Subgroups — C₄×GL(2,𝔽₃) #5392

Order 192.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₄×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#859	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×SD₁₆	—	3	1	no	maximal, Sylow
5	C₂×C₄×D₃	—	4	1	no	maximal
6	C₂×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	GL(2,𝔽₃)	C₄	1	2	yes	direct product factor
8	C₂×SD₁₆	—	3	1	no	—
9	C₄×Q₈	D₃	1	1	yes	characteristic, complemented
10	C₄×D₄	—	3	1	no	—
11	#106	—	3	1	no	—
12	#103	—	3	1	no	—
13	#102	—	3	1	no	—
14	C₄×C₈	—	3	1	no	—
15	C₂²×D₃	—	4	1	no	—
16	C₂×C₃×C₄	—	4	1	no	—
17	C₂×Dic₃	—	4	1	no	—
18	C₄×D₃	—	4	4	no	—
19	SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic, complemented
20	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
21	C₂×D₄	—	3	1	no	—
22	C₂²×C₄	—	6	1	no	—
23	SD₁₆	—	3	2	no	—
24	SD₁₆	—	3	2	no	—
25	C₂×C₈	—	3	1	no	—
26	C₂×C₈	—	3	1	no	—
27	C₄⋊C₄	—	3	1	no	—
28	C₄⋊C₄	—	3	1	no	—
29	C₂²⋊C₄	—	6	1	no	—
30	C₄²	—	3	1	no	—
31	C₂²×C₃	—	4	1	no	—
32	C₂×D₃	—	4	2	no	—
33	C₂×D₃	—	4	2	no	—
34	C₂×D₃	—	4	2	no	—
35	C₃×C₄	—	4	2	no	—
36	Dic₃	—	4	2	no	—
37	C₂³	—	6	1	no	—
38	Q₈	—	3	1	no	—
39	Q₈	C₄×D₃	1	1	yes	characteristic, complemented
40	D₄	—	3	2	no	—
41	D₄	—	6	1	no	—
42	C₂×C₄	—	3	1	no	—
43	C₂×C₄	—	3	1	no	—
44	C₂×C₄	—	6	1	no	—
45	C₂×C₄	—	12	2	no	—
46	C₂×C₄	S₄	1	1	yes	characteristic
47	C₈	—	3	2	no	—
48	C₈	—	6	1	no	—
49	C₂×C₃	—	4	1	no	—
50	C₂×C₃	—	4	1	no	—
51	C₂×C₃	—	4	1	no	—
52	D₃	—	4	4	no	—
53	C₂²	—	6	2	no	—
54	C₂²	—	12	1	no	—
55	C₂²	—	12	1	no	—
56	C₂²	C₂×S₄	1	1	yes	characteristic
57	C₄	—	3	1	no	—
58	C₄	—	3	1	no	—
59	C₄	—	6	1	no	—
60	C₄	—	12	1	no	—
61	C₄	GL(2,𝔽₃)	1	2	yes	direct product factor
62	C₃	—	4	1	no	Sylow
63	C₂	—	12	2	no	—
64	C₂	C₄×S₄	1	1	yes	characteristic
65	C₂	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
66	C₂	#985	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₄
Frattini, socle	C₂²	C₂×S₄
Fitting	C₄×Q₈	D₃
prime radical	C₂×GL(2,𝔽₃)	C₂
cosocle	C₂×SL(2,𝔽₃)	C₂²
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×SL(2,𝔽₃)	C₂	semidirect
3. #859	C₂	semidirect
7. GL(2,𝔽₃)	C₄	direct (N × H)
9. C₄×Q₈	D₃	semidirect
19. SL(2,𝔽₃)	C₂×C₄	semidirect
39. Q₈	C₄×D₃	semidirect
61. C₄	GL(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×GL(2,𝔽₃)	2
C₄×SL(2,𝔽₃)	2
#859	2
C₄×SD₁₆	3
C₂×C₄×D₃	4

Subgroups — C4×GL(2,𝔽3) #5392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×GL(2,𝔽₃) #5392