Subgroups — Q₈×C₃²⋊H(3,𝔽₃) #14517835

Order 1944.

1134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃²⋊H(3,𝔽₃)	C₂²
LCS	G ▷ C₂×C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃⁴×Q₈	C₃	semidirect
3. C₃×Q₈×H(3,𝔽₃)	C₃	semidirect
7. C₃²⋊H(3,𝔽₃)	Q₈	direct (N × H)
10. C₃³×Q₈	C₃²	semidirect
18. C₃⁴	C₃×Q₈	semidirect
19. C₃×H(3,𝔽₃)	C₃×Q₈	semidirect
23. C₃²×Q₈	H(3,𝔽₃)	semidirect
36. C₃³	C₃²×Q₈	semidirect
52. C₃²	Q₈×H(3,𝔽₃)	semidirect
54. Q₈	C₃²⋊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
C₄×C₃²⋊H(3,𝔽₃)	2
C₃⁴×Q₈	3
C₃×Q₈×H(3,𝔽₃)	3

Subgroups — Q8×C32⋊H(3,𝔽3) #14517835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃²⋊H(3,𝔽₃) #14517835