Subgroups — D₃×GL(2,𝔽₃) #64203

Order 288.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×GL(2,𝔽₃)	—	3	1	no	maximal
5	D₃×SD₁₆	—	3	1	no	maximal
6	C₂×D₃²	—	4	1	no	maximal
7	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
8	D₃×Q₈	D₃	1	1	yes	characteristic, complemented
9	D₃×D₄	—	3	1	no	—
10	C₂×SL(2,𝔽₃)	—	3	1	no	—
11	GL(2,𝔽₃)	—	3	1	no	—
12	GL(2,𝔽₃)	—	6	1	no	—
13	GL(2,𝔽₃)	D₃	1	1	yes	characteristic, direct product factor
14	C₃×SD₁₆	—	3	1	no	—
15	#215	—	3	1	no	—
16	#214	—	3	1	no	—
17	#204	—	3	1	no	—
18	D₃×C₈	—	3	1	no	—
19	C₂×C₃⋊D₃	—	4	1	no	—
20	C₂×C₃×D₃	—	4	1	no	—
21	C₂×C₃×D₃	—	4	1	no	—
22	D₃²	—	4	4	no	—
23	C₂×SD₁₆	—	9	1	no	Sylow
24	C₂²×D₃	—	6	1	no	—
25	C₂²×D₃	—	12	1	no	—
26	C₃×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
27	C₃×D₄	—	3	1	no	—
28	#67	—	6	1	no	—
29	D₁₂	—	3	1	no	—
30	C₄×D₃	—	3	1	no	—
31	Dic₆	—	3	1	no	—
32	SL(2,𝔽₃)	—	2	1	no	—
33	SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic, complemented
34	C₃×C₈	—	3	1	no	—
35	C₃⋊C₈	—	3	1	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₃⋊D₃	—	4	2	no	—
38	C₃×D₃	—	4	2	no	—
39	C₃×D₃	—	4	2	no	—
40	C₂×Q₈	—	3	1	no	—
41	C₂×D₄	—	9	1	no	—
42	SD₁₆	—	3	1	no	—
43	SD₁₆	—	9	1	no	—
44	SD₁₆	—	9	1	no	—
45	SD₁₆	—	9	1	no	—
46	C₂×C₈	—	9	1	no	—
47	C₂²×C₃	—	6	1	no	—
48	C₂²×C₃	—	12	1	no	—
49	C₂×D₃	—	4	1	no	—
50	C₂×D₃	—	6	1	no	—
51	C₂×D₃	—	12	1	no	—
52	C₂×D₃	—	12	4	no	—
53	C₂×D₃	—	12	2	no	—
54	C₂×D₃	—	24	1	no	—
55	C₂×D₃	S₄	1	1	yes	characteristic
56	C₃×C₄	—	3	1	no	—
57	Dic₃	—	3	1	no	—
58	C₃²	—	4	1	no	Sylow
59	C₂³	—	18	1	no	—
60	Q₈	—	9	1	no	—
61	Q₈	D₃²	1	1	yes	characteristic, complemented
62	D₄	—	3	1	no	—
63	D₄	—	9	1	no	—
64	D₄	—	18	1	no	—
65	C₂×C₄	—	9	1	no	—
66	C₈	—	3	1	no	—
67	C₈	—	9	1	no	—
68	C₂×C₃	—	4	1	no	—
69	C₂×C₃	—	8	1	no	—
70	C₂×C₃	—	12	2	no	—
71	C₂×C₃	—	12	1	no	—
72	C₂×C₃	C₂×S₄	1	1	yes	characteristic
73	D₃	—	4	2	no	—
74	D₃	—	12	2	no	—
75	D₃	—	12	1	no	—
76	D₃	—	24	2	no	—
77	D₃	GL(2,𝔽₃)	1	2	yes	direct product factor
78	C₂²	—	3	1	no	—
79	C₂²	—	6	1	no	—
80	C₂²	—	18	1	no	—
81	C₂²	—	36	2	no	—
82	C₄	—	3	1	no	—
83	C₄	—	9	1	no	—
84	C₃	—	4	1	no	—
85	C₃	—	8	1	no	—
86	C₃	C₂×GL(2,𝔽₃)	1	1	yes	characteristic, complemented
87	C₂	—	3	2	no	—
88	C₂	—	12	1	no	—
89	C₂	—	36	1	no	—
90	C₂	D₃×S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×S₄
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂²
Fitting	C₃×Q₈	C₂×D₃
socle	C₂×C₃	C₂×S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. D₃×SL(2,𝔽₃)	C₂	semidirect
2. C₃⋊GL(2,𝔽₃)	C₂	semidirect
3. C₃×GL(2,𝔽₃)	C₂	semidirect
7. C₃×SL(2,𝔽₃)	C₂²	semidirect
8. D₃×Q₈	D₃	semidirect
13. GL(2,𝔽₃)	D₃	direct (N × H)
26. C₃×Q₈	C₂×D₃	semidirect
33. SL(2,𝔽₃)	C₂×D₃	semidirect
61. Q₈	D₃²	semidirect
77. D₃	GL(2,𝔽₃)	direct (N × H)
86. C₃	C₂×GL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
D₃×SL(2,𝔽₃)	2
C₃⋊GL(2,𝔽₃)	2
C₃×GL(2,𝔽₃)	2
C₂×GL(2,𝔽₃)	3
D₃×SD₁₆	3
C₂×D₃²	4

Subgroups — D3×GL(2,𝔽3) #64203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×GL(2,𝔽₃) #64203