Subgroups — H(3,𝔽₃)×#126 #11721914

Order 864.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×H(3,𝔽₃)	C₂	1	3	yes	maximal, complemented
2	C₂²⋊C₄×H(3,𝔽₃)	C₂	1	3	yes	maximal
3	C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#126	C₃	1	4	yes	maximal, complemented
5	C₂³×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
7	C₂×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	C₃²×C₄⋊C₄	C₂×C₃	1	12	yes	complemented
9	C₃²×C₂²⋊C₄	C₂×C₃	1	12	yes	—
10	C₃²×C₄²	C₂×C₃	1	4	yes	complemented
11	C₂²×H(3,𝔽₃)	—	2	3	no	—
12	C₂²×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	—	2	3	no	—
14	C₄×H(3,𝔽₃)	—	2	3	no	—
15	C₃×#126	—	3	4	no	—
16	C₃×#126	C₃²	1	1	yes	characteristic
17	C₂³×C₃²	C₂²×C₃	1	4	yes	—
18	C₂×C₃²×C₄	C₂²×C₃	1	12	yes	—
19	C₂×C₃²×C₄	C₂²×C₃	1	12	yes	—
20	C₂×H(3,𝔽₃)	—	4	1	no	—
21	C₂×H(3,𝔽₃)	#41	1	3	yes	—
22	C₃×C₄⋊C₄	—	3	12	no	—
23	C₃×C₄⋊C₄	C₂×C₃²	1	3	yes	—
24	C₃×C₂²⋊C₄	—	3	12	no	—
25	C₃×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
26	C₃×C₄²	—	3	4	no	—
27	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
28	C₂²×C₃²	—	2	12	no	—
29	C₂²×C₃²	C₂³×C₃	1	4	yes	—
30	C₃²×C₄	—	2	12	no	—
31	C₃²×C₄	—	2	12	no	—
32	#126	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
33	H(3,𝔽₃)	#126	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×C₃	—	3	4	no	—
35	C₂³×C₃	C₂²×C₃²	1	1	yes	characteristic
36	C₂×C₃×C₄	—	3	12	no	—
37	C₂×C₃×C₄	—	3	12	no	—
38	C₂×C₃×C₄	C₂²×C₃²	1	3	yes	—
39	C₂×C₃×C₄	C₂²×C₃²	1	3	yes	—
40	C₂×C₃²	—	4	4	no	—
41	C₂×C₃²	C₃×#41	1	12	yes	—
42	C₄⋊C₄	C₂×H(3,𝔽₃)	1	3	yes	complemented
43	C₂²⋊C₄	C₂×H(3,𝔽₃)	1	3	yes	—
44	C₄²	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
45	C₂²×C₃	—	2	3	no	—
46	C₂²×C₃	—	3	4	no	—
47	C₂²×C₃	—	6	12	no	—
48	C₂²×C₃	C₂³×C₃²	1	1	yes	characteristic
49	C₃×C₄	—	2	3	no	—
50	C₃×C₄	—	2	3	no	—
51	C₃×C₄	—	6	12	no	—
52	C₃×C₄	—	6	12	no	—
53	C₃²	C₃×#126	1	4	yes	complemented
54	C₂³	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
55	C₂×C₄	C₂²×H(3,𝔽₃)	1	3	yes	—
56	C₂×C₄	C₂²×H(3,𝔽₃)	1	3	yes	—
57	C₂×C₃	—	3	12	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂×C₃	—	12	4	no	—
60	C₂×C₃	C₃²×#41	1	3	yes	—
61	C₂²	—	2	3	no	—
62	C₂²	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
63	C₄	—	2	3	no	—
64	C₄	—	2	3	no	—
65	C₃	—	3	4	no	—
66	C₃	C₃²×#126	1	1	yes	characteristic
67	C₂	—	4	1	no	—
68	C₂	#41×H(3,𝔽₃)	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₄⋊C₄×H(3,𝔽₃)	C₂	semidirect
3. C₄²×H(3,𝔽₃)	C₂	semidirect
4. C₃²×#126	C₃	semidirect
8. C₃²×C₄⋊C₄	C₂×C₃	semidirect
10. C₃²×C₄²	C₂×C₃	semidirect
32. #126	H(3,𝔽₃)	direct (N × H)
33. H(3,𝔽₃)	#126	direct (N × H)
42. C₄⋊C₄	C₂×H(3,𝔽₃)	semidirect
44. C₄²	C₂×H(3,𝔽₃)	semidirect
53. C₃²	C₃×#126	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×H(3,𝔽₃)	2
C₂²⋊C₄×H(3,𝔽₃)	2
C₄²×H(3,𝔽₃)	2
C₃²×#126	3

Subgroups — H(3,𝔽3)×#126 #11721914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#126 #11721914