Subgroups — Q₈×#1094 #11722569

Order 864.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1094	C₂	1	3	yes	maximal
2	C₂²×C₃²×Q₈	C₃	1	1	yes	maximal, characteristic
3	Q₈×C₂²⋊C₉	C₃	1	3	yes	maximal, complemented
4	C₂×#1094	C₂²	1	1	yes	characteristic
5	Q₈×3₋¹⁺²	—	4	1	no	maximal
6	C₂×C₃²×Q₈	—	3	1	no	—
7	C₂×C₃²×Q₈	—	3	3	no	—
8	C₂²×C₃²×C₄	C₂×C₃	1	3	yes	—
9	C₄×C₂²⋊C₉	C₂×C₃	1	9	yes	—
10	#1094	Q₈	1	1	yes	characteristic, direct product factor
11	C₄×3₋¹⁺²	—	4	3	no	—
12	C₂²×C₃×Q₈	—	3	1	no	—
13	C₂²×C₃×Q₈	C₃²	1	1	yes	characteristic
14	C₂³×C₃²	C₂²×C₃	1	1	yes	characteristic
15	C₃²×Q₈	—	3	2	no	—
16	C₃²×Q₈	—	3	3	no	—
17	C₃²×Q₈	A₄	1	1	yes	characteristic
18	C₂×C₃²×C₄	—	3	3	no	—
19	C₂×C₃²×C₄	—	3	3	no	—
20	C₂×C₂²⋊C₉	C₂²×C₃	1	3	yes	—
21	Q₈×C₉	—	4	3	no	—
22	C₂×3₋¹⁺²	—	4	1	no	—
23	C₂×C₃×Q₈	—	3	3	no	—
24	C₂×C₃×Q₈	—	3	9	no	—
25	C₂×C₃×Q₈	—	3	3	no	—
26	C₂×C₃×Q₈	—	3	1	no	—
27	C₂²×C₃×C₄	—	3	3	no	—
28	C₂²×C₃×C₄	C₂×C₃²	1	3	yes	—
29	C₂²×C₃²	—	3	1	no	—
30	C₂²×C₃²	—	3	1	no	—
31	C₂²×C₃²	C₃×Q₈	1	1	yes	characteristic
32	C₃²×C₄	—	3	3	no	—
33	C₃²×C₄	C₂×A₄	1	3	yes	—
34	C₂²⋊C₉	C₃×Q₈	1	3	yes	complemented
35	C₄×C₉	—	4	9	no	—
36	C₂²×Q₈	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
37	3₋¹⁺²	—	4	1	no	Sylow
38	C₂³×C₃	—	3	1	no	—
39	C₂³×C₃	C₂²×C₃²	1	1	yes	characteristic
40	C₃×Q₈	—	3	6	no	—
41	C₃×Q₈	—	3	2	no	—
42	C₃×Q₈	—	3	9	no	—
43	C₃×Q₈	—	3	3	no	—
44	C₃×Q₈	—	3	1	no	—
45	C₃×Q₈	C₃×A₄	1	1	yes	characteristic
46	C₂×C₃×C₄	—	3	9	no	—
47	C₂×C₃×C₄	—	3	9	no	—
48	C₂×C₃×C₄	—	3	3	no	—
49	C₂×C₃×C₄	—	3	3	no	—
50	C₂×C₃²	—	3	1	no	—
51	C₂×C₃²	—	3	1	no	—
52	C₂×C₃²	C₂²×A₄	1	1	yes	characteristic
53	C₂×C₉	—	4	3	no	—
54	C₂×Q₈	—	3	3	no	—
55	C₂×Q₈	—	3	1	no	—
56	C₂²×C₄	C₂×3₋¹⁺²	1	3	yes	—
57	C₂²×C₃	—	3	1	no	—
58	C₂²×C₃	—	3	3	no	—
59	C₂²×C₃	—	3	3	no	—
60	C₂²×C₃	—	3	1	no	—
61	C₂²×C₃	—	3	1	no	—
62	C₂²×C₃	C₃²×Q₈	1	1	yes	characteristic
63	C₃×C₄	—	3	9	no	—
64	C₃×C₄	—	3	3	no	—
65	C₃×C₄	—	3	3	no	—
66	C₃×C₄	C₂×C₃×A₄	1	3	yes	—
67	C₃²	Q₈×A₄	1	1	yes	characteristic
68	C₉	—	4	3	no	—
69	C₂³	C₂²×3₋¹⁺²	1	1	yes	characteristic
70	Q₈	—	3	2	no	—
71	Q₈	—	3	3	no	—
72	Q₈	#1094	1	1	yes	characteristic, direct product factor
73	C₂×C₄	—	3	3	no	—
74	C₂×C₄	—	3	3	no	—
75	C₂×C₃	—	3	3	no	—
76	C₂×C₃	—	3	3	no	—
77	C₂×C₃	—	3	1	no	—
78	C₂×C₃	—	3	1	no	—
79	C₂×C₃	—	3	1	no	—
80	C₂×C₃	C₂²×C₃×A₄	1	1	yes	characteristic
81	C₂²	—	3	1	no	—
82	C₂²	—	3	1	no	—
83	C₂²	Q₈×3₋¹⁺²	1	1	yes	characteristic, complemented
84	C₄	—	3	3	no	—
85	C₄	C₂×#1094	1	3	yes	—
86	C₃	—	3	1	no	—
87	C₃	C₃×Q₈×A₄	1	1	yes	characteristic
88	C₂	—	3	1	no	—
89	C₂	—	3	1	no	—
90	C₂	C₂²×#1094	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃	C₂²×C₃×A₄
commutator, socle, cosocle	C₂³×C₃	C₂²×C₃²
Fitting	C₂²×C₃²×Q₈	C₃
prime radical	C₂³×C₃²	C₂²×C₃
hypercenter	C₃²×Q₈	A₄
hypocenter	C₂²	Q₈×3₋¹⁺²
LCS	G ▷ C₂³×C₃ ▷ C₂²
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈×C₂²⋊C₉	C₃	semidirect
10. #1094	Q₈	direct (N × H)
34. C₂²⋊C₉	C₃×Q₈	semidirect
36. C₂²×Q₈	3₋¹⁺²	semidirect
72. Q₈	#1094	direct (N × H)
83. C₂²	Q₈×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₄×#1094	2
C₂²×C₃²×Q₈	3
Q₈×C₂²⋊C₉	3
Q₈×3₋¹⁺²	4

Subgroups — Q8×#1094 #11722569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#1094 #11722569