Subgroups — C₂³×GL(2,𝔽₃) #87479

Order 384.

1187 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×GL(2,𝔽₃)	C₂	1	14	yes	maximal, direct product factor
3	C₂³×SD₁₆	—	3	1	no	maximal, Sylow
4	C₂⁴×D₃	—	4	1	no	maximal
5	C₂²×SL(2,𝔽₃)	C₂²	1	7	yes	complemented
6	C₂×GL(2,𝔽₃)	C₂²	1	28	yes	direct product factor
7	C₂³×Q₈	D₃	1	1	yes	characteristic, complemented
8	C₂³×D₄	—	3	1	no	—
9	C₂²×SD₁₆	—	3	14	no	—
10	C₂²×SD₁₆	—	3	14	no	—
11	C₂³×C₈	—	3	1	no	—
12	C₂⁴×C₃	—	4	1	no	—
13	C₂³×D₃	—	4	14	no	—
14	C₂³×D₃	—	4	16	no	—
15	C₂×SL(2,𝔽₃)	C₂³	1	7	yes	complemented
16	GL(2,𝔽₃)	C₂³	1	8	yes	direct product factor
17	C₂⁵	—	6	1	no	—
18	C₂²×Q₈	—	3	7	no	—
19	C₂²×Q₈	C₂×D₃	1	7	yes	complemented
20	C₂²×D₄	—	3	14	no	—
21	C₂²×D₄	—	6	7	no	—
22	C₂³×C₄	—	3	1	no	—
23	C₂×SD₁₆	—	3	84	no	—
24	C₂×SD₁₆	—	3	28	no	—
25	C₂²×C₈	—	3	14	no	—
26	C₂³×C₃	—	4	8	no	—
27	C₂³×C₃	—	4	7	no	—
28	C₂²×D₃	—	4	28	no	—
29	C₂²×D₃	—	4	112	no	—
30	SL(2,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented
31	C₂⁴	—	6	14	no	—
32	C₂⁴	—	12	8	no	—
33	C₂⁴	S₄	1	1	yes	characteristic
34	C₂×Q₈	—	3	21	no	—
35	C₂×Q₈	—	6	7	no	—
36	C₂×Q₈	C₂²×D₃	1	7	yes	complemented
37	C₂×D₄	—	3	28	no	—
38	C₂×D₄	—	6	42	no	—
39	C₂²×C₄	—	3	7	no	—
40	C₂²×C₄	—	3	7	no	—
41	SD₁₆	—	3	56	no	—
42	SD₁₆	—	3	8	no	—
43	C₂×C₈	—	3	28	no	—
44	C₂²×C₃	—	4	28	no	—
45	C₂²×C₃	—	4	7	no	—
46	C₂×D₃	—	4	8	no	—
47	C₂×D₃	—	4	112	no	—
48	C₂³	—	6	28	no	—
49	C₂³	—	12	56	no	—
50	C₂³	GL(2,𝔽₃)	1	8	yes	direct product factor
51	C₂³	C₂×S₄	1	7	yes	—
52	Q₈	—	3	7	no	—
53	Q₈	—	6	7	no	—
54	Q₈	C₂³×D₃	1	1	yes	characteristic, complemented
55	D₄	—	3	8	no	—
56	D₄	—	6	28	no	—
57	C₂×C₄	—	3	7	no	—
58	C₂×C₄	—	3	21	no	—
59	C₈	—	3	8	no	—
60	C₂×C₃	—	4	14	no	—
61	C₂×C₃	—	4	1	no	—
62	D₃	—	4	16	no	—
63	C₂²	—	6	8	no	—
64	C₂²	—	12	56	no	—
65	C₂²	C₂×GL(2,𝔽₃)	1	28	yes	direct product factor
66	C₂²	C₂²×S₄	1	7	yes	—
67	C₄	—	3	1	no	—
68	C₄	—	3	7	no	—
69	C₃	—	4	1	no	Sylow
70	C₂	—	12	8	no	—
71	C₂	C₂²×GL(2,𝔽₃)	1	14	yes	direct product factor
72	C₂	C₂³×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂⁴	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴
Frattini	C₂	C₂³×S₄
Fitting	C₂³×Q₈	D₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×SL(2,𝔽₃)	C₂	semidirect
2. C₂²×GL(2,𝔽₃)	C₂	direct (N × H)
5. C₂²×SL(2,𝔽₃)	C₂²	semidirect
6. C₂×GL(2,𝔽₃)	C₂²	direct (N × H)
7. C₂³×Q₈	D₃	semidirect
15. C₂×SL(2,𝔽₃)	C₂³	semidirect
16. GL(2,𝔽₃)	C₂³	direct (N × H)
19. C₂²×Q₈	C₂×D₃	semidirect
30. SL(2,𝔽₃)	C₂⁴	semidirect
36. C₂×Q₈	C₂²×D₃	semidirect
50. C₂³	GL(2,𝔽₃)	direct (N × H)
54. Q₈	C₂³×D₃	semidirect
65. C₂²	C₂×GL(2,𝔽₃)	direct (N × H)
71. C₂	C₂²×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
C₂³×SL(2,𝔽₃)	2
C₂²×GL(2,𝔽₃)	2
C₂³×SD₁₆	3
C₂⁴×D₃	4

Subgroups — C23×GL(2,𝔽3) #87479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×GL(2,𝔽₃) #87479