Subgroups — #14517263

Order 1944.

416 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#6204	C₃	1	2	yes	maximal, complemented
4	(C₃×SL(2,𝔽₃))⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₂×#6891	—	4	1	no	maximal
6	C₃×C₄×3₋¹⁺²	—	3	1	no	—
7	#6891	—	4	1	no	Sylow
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	C₃²×SL(2,𝔽₃)	—	3	3	no	—
10	C₃²×SL(2,𝔽₃)	—	3	9	no	—
11	C₃²×SL(2,𝔽₃)	C₃²	1	3	yes	—
12	Q₈×3₋¹⁺²	—	3	3	no	—
13	C₃×Q₈×C₉	C₃²	1	3	yes	complemented
14	#6204	—	3	6	no	—
15	C₃×Q₈⋊C₉	C₃²	1	6	yes	complemented
16	C₂×C₃⁴	—	4	1	no	—
17	C₂×C₃×3₋¹⁺²	—	4	2	no	—
18	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
19	C₂×C₃²⋊C₉	—	4	9	no	—
20	C₃³×C₄	—	3	1	no	—
21	C₄×3₋¹⁺²	—	3	9	no	—
22	C₃×C₄×C₉	—	3	3	no	—
23	C₃⁴	—	4	1	no	—
24	C₃×3₋¹⁺²	—	4	2	no	—
25	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
26	C₃²⋊C₉	—	4	9	no	—
27	C₃²×Q₈	—	3	3	no	—
28	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
29	C₃²×Q₈	C₃³	1	1	yes	characteristic
30	C₃×SL(2,𝔽₃)	—	3	27	no	—
31	C₃×SL(2,𝔽₃)	—	3	3	no	—
32	C₃×SL(2,𝔽₃)	—	3	6	no	—
33	C₃×SL(2,𝔽₃)	3₋¹⁺²	1	9	yes	complemented
34	Q₈×C₉	—	3	3	no	—
35	Q₈⋊C₉	—	3	6	no	—
36	C₂×C₃³	—	4	3	no	—
37	C₂×C₃³	—	12	9	no	—
38	C₂×C₃³	—	12	3	no	—
39	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
40	C₂×3₋¹⁺²	—	3	3	no	—
41	C₂×3₋¹⁺²	—	12	6	no	—
42	C₂×C₃×C₉	—	4	6	no	—
43	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
44	C₃²×C₄	—	3	1	no	—
45	C₃²×C₄	—	3	3	no	—
46	C₃²×C₄	—	9	3	no	—
47	C₄×C₉	—	3	9	no	—
48	C₃³	—	4	3	no	—
49	C₃³	—	12	9	no	—
50	C₃³	—	12	3	no	—
51	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
52	3₋¹⁺²	—	3	3	no	—
53	3₋¹⁺²	—	12	6	no	—
54	C₃×C₉	—	4	6	no	—
55	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	complemented
56	C₃×Q₈	—	3	3	no	—
57	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×Q₈	C₃×3₋¹⁺²	1	3	yes	—
59	SL(2,𝔽₃)	—	3	9	no	—
60	C₂×C₃²	—	3	3	no	—
61	C₂×C₃²	—	4	9	no	—
62	C₂×C₃²	—	12	27	no	—
63	C₂×C₃²	—	12	6	no	—
64	C₂×C₃²	—	12	3	no	—
65	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
66	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
67	C₂×C₉	—	3	3	no	—
68	C₂×C₉	—	12	6	no	—
69	C₃×C₄	—	3	3	no	—
70	C₃×C₄	—	3	1	no	—
71	C₃×C₄	—	9	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	4	9	no	—
74	C₃²	—	12	27	no	—
75	C₃²	—	12	6	no	—
76	C₃²	—	12	3	no	—
77	C₃²	Q₈⋊H(3,𝔽₃)	1	3	yes	—
78	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
79	C₉	—	3	3	no	—
80	C₉	—	12	6	no	—
81	Q₈	#6891	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	—	3	3	no	—
83	C₂×C₃	—	12	9	no	—
84	C₂×C₃	A₄×3₋¹⁺²	1	3	yes	—
85	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
86	C₄	—	3	1	no	—
87	C₃	—	3	3	no	—
88	C₃	—	12	9	no	—
89	C₃	SL(2,𝔽₃)×3₋¹⁺²	1	3	yes	—
90	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
91	C₂	#11757856	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	C₃×Q₈×3₋¹⁺²	C₃
prime radical	C₃³×SL(2,𝔽₃)	C₃
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6891
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×3₋¹⁺²	C₃	semidirect
3. C₃×#6204	C₃	semidirect
4. (C₃×SL(2,𝔽₃))⋊C₉	C₃	semidirect
13. C₃×Q₈×C₉	C₃²	semidirect
15. C₃×Q₈⋊C₉	C₃²	semidirect
25. C₃×3₋¹⁺²	SL(2,𝔽₃)	semidirect
33. C₃×SL(2,𝔽₃)	3₋¹⁺²	semidirect
55. C₃×C₉	C₃×SL(2,𝔽₃)	semidirect
81. Q₈	#6891	semidirect

Maximal subgroups

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

subgroup	index
C₃³×SL(2,𝔽₃)	3
C₃×Q₈×3₋¹⁺²	3
C₃×#6204	3
(C₃×SL(2,𝔽₃))⋊C₉	3
C₂×#6891	4