Subgroups — #14515889

Order 1944.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×#6204	C₃	1	2	yes	maximal, complemented
3	#10619584	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#6857	—	4	1	no	maximal
5	C₃×C₄×H(3,𝔽₃)	—	3	1	no	—
6	#6857	—	4	1	no	Sylow
7	C₃³×Q₈	—	3	1	no	—
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	Q₈×H(3,𝔽₃)	—	3	3	no	—
10	#6204	—	3	6	no	—
11	C₃×Q₈⋊C₉	—	3	2	no	—
12	C₃×Q₈⋊C₉	—	3	1	no	—
13	C₂×C₃×3₋¹⁺²	—	4	2	no	—
14	C₂×C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
15	C₂×C₃²⋊C₉	—	4	1	no	—
16	C₃³×C₄	—	3	3	no	—
17	C₃³×C₄	—	3	1	no	—
18	C₄×H(3,𝔽₃)	—	3	9	no	—
19	C₃×3₋¹⁺²	—	4	2	no	—
20	C₃×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic
21	C₃²⋊C₉	—	4	1	no	—
22	C₃²×Q₈	—	3	1	no	—
23	C₃²×Q₈	—	3	3	no	—
24	C₃²×Q₈	—	3	2	no	—
25	C₃²×Q₈	—	3	1	no	—
26	C₃²×Q₈	—	9	2	no	—
27	C₃²×Q₈	—	9	1	no	—
28	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
29	Q₈⋊C₉	—	3	6	no	—
30	Q₈⋊C₉	—	9	1	no	—
31	C₂×C₃³	—	3	1	no	—
32	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
33	C₂×3₋¹⁺²	—	12	6	no	—
34	C₂×H(3,𝔽₃)	—	3	3	no	—
35	C₂×C₃×C₉	—	12	2	no	—
36	C₂×C₃×C₉	—	12	1	no	—
37	C₃²×C₄	—	3	1	no	—
38	C₃²×C₄	—	3	3	no	—
39	C₃²×C₄	—	3	9	no	—
40	C₃²×C₄	—	9	1	no	—
41	C₃²×C₄	—	9	2	no	—
42	C₃²×C₄	—	9	3	no	—
43	C₃²×C₄	—	9	6	no	—
44	C₃³	—	3	1	no	—
45	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
46	3₋¹⁺²	—	12	6	no	—
47	H(3,𝔽₃)	—	3	3	no	—
48	C₃×C₉	—	12	2	no	—
49	C₃×C₉	—	12	1	no	—
50	C₃×Q₈	—	9	1	no	—
51	C₃×Q₈	—	9	3	no	—
52	C₃×Q₈	AΣL(1,𝔽₂₇)	1	2	yes	—
53	C₃×Q₈	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
54	C₃×Q₈	#731	1	1	yes	characteristic
55	C₂×C₃²	—	3	1	no	—
56	C₂×C₃²	—	3	2	no	—
57	C₂×C₃²	—	3	1	no	—
58	C₂×C₃²	—	3	3	no	—
59	C₂×C₃²	—	9	1	no	—
60	C₂×C₃²	—	9	2	no	—
61	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
62	C₂×C₉	—	12	6	no	—
63	C₂×C₉	—	36	1	no	—
64	C₃×C₄	—	3	1	no	—
65	C₃×C₄	—	3	2	no	—
66	C₃×C₄	—	3	1	no	—
67	C₃×C₄	—	9	3	no	—
68	C₃×C₄	—	9	9	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	9	2	no	—
75	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
76	C₉	—	12	6	no	—
77	C₉	—	36	1	no	—
78	Q₈	#6857	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	9	1	no	—
80	C₂×C₃	—	9	3	no	—
81	C₂×C₃	#66361	1	1	yes	characteristic
82	C₂×C₃	#66363	1	2	yes	—
83	C₂×C₃	#66368	1	1	yes	characteristic
84	C₄	—	3	1	no	—
85	C₃	—	9	1	no	—
86	C₃	—	9	3	no	—
87	C₃	#10619578	1	1	yes	characteristic
88	C₃	#10619580	1	2	yes	—
89	C₃	#10619585	1	1	yes	characteristic
90	C₂	#11757451	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₃³×Q₈	C₃²
Frattini	C₂×C₃³	C₃×A₄
Fitting	C₃×Q₈×H(3,𝔽₃)	C₃
prime radical, hypercenter	C₂×C₃×H(3,𝔽₃)	A₄
hypocenter	Q₈	#6857
LCS	G ▷ C₃³×Q₈ ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃³ ◁ C₂×C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃×#6204	C₃	semidirect
3. #10619584	C₃	semidirect
78. Q₈	#6857	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×H(3,𝔽₃)	3
C₃×#6204	3
#10619584	3
C₂×#6857	4