Subgroups — #11720584

Order 864.

209 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#88538	C₂	1	2	yes	maximal
3	C₃×#836	—	3	3	no	maximal
4	C₃×#836	—	3	1	no	maximal
5	Q₈×H(3,𝔽₃)	—	2	2	no	—
6	C₂×C₄×H(3,𝔽₃)	C₄	1	2	yes	—
7	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	#6180	—	2	2	no	—
9	C₂×C₃²×Q₈	D₃	1	3	yes	—
10	C₂×C₃²×Q₈	D₃	1	1	yes	characteristic
11	C₃×#208	—	3	6	no	—
12	C₃×#208	—	3	2	no	—
13	C₂²×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	C₄×H(3,𝔽₃)	—	2	2	no	—
15	C₄×H(3,𝔽₃)	D₄	1	2	yes	—
16	C₃×#101	—	9	1	no	—
17	#836	—	3	3	no	—
18	#836	—	3	1	no	—
19	C₃²×Q₈	—	2	6	no	—
20	C₃²×Q₈	—	2	2	no	—
21	C₂×C₃²×C₄	Dic₃	1	6	yes	—
22	C₂×C₃²×C₄	Dic₃	1	2	yes	—
23	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
24	C₂×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃×C₃⋊C₈	—	6	6	no	—
26	C₃×C₃⋊C₈	—	6	2	no	—
27	C₂×H(3,𝔽₃)	—	2	1	no	—
28	C₂×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₃×Q₈	—	3	3	no	—
30	C₂×C₃×Q₈	—	6	1	no	—
31	C₂×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
32	C₃×M₁₆	—	9	2	no	—
33	#208	—	3	6	no	—
34	#208	—	3	2	no	—
35	C₂²×C₃²	C₂×Dic₃	1	3	yes	—
36	C₂²×C₃²	C₂×Dic₃	1	1	yes	characteristic
37	C₃²×C₄	—	2	6	no	—
38	C₃²×C₄	—	2	2	no	—
39	C₃²×C₄	#67	1	6	yes	—
40	C₃²×C₄	#67	1	2	yes	—
41	#101	—	9	1	no	Sylow
42	H(3,𝔽₃)	#101	1	1	yes	characteristic, complemented, Sylow
43	C₃×Q₈	—	2	2	no	—
44	C₃×Q₈	—	6	6	no	—
45	C₃×Q₈	—	6	2	no	—
46	C₂×C₃×C₄	—	3	6	no	—
47	C₂×C₃×C₄	—	3	3	no	—
48	C₂×C₃×C₄	—	6	2	no	—
49	C₂×C₃×C₄	—	6	1	no	—
50	C₂×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
51	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
52	C₃×C₈	—	18	2	no	—
53	C₃⋊C₈	—	6	6	no	—
54	C₃⋊C₈	—	6	2	no	—
55	C₂×C₃²	—	2	3	no	—
56	C₂×C₃²	—	2	1	no	—
57	C₂×C₃²	#217	1	3	yes	—
58	C₂×C₃²	#217	1	1	yes	characteristic
59	C₂×Q₈	#272	1	1	yes	characteristic
60	M₁₆	—	9	2	no	—
61	C₂²×C₃	—	3	3	no	—
62	C₂²×C₃	—	6	1	no	—
63	C₂²×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
64	C₃×C₄	—	2	2	no	—
65	C₃×C₄	—	3	6	no	—
66	C₃×C₄	—	6	6	no	—
67	C₃×C₄	—	6	2	no	—
68	C₃×C₄	—	6	2	no	—
69	C₃×C₄	#638	1	2	yes	—
70	C₃²	#836	1	3	yes	complemented
71	C₃²	#836	1	1	yes	characteristic, complemented
72	Q₈	—	2	2	no	—
73	C₂×C₄	#1084	1	2	yes	—
74	C₂×C₄	C₂×#272	1	1	yes	characteristic
75	C₈	—	18	2	no	—
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	3	3	no	—
78	C₂×C₃	—	6	3	no	—
79	C₂×C₃	—	6	1	no	—
80	C₂×C₃	—	6	1	no	—
81	C₂×C₃	#3757	1	1	yes	characteristic
82	C₂²	C₂×#1084	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	#6235	1	2	yes	—
85	C₃	—	3	3	no	—
86	C₃	—	6	1	no	—
87	C₃	#63663	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	#88552	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#3757
commutator, prime radical	C₂²×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×Q₈×H(3,𝔽₃)	C₂
cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×Q₈	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 ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
42. H(3,𝔽₃)	#101	semidirect
70. C₃²	#836	semidirect
71. C₃²	#836	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×H(3,𝔽₃)	2
#88538	2
C₃×#836	3