Subgroups — #14517262

Order 1944.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#6204	C₃	1	3	yes	maximal
3	#10619576	C₃	1	9	yes	maximal, complemented
4	C₂×#6912	—	4	1	no	maximal
5	C₃×C₄×3₋¹⁺²	—	3	1	no	—
6	#6912	—	4	1	no	Sylow
7	C₃³×Q₈	C₃²	1	1	yes	characteristic
8	Q₈×3₋¹⁺²	—	3	3	no	—
9	C₃×Q₈×C₉	C₃²	1	3	yes	—
10	#6204	—	3	6	no	—
11	#6204	C₃²	1	9	yes	—
12	C₃×Q₈⋊C₉	—	3	3	no	—
13	C₂×C₃×3₋¹⁺²	—	4	3	no	—
14	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
15	C₂×#733	—	4	9	no	—
16	C₃³×C₄	—	3	1	no	—
17	C₄×3₋¹⁺²	—	3	9	no	—
18	C₃×C₄×C₉	—	3	3	no	—
19	C₃×3₋¹⁺²	—	4	3	no	—
20	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic
21	#733	—	4	9	no	—
22	C₃²×Q₈	—	9	1	no	—
23	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
24	C₃²×Q₈	C₃³	1	1	yes	characteristic
25	Q₈×C₉	—	3	3	no	—
26	Q₈⋊C₉	—	3	9	no	—
27	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
28	C₂×3₋¹⁺²	—	3	3	no	—
29	C₂×3₋¹⁺²	—	4	9	no	—
30	C₂×3₋¹⁺²	—	12	6	no	—
31	C₂×C₃×C₉	—	12	3	no	—
32	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
33	C₃²×C₄	—	3	1	no	—
34	C₃²×C₄	—	3	3	no	—
35	C₃²×C₄	—	9	3	no	—
36	C₄×C₉	—	3	9	no	—
37	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
38	3₋¹⁺²	—	3	3	no	—
39	3₋¹⁺²	—	4	9	no	—
40	3₋¹⁺²	—	12	6	no	—
41	C₃×C₉	—	12	3	no	—
42	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	—
43	C₃×Q₈	—	3	1	no	—
44	C₃×Q₈	—	3	3	no	—
45	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
46	C₂×C₃²	—	9	1	no	—
47	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
48	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
49	C₂×C₉	—	3	3	no	—
50	C₂×C₉	—	12	9	no	—
51	C₃×C₄	—	3	3	no	—
52	C₃×C₄	—	3	1	no	—
53	C₃×C₄	—	9	3	no	—
54	C₃²	—	9	1	no	—
55	C₃²	Q₈⋊H(3,𝔽₃)	1	3	yes	—
56	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
57	C₉	—	3	3	no	—
58	C₉	—	12	9	no	—
59	Q₈	#6912	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	3	3	no	—
62	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
63	C₄	—	3	1	no	—
64	C₃	—	3	3	no	—
65	C₃	—	3	1	no	—
66	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
67	C₂	#11757854	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	C₃×Q₈×3₋¹⁺²	C₃
prime radical	C₂×C₃³	C₃×A₄
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6912
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ C₂×C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. #10619576	C₃	semidirect
59. Q₈	#6912	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×3₋¹⁺²	3
C₃×#6204	3
#10619576	3
C₂×#6912	4