Subgroups — D₃×#226 #64200

Order 288.

83 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
2	C₃⋊#226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#226	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#226	—	3	1	no	maximal
5	D₃×Q₁₆	—	3	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	D₃×Dic₃	—	4	1	no	maximal
8	D₃×Q₈	—	3	1	no	—
9	D₃×Q₈	D₃	1	1	yes	characteristic
10	C₂×SL(2,𝔽₃)	—	3	1	no	—
11	#226	—	3	1	no	—
12	#226	—	6	1	no	—
13	#226	D₃	1	1	yes	characteristic, direct product factor
14	C₃×Q₁₆	—	3	1	no	—
15	#216	—	3	1	no	—
16	#216	—	3	1	no	—
17	Dic₁₂	—	3	1	no	—
18	D₃×C₈	—	3	1	no	—
19	C₂×C₃×D₃	—	4	1	no	—
20	C₃⋊Dic₃	—	4	1	no	—
21	C₃×Dic₃	—	4	1	no	—
22	C₂×Q₁₆	—	9	1	no	Sylow
23	C₃×Q₈	—	3	1	no	—
24	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
25	C₂×Dic₃	—	12	1	no	—
26	C₄×D₃	—	3	1	no	—
27	C₄×D₃	—	6	1	no	—
28	Dic₆	—	3	1	no	—
29	Dic₆	—	3	1	no	—
30	Dic₆	—	6	1	no	—
31	SL(2,𝔽₃)	—	2	1	no	—
32	SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic
33	C₃×C₈	—	3	1	no	—
34	C₃⋊C₈	—	3	1	no	—
35	C₂×C₃²	—	4	1	no	—
36	C₃×D₃	—	4	2	no	—
37	C₂×Q₈	—	3	1	no	—
38	C₂×Q₈	—	9	1	no	—
39	Q₁₆	—	3	1	no	—
40	Q₁₆	—	9	1	no	—
41	Q₁₆	—	9	1	no	—
42	Q₁₆	—	9	1	no	—
43	C₂×C₈	—	9	1	no	—
44	C₂²×C₃	—	12	1	no	—
45	C₂×D₃	S₄	1	1	yes	characteristic
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	6	1	no	—
48	Dic₃	—	3	1	no	—
49	Dic₃	—	4	1	no	—
50	Dic₃	—	6	1	no	—
51	Dic₃	—	12	1	no	—
52	Dic₃	—	24	1	no	—
53	C₃²	—	4	1	no	Sylow
54	Q₈	—	3	1	no	—
55	Q₈	—	9	1	no	—
56	Q₈	—	9	1	no	—
57	Q₈	—	18	1	no	—
58	Q₈	D₃²	1	1	yes	characteristic
59	C₂×C₄	—	9	1	no	—
60	C₂×C₄	—	18	1	no	—
61	C₈	—	3	1	no	—
62	C₈	—	9	1	no	—
63	C₂×C₃	—	4	1	no	—
64	C₂×C₃	—	8	1	no	—
65	C₂×C₃	—	12	2	no	—
66	C₂×C₃	C₂×S₄	1	1	yes	characteristic
67	D₃	#226	1	2	yes	direct product factor
68	C₂²	—	3	1	no	—
69	C₄	—	3	1	no	—
70	C₄	—	6	1	no	—
71	C₄	—	9	1	no	—
72	C₄	—	18	1	no	—
73	C₃	—	4	1	no	—
74	C₃	—	8	1	no	—
75	C₃	C₂×#226	1	1	yes	characteristic, complemented
76	C₂	—	3	2	no	—
77	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
prime radical	D₃×SL(2,𝔽₃)	C₂
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
2. C₃⋊#226	C₂	semidirect
3. C₃×#226	C₂	semidirect
13. #226	D₃	direct (N × H)
67. D₃	#226	direct (N × H)
75. C₃	C₂×#226	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⋊#226	2
C₃×#226	2
C₂×#226	3
D₃×Q₁₆	3
D₃×Dic₃	4

Subgroups — D3×#226 #64200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#226 #64200