Subgroups — Q₈×#6893 #14517838

Order 1944.

468 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6893	C₂	1	3	yes	maximal
2	C₃×Q₈×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
3	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃²×Q₈×C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	Q₈×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
6	C₂×#6893	C₂²	1	1	yes	characteristic
7	C₃×C₄×3₋¹⁺²	C₂×C₃	1	6	yes	—
8	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	3	yes	—
9	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
10	C₄×C₃²⋊C₉	C₂×C₃	1	27	yes	—
11	#6893	Q₈	1	1	yes	characteristic, direct product factor, Sylow
12	C₃³×Q₈	C₃²	1	3	yes	—
13	C₃³×Q₈	C₃²	1	1	yes	characteristic
14	Q₈×3₋¹⁺²	—	3	6	no	—
15	Q₈×H(3,𝔽₃)	—	3	3	no	—
16	C₃×Q₈×C₉	—	3	3	no	—
17	C₃×Q₈×C₉	C₃²	1	3	yes	—
18	C₃×Q₈×C₉	C₃²	1	6	yes	—
19	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	2	yes	—
20	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
21	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
22	C₂×C₃²⋊C₉	C₂²×C₃	1	9	yes	—
23	C₃³×C₄	C₂×C₃²	1	9	yes	—
24	C₃³×C₄	C₂×C₃²	1	3	yes	—
25	C₄×3₋¹⁺²	—	3	18	no	—
26	C₄×H(3,𝔽₃)	—	3	9	no	—
27	C₃×C₄×C₉	—	3	9	no	—
28	C₃×C₄×C₉	C₂×C₃²	1	9	yes	—
29	C₃×C₄×C₉	C₂×C₃²	1	18	yes	—
30	C₃×3₋¹⁺²	C₃×Q₈	1	2	yes	complemented
31	C₃×H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
32	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic, complemented
33	C₃²⋊C₉	C₃×Q₈	1	9	yes	complemented
34	C₃²×Q₈	—	3	3	no	—
35	C₃²×Q₈	—	3	3	no	—
36	C₃²×Q₈	—	3	9	no	—
37	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
38	C₃²×Q₈	C₃³	1	1	yes	characteristic
39	Q₈×C₉	—	3	3	no	—
40	Q₈×C₉	—	3	6	no	—
41	C₂×C₃³	C₂²×C₃²	1	3	yes	—
42	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
43	C₂×3₋¹⁺²	—	3	6	no	—
44	C₂×H(3,𝔽₃)	—	3	3	no	—
45	C₂×C₃×C₉	—	3	3	no	—
46	C₂×C₃×C₉	C₂²×C₃²	1	3	yes	—
47	C₂×C₃×C₉	C₂²×C₃²	1	6	yes	—
48	C₃²×C₄	—	3	9	no	—
49	C₃²×C₄	—	3	9	no	—
50	C₃²×C₄	—	3	27	no	—
51	C₃²×C₄	C₂×H(3,𝔽₃)	1	9	yes	—
52	C₃²×C₄	C₂×C₃³	1	3	yes	—
53	C₄×C₉	—	3	9	no	—
54	C₄×C₉	—	3	18	no	—
55	C₃³	C₃²×Q₈	1	3	yes	—
56	C₃³	C₃²×Q₈	1	1	yes	characteristic
57	3₋¹⁺²	—	3	6	no	—
58	H(3,𝔽₃)	—	3	3	no	—
59	C₃×C₉	—	3	3	no	—
60	C₃×C₉	C₃²×Q₈	1	3	yes	—
61	C₃×C₉	C₃²×Q₈	1	6	yes	—
62	C₃×Q₈	—	3	3	no	—
63	C₃×Q₈	—	9	3	no	—
64	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃×Q₈	#737	1	3	yes	—
66	C₂×C₃²	—	3	3	no	—
67	C₂×C₃²	—	3	3	no	—
68	C₂×C₃²	—	3	9	no	—
69	C₂×C₃²	C₂²×H(3,𝔽₃)	1	3	yes	—
70	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
71	C₂×C₉	—	3	3	no	—
72	C₂×C₉	—	3	6	no	—
73	C₃×C₄	—	3	9	no	—
74	C₃×C₄	—	9	9	no	—
75	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	3	yes	—
76	C₃×C₄	C₂×#737	1	9	yes	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	9	no	—
80	C₃²	Q₈×H(3,𝔽₃)	1	3	yes	—
81	C₃²	C₃³×Q₈	1	1	yes	characteristic
82	C₉	—	3	3	no	—
83	C₉	—	3	6	no	—
84	Q₈	#6893	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×C₃	—	3	3	no	—
86	C₂×C₃	—	9	3	no	—
87	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
88	C₂×C₃	C₂²×#737	1	3	yes	—
89	C₄	C₂×#6893	1	3	yes	—
90	C₃	—	3	3	no	—
91	C₃	—	9	3	no	—
92	C₃	C₃×Q₈×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃	Q₈×#737	1	3	yes	—
94	C₂	C₂²×#6893	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	C₂²×C₃
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. C₃×Q₈×3₋¹⁺²	C₃	semidirect
4. C₃²×Q₈×C₉	C₃	semidirect
5. Q₈×C₃²⋊C₉	C₃	semidirect
11. #6893	Q₈	direct (N × H)
30. C₃×3₋¹⁺²	C₃×Q₈	semidirect
32. C₃²×C₉	C₃×Q₈	semidirect
33. C₃²⋊C₉	C₃×Q₈	semidirect
84. Q₈	#6893	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×#6893	2
C₃×Q₈×3₋¹⁺²	3
C₃×Q₈×H(3,𝔽₃)	3
C₃²×Q₈×C₉	3
Q₈×C₃²⋊C₉	3

Subgroups — Q8×#6893 #14517838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6893 #14517838