Subgroups — D₃×S₄ #3840

Order 144.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊S₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×S₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×S₄	—	3	1	no	maximal
5	D₃×D₄	—	3	1	no	maximal
6	C₃×A₄	C₂²	1	1	yes	characteristic, complemented
7	D₃²	—	4	1	no	maximal
8	C₂²×D₃	—	3	1	no	—
9	C₂²×D₃	D₃	1	1	yes	characteristic, complemented
10	C₂×A₄	—	3	1	no	—
11	S₄	—	3	1	no	—
12	S₄	—	6	1	no	—
13	S₄	D₃	1	1	yes	characteristic, direct product factor
14	C₃×D₄	—	3	1	no	—
15	#67	—	3	1	no	—
16	#67	—	3	1	no	—
17	D₁₂	—	3	1	no	—
18	C₄×D₃	—	3	1	no	—
19	C₃⋊D₃	—	4	1	no	—
20	C₃×D₃	—	4	1	no	—
21	C₃×D₃	—	4	1	no	—
22	C₂×D₄	—	9	1	no	Sylow
23	C₂²×C₃	—	3	1	no	—
24	C₂²×C₃	C₂×D₃	1	1	yes	characteristic, complemented
25	C₂×D₃	—	3	1	no	—
26	C₂×D₃	—	3	1	no	—
27	C₂×D₃	—	3	1	no	—
28	C₂×D₃	—	6	1	no	—
29	C₂×D₃	—	6	1	no	—
30	C₂×D₃	—	12	1	no	—
31	A₄	—	2	1	no	—
32	A₄	C₂×D₃	1	1	yes	characteristic, complemented
33	C₃×C₄	—	3	1	no	—
34	Dic₃	—	3	1	no	—
35	C₃²	—	4	1	no	Sylow
36	C₂³	—	3	1	no	—
37	C₂³	—	9	1	no	—
38	D₄	—	3	1	no	—
39	D₄	—	9	1	no	—
40	D₄	—	9	1	no	—
41	D₄	—	9	1	no	—
42	C₂×C₄	—	9	1	no	—
43	C₂×C₃	—	3	1	no	—
44	C₂×C₃	—	6	1	no	—
45	C₂×C₃	—	12	1	no	—
46	D₃	—	3	1	no	—
47	D₃	—	4	1	no	—
48	D₃	—	6	1	no	—
49	D₃	—	12	1	no	—
50	D₃	—	24	1	no	—
51	D₃	S₄	1	1	yes	characteristic, direct product factor
52	C₂²	—	3	1	no	—
53	C₂²	—	9	1	no	—
54	C₂²	—	9	1	no	—
55	C₂²	—	9	1	no	—
56	C₂²	—	18	1	no	—
57	C₂²	—	18	1	no	—
58	C₂²	D₃²	1	1	yes	characteristic, complemented
59	C₄	—	3	1	no	—
60	C₄	—	9	1	no	—
61	C₃	—	4	1	no	—
62	C₃	—	8	1	no	—
63	C₃	C₂×S₄	1	1	yes	characteristic, complemented
64	C₂	—	3	1	no	—
65	C₂	—	3	1	no	—
66	C₂	—	6	1	no	—
67	C₂	—	9	1	no	—
68	C₂	—	18	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×A₄	C₂²
Fitting, socle	C₂²×C₃	C₂×D₃
LCS	G ▷ C₃×A₄
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. D₃×A₄	C₂	semidirect
2. C₃⋊S₄	C₂	semidirect
3. C₃×S₄	C₂	semidirect
6. C₃×A₄	C₂²	semidirect
9. C₂²×D₃	D₃	semidirect
13. S₄	D₃	direct (N × H)
24. C₂²×C₃	C₂×D₃	semidirect
32. A₄	C₂×D₃	semidirect
51. D₃	S₄	direct (N × H)
58. C₂²	D₃²	semidirect
63. C₃	C₂×S₄	semidirect

Maximal subgroups

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

subgroup	index
D₃×A₄	2
C₃⋊S₄	2
C₃×S₄	2
C₂×S₄	3
D₃×D₄	3
D₃²	4

Subgroups — D3×S4 #3840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×S₄ #3840