Subgroups — 3₋¹⁺²×#226 #13048364

Order 1296.

137 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₃)×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃²×#226	C₃	1	1	yes	maximal, characteristic
3	C₉×#226	C₃	1	3	yes	maximal, complemented
4	Q₁₆×3₋¹⁺²	—	3	1	no	maximal
5	Dic₃×3₋¹⁺²	—	4	1	no	maximal
6	C₃²×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic
7	Q₈×3₋¹⁺²	—	3	1	no	—
8	Q₈×3₋¹⁺²	D₃	1	1	yes	characteristic
9	#6204	—	2	1	no	—
10	C₉⋊SL(2,𝔽₃)	—	2	3	no	—
11	C₉×SL(2,𝔽₃)	C₂×C₃	1	3	yes	—
12	C₈×3₋¹⁺²	—	3	1	no	—
13	C₂×C₃×3₋¹⁺²	—	4	1	no	—
14	C₃×#226	—	3	1	no	—
15	C₃×#226	C₃²	1	1	yes	characteristic
16	C₃²×Q₁₆	—	3	1	no	—
17	C₉×Q₁₆	—	3	3	no	—
18	C₃²×Dic₃	—	4	1	no	—
19	C₄×3₋¹⁺²	—	3	1	no	—
20	C₄×3₋¹⁺²	—	6	1	no	—
21	C₉×Dic₃	—	4	3	no	—
22	C₃×3₋¹⁺²	—	4	1	no	Sylow
23	C₃²×Q₈	—	3	1	no	—
24	C₃²×Q₈	C₃×D₃	1	1	yes	characteristic
25	C₃×SL(2,𝔽₃)	—	2	1	no	—
26	C₃×SL(2,𝔽₃)	—	3	1	no	—
27	C₃×SL(2,𝔽₃)	—	6	1	no	—
28	C₃×SL(2,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
29	C₃²×C₈	—	3	1	no	—
30	Q₈×C₉	—	3	3	no	—
31	Q₈×C₉	C₃×D₃	1	3	yes	—
32	Q₈⋊C₉	—	2	3	no	—
33	C₈×C₉	—	3	3	no	—
34	C₂×C₃³	—	4	1	no	—
35	C₂×3₋¹⁺²	—	8	1	no	—
36	C₂×3₋¹⁺²	—	8	3	no	—
37	C₂×3₋¹⁺²	S₄	1	1	yes	characteristic
38	C₂×C₃×C₉	—	4	3	no	—
39	#226	3₋¹⁺²	1	1	yes	characteristic, direct product factor
40	C₃×Q₁₆	—	3	1	no	—
41	C₃×Q₁₆	—	9	1	no	—
42	C₃²×C₄	—	3	1	no	—
43	C₃²×C₄	—	6	1	no	—
44	C₃×Dic₃	—	4	1	no	—
45	C₃×Dic₃	—	12	1	no	—
46	C₄×C₉	—	3	3	no	—
47	C₄×C₉	—	6	3	no	—
48	C₃³	—	4	1	no	—
49	3₋¹⁺²	—	8	1	no	—
50	3₋¹⁺²	—	8	3	no	—
51	3₋¹⁺²	#226	1	1	yes	characteristic, direct product factor
52	C₃×C₉	—	4	3	no	—
53	C₃×Q₈	—	3	1	no	—
54	C₃×Q₈	—	3	1	no	—
55	C₃×Q₈	—	9	1	no	—
56	C₃×Q₈	C₃²×D₃	1	1	yes	characteristic
57	SL(2,𝔽₃)	—	2	1	no	—
58	SL(2,𝔽₃)	—	6	1	no	—
59	SL(2,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic
60	C₃×C₈	—	3	1	no	—
61	C₃×C₈	—	9	1	no	—
62	C₂×C₃²	—	4	1	no	—
63	C₂×C₃²	—	8	1	no	—
64	C₂×C₃²	—	12	1	no	—
65	C₂×C₃²	—	24	1	no	—
66	C₂×C₃²	C₃×S₄	1	1	yes	characteristic
67	C₂×C₉	—	8	3	no	—
68	C₂×C₉	C₃×S₄	1	3	yes	—
69	Q₁₆	—	3	1	no	Sylow
70	C₃×C₄	—	3	1	no	—
71	C₃×C₄	—	6	1	no	—
72	C₃×C₄	—	9	1	no	—
73	C₃×C₄	—	18	1	no	—
74	Dic₃	—	4	1	no	—
75	C₃²	—	4	1	no	—
76	C₃²	—	8	1	no	—
77	C₃²	—	12	1	no	—
78	C₃²	—	24	1	no	—
79	C₃²	C₃×#226	1	1	yes	characteristic
80	C₉	—	8	3	no	—
81	C₉	C₃×#226	1	3	yes	complemented
82	Q₈	—	3	1	no	—
83	Q₈	D₃×3₋¹⁺²	1	1	yes	characteristic
84	C₈	—	3	1	no	—
85	C₂×C₃	—	3	1	no	—
86	C₂×C₃	—	4	1	no	—
87	C₂×C₃	—	8	1	no	—
88	C₂×C₃	—	24	1	no	—
89	C₂×C₃	C₃²×S₄	1	1	yes	characteristic
90	C₄	—	3	1	no	—
91	C₄	—	6	1	no	—
92	C₃	—	3	1	no	—
93	C₃	—	4	1	no	—
94	C₃	—	8	1	no	—
95	C₃	—	24	1	no	—
96	C₃	C₃²×#226	1	1	yes	characteristic
97	C₂	S₄×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃²×S₄
commutator, cosocle	C₃×SL(2,𝔽₃)	C₂×C₃²
Fitting	Q₈×3₋¹⁺²	D₃
prime radical	C₃²×SL(2,𝔽₃)	C₂×C₃
hypercenter	C₂×3₋¹⁺²	S₄
hypocenter	SL(2,𝔽₃)	C₂×3₋¹⁺²
LCS	G ▷ C₃×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. C₉×#226	C₃	semidirect
39. #226	3₋¹⁺²	direct (N × H)
51. 3₋¹⁺²	#226	direct (N × H)
81. C₉	C₃×#226	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)×3₋¹⁺²	2
C₃²×#226	3
C₉×#226	3
Q₁₆×3₋¹⁺²	3
Dic₃×3₋¹⁺²	4

Subgroups — 3−1+2×#226 #13048364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#226 #13048364