Subgroups — C₂×Q₈⋊H(3,𝔽₃) #88689

Order 432.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×C₃²×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×SL(2,𝔽₃)	C₃	1	3	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	—	4	1	no	maximal
5	C₃²×Q₈	—	3	1	no	—
6	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃²×C₄	—	3	1	no	—
8	C₃×SL(2,𝔽₃)	C₂×C₃	1	3	yes	complemented
9	C₂×H(3,𝔽₃)	—	4	2	no	—
10	C₂×H(3,𝔽₃)	—	4	1	no	—
11	C₂×C₃×Q₈	—	3	1	no	—
12	C₂×C₃×Q₈	C₃²	1	1	yes	characteristic
13	C₂×SL(2,𝔽₃)	—	3	3	no	—
14	C₂²×C₃²	—	4	3	no	—
15	C₂²×C₃²	A₄	1	1	yes	characteristic
16	C₃²×C₄	—	3	1	no	—
17	C₃²×C₄	—	3	1	no	—
18	H(3,𝔽₃)	—	4	1	no	Sylow
19	C₃×Q₈	—	3	1	no	—
20	C₃×Q₈	—	3	3	no	—
21	C₃×Q₈	—	3	1	no	—
22	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
23	C₂×C₃×C₄	—	3	3	no	—
24	C₂×C₃×C₄	—	3	1	no	—
25	SL(2,𝔽₃)	—	3	3	no	—
26	C₂×C₃²	—	4	6	no	—
27	C₂×C₃²	—	4	3	no	—
28	C₂×C₃²	SL(2,𝔽₃)	1	2	yes	complemented
29	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
30	C₂×Q₈	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
31	C₂²×C₃	—	3	1	no	—
32	C₂²×C₃	—	12	3	no	—
33	C₂²×C₃	C₃×A₄	1	1	yes	characteristic
34	C₃×C₄	—	3	3	no	—
35	C₃×C₄	—	3	3	no	—
36	C₃×C₄	—	3	1	no	—
37	C₃×C₄	—	3	1	no	—
38	C₃²	—	4	3	no	—
39	C₃²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
40	Q₈	—	3	1	no	—
41	Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
42	C₂×C₄	—	3	1	no	—
43	C₂×C₃	—	3	2	no	—
44	C₂×C₃	—	3	1	no	—
45	C₂×C₃	—	12	6	no	—
46	C₂×C₃	—	12	3	no	—
47	C₂×C₃	C₃×SL(2,𝔽₃)	1	2	yes	—
48	C₂×C₃	C₂×C₃×A₄	1	1	yes	characteristic
49	C₂²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
50	C₄	—	3	1	no	—
51	C₄	—	3	1	no	—
52	C₃	—	3	1	no	—
53	C₃	—	12	3	no	—
54	C₃	C₂×C₃×SL(2,𝔽₃)	1	1	yes	characteristic
55	C₂	Q₈⋊H(3,𝔽₃)	1	2	yes	direct product factor
56	C₂	C₂×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₃×A₄
commutator, cosocle	C₃×Q₈	C₂×C₃²
Frattini	C₂×C₃	C₂×C₃×A₄
Fitting	C₂×C₃²×Q₈	C₃
hypercenter	C₂²×C₃²	A₄
hypocenter	Q₈	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
1. Q₈⋊H(3,𝔽₃)	C₂	direct (N × H)
2. C₂×C₃²×Q₈	C₃	semidirect
3. C₂×C₃×SL(2,𝔽₃)	C₃	semidirect
6. C₃²×Q₈	C₂×C₃	semidirect
8. C₃×SL(2,𝔽₃)	C₂×C₃	semidirect
28. C₂×C₃²	SL(2,𝔽₃)	semidirect
30. C₂×Q₈	H(3,𝔽₃)	semidirect
39. C₃²	C₂×SL(2,𝔽₃)	semidirect
41. Q₈	C₂×H(3,𝔽₃)	semidirect
55. C₂	Q₈⋊H(3,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
Q₈⋊H(3,𝔽₃)	2
C₂×C₃²×Q₈	3
C₂×C₃×SL(2,𝔽₃)	3
C₂²×H(3,𝔽₃)	4

Subgroups — C2×Q8⋊H(3,𝔽3) #88689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈⋊H(3,𝔽₃) #88689