Subgroups — #14517216

Order 1944.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₃)×H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	C₃×C₉⋊SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	Q₈×#732	C₃	1	1	yes	maximal, characteristic, complemented
4	#10619579	C₃	1	6	yes	maximal, complemented
5	#10619575	C₃	1	2	yes	maximal, complemented
6	C₂×#6911	—	4	1	no	maximal
7	C₄×#732	—	3	1	no	—
8	#6911	—	4	1	no	Sylow
9	C₃²×SL(2,𝔽₃)	—	3	3	no	—
10	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
11	Q₈×H(3,𝔽₃)	C₃²	1	3	yes	complemented
12	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic, complemented
13	Q₈⋊H(3,𝔽₃)	—	3	6	no	—
14	Q₈⋊H(3,𝔽₃)	C₃²	1	6	yes	complemented
15	C₃×Q₈⋊C₉	C₃²	1	2	yes	complemented
16	C₉⋊SL(2,𝔽₃)	—	3	3	no	—
17	C₂×C₃×3₋¹⁺²	—	4	1	no	—
18	C₂×C₃×H(3,𝔽₃)	—	4	3	no	—
19	C₂×#732	—	4	6	no	—
20	C₂×#732	—	4	2	no	—
21	C₂×#732	A₄	1	1	yes	characteristic
22	C₄×H(3,𝔽₃)	—	3	3	no	—
23	C₃×C₄×C₉	—	3	1	no	—
24	C₃×3₋¹⁺²	—	4	1	no	—
25	C₃×H(3,𝔽₃)	—	4	3	no	—
26	#732	—	4	6	no	—
27	#732	—	4	2	no	—
28	#732	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
29	C₃²×Q₈	—	3	3	no	—
30	C₃²×Q₈	C₃³	1	1	yes	characteristic
31	C₃×SL(2,𝔽₃)	—	3	6	no	—
32	C₃×SL(2,𝔽₃)	—	9	1	no	—
33	C₃×SL(2,𝔽₃)	—	9	9	no	—
34	C₃×SL(2,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
35	Q₈×C₉	—	3	1	no	—
36	Q₈⋊C₉	—	3	2	no	—
37	C₂×C₃³	—	4	1	no	—
38	C₂×C₃³	—	12	3	no	—
39	C₂×3₋¹⁺²	—	12	3	no	—
40	C₂×H(3,𝔽₃)	—	4	6	no	—
41	C₂×H(3,𝔽₃)	—	12	6	no	—
42	C₂×H(3,𝔽₃)	C₃×A₄	1	3	yes	—
43	C₂×C₃×C₉	—	4	2	no	—
44	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
45	C₃²×C₄	—	3	1	no	—
46	C₃²×C₄	—	9	3	no	—
47	C₄×C₉	—	9	1	no	—
48	C₃³	—	4	1	no	—
49	C₃³	—	12	3	no	—
50	3₋¹⁺²	—	12	3	no	—
51	H(3,𝔽₃)	—	4	6	no	—
52	H(3,𝔽₃)	—	12	6	no	—
53	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	3	yes	complemented
54	C₃×C₉	—	4	2	no	—
55	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
56	C₃×Q₈	—	3	1	no	—
57	C₃×Q₈	—	9	3	no	—
58	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	SL(2,𝔽₃)	—	3	3	no	—
60	SL(2,𝔽₃)	—	9	6	no	—
61	C₂×C₃²	—	3	3	no	—
62	C₂×C₃²	—	4	3	no	—
63	C₂×C₃²	—	12	6	no	—
64	C₂×C₃²	—	36	1	no	—
65	C₂×C₃²	—	36	9	no	—
66	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
67	C₂×C₉	—	3	1	no	—
68	C₂×C₉	—	12	2	no	—
69	C₃×C₄	—	3	1	no	—
70	C₃×C₄	—	9	1	no	—
71	C₃×C₄	—	27	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	4	3	no	—
74	C₃²	—	12	6	no	—
75	C₃²	—	36	1	no	—
76	C₃²	—	36	9	no	—
77	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
78	C₉	—	3	1	no	—
79	C₉	—	12	2	no	—
80	Q₈	#6911	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	—	3	1	no	—
82	C₂×C₃	—	9	3	no	—
83	C₂×C₃	—	12	3	no	—
84	C₂×C₃	—	36	6	no	—
85	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
86	C₄	—	3	1	no	—
87	C₃	—	3	1	no	—
88	C₃	—	9	3	no	—
89	C₃	—	12	3	no	—
90	C₃	—	36	6	no	—
91	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
92	C₂	#11757809	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	Q₈×#732	C₃
hypercenter	C₂×#732	A₄
hypocenter	Q₈	#6911
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×#732

Complements

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

N	H ≅ G/N	type
1. SL(2,𝔽₃)×H(3,𝔽₃)	C₃	semidirect
2. C₃×C₉⋊SL(2,𝔽₃)	C₃	semidirect
3. Q₈×#732	C₃	semidirect
4. #10619579	C₃	semidirect
5. #10619575	C₃	semidirect
11. Q₈×H(3,𝔽₃)	C₃²	semidirect
12. C₃×Q₈×C₉	C₃²	semidirect
14. Q₈⋊H(3,𝔽₃)	C₃²	semidirect
15. C₃×Q₈⋊C₉	C₃²	semidirect
28. #732	SL(2,𝔽₃)	semidirect
53. H(3,𝔽₃)	C₃×SL(2,𝔽₃)	semidirect
55. C₃×C₉	C₃×SL(2,𝔽₃)	semidirect
80. Q₈	#6911	semidirect

Maximal subgroups

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

subgroup	index
SL(2,𝔽₃)×H(3,𝔽₃)	3
C₃×C₉⋊SL(2,𝔽₃)	3
Q₈×#732	3
#10619579	3
#10619575	3
C₂×#6911	4