Subgroups — Q₁₆×SL(2,𝔽₃) #73269

Order 384.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SL(2,𝔽₃)	C₂	1	2	yes	maximal
2	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	Q₈×Q₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂×C₃×Q₁₆	—	4	1	no	maximal
5	C₄×SL(2,𝔽₃)	—	2	2	no	—
6	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	Q₈²	C₂×C₃	1	2	yes	—
8	#500	—	3	1	no	—
9	#477	—	3	2	no	—
10	C₈×Q₈	C₂×C₃	1	1	yes	characteristic
11	C₄×Q₁₆	—	3	1	no	—
12	C₂×C₃×Q₈	—	4	2	no	—
13	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
14	C₃×Q₁₆	—	4	4	no	—
15	C₂×C₃×C₈	—	4	1	no	—
16	C₂×Q₁₆	A₄	1	1	yes	characteristic
17	C₄⋊Q₈	—	3	2	no	—
18	C₄⋊Q₈	—	6	2	no	—
19	C₄×Q₈	—	2	2	no	—
20	C₄×Q₈	—	3	2	no	—
21	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
22	#107	—	3	2	no	—
23	#107	—	3	1	no	—
24	C₄⋊C₈	—	3	1	no	—
25	#103	—	3	2	no	—
26	C₄×C₈	—	3	1	no	—
27	C₃×Q₈	—	4	4	no	—
28	C₃×Q₈	—	8	2	no	—
29	C₂×C₃×C₄	—	4	1	no	—
30	C₂×C₃×C₄	—	8	2	no	—
31	SL(2,𝔽₃)	Q₁₆	1	1	yes	characteristic, direct product factor
32	C₃×C₈	—	4	2	no	—
33	C₂×Q₈	—	6	2	no	—
34	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
35	C₂×Q₈	C₂×A₄	1	2	yes	—
36	Q₁₆	SL(2,𝔽₃)	1	4	yes	direct product factor
37	C₂×C₈	—	3	1	no	—
38	C₂×C₈	C₂×A₄	1	1	yes	characteristic
39	C₄⋊C₄	—	3	1	no	—
40	C₄⋊C₄	—	3	2	no	—
41	C₄⋊C₄	—	6	2	no	—
42	C₄⋊C₄	—	6	2	no	—
43	C₄²	—	3	1	no	—
44	C₄²	—	6	2	no	—
45	C₂²×C₃	—	4	1	no	—
46	C₃×C₄	—	4	1	no	—
47	C₃×C₄	—	4	1	no	—
48	C₃×C₄	—	8	4	no	—
49	Q₈	—	2	2	no	—
50	Q₈	—	3	1	no	—
51	Q₈	—	24	2	no	—
52	Q₈	C₃×Q₁₆	1	1	yes	characteristic, complemented
53	Q₈	C₂×SL(2,𝔽₃)	1	4	yes	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	3	1	no	—
57	C₂×C₄	—	6	2	no	—
58	C₂×C₄	C₂²×A₄	1	1	yes	characteristic
59	C₈	—	6	1	no	—
60	C₈	C₂×SL(2,𝔽₃)	1	2	yes	—
61	C₂×C₃	—	4	1	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	—	4	1	no	—
64	C₂²	D₄×A₄	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	3	1	no	—
67	C₄	—	3	1	no	—
68	C₄	—	6	1	no	—
69	C₄	—	12	2	no	—
70	C₄	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic
71	C₄	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic
72	C₃	—	4	1	no	Sylow
73	C₂	D₄×SL(2,𝔽₃)	1	1	yes	characteristic
74	C₂	A₄×Q₁₆	1	1	yes	characteristic
75	C₂	#576⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄×A₄
commutator, cosocle	C₄×Q₈	C₂²×C₃
Frattini	C₂×C₄	C₂²×A₄
Fitting	Q₈×Q₁₆	C₃
prime radical	C₂×SL(2,𝔽₃)	D₄
hypercenter	C₂×Q₁₆	A₄
hypocenter	Q₈	C₃×Q₁₆
LCS	G ▷ C₄×Q₈ ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. Q₈×Q₁₆	C₃	semidirect
31. SL(2,𝔽₃)	Q₁₆	direct (N × H)
36. Q₁₆	SL(2,𝔽₃)	direct (N × H)
52. Q₈	C₃×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×SL(2,𝔽₃)	2
C₈×SL(2,𝔽₃)	2
Q₈×Q₁₆	3
C₂×C₃×Q₁₆	4

Subgroups — Q16×SL(2,𝔽3) #73269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×SL(2,𝔽₃) #73269