Subgroups — C₂⁴×SL(2,𝔽₃) #87527

Order 384.

1779 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SL(2,𝔽₃)	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
3	C₂⁵×C₃	—	4	1	no	maximal
4	C₂²×SL(2,𝔽₃)	C₂²	1	35	yes	direct product factor
5	C₂³×Q₈	—	3	15	no	—
6	C₂³×Q₈	C₂×C₃	1	15	yes	complemented
7	C₂⁴×C₄	—	3	1	no	—
8	C₂⁴×C₃	—	4	16	no	—
9	C₂⁴×C₃	—	4	15	no	—
10	C₂×SL(2,𝔽₃)	C₂³	1	15	yes	direct product factor
11	C₂⁵	A₄	1	1	yes	characteristic
12	C₂²×Q₈	—	3	70	no	—
13	C₂²×Q₈	—	3	105	no	—
14	C₂²×Q₈	C₂²×C₃	1	35	yes	complemented
15	C₂³×C₄	—	3	15	no	—
16	C₂³×C₄	—	3	15	no	—
17	C₂³×C₃	—	4	120	no	—
18	C₂³×C₃	—	4	35	no	—
19	SL(2,𝔽₃)	C₂⁴	1	1	yes	characteristic, direct product factor
20	C₂⁴	SL(2,𝔽₃)	1	16	yes	direct product factor
21	C₂⁴	C₂×A₄	1	15	yes	—
22	C₂×Q₈	—	3	210	no	—
23	C₂×Q₈	—	3	105	no	—
24	C₂×Q₈	C₂³×C₃	1	15	yes	complemented
25	C₂²×C₄	—	3	105	no	—
26	C₂²×C₄	—	3	35	no	—
27	C₂²×C₃	—	4	140	no	—
28	C₂²×C₃	—	4	15	no	—
29	C₂³	C₂×SL(2,𝔽₃)	1	120	yes	direct product factor
30	C₂³	C₂²×A₄	1	35	yes	—
31	Q₈	—	3	15	no	—
32	Q₈	—	3	70	no	—
33	Q₈	C₂⁴×C₃	1	1	yes	characteristic, complemented
34	C₂×C₄	—	3	105	no	—
35	C₂×C₄	—	3	15	no	—
36	C₂×C₃	—	4	30	no	—
37	C₂×C₃	—	4	1	no	—
38	C₂²	C₂²×SL(2,𝔽₃)	1	140	yes	direct product factor
39	C₂²	C₂³×A₄	1	15	yes	—
40	C₄	—	3	15	no	—
41	C₄	—	3	1	no	—
42	C₃	—	4	1	no	Sylow
43	C₂	C₂³×SL(2,𝔽₃)	1	30	yes	direct product factor
44	C₂	C₂⁴×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂⁵	A₄
commutator, cosocle, hypocenter	Q₈	C₂⁴×C₃
Frattini	C₂	C₂⁴×A₄
Fitting	C₂⁴×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂³×SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂⁴×Q₈	C₃	semidirect
4. C₂²×SL(2,𝔽₃)	C₂²	direct (N × H)
6. C₂³×Q₈	C₂×C₃	semidirect
10. C₂×SL(2,𝔽₃)	C₂³	direct (N × H)
14. C₂²×Q₈	C₂²×C₃	semidirect
19. SL(2,𝔽₃)	C₂⁴	direct (N × H)
20. C₂⁴	SL(2,𝔽₃)	direct (N × H)
24. C₂×Q₈	C₂³×C₃	semidirect
29. C₂³	C₂×SL(2,𝔽₃)	direct (N × H)
33. Q₈	C₂⁴×C₃	semidirect
38. C₂²	C₂²×SL(2,𝔽₃)	direct (N × H)
43. C₂	C₂³×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
C₂³×SL(2,𝔽₃)	2
C₂⁴×Q₈	3
C₂⁵×C₃	4

Subgroups — C24×SL(2,𝔽3) #87527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×SL(2,𝔽₃) #87527