Subgroups — C₃²×Q₈⋊H(3,𝔽₃) #14518871

Order 1944.

1562 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃³×SL(2,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	36	yes	maximal, direct product factor
4	C₂×C₃²×H(3,𝔽₃)	—	4	1	no	maximal
5	C₃⁴×C₄	—	3	1	no	—
6	C₃²×H(3,𝔽₃)	—	4	1	no	Sylow
7	C₃³×Q₈	—	3	9	no	—
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	C₃³×Q₈	C₃²	1	12	yes	complemented
10	C₃²×SL(2,𝔽₃)	—	3	27	no	—
11	C₃²×SL(2,𝔽₃)	C₃²	1	36	yes	complemented
12	Q₈⋊H(3,𝔽₃)	C₃²	1	81	yes	direct product factor
13	C₂×C₃⁴	—	4	3	no	—
14	C₂×C₃⁴	A₄	1	1	yes	characteristic
15	C₂×C₃×H(3,𝔽₃)	—	4	36	no	—
16	C₃³×C₄	—	3	27	no	—
17	C₃³×C₄	—	3	12	no	—
18	C₃³×C₄	—	3	1	no	—
19	C₃⁴	—	4	3	no	—
20	C₃⁴	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
21	C₃×H(3,𝔽₃)	—	4	36	no	—
22	C₃²×Q₈	—	3	36	no	—
23	C₃²×Q₈	H(3,𝔽₃)	1	9	yes	complemented
24	C₃²×Q₈	C₃³	1	4	yes	—
25	C₃²×Q₈	C₃³	1	9	yes	complemented
26	C₃×SL(2,𝔽₃)	—	3	108	no	—
27	C₃×SL(2,𝔽₃)	C₃³	1	27	yes	complemented
28	C₂×C₃³	—	3	9	no	—
29	C₂×C₃³	—	4	36	no	—
30	C₂×C₃³	—	12	27	no	—
31	C₂×C₃³	C₃×A₄	1	12	yes	—
32	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
33	C₂×H(3,𝔽₃)	—	4	81	no	—
34	C₃²×C₄	—	3	108	no	—
35	C₃²×C₄	—	3	9	no	—
36	C₃²×C₄	—	3	4	no	—
37	C₃²×C₄	—	3	9	no	—
38	C₃³	—	3	9	no	—
39	C₃³	—	4	36	no	—
40	C₃³	—	12	27	no	—
41	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₃³	C₃×SL(2,𝔽₃)	1	12	yes	complemented
43	H(3,𝔽₃)	—	4	81	no	—
44	C₃×Q₈	—	3	9	no	—
45	C₃×Q₈	C₃×H(3,𝔽₃)	1	12	yes	complemented
46	C₃×Q₈	C₃⁴	1	1	yes	characteristic
47	SL(2,𝔽₃)	—	3	27	no	—
48	C₂×C₃²	—	3	36	no	—
49	C₂×C₃²	—	4	27	no	—
50	C₂×C₃²	—	12	108	no	—
51	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	9	yes	—
52	C₂×C₃²	C₃²×A₄	1	9	yes	—
53	C₂×C₃²	C₃²×A₄	1	4	yes	—
54	C₃×C₄	—	3	27	no	—
55	C₃×C₄	—	3	12	no	—
56	C₃×C₄	—	3	1	no	—
57	C₃²	—	3	36	no	—
58	C₃²	—	4	27	no	—
59	C₃²	—	12	108	no	—
60	C₃²	Q₈⋊H(3,𝔽₃)	1	9	yes	direct product factor
61	C₃²	C₃²×SL(2,𝔽₃)	1	4	yes	—
62	C₃²	C₃²×SL(2,𝔽₃)	1	9	yes	complemented
63	Q₈	C₃²×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	3	9	no	—
65	C₂×C₃	—	12	27	no	—
66	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	12	yes	—
67	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
68	C₄	—	3	1	no	—
69	C₃	—	3	9	no	—
70	C₃	—	12	27	no	—
71	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	12	yes	direct product factor
72	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
73	C₂	C₃²×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃³	C₃×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Frattini	C₂×C₃	C₃³×A₄
Fitting	C₃⁴×Q₈	C₃
hypercenter	C₂×C₃⁴	A₄
hypocenter	Q₈	C₃²×H(3,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃³ ◁ C₂×C₃⁴

Complements

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

N	H ≅ G/N	type
1. C₃⁴×Q₈	C₃	semidirect
2. C₃³×SL(2,𝔽₃)	C₃	semidirect
3. C₃×Q₈⋊H(3,𝔽₃)	C₃	direct (N × H)
9. C₃³×Q₈	C₃²	semidirect
11. C₃²×SL(2,𝔽₃)	C₃²	semidirect
12. Q₈⋊H(3,𝔽₃)	C₃²	direct (N × H)
20. C₃⁴	SL(2,𝔽₃)	semidirect
23. C₃²×Q₈	H(3,𝔽₃)	semidirect
25. C₃²×Q₈	C₃³	semidirect
27. C₃×SL(2,𝔽₃)	C₃³	semidirect
42. C₃³	C₃×SL(2,𝔽₃)	semidirect
45. C₃×Q₈	C₃×H(3,𝔽₃)	semidirect
60. C₃²	Q₈⋊H(3,𝔽₃)	direct (N × H)
62. C₃²	C₃²×SL(2,𝔽₃)	semidirect
63. Q₈	C₃²×H(3,𝔽₃)	semidirect
71. C₃	C₃×Q₈⋊H(3,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×Q₈	3
C₃³×SL(2,𝔽₃)	3
C₃×Q₈⋊H(3,𝔽₃)	3
C₂×C₃²×H(3,𝔽₃)	4

Subgroups — C32×Q8⋊H(3,𝔽3) #14518871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Q₈⋊H(3,𝔽₃) #14518871