Subgroups — 3₋¹⁺²×D₃₂ #13136116

Order 1728.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×D₁₆	C₂	1	2	yes	maximal, complemented
2	3₋¹⁺²×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₃₂	C₃	1	1	yes	maximal, characteristic
4	C₉×D₃₂	C₃	1	3	yes	maximal, complemented
5	D₈×3₋¹⁺²	—	2	2	no	—
6	C₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₃²×D₁₆	C₂×C₃	1	2	yes	—
8	C₃²×C₃₂	C₂×C₃	1	1	yes	characteristic
9	C₉×D₁₆	C₂×C₃	1	6	yes	complemented
10	C₉×C₃₂	C₂×C₃	1	3	yes	complemented
11	D₄×3₋¹⁺²	—	4	2	no	—
12	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
13	C₃×D₃₂	—	3	1	no	—
14	C₃×D₃₂	C₃²	1	1	yes	characteristic
15	C₃²×D₈	—	2	2	no	—
16	C₃²×C₁₆	C₂²×C₃	1	1	yes	characteristic
17	C₉×D₈	—	2	6	no	—
18	C₉×C₁₆	C₂²×C₃	1	3	yes	—
19	C₂²×3₋¹⁺²	—	8	2	no	—
20	C₄×3₋¹⁺²	D₈	1	1	yes	characteristic
21	C₃×D₁₆	—	3	2	no	—
22	C₃×D₁₆	C₂×C₃²	1	2	yes	—
23	C₃×C₃₂	—	3	1	no	—
24	C₃×C₃₂	C₂×C₃²	1	1	yes	characteristic
25	C₃²×D₄	—	4	2	no	—
26	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
27	D₄×C₉	—	4	6	no	—
28	C₈×C₉	C₃×D₄	1	3	yes	—
29	D₃₂	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×3₋¹⁺²	—	16	2	no	—
31	C₂×3₋¹⁺²	D₁₆	1	1	yes	characteristic
32	C₃×D₈	—	2	2	no	—
33	C₃×D₈	—	6	2	no	—
34	C₃×C₁₆	—	3	1	no	—
35	C₃×C₁₆	C₂²×C₃²	1	1	yes	characteristic
36	C₂²×C₃²	—	8	2	no	—
37	C₃²×C₄	C₃×D₈	1	1	yes	characteristic
38	C₂²×C₉	—	8	6	no	—
39	C₄×C₉	C₃×D₈	1	3	yes	—
40	D₁₆	C₂×3₋¹⁺²	1	2	yes	complemented
41	C₃₂	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
42	3₋¹⁺²	D₃₂	1	1	yes	characteristic, direct product factor, Sylow
43	C₃×D₄	—	4	2	no	—
44	C₃×D₄	—	12	2	no	—
45	C₃×C₈	—	3	1	no	—
46	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
47	C₂×C₃²	—	16	2	no	—
48	C₂×C₃²	C₃×D₁₆	1	1	yes	characteristic
49	C₂×C₉	—	16	6	no	—
50	C₂×C₉	C₃×D₁₆	1	3	yes	—
51	D₈	—	2	2	no	—
52	C₁₆	C₂²×3₋¹⁺²	1	1	yes	characteristic
53	C₂²×C₃	—	8	2	no	—
54	C₂²×C₃	—	24	2	no	—
55	C₃×C₄	—	3	1	no	—
56	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
57	C₃²	C₃×D₃₂	1	1	yes	characteristic
58	C₉	C₃×D₃₂	1	3	yes	complemented
59	D₄	—	4	2	no	—
60	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	16	2	no	—
63	C₂×C₃	—	48	2	no	—
64	C₂×C₃	C₃²×D₁₆	1	1	yes	characteristic
65	C₂²	—	8	2	no	—
66	C₄	D₈×3₋¹⁺²	1	1	yes	characteristic
67	C₃	—	3	1	no	—
68	C₃	C₃²×D₃₂	1	1	yes	characteristic
69	C₂	—	16	2	no	—
70	C₂	3₋¹⁺²×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×D₁₆
commutator, Frattini, cosocle	C₃×C₁₆	C₂²×C₃²
prime radical	C₃²×D₃₂	C₃
LCS	G ▷ C₃×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺² ◁ C₈×3₋¹⁺² ◁ C₁₆×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×D₁₆	C₂	semidirect
2. 3₋¹⁺²×C₃₂	C₂	semidirect
4. C₉×D₃₂	C₃	semidirect
9. C₉×D₁₆	C₂×C₃	semidirect
10. C₉×C₃₂	C₂×C₃	semidirect
29. D₃₂	3₋¹⁺²	direct (N × H)
40. D₁₆	C₂×3₋¹⁺²	semidirect
41. C₃₂	C₂×3₋¹⁺²	semidirect
42. 3₋¹⁺²	D₃₂	direct (N × H)
58. C₉	C₃×D₃₂	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×D₁₆	2
3₋¹⁺²×C₃₂	2
C₃²×D₃₂	3
C₉×D₃₂	3

Subgroups — 3−1+2×D32 #13136116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×D₃₂ #13136116