Subgroups — SL(2,𝔽₃)⋊C₁₆ #68004

Order 384.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	Q₈⋊C₁₆	—	3	1	no	maximal, Sylow
3	C₄×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
4	C₂×C₃⋊C₁₆	—	4	1	no	maximal
5	C₈×Q₈	D₃	1	1	yes	characteristic
6	C₄⋊C₁₆	—	3	1	no	—
7	C₄×C₁₆	—	3	1	no	—
8	C₂×SL(2,𝔽₃)	C₈	1	1	yes	characteristic
9	C₂×C₃×C₈	—	4	1	no	—
10	C₃⋊C₁₆	—	4	2	no	—
11	C₄×Q₈	Dic₃	1	1	yes	characteristic
12	C₂×C₁₆	—	6	1	no	—
13	C₂×C₁₆	—	6	1	no	—
14	C₄⋊C₈	—	3	1	no	—
15	C₄×C₈	—	3	1	no	—
16	C₂×C₃×C₄	—	4	1	no	—
17	SL(2,𝔽₃)	C₁₆	1	1	yes	characteristic, complemented
18	C₃×C₈	—	4	1	no	—
19	C₃×C₈	—	4	1	no	—
20	C₂×Q₈	C₃⋊C₈	1	1	yes	characteristic
21	C₂×C₈	—	3	1	no	—
22	C₂×C₈	S₄	1	1	yes	characteristic
23	C₄⋊C₄	—	3	1	no	—
24	C₄²	—	3	1	no	—
25	C₁₆	—	6	2	no	—
26	C₁₆	—	12	1	no	—
27	C₂²×C₃	—	4	1	no	—
28	C₃×C₄	—	4	1	no	—
29	C₃×C₄	—	4	1	no	—
30	Q₈	—	3	1	no	—
31	Q₈	C₃⋊C₁₆	1	1	yes	characteristic, complemented
32	C₂×C₄	—	3	1	no	—
33	C₂×C₄	—	3	1	no	—
34	C₂×C₄	SL(2,ℤ/4)	1	1	yes	characteristic
35	C₈	—	6	1	no	—
36	C₈	#226	1	1	yes	characteristic
37	C₈	GL(2,𝔽₃)	1	1	yes	characteristic
38	C₂×C₃	—	4	1	no	—
39	C₂×C₃	—	4	1	no	—
40	C₂×C₃	—	4	1	no	—
41	C₂²	A₄⋊C₈	1	1	yes	characteristic
42	C₄	—	3	1	no	—
43	C₄	—	3	1	no	—
44	C₄	—	6	1	no	—
45	C₄	#859	1	1	yes	characteristic
46	C₄	GU(2,𝔽₃)	1	1	yes	characteristic
47	C₃	—	4	1	no	Sylow
48	C₂	SL(2,𝔽₃)⋊C₈	1	1	yes	characteristic
49	C₂	A₄⋊C₁₆	1	1	yes	characteristic
50	C₂	#4628	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₈	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₁₆
Fitting	C₈×Q₈	D₃
socle	C₂²	A₄⋊C₈
prime radical	C₂×SL(2,𝔽₃)	C₈
cosocle	C₈×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₈

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

N	H ≅ G/N	type
17. SL(2,𝔽₃)	C₁₆	semidirect
31. Q₈	C₃⋊C₁₆	semidirect

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