Subgroups — #14517214

Order 1944.

205 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619913	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	#10619580	C₃	1	3	yes	maximal, complemented
5	#10619578	C₃	1	3	yes	maximal, complemented
6	#10619576	C₃	1	2	yes	maximal, complemented
7	C₂×#6913	—	4	1	no	maximal
8	C₄×#732	—	3	1	no	—
9	#6913	—	4	1	no	Sylow
10	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
11	Q₈×H(3,𝔽₃)	C₃²	1	3	yes	—
12	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
13	#6204	—	3	6	no	—
14	#6204	C₃²	1	6	yes	—
15	C₃×Q₈⋊C₉	—	3	3	no	—
16	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
18	C₉⋊SL(2,𝔽₃)	—	3	3	no	—
19	C₂×#737	—	4	3	no	—
20	C₂×C₃×3₋¹⁺²	—	4	1	no	—
21	C₂×#733	—	4	2	no	—
22	C₂×#732	A₄	1	1	yes	characteristic
23	C₂×#731	—	4	3	no	—
24	C₂×AΣL(1,𝔽₂₇)	—	4	3	no	—
25	C₄×H(3,𝔽₃)	—	3	3	no	—
26	C₃×C₄×C₉	—	3	1	no	—
27	#737	—	4	3	no	—
28	C₃×3₋¹⁺²	—	4	1	no	—
29	#733	—	4	2	no	—
30	#732	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
31	#731	—	4	3	no	—
32	AΣL(1,𝔽₂₇)	—	4	3	no	—
33	C₃²×Q₈	—	3	3	no	—
34	C₃²×Q₈	C₃³	1	1	yes	characteristic
35	C₃×SL(2,𝔽₃)	—	3	1	no	—
36	C₃×SL(2,𝔽₃)	—	9	1	no	—
37	Q₈×C₉	—	3	1	no	—
38	Q₈⋊C₉	—	3	1	no	—
39	Q₈⋊C₉	—	3	6	no	—
40	Q₈⋊C₉	H(3,𝔽₃)	1	3	yes	—
41	C₂×C₃³	—	4	1	no	—
42	C₂×3₋¹⁺²	—	4	6	no	—
43	C₂×3₋¹⁺²	—	12	3	no	—
44	C₂×3₋¹⁺²	—	12	6	no	—
45	C₂×H(3,𝔽₃)	C₃×A₄	1	3	yes	—
46	C₂×C₃×C₉	—	4	1	no	—
47	C₂×C₃×C₉	—	4	1	no	—
48	C₂×C₃×C₉	—	12	3	no	—
49	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
50	C₃²×C₄	—	3	1	no	—
51	C₃²×C₄	—	9	3	no	—
52	C₄×C₉	—	9	1	no	—
53	C₃³	—	4	1	no	—
54	3₋¹⁺²	—	4	6	no	—
55	3₋¹⁺²	—	12	3	no	—
56	3₋¹⁺²	—	12	6	no	—
57	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	3	yes	—
58	C₃×C₉	—	4	1	no	—
59	C₃×C₉	—	4	1	no	—
60	C₃×C₉	—	12	3	no	—
61	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
62	C₃×Q₈	—	3	1	no	—
63	C₃×Q₈	—	9	3	no	—
64	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	SL(2,𝔽₃)	—	9	1	no	—
66	C₂×C₃²	—	3	3	no	—
67	C₂×C₃²	—	12	1	no	—
68	C₂×C₃²	—	36	1	no	—
69	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
70	C₂×C₉	—	3	1	no	—
71	C₂×C₉	—	4	3	no	—
72	C₂×C₉	—	12	1	no	—
73	C₂×C₉	—	12	6	no	—
74	C₃×C₄	—	3	1	no	—
75	C₃×C₄	—	9	1	no	—
76	C₃×C₄	—	27	3	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	12	1	no	—
79	C₃²	—	36	1	no	—
80	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
81	C₉	—	3	1	no	—
82	C₉	—	4	3	no	—
83	C₉	—	12	1	no	—
84	C₉	—	12	6	no	—
85	Q₈	#6913	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	3	1	no	—
87	C₂×C₃	—	9	3	no	—
88	C₂×C₃	—	36	1	no	—
89	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
90	C₄	—	3	1	no	—
91	C₃	—	3	1	no	—
92	C₃	—	9	3	no	—
93	C₃	—	36	1	no	—
94	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
95	C₂	#11757807	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	#6913
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. #10619913	C₃	semidirect
2. C₃×C₉⋊SL(2,𝔽₃)	C₃	semidirect
3. Q₈×#732	C₃	semidirect
4. #10619580	C₃	semidirect
5. #10619578	C₃	semidirect
6. #10619576	C₃	semidirect
30. #732	SL(2,𝔽₃)	semidirect
85. Q₈	#6913	semidirect

Maximal subgroups

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

subgroup	index
#10619913	3
C₃×C₉⋊SL(2,𝔽₃)	3
Q₈×#732	3
#10619580	3
#10619578	3
#10619576	3
C₂×#6913	4