Subgroups — H(3,𝔽₃)×SD₃₂ #11720639

Order 864.

143 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	C₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×SD₃₂	C₃	1	4	yes	maximal, complemented
5	Q₈×H(3,𝔽₃)	—	2	1	no	—
6	D₄×H(3,𝔽₃)	—	2	1	no	—
7	C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₃²×Q₁₆	C₂×C₃	1	4	yes	complemented
9	C₃²×D₈	C₂×C₃	1	4	yes	—
10	C₃²×C₁₆	C₂×C₃	1	4	yes	complemented
11	C₂²×H(3,𝔽₃)	—	4	1	no	—
12	C₄×H(3,𝔽₃)	—	4	1	no	—
13	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
14	C₃×SD₃₂	—	3	4	no	—
15	C₃×SD₃₂	C₃²	1	1	yes	characteristic
16	C₃²×Q₈	—	2	4	no	—
17	C₃²×D₄	—	2	4	no	—
18	C₃²×C₈	C₂²×C₃	1	4	yes	—
19	C₂×H(3,𝔽₃)	—	8	1	no	—
20	C₂×H(3,𝔽₃)	D₈	1	1	yes	characteristic
21	C₃×Q₁₆	—	3	4	no	—
22	C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
23	C₃×D₈	—	3	4	no	—
24	C₃×D₈	C₂×C₃²	1	1	yes	characteristic
25	C₃×C₁₆	—	3	4	no	—
26	C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
27	C₂²×C₃²	—	4	4	no	—
28	C₃²×C₄	—	4	4	no	—
29	C₃²×C₄	C₃×D₄	1	4	yes	—
30	SD₃₂	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
31	H(3,𝔽₃)	SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
32	C₃×Q₈	—	2	1	no	—
33	C₃×Q₈	—	6	4	no	—
34	C₃×D₄	—	2	1	no	—
35	C₃×D₄	—	6	4	no	—
36	C₃×C₈	—	3	4	no	—
37	C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
38	C₂×C₃²	—	8	4	no	—
39	C₂×C₃²	C₃×D₈	1	4	yes	—
40	Q₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
41	D₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
43	C₂²×C₃	—	4	1	no	—
44	C₂²×C₃	—	12	4	no	—
45	C₃×C₄	—	3	4	no	—
46	C₃×C₄	—	4	1	no	—
47	C₃×C₄	—	12	4	no	—
48	C₃×C₄	C₃²×D₄	1	1	yes	characteristic
49	C₃²	C₃×SD₃₂	1	4	yes	complemented
50	Q₈	—	2	1	no	—
51	D₄	—	2	1	no	—
52	C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
53	C₂×C₃	—	3	4	no	—
54	C₂×C₃	—	8	1	no	—
55	C₂×C₃	—	24	4	no	—
56	C₂×C₃	C₃²×D₈	1	1	yes	characteristic
57	C₂²	—	4	1	no	—
58	C₄	—	4	1	no	—
59	C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃	—	3	4	no	—
61	C₃	C₃²×SD₃₂	1	1	yes	characteristic
62	C₂	—	8	1	no	—
63	C₂	D₈×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×D₈
commutator, Frattini, cosocle	C₃×C₈	C₂²×C₃²
prime radical	D₈×H(3,𝔽₃)	C₂
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₄×H(3,𝔽₃) ◁ C₈×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₁₆×H(3,𝔽₃)	C₂	semidirect
3. C₁₆×H(3,𝔽₃)	C₂	semidirect
4. C₃²×SD₃₂	C₃	semidirect
8. C₃²×Q₁₆	C₂×C₃	semidirect
10. C₃²×C₁₆	C₂×C₃	semidirect
30. SD₃₂	H(3,𝔽₃)	direct (N × H)
31. H(3,𝔽₃)	SD₃₂	direct (N × H)
40. Q₁₆	C₂×H(3,𝔽₃)	semidirect
42. C₁₆	C₂×H(3,𝔽₃)	semidirect
49. C₃²	C₃×SD₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₁₆×H(3,𝔽₃)	2
D₈×H(3,𝔽₃)	2
C₁₆×H(3,𝔽₃)	2
C₃²×SD₃₂	3

Subgroups — H(3,𝔽3)×SD32 #11720639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×SD₃₂ #11720639