Subgroups — C₃₂×C₃⋊(C₃⋊D₃) #13145992

Order 1728.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₃⋊(C₃⋊C₃₂))	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃⋊D₃×C₃₂	—	3	13	no	maximal
5	C₈×C₃⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
6	C₃³×C₁₆	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊(C₃⋊C₁₆))	C₄	1	1	yes	characteristic
8	C₁₆×C₃⋊D₃	—	3	13	no	—
9	C₃²×C₃₂	D₃	1	13	yes	complemented
10	C₃⋊(C₃⋊C₃₂)	—	3	13	no	—
11	C₄×C₃⋊(C₃⋊D₃)	C₈	1	1	yes	characteristic
12	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
13	C₃⋊(C₃⋊(C₃⋊C₈))	C₈	1	1	yes	characteristic
14	D₃×C₃₂	—	9	13	no	—
15	C₈×C₃⋊D₃	—	3	13	no	—
16	C₃²×C₁₆	C₂×D₃	1	13	yes	—
17	C₃⋊(C₃⋊C₁₆)	—	3	13	no	—
18	C₂×C₃⋊(C₃⋊D₃)	C₁₆	1	1	yes	characteristic
19	C₃³×C₄	C₂×C₈	1	1	yes	characteristic
20	C₃⋊(C₃⋊Dic₃)	C₁₆	1	1	yes	characteristic
21	D₃×C₁₆	—	9	13	no	—
22	C₃×C₃₂	C₃⋊D₃	1	13	yes	complemented
23	C₃⋊C₃₂	—	9	13	no	—
24	C₄×C₃⋊D₃	—	3	13	no	—
25	C₃²×C₈	C₄×D₃	1	13	yes	—
26	C₃⋊(C₃⋊C₈)	—	3	13	no	—
27	C₂×C₃₂	—	27	1	no	Sylow
28	C₂×C₃³	C₂×C₁₆	1	1	yes	characteristic
29	C₃⋊(C₃⋊D₃)	C₃₂	1	2	yes	direct product factor
30	D₃×C₈	—	9	13	no	—
31	C₃×C₁₆	C₂×C₃⋊D₃	1	13	yes	—
32	C₃⋊C₁₆	—	9	13	no	—
33	C₂×C₃⋊D₃	—	3	13	no	—
34	C₃²×C₄	D₃×C₈	1	13	yes	—
35	C₃⋊Dic₃	—	3	13	no	—
36	C₂×C₁₆	—	27	1	no	—
37	C₃₂	—	27	1	no	—
38	C₃₂	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
39	C₃³	C₂×C₃₂	1	1	yes	characteristic, complemented, Sylow
40	C₄×D₃	—	9	13	no	—
41	C₃×C₈	C₄×C₃⋊D₃	1	13	yes	—
42	C₃⋊C₈	—	9	13	no	—
43	C₂×C₃²	D₃×C₁₆	1	13	yes	—
44	C₃⋊D₃	—	3	26	no	—
45	C₂×C₈	—	27	1	no	—
46	C₁₆	—	27	1	no	—
47	C₁₆	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
48	C₂×D₃	—	9	13	no	—
49	C₃×C₄	C₈×C₃⋊D₃	1	13	yes	—
50	Dic₃	—	9	13	no	—
51	C₃²	D₃×C₃₂	1	13	yes	complemented
52	C₂×C₄	—	27	1	no	—
53	C₈	—	27	1	no	—
54	C₈	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
55	C₂×C₃	C₁₆×C₃⋊D₃	1	13	yes	—
56	D₃	—	9	26	no	—
57	C₂²	—	27	1	no	—
58	C₄	—	27	1	no	—
59	C₄	C₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
60	C₃	C₃⋊D₃×C₃₂	1	13	yes	complemented
61	C₂	—	27	2	no	—
62	C₂	C₁₆×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃₂	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₂×C₃₂
Frattini	C₁₆	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₃₂	C₂
socle	C₂×C₃³	C₂×C₁₆
prime radical	C₂×C₃⋊(C₃⋊D₃)	C₁₆
cosocle	C₃³×C₁₆	C₂²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
2. C₃³×C₃₂	C₂	semidirect
3. C₃⋊(C₃⋊(C₃⋊C₃₂))	C₂	semidirect
9. C₃²×C₃₂	D₃	semidirect
22. C₃×C₃₂	C₃⋊D₃	semidirect
29. C₃⋊(C₃⋊D₃)	C₃₂	direct (N × H)
38. C₃₂	C₃⋊(C₃⋊D₃)	direct (N × H)
39. C₃³	C₂×C₃₂	semidirect
51. C₃²	D₃×C₃₂	semidirect
60. C₃	C₃⋊D₃×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×C₃⋊(C₃⋊D₃)	2
C₃³×C₃₂	2
C₃⋊(C₃⋊(C₃⋊C₃₂))	2
C₃⋊D₃×C₃₂	3

Subgroups — C32×C3⋊(C3⋊D3) #13145992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₂×C₃⋊(C₃⋊D₃) #13145992