Subgroups — D₄×SL(2,𝔽₃) #5445

Order 192.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
2	C₄×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂×C₃×D₄	—	4	1	no	maximal
5	C₂×SL(2,𝔽₃)	—	2	2	no	—
6	C₂×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂²×Q₈	C₂×C₃	1	2	yes	complemented
8	C₄⋊Q₈	—	3	1	no	—
9	C₂²⋊Q₈	—	3	2	no	—
10	C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
11	C₄×D₄	—	3	1	no	—
12	C₂³×C₃	—	4	2	no	—
13	C₃×D₄	—	4	4	no	—
14	C₂×C₃×C₄	—	4	1	no	—
15	SL(2,𝔽₃)	D₄	1	1	yes	characteristic, direct product factor
16	C₂×Q₈	—	2	2	no	—
17	C₂×Q₈	—	3	2	no	—
18	C₂×Q₈	—	6	2	no	—
19	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic
20	C₂×D₄	A₄	1	1	yes	characteristic
21	C₂²×C₄	—	3	2	no	—
22	C₄⋊C₄	—	3	1	no	—
23	C₄⋊C₄	—	3	2	no	—
24	C₄⋊C₄	—	3	1	no	—
25	C₂²⋊C₄	—	3	2	no	—
26	C₄²	—	3	1	no	—
27	C₂²×C₃	—	4	4	no	—
28	C₂²×C₃	—	4	1	no	—
29	C₂²×C₃	—	8	2	no	—
30	C₂²×C₃	—	8	2	no	—
31	C₃×C₄	—	4	2	no	—
32	C₂³	C₂×A₄	1	2	yes	—
33	Q₈	—	3	1	no	—
34	Q₈	—	6	2	no	—
35	Q₈	—	6	2	no	—
36	Q₈	C₃×D₄	1	1	yes	characteristic, complemented
37	D₄	SL(2,𝔽₃)	1	4	yes	direct product factor
38	C₂×C₄	—	3	1	no	—
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	3	2	no	—
41	C₂×C₄	—	6	2	no	—
42	C₂×C₄	—	6	2	no	—
43	C₂×C₄	C₂×A₄	1	1	yes	characteristic
44	C₂×C₃	—	4	1	no	—
45	C₂×C₃	—	4	1	no	—
46	C₂×C₃	—	4	1	no	—
47	C₂×C₃	—	8	4	no	—
48	C₂²	—	2	2	no	—
49	C₂²	—	2	2	no	—
50	C₂²	C₂×SL(2,𝔽₃)	1	4	yes	complemented
51	C₂²	C₂²×A₄	1	1	yes	characteristic
52	C₄	—	3	1	no	—
53	C₄	—	3	1	no	—
54	C₄	—	6	2	no	—
55	C₄	—	6	1	no	—
56	C₄	C₂×SL(2,𝔽₃)	1	2	yes	complemented
57	C₃	—	4	1	no	Sylow
58	C₂	—	2	4	no	—
59	C₂	D₄×A₄	1	1	yes	characteristic
60	C₂	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic
61	C₂	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²	C₂²×A₄
commutator, cosocle	C₂×Q₈	C₂²×C₃
Fitting	D₄×Q₈	C₃
hypercenter	C₂×D₄	A₄
hypocenter	Q₈	C₃×D₄
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×SL(2,𝔽₃)	C₂	semidirect
2. C₄×SL(2,𝔽₃)	C₂	semidirect
3. D₄×Q₈	C₃	semidirect
7. C₂²×Q₈	C₂×C₃	semidirect
10. C₄×Q₈	C₂×C₃	semidirect
15. SL(2,𝔽₃)	D₄	direct (N × H)
36. Q₈	C₃×D₄	semidirect
37. D₄	SL(2,𝔽₃)	direct (N × H)
50. C₂²	C₂×SL(2,𝔽₃)	semidirect
56. C₄	C₂×SL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SL(2,𝔽₃)	2
C₄×SL(2,𝔽₃)	2
D₄×Q₈	3
C₂×C₃×D₄	4

Subgroups — D4×SL(2,𝔽3) #5445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×SL(2,𝔽₃) #5445