₂	—	6	6	no	—
33	C₄×D₃	—	6	6	no	—
34	C₃×C₈	—	9	1	no	—
35	C₃×C₈	—	9	3	no	—
36	C₂×C₃²	—	2	2	no	—
37	C₃×D₃	—	12	6	no	—
38	#41	—	9	1	no	—
39	M₁₆	—	9	3	no	—
40	C₂×C₈	—	9	3	no	—
41	C₂²×C₃	—	9	3	no	—
42	C₂×D₃	—	6	6	no	—
43	C₃×C₄	—	6	6	no	—
44	C₃×C₄	—	9	1	no	—
45	C₃×C₄	C₂×#157	1	3	yes	—
46	Dic₃	—	6	2	no	—
47	C₃²	—	2	2	no	—
48	Q₈	#1088	1	1	yes	characteristic
49	D₄	—	9	3	no	—
50	C₂×C₄	—	9	3	no	—
51	C₈	—	9	1	no	—
52	C₈	—	9	3	no	—
53	C₂×C₃	—	6	2	no	—
54	C₂×C₃	—	18	3	no	—
55	C₂×C₃	C₂²×#157	1	1	yes	characteristic
56	D₃	—	12	6	no	—
57	C₂²	—	9	3	no	—
58	C₄	—	9	1	no	—
59	C₄	C₂×#1088	1	3	yes	—
60	C₃	—	6	2	no	—
61	C₃	#64289	1	1	yes	characteristic
62	C₂	—	18	3	no	—
63	C₂	C₂²×#1088	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×#157
commutator	C₂×H(3,𝔽₃)	C₂²×C₄
Fitting	Q₈×H(3,𝔽₃)	C₄
prime radical	#88748	C₂
cosocle	#1084	C₂³
hypercenter	C₃×Q₈	#157
hypocenter	H(3,𝔽₃)	#131
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

Complements

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

N	H ≅ G/N	type
2. #88627	C₂	semidirect
3. #88626	C₂	semidirect
27. H(3,𝔽₃)	#131	semidirect

Maximal subgroups

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

subgroup	index
#88748	2
#88627	2
#88626	2
C₃×#131	9