Subgroups — Q₈×#6896 #14517841

Order 1944.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6896	C₂	1	3	yes	maximal
2	C₃×Q₈×3₋¹⁺²	C₃	1	2	yes	maximal
3	Q₈×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
4	Q₈×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
5	Q₈×C₃²⋊C₉	C₃	1	2	yes	maximal
6	Q₈×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×#6896	C₂²	1	1	yes	characteristic
8	C₃×C₄×3₋¹⁺²	C₂×C₃	1	6	yes	—
9	C₄×C₉⋊C₉	C₂×C₃	1	12	yes	—
10	C₄×C₉⋊C₉	C₂×C₃	1	12	yes	—
11	C₄×C₃²⋊C₉	C₂×C₃	1	6	yes	—
12	C₄×C₉²	C₂×C₃	1	3	yes	—
13	#6896	Q₈	1	1	yes	characteristic, direct product factor, Sylow
14	C₃³×Q₈	C₃²	1	1	yes	characteristic
15	Q₈×3₋¹⁺²	—	3	6	no	—
16	C₃×Q₈×C₉	C₃²	1	2	yes	—
17	C₃×Q₈×C₉	C₃²	1	4	yes	—
18	C₃×Q₈×C₉	C₃²	1	4	yes	—
19	C₃×Q₈×C₉	C₃²	1	2	yes	—
20	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	2	yes	—
21	C₂×C₉⋊C₉	C₂²×C₃	1	4	yes	—
22	C₂×C₉⋊C₉	C₂²×C₃	1	4	yes	—
23	C₂×C₃²⋊C₉	C₂²×C₃	1	2	yes	—
24	C₂×C₉²	C₂²×C₃	1	1	yes	characteristic
25	C₃³×C₄	C₂×C₃²	1	3	yes	—
26	C₄×3₋¹⁺²	—	3	18	no	—
27	C₃×C₄×C₉	C₂×C₃²	1	6	yes	—
28	C₃×C₄×C₉	C₂×C₃²	1	12	yes	—
29	C₃×C₄×C₉	C₂×C₃²	1	12	yes	—
30	C₃×C₄×C₉	C₂×C₃²	1	6	yes	—
31	C₃×3₋¹⁺²	C₃×Q₈	1	2	yes	—
32	C₉⋊C₉	C₃×Q₈	1	4	yes	complemented
33	C₉⋊C₉	C₃×Q₈	1	4	yes	complemented
34	C₃²⋊C₉	C₃×Q₈	1	2	yes	—
35	C₉²	C₃×Q₈	1	1	yes	characteristic, complemented
36	C₃²×Q₈	—	3	2	no	—
37	C₃²×Q₈	—	3	2	no	—
38	C₃²×Q₈	C₃³	1	1	yes	characteristic
39	Q₈×C₉	—	3	4	no	—
40	Q₈×C₉	—	3	2	no	—
41	Q₈×C₉	—	3	4	no	—
42	Q₈×C₉	3₋¹⁺²	1	6	yes	complemented
43	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
44	C₂×3₋¹⁺²	—	3	6	no	—
45	C₂×C₃×C₉	C₂²×C₃²	1	2	yes	—
46	C₂×C₃×C₉	C₂²×C₃²	1	4	yes	—
47	C₂×C₃×C₉	C₂²×C₃²	1	4	yes	—
48	C₂×C₃×C₉	C₂²×C₃²	1	2	yes	—
49	C₃²×C₄	—	3	6	no	—
50	C₃²×C₄	—	3	6	no	—
51	C₃²×C₄	C₂×C₃³	1	3	yes	—
52	C₄×C₉	—	3	12	no	—
53	C₄×C₉	—	3	6	no	—
54	C₄×C₉	—	3	12	no	—
55	C₄×C₉	C₂×3₋¹⁺²	1	18	yes	—
56	C₃³	C₃²×Q₈	1	1	yes	characteristic
57	3₋¹⁺²	—	3	6	no	—
58	C₃×C₉	C₃²×Q₈	1	2	yes	—
59	C₃×C₉	C₃²×Q₈	1	4	yes	—
60	C₃×C₉	C₃²×Q₈	1	4	yes	—
61	C₃×C₉	C₃²×Q₈	1	2	yes	—
62	C₃×Q₈	—	9	1	no	—
63	C₃×Q₈	C₃×3₋¹⁺²	1	2	yes	—
64	C₃×Q₈	#737	1	2	yes	—
65	C₂×C₃²	—	3	2	no	—
66	C₂×C₃²	—	3	2	no	—
67	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
68	C₂×C₉	—	3	4	no	—
69	C₂×C₉	—	3	2	no	—
70	C₂×C₉	—	3	4	no	—
71	C₂×C₉	C₂²×3₋¹⁺²	1	6	yes	—
72	C₃×C₄	—	9	3	no	—
73	C₃×C₄	C₂×C₃×3₋¹⁺²	1	6	yes	—
74	C₃×C₄	C₂×#737	1	6	yes	—
75	C₃²	—	3	2	no	—
76	C₃²	—	3	2	no	—
77	C₃²	C₃³×Q₈	1	1	yes	characteristic
78	C₉	—	3	4	no	—
79	C₉	—	3	2	no	—
80	C₉	—	3	4	no	—
81	C₉	Q₈×3₋¹⁺²	1	6	yes	complemented
82	Q₈	#6896	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	—	9	1	no	—
84	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	2	yes	—
85	C₂×C₃	C₂²×#737	1	2	yes	—
86	C₄	C₂×#6896	1	3	yes	—
87	C₃	—	9	1	no	—
88	C₃	C₃×Q₈×3₋¹⁺²	1	2	yes	—
89	C₃	Q₈×#737	1	2	yes	—
90	C₂	C₂²×#6896	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃³	C₂²×C₃²
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
3. Q₈×C₉⋊C₉	C₃	semidirect
4. Q₈×C₉⋊C₉	C₃	semidirect
6. Q₈×C₉²	C₃	semidirect
13. #6896	Q₈	direct (N × H)
32. C₉⋊C₉	C₃×Q₈	semidirect
33. C₉⋊C₉	C₃×Q₈	semidirect
35. C₉²	C₃×Q₈	semidirect
42. Q₈×C₉	3₋¹⁺²	semidirect
81. C₉	Q₈×3₋¹⁺²	semidirect
82. Q₈	#6896	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×#6896	2
C₃×Q₈×3₋¹⁺²	3
Q₈×C₉⋊C₉	3
Q₈×C₃²⋊C₉	3
Q₈×C₉²	3

Subgroups — Q8×#6896 #14517841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6896 #14517841