Subgroups — Q₈×C₉×A₄ #11722551

Order 864.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₉	C₂²×A₄
commutator	C₂³	C₂²×C₃×C₉
Frattini	C₂×C₃	C₂²×C₃×A₄
Fitting	C₂²×Q₈×C₉	C₃
socle, cosocle	C₂³×C₃	C₂²×C₃²
prime radical	C₂×C₃×A₄	C₂²×C₃
hypercenter	Q₈×C₉	A₄
hypocenter	C₂²	C₃×Q₈×C₉
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
3. C₂²×Q₈×C₉	C₃	semidirect
4. Q₈×C₂²⋊C₉	C₃	semidirect
12. C₉×A₄	Q₈	direct (N × H)
15. Q₈×A₄	C₉	direct (N × H)
23. Q₈×C₉	A₄	direct (N × H)
35. C₂²×C₉	C₃×Q₈	semidirect
36. C₂²⋊C₉	C₃×Q₈	semidirect
40. C₂²×Q₈	C₃×C₉	semidirect
61. A₄	Q₈×C₉	direct (N × H)
67. C₉	Q₈×A₄	direct (N × H)
71. Q₈	C₉×A₄	direct (N × H)
80. C₂²	C₃×Q₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×A₄	2
C₃×Q₈×A₄	3
C₂²×Q₈×C₉	3
Q₈×C₂²⋊C₉	3
C₃×Q₈×C₉	4

Subgroups — Q8×C9×A4 #11722551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉×A₄ #11722551