Subgroups — C₄²×SL(2,𝔽₃) #73164

Order 384.

229 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SL(2,𝔽₃)	C₂	1	3	yes	maximal
2	C₄²×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
3	C₂²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
4	C₂×C₃×C₄²	—	4	1	no	maximal
5	C₄×SL(2,𝔽₃)	C₄	1	6	yes	direct product factor
6	C₂×C₄×Q₈	C₂×C₃	1	3	yes	—
7	C₄×C₄⋊C₄	—	3	3	no	—
8	C₄³	—	3	1	no	—
9	C₂²×C₃×C₄	—	4	3	no	—
10	C₂×SL(2,𝔽₃)	C₂×C₄	1	3	yes	—
11	C₃×C₄²	—	4	4	no	—
12	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
13	C₄×Q₈	—	3	6	no	—
14	C₄×Q₈	C₃×C₄	1	6	yes	complemented
15	C₂×C₄⋊C₄	—	3	3	no	—
16	C₂×C₄²	—	3	3	no	—
17	C₂×C₄²	—	3	3	no	—
18	C₂×C₄²	A₄	1	1	yes	characteristic
19	#95	—	3	2	no	—
20	C₂³×C₃	—	4	1	no	—
21	C₂×C₃×C₄	—	4	6	no	—
22	C₂×C₃×C₄	—	4	6	no	—
23	C₂×C₃×C₄	—	4	6	no	—
24	SL(2,𝔽₃)	C₄²	1	1	yes	characteristic, direct product factor
25	C₂×Q₈	—	3	3	no	—
26	C₂×Q₈	C₂×C₃×C₄	1	3	yes	—
27	C₂²×C₄	—	3	1	no	—
28	C₂²×C₄	—	3	3	no	—
29	C₂²×C₄	C₂×A₄	1	3	yes	—
30	C₄⋊C₄	—	3	6	no	—
31	C₄⋊C₄	—	3	6	no	—
32	C₄²	—	3	6	no	—
33	C₄²	—	3	6	no	—
34	C₄²	—	6	6	no	—
35	C₄²	SL(2,𝔽₃)	1	4	yes	direct product factor
36	C₂²×C₃	—	4	3	no	—
37	C₂²×C₃	—	4	1	no	—
38	C₂²×C₃	—	4	3	no	—
39	C₃×C₄	—	4	12	no	—
40	C₂³	C₂²×A₄	1	1	yes	characteristic
41	Q₈	—	3	3	no	—
42	Q₈	—	3	2	no	—
43	Q₈	C₃×C₄²	1	1	yes	characteristic, complemented
44	C₂×C₄	—	3	3	no	—
45	C₂×C₄	—	3	3	no	—
46	C₂×C₄	—	3	6	no	—
47	C₂×C₄	—	6	6	no	—
48	C₂×C₄	C₄×A₄	1	6	yes	—
49	C₂×C₄	C₂×SL(2,𝔽₃)	1	6	yes	—
50	C₂×C₄	#41⋊C₃	1	6	yes	—
51	C₂×C₃	—	4	1	no	—
52	C₂×C₃	—	4	3	no	—
53	C₂×C₃	—	4	3	no	—
54	C₂²	C₂×C₄×A₄	1	3	yes	—
55	C₂²	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic
56	C₂²	C₂×#41⋊C₃	1	3	yes	—
57	C₄	—	3	3	no	—
58	C₄	—	3	1	no	—
59	C₄	—	6	6	no	—
60	C₄	C₄×SL(2,𝔽₃)	1	12	yes	direct product factor
61	C₃	—	4	1	no	Sylow
62	C₂	C₄²×A₄	1	1	yes	characteristic
63	C₂	C₂×C₄×SL(2,𝔽₃)	1	3	yes	—
64	C₂	C₄×#41⋊C₃	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	A₄
commutator, hypocenter	Q₈	C₃×C₄²
Frattini, socle	C₂³	C₂²×A₄
Fitting	C₄²×Q₈	C₃
prime radical	C₂²×SL(2,𝔽₃)	C₂²
cosocle	C₂²×Q₈	C₂²×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈	C₃	semidirect
5. C₄×SL(2,𝔽₃)	C₄	direct (N × H)
14. C₄×Q₈	C₃×C₄	semidirect
24. SL(2,𝔽₃)	C₄²	direct (N × H)
35. C₄²	SL(2,𝔽₃)	direct (N × H)
43. Q₈	C₃×C₄²	semidirect
60. C₄	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
C₂×C₄×SL(2,𝔽₃)	2
C₄²×Q₈	3
C₂×C₃×C₄²	4

Subgroups — C42×SL(2,𝔽3) #73164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×SL(2,𝔽₃) #73164