Subgroups — Q₈×#6912 #14517857

Order 1944.

372 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6912	C₂	1	3	yes	maximal
2	C₃×Q₈×3₋¹⁺²	C₃	1	3	yes	maximal
3	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
4	Q₈×#733	C₃	1	9	yes	maximal, complemented
5	C₂×#6912	C₂²	1	1	yes	characteristic
6	C₃×C₄×3₋¹⁺²	C₂×C₃	1	9	yes	—
7	C₃×C₄×3₋¹⁺²	C₂×C₃	1	3	yes	—
8	C₄×#733	C₂×C₃	1	27	yes	—
9	#6912	Q₈	1	1	yes	characteristic, direct product factor, Sylow
10	C₃³×Q₈	C₃²	1	1	yes	characteristic
11	Q₈×3₋¹⁺²	—	3	6	no	—
12	Q₈×3₋¹⁺²	—	3	3	no	—
13	Q₈×3₋¹⁺²	C₃²	1	9	yes	—
14	C₃×Q₈×C₉	—	3	3	no	—
15	C₃×Q₈×C₉	C₃²	1	3	yes	—
16	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	3	yes	—
17	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	1	yes	characteristic
18	C₂×#733	C₂²×C₃	1	9	yes	—
19	C₃³×C₄	C₂×C₃²	1	3	yes	—
20	C₄×3₋¹⁺²	—	3	18	no	—
21	C₄×3₋¹⁺²	—	3	9	no	—
22	C₄×3₋¹⁺²	C₂×C₃²	1	27	yes	—
23	C₃×C₄×C₉	—	3	9	no	—
24	C₃×C₄×C₉	C₂×C₃²	1	9	yes	—
25	C₃×3₋¹⁺²	C₃×Q₈	1	3	yes	—
26	C₃×3₋¹⁺²	C₃×Q₈	1	1	yes	characteristic
27	#733	C₃×Q₈	1	9	yes	complemented
28	C₃²×Q₈	—	9	1	no	—
29	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
30	C₃²×Q₈	C₃³	1	1	yes	characteristic
31	Q₈×C₉	—	3	9	no	—
32	Q₈×C₉	—	3	3	no	—
33	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
34	C₂×3₋¹⁺²	—	3	6	no	—
35	C₂×3₋¹⁺²	—	3	3	no	—
36	C₂×3₋¹⁺²	C₂²×C₃²	1	9	yes	—
37	C₂×C₃×C₉	—	3	3	no	—
38	C₂×C₃×C₉	C₂²×C₃²	1	3	yes	—
39	C₃²×C₄	—	9	3	no	—
40	C₃²×C₄	C₂×H(3,𝔽₃)	1	9	yes	—
41	C₃²×C₄	C₂×C₃³	1	3	yes	—
42	C₄×C₉	—	3	27	no	—
43	C₄×C₉	—	3	9	no	—
44	C₃³	C₃²×Q₈	1	1	yes	characteristic
45	3₋¹⁺²	—	3	6	no	—
46	3₋¹⁺²	—	3	3	no	—
47	3₋¹⁺²	C₃²×Q₈	1	9	yes	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	C₃²×Q₈	1	3	yes	—
50	C₃×Q₈	—	3	3	no	—
51	C₃×Q₈	—	3	1	no	—
52	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₂×C₃²	—	9	1	no	—
54	C₂×C₃²	C₂²×H(3,𝔽₃)	1	3	yes	—
55	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
56	C₂×C₉	—	3	9	no	—
57	C₂×C₉	—	3	3	no	—
58	C₃×C₄	—	3	9	no	—
59	C₃×C₄	—	3	3	no	—
60	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	3	yes	—
61	C₃²	—	9	1	no	—
62	C₃²	Q₈×H(3,𝔽₃)	1	3	yes	—
63	C₃²	C₃³×Q₈	1	1	yes	characteristic
64	C₉	—	3	9	no	—
65	C₉	—	3	3	no	—
66	Q₈	#6912	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×C₃	—	3	3	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₄	C₂×#6912	1	3	yes	—
71	C₃	—	3	3	no	—
72	C₃	—	3	1	no	—
73	C₃	C₃×Q₈×H(3,𝔽₃)	1	1	yes	characteristic
74	C₂	C₂²×#6912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃³	C₂²×C₃²
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃³×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. Q₈×#733	C₃	semidirect
9. #6912	Q₈	direct (N × H)
27. #733	C₃×Q₈	semidirect
66. Q₈	#6912	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×#6912	2
C₃×Q₈×3₋¹⁺²	3
Q₈×#733	3

Subgroups — Q8×#6912 #14517857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6912 #14517857