Subgroups — C₂×C₂²⋊SL(2,𝔽₃) #5947

Order 192.

175 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, direct product factor
2	C₂³×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
3	C₂²×A₄	—	4	1	no	maximal
4	C₂×SL(2,𝔽₃)	—	4	4	no	maximal
5	C₂²×Q₈	—	3	4	no	—
6	C₂²×Q₈	—	3	4	no	—
7	C₂²×Q₈	—	3	1	no	—
8	C₂²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂³×C₄	—	3	1	no	—
10	C₂×A₄	—	4	2	no	—
11	C₂×A₄	—	4	1	no	—
12	SL(2,𝔽₃)	—	4	4	no	—
13	C₂⁴	A₄	1	1	yes	characteristic
14	C₂×Q₈	—	3	12	no	—
15	C₂×Q₈	—	3	12	no	—
16	C₂×Q₈	—	3	4	no	—
17	C₂×Q₈	—	3	4	no	—
18	C₂×Q₈	—	3	4	no	—
19	C₂×Q₈	A₄	1	4	yes	complemented
20	C₂²×C₄	—	3	6	no	—
21	C₂²×C₄	—	3	6	no	—
22	C₂²×C₄	—	3	1	no	—
23	C₂²×C₄	—	3	1	no	—
24	C₂²×C₃	—	16	1	no	—
25	A₄	—	4	1	no	—
26	C₂³	—	3	1	no	—
27	C₂³	—	3	2	no	—
28	C₂³	—	3	1	no	—
29	C₂³	SL(2,𝔽₃)	1	2	yes	complemented
30	C₂³	C₂×A₄	1	1	yes	characteristic
31	Q₈	—	3	4	no	—
32	Q₈	—	3	12	no	—
33	Q₈	—	3	4	no	—
34	Q₈	C₂×A₄	1	4	yes	complemented
35	C₂×C₄	—	3	6	no	—
36	C₂×C₄	—	3	12	no	—
37	C₂×C₄	—	3	6	no	—
38	C₂×C₄	—	3	4	no	—
39	C₂×C₃	—	16	2	no	—
40	C₂×C₃	—	16	1	no	—
41	C₂²	—	3	2	no	—
42	C₂²	—	3	1	no	—
43	C₂²	—	3	2	no	—
44	C₂²	—	3	2	no	—
45	C₂²	—	3	2	no	—
46	C₂²	—	3	1	no	—
47	C₂²	—	3	1	no	—
48	C₂²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
49	C₂²	C₂²⋊A₄	1	1	yes	characteristic
50	C₄	—	3	4	no	—
51	C₄	—	3	4	no	—
52	C₃	—	16	1	no	Sylow
53	C₂	—	3	2	no	—
54	C₂	—	3	1	no	—
55	C₂	—	3	1	no	—
56	C₂	C₂²⋊SL(2,𝔽₃)	1	2	yes	direct product factor
57	C₂	C₂×C₂²⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	C₂²⋊A₄
commutator, cosocle, hypocenter	C₂²×Q₈	C₂×C₃
Frattini	C₂	C₂×C₂²⋊A₄
Fitting	C₂³×Q₈	C₃
socle	C₂⁴	A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²⋊SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂³×Q₈	C₃	semidirect
8. C₂²×Q₈	C₂×C₃	semidirect
19. C₂×Q₈	A₄	semidirect
29. C₂³	SL(2,𝔽₃)	semidirect
34. Q₈	C₂×A₄	semidirect
48. C₂²	C₂×SL(2,𝔽₃)	semidirect
56. C₂	C₂²⋊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
C₂³×Q₈	3
C₂²×A₄	4
C₂×SL(2,𝔽₃)	4

Subgroups — C2×C22⋊SL(2,𝔽3) #5947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊SL(2,𝔽₃) #5947