Subgroups — M₁₆×H(3,𝔽₃) #88566

Order 432.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₈×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	C₃²×M₁₆	C₃	1	4	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄	C₂×C₃	1	4	yes	—
8	C₃²×C₈	C₂×C₃	1	8	yes	complemented
9	C₂×H(3,𝔽₃)	—	2	1	no	—
10	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
11	C₃×M₁₆	—	3	4	no	—
12	C₃×M₁₆	C₃²	1	1	yes	characteristic
13	C₂²×C₃²	C₃×C₄	1	4	yes	—
14	C₃²×C₄	C₃×C₄	1	4	yes	—
15	C₃²×C₄	C₂²×C₃	1	4	yes	—
16	H(3,𝔽₃)	M₁₆	1	1	yes	characteristic, direct product factor, Sylow
17	C₂×C₃×C₄	—	3	4	no	—
18	C₂×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
19	C₃×C₈	—	3	8	no	—
20	C₃×C₈	C₂×C₃²	1	2	yes	—
21	C₂×C₃²	—	2	4	no	—
22	C₂×C₃²	C₂×C₃×C₄	1	4	yes	—
23	M₁₆	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂²×C₃	—	3	4	no	—
25	C₂²×C₃	C₃²×C₄	1	1	yes	characteristic
26	C₃×C₄	—	3	4	no	—
27	C₃×C₄	—	3	4	no	—
28	C₃×C₄	C₃²×C₄	1	1	yes	characteristic
29	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
30	C₃²	C₃×M₁₆	1	4	yes	complemented
31	C₂×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
32	C₈	C₂×H(3,𝔽₃)	1	2	yes	complemented
33	C₂×C₃	—	2	1	no	—
34	C₂×C₃	—	3	4	no	—
35	C₂×C₃	—	6	4	no	—
36	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
37	C₂²	C₄×H(3,𝔽₃)	1	1	yes	characteristic
38	C₄	C₄×H(3,𝔽₃)	1	1	yes	characteristic
39	C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃	—	3	4	no	—
41	C₃	C₃²×M₁₆	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₈×H(3,𝔽₃)	C₂	semidirect
3. C₃²×M₁₆	C₃	semidirect
8. C₃²×C₈	C₂×C₃	semidirect
16. H(3,𝔽₃)	M₁₆	direct (N × H)
23. M₁₆	H(3,𝔽₃)	direct (N × H)
30. C₃²	C₃×M₁₆	semidirect
32. C₈	C₂×H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×H(3,𝔽₃)	2
C₈×H(3,𝔽₃)	2
C₃²×M₁₆	3

Subgroups — M16×H(3,𝔽3) #88566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×H(3,𝔽₃) #88566