Subgroups — C₂×C₄×SL(2,𝔽₃) #5437

Order 192.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂²×C₄	A₄
commutator, hypocenter	Q₈	C₂×C₃×C₄
Frattini	C₂²	C₂²×A₄
Fitting	C₂×C₄×Q₈	C₃
socle	C₂³	C₂×A₄
prime radical	C₂²×SL(2,𝔽₃)	C₂
cosocle	C₂×Q₈	C₂²×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×SL(2,𝔽₃)	C₂	direct (N × H)
3. C₂×C₄×Q₈	C₃	semidirect
5. C₂×SL(2,𝔽₃)	C₄	direct (N × H)
9. C₄×Q₈	C₂×C₃	semidirect
15. SL(2,𝔽₃)	C₂×C₄	direct (N × H)
18. C₂×Q₈	C₃×C₄	semidirect
36. Q₈	C₂×C₃×C₄	semidirect
43. C₂×C₄	SL(2,𝔽₃)	direct (N × H)
57. C₄	C₂×SL(2,𝔽₃)	direct (N × H)
59. C₂	C₄×SL(2,𝔽₃)	direct (N × H)

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
C₂×C₄×Q₈	3
C₂²×C₃×C₄	4

Subgroups — C2×C4×SL(2,𝔽3) #5437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×SL(2,𝔽₃) #5437