Subgroups — #11720561

Order 864.

209 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6229	C₂	1	1	yes	maximal, characteristic
2	M₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	#88538	C₂	1	1	yes	maximal, characteristic
4	C₃×#824	—	3	4	no	maximal
5	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×#1084	C₄	1	2	yes	—
7	#6229	—	2	1	no	—
8	#6229	—	2	1	no	—
9	C₈×H(3,𝔽₃)	—	2	1	no	—
10	#6180	—	2	1	no	—
11	C₂×C₃×Dic₆	—	3	4	no	—
12	C₃²×M₁₆	D₃	1	4	yes	—
13	C₃×#208	—	3	4	no	—
14	C₂²×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
16	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	#1084	—	2	2	no	—
18	C₃×#101	—	9	1	no	—
19	#824	—	3	4	no	—
20	C₂×C₃²×C₄	C₂×D₃	1	4	yes	—
21	C₂×C₃×Dic₃	—	3	8	no	—
22	C₃×Dic₆	—	6	4	no	—
23	C₃×Dic₆	—	6	4	no	—
24	C₃²×C₈	—	2	4	no	—
25	C₃×C₃⋊C₈	—	6	4	no	—
26	C₂×H(3,𝔽₃)	—	2	1	no	—
27	C₂×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×Q₈	—	9	1	no	—
29	C₂×Dic₆	—	3	4	no	—
30	C₃×M₁₆	—	6	4	no	—
31	C₃×M₁₆	—	9	1	no	—
32	C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
33	#208	—	3	4	no	—
34	C₂²×C₃²	C₄×D₃	1	4	yes	—
35	C₃²×C₄	D₁₂	1	4	yes	—
36	C₃²×C₄	#67	1	4	yes	—
37	C₃×Dic₃	—	6	8	no	—
38	#101	—	9	1	no	Sylow
39	H(3,𝔽₃)	#101	1	1	yes	characteristic, complemented, Sylow
40	C₃×Q₈	—	18	1	no	—
41	C₃×Q₈	—	18	1	no	—
42	C₂×C₃×C₄	—	6	4	no	—
43	C₂×C₃×C₄	—	9	2	no	—
44	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₂×Dic₃	—	3	8	no	—
46	Dic₆	—	6	4	no	—
47	Dic₆	—	6	4	no	—
48	C₃×C₈	—	2	1	no	—
49	C₃×C₈	—	6	4	no	—
50	C₃×C₈	—	18	1	no	—
51	C₃⋊C₈	—	6	4	no	—
52	C₂×C₃²	—	2	4	no	—
53	C₂×C₃²	(C₂×D₃)⋊C₄	1	4	yes	—
54	C₂×Q₈	—	9	1	no	—
55	M₁₆	—	9	1	no	—
56	M₁₆	#272	1	1	yes	characteristic
57	C₂²×C₃	—	6	4	no	—
58	C₂²×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
59	C₃×C₄	—	6	4	no	—
60	C₃×C₄	—	6	4	no	—
61	C₃×C₄	—	18	2	no	—
62	C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
63	C₃×C₄	#638	1	1	yes	characteristic
64	Dic₃	—	6	8	no	—
65	C₃²	#824	1	4	yes	complemented
66	Q₈	—	18	1	no	—
67	Q₈	—	18	1	no	—
68	C₂×C₄	—	9	2	no	—
69	C₂×C₄	C₂×#272	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	18	1	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	6	4	no	—
74	C₂×C₃	—	12	4	no	—
75	C₂×C₃	#3752	1	1	yes	characteristic
76	C₂²	C₄×#272	1	1	yes	characteristic
77	C₄	—	18	2	no	—
78	C₄	C₄⋊#272	1	1	yes	characteristic
79	C₄	#6235	1	1	yes	characteristic
80	C₃	—	6	4	no	—
81	C₃	#63651	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	#88542	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#3752
commutator, prime radical	C₂²×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×C₃⋊D₃
Fitting	M₁₆×H(3,𝔽₃)	C₂
cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₃×M₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#101
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×M₁₆

Complements

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

N	H ≅ G/N	type
39. H(3,𝔽₃)	#101	semidirect
65. C₃²	#824	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6229	2
M₁₆×H(3,𝔽₃)	2
#88538	2
C₃×#824	3