Subgroups — H(3,𝔽₃)×#131 #11721929

Order 864.

341 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	M₁₆×H(3,𝔽₃)	C₂	1	3	yes	maximal, complemented
3	C₂×C₈×H(3,𝔽₃)	C₂	1	3	yes	maximal, complemented
4	C₃²×#131	C₃	1	4	yes	maximal, complemented
5	Q₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	D₄×H(3,𝔽₃)	C₄	1	3	yes	—
7	C₂×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₈×H(3,𝔽₃)	C₂²	1	3	yes	—
10	C₃²×#41	C₂×C₃	1	4	yes	—
11	C₃²×M₁₆	C₂×C₃	1	12	yes	complemented
12	C₂×C₃²×C₈	C₂×C₃	1	12	yes	complemented
13	C₂²×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
14	C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
15	C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₃×#131	—	3	4	no	—
17	C₃×#131	C₃²	1	1	yes	characteristic
18	C₃²×Q₈	C₃×C₄	1	4	yes	—
19	C₃²×D₄	C₃×C₄	1	12	yes	—
20	C₂×C₃²×C₄	C₂²×C₃	1	12	yes	—
21	C₃²×C₈	C₂²×C₃	1	12	yes	—
22	C₃²×C₈	C₂²×C₃	1	4	yes	—
23	C₂×H(3,𝔽₃)	—	2	3	no	—
24	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
25	C₃×#41	—	3	4	no	—
26	C₃×#41	C₂×C₃²	1	1	yes	characteristic
27	C₃×M₁₆	—	3	12	no	—
28	C₃×M₁₆	C₂×C₃²	1	3	yes	—
29	C₂×C₃×C₈	—	3	12	no	—
30	C₂×C₃×C₈	C₂×C₃²	1	3	yes	—
31	C₂²×C₃²	C₂×C₃×C₄	1	12	yes	—
32	C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
33	C₃²×C₄	C₂³×C₃	1	4	yes	—
34	#131	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
35	H(3,𝔽₃)	#131	1	1	yes	characteristic, direct product factor, Sylow
36	C₃×Q₈	—	3	4	no	—
37	C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
38	C₃×D₄	—	3	12	no	—
39	C₃×D₄	C₃²×C₄	1	3	yes	—
40	C₂×C₃×C₄	—	3	12	no	—
41	C₂×C₃×C₄	C₂²×C₃²	1	3	yes	—
42	C₃×C₈	—	3	4	no	—
43	C₃×C₈	—	3	12	no	—
44	C₃×C₈	C₂²×C₃²	1	3	yes	—
45	C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
46	C₂×C₃²	—	2	12	no	—
47	C₂×C₃²	C₂²×C₃×C₄	1	4	yes	—
48	#41	C₂×H(3,𝔽₃)	1	1	yes	characteristic
49	M₁₆	C₂×H(3,𝔽₃)	1	3	yes	complemented
50	C₂×C₈	C₂×H(3,𝔽₃)	1	3	yes	complemented
51	C₂²×C₃	—	3	12	no	—
52	C₂²×C₃	C₂×C₃²×C₄	1	3	yes	—
53	C₃×C₄	—	3	12	no	—
54	C₃×C₄	—	3	4	no	—
55	C₃×C₄	C₂×C₃²×C₄	1	3	yes	—
56	C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
57	C₃²	C₃×#131	1	4	yes	complemented
58	Q₈	C₄×H(3,𝔽₃)	1	1	yes	characteristic
59	D₄	C₄×H(3,𝔽₃)	1	3	yes	—
60	C₂×C₄	C₂²×H(3,𝔽₃)	1	3	yes	—
61	C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
62	C₈	C₂²×H(3,𝔽₃)	1	3	yes	—
63	C₂×C₃	—	2	3	no	—
64	C₂×C₃	—	3	4	no	—
65	C₂×C₃	—	6	12	no	—
66	C₂×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
67	C₂²	C₂×C₄×H(3,𝔽₃)	1	3	yes	—
68	C₄	C₂×C₄×H(3,𝔽₃)	1	3	yes	—
69	C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃	—	3	4	no	—
71	C₃	C₃²×#131	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	C₂²×C₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂²×C₃²
commutator, socle	C₂×C₃	C₂²×C₃²×C₄
Frattini, cosocle	C₃×C₄	C₂³×C₃²
prime radical	#41×H(3,𝔽₃)	C₂
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₁₆×H(3,𝔽₃)	C₂	semidirect
3. C₂×C₈×H(3,𝔽₃)	C₂	semidirect
4. C₃²×#131	C₃	semidirect
11. C₃²×M₁₆	C₂×C₃	semidirect
12. C₂×C₃²×C₈	C₂×C₃	semidirect
34. #131	H(3,𝔽₃)	direct (N × H)
35. H(3,𝔽₃)	#131	direct (N × H)
49. M₁₆	C₂×H(3,𝔽₃)	semidirect
50. C₂×C₈	C₂×H(3,𝔽₃)	semidirect
57. C₃²	C₃×#131	semidirect

Maximal subgroups

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

subgroup	index
#41×H(3,𝔽₃)	2
M₁₆×H(3,𝔽₃)	2
C₂×C₈×H(3,𝔽₃)	2
C₃²×#131	3

Subgroups — H(3,𝔽3)×#131 #11721929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#131 #11721929