Subgroups — #14517266

Order 1944.

272 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, complemented
2	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#6204	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₃×SL(2,𝔽₃))⋊C₉	C₃	1	9	yes	maximal, complemented
6	C₂×#6893	—	4	1	no	maximal
7	C₃×C₄×3₋¹⁺²	—	3	1	no	—
8	#6893	—	4	1	no	Sylow
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	C₃²×SL(2,𝔽₃)	C₃²	1	3	yes	—
11	Q₈×3₋¹⁺²	—	3	3	no	—
12	C₃×Q₈×C₉	C₃²	1	3	yes	—
13	Q₈⋊H(3,𝔽₃)	—	3	3	no	—
14	#6204	—	3	3	no	—
15	C₃×Q₈⋊C₉	—	3	3	no	—
16	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
17	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
18	C₂×C₃×3₋¹⁺²	—	4	1	no	—
19	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
20	C₂×C₃×H(3,𝔽₃)	—	4	1	no	—
21	C₂×C₃²×C₉	—	4	1	no	—
22	C₂×C₃²⋊C₉	—	4	9	no	—
23	C₃³×C₄	—	3	1	no	—
24	C₄×3₋¹⁺²	—	3	9	no	—
25	C₃×C₄×C₉	—	3	3	no	—
26	C₃×3₋¹⁺²	—	4	1	no	—
27	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
28	C₃×H(3,𝔽₃)	—	4	1	no	—
29	C₃²×C₉	—	4	1	no	—
30	C₃²⋊C₉	—	4	9	no	—
31	C₃²×Q₈	—	3	3	no	—
32	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
33	C₃²×Q₈	C₃³	1	1	yes	characteristic
34	C₃×SL(2,𝔽₃)	—	3	3	no	—
35	C₃×SL(2,𝔽₃)	—	3	9	no	—
36	Q₈×C₉	—	3	3	no	—
37	Q₈⋊C₉	—	3	3	no	—
38	Q₈⋊C₉	—	3	3	no	—
39	C₂×C₃³	—	4	3	no	—
40	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
41	C₂×3₋¹⁺²	—	3	3	no	—
42	C₂×3₋¹⁺²	—	12	3	no	—
43	C₂×H(3,𝔽₃)	—	12	3	no	—
44	C₂×C₃×C₉	—	4	3	no	—
45	C₂×C₃×C₉	—	4	3	no	—
46	C₂×C₃×C₉	—	12	3	no	—
47	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
48	C₃²×C₄	—	3	1	no	—
49	C₃²×C₄	—	3	3	no	—
50	C₃²×C₄	—	9	3	no	—
51	C₄×C₉	—	3	9	no	—
52	C₃³	—	4	3	no	—
53	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
54	3₋¹⁺²	—	3	3	no	—
55	3₋¹⁺²	—	12	3	no	—
56	H(3,𝔽₃)	—	12	3	no	—
57	C₃×C₉	—	4	3	no	—
58	C₃×C₉	—	4	3	no	—
59	C₃×C₉	—	12	3	no	—
60	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	—
61	C₃×Q₈	—	3	3	no	—
62	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×Q₈	#737	1	3	yes	—
64	SL(2,𝔽₃)	—	9	3	no	—
65	C₂×C₃²	—	3	3	no	—
66	C₂×C₃²	—	12	9	no	—
67	C₂×C₃²	—	12	3	no	—
68	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
69	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
70	C₂×C₉	—	3	3	no	—
71	C₂×C₉	—	12	3	no	—
72	C₂×C₉	—	12	3	no	—
73	C₃×C₄	—	3	3	no	—
74	C₃×C₄	—	3	1	no	—
75	C₃×C₄	—	9	3	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	12	9	no	—
78	C₃²	—	12	3	no	—
79	C₃²	Q₈⋊H(3,𝔽₃)	1	3	yes	—
80	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
81	C₉	—	3	3	no	—
82	C₉	—	12	3	no	—
83	C₉	—	12	3	no	—
84	Q₈	#6893	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₃	—	3	3	no	—
86	C₂×C₃	—	36	3	no	—
87	C₂×C₃	#66440	1	3	yes	—
88	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
89	C₄	—	3	1	no	—
90	C₃	—	3	3	no	—
91	C₃	—	36	3	no	—
92	C₃	#10619916	1	3	yes	—
93	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
94	C₂	#11757859	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	C₃×Q₈⋊H(3,𝔽₃)	C₃
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6893
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
1. C₃×Q₈×3₋¹⁺²	C₃	semidirect
3. C₃×#6204	C₃	semidirect
4. C₃²×Q₈⋊C₉	C₃	semidirect
5. (C₃×SL(2,𝔽₃))⋊C₉	C₃	semidirect
27. C₃×3₋¹⁺²	SL(2,𝔽₃)	semidirect
84. Q₈	#6893	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×Q₈⋊H(3,𝔽₃)	3
C₃×#6204	3
C₃²×Q₈⋊C₉	3
(C₃×SL(2,𝔽₃))⋊C₉	3
C₂×#6893	4