Subgroups — C₃²×Q₈×A₄ #11724637

Order 864.

468 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×A₄	C₂	1	3	yes	maximal
2	C₂²×C₃²×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×A₄	C₃	1	12	yes	maximal, direct product factor
4	C₂×C₃²×A₄	C₂²	1	1	yes	characteristic
5	C₃³×Q₈	—	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	C₃×C₄×A₄	C₂×C₃	1	36	yes	—
10	C₃²×A₄	Q₈	1	1	yes	characteristic, direct product factor
11	C₃³×C₄	—	4	3	no	—
12	C₂²×C₃×Q₈	C₃²	1	4	yes	complemented
13	Q₈×A₄	C₃²	1	9	yes	direct product factor
14	C₂³×C₃²	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×A₄	C₂²×C₃	1	12	yes	—
16	C₃²×Q₈	—	3	2	no	—
17	C₃²×Q₈	—	3	3	no	—
18	C₃²×Q₈	—	4	12	no	—
19	C₃²×Q₈	A₄	1	1	yes	characteristic, direct product factor
20	C₂×C₃²×C₄	—	3	3	no	—
21	C₂×C₃²×C₄	—	3	3	no	—
22	C₂×C₃³	—	4	1	no	—
23	C₂×C₃×Q₈	—	3	12	no	—
24	C₂×C₃×Q₈	—	3	4	no	—
25	C₂²×C₃×C₄	C₂×C₃²	1	12	yes	—
26	C₄×A₄	C₂×C₃²	1	27	yes	—
27	C₂²×C₃²	—	3	1	no	—
28	C₂²×C₃²	—	3	1	no	—
29	C₂²×C₃²	C₃×Q₈	1	1	yes	characteristic, complemented
30	C₃×A₄	C₃×Q₈	1	12	yes	direct product factor
31	C₃²×C₄	—	3	3	no	—
32	C₃²×C₄	—	4	36	no	—
33	C₃²×C₄	C₂×A₄	1	3	yes	—
34	C₂²×Q₈	C₃³	1	1	yes	characteristic, complemented, Sylow
35	C₃³	—	4	1	no	Sylow
36	C₂³×C₃	C₂²×C₃²	1	4	yes	—
37	C₂×A₄	C₂²×C₃²	1	9	yes	—
38	C₃×Q₈	—	3	8	no	—
39	C₃×Q₈	—	3	12	no	—
40	C₃×Q₈	—	4	9	no	—
41	C₃×Q₈	C₃×A₄	1	4	yes	direct product factor
42	C₂×C₃×C₄	—	3	12	no	—
43	C₂×C₃×C₄	—	3	12	no	—
44	C₂×C₃²	—	3	1	no	—
45	C₂×C₃²	—	3	1	no	—
46	C₂×C₃²	—	4	12	no	—
47	C₂×C₃²	C₂²×A₄	1	1	yes	characteristic
48	C₂×Q₈	—	3	3	no	—
49	C₂×Q₈	—	3	1	no	—
50	C₂²×C₄	C₂×C₃³	1	3	yes	—
51	C₂²×C₃	—	3	4	no	—
52	C₂²×C₃	—	3	4	no	—
53	C₂²×C₃	C₃²×Q₈	1	4	yes	complemented
54	A₄	C₃²×Q₈	1	9	yes	direct product factor
55	C₃×C₄	—	3	12	no	—
56	C₃×C₄	—	4	27	no	—
57	C₃×C₄	C₂×C₃×A₄	1	12	yes	—
58	C₃²	—	4	12	no	—
59	C₃²	Q₈×A₄	1	1	yes	characteristic, direct product factor
60	C₂³	C₂²×C₃³	1	1	yes	characteristic
61	Q₈	—	3	2	no	—
62	Q₈	—	3	3	no	—
63	Q₈	C₃²×A₄	1	1	yes	characteristic, direct product factor
64	C₂×C₄	—	3	3	no	—
65	C₂×C₄	—	3	3	no	—
66	C₂×C₃	—	3	4	no	—
67	C₂×C₃	—	3	4	no	—
68	C₂×C₃	—	4	9	no	—
69	C₂×C₃	C₂²×C₃×A₄	1	4	yes	—
70	C₂²	—	3	1	no	—
71	C₂²	—	3	1	no	—
72	C₂²	C₃³×Q₈	1	1	yes	characteristic, complemented
73	C₄	—	3	3	no	—
74	C₄	C₂×C₃²×A₄	1	3	yes	—
75	C₃	—	4	9	no	—
76	C₃	C₃×Q₈×A₄	1	4	yes	direct product factor
77	C₂	—	3	1	no	—
78	C₂	—	3	1	no	—
79	C₂	C₂²×C₃²×A₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂²×A₄
commutator, cosocle	C₂³	C₂²×C₃³
Frattini	C₂	C₂²×C₃²×A₄
Fitting	C₂²×C₃²×Q₈	C₃
socle	C₂³×C₃²	C₂²×C₃
prime radical	C₂×C₃²×A₄	C₂²
hypercenter	C₃²×Q₈	A₄
hypocenter	C₂²	C₃³×Q₈
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×Q₈	C₃	semidirect
3. C₃×Q₈×A₄	C₃	direct (N × H)
10. C₃²×A₄	Q₈	direct (N × H)
12. C₂²×C₃×Q₈	C₃²	semidirect
13. Q₈×A₄	C₃²	direct (N × H)
19. C₃²×Q₈	A₄	direct (N × H)
29. C₂²×C₃²	C₃×Q₈	semidirect
30. C₃×A₄	C₃×Q₈	direct (N × H)
34. C₂²×Q₈	C₃³	semidirect
41. C₃×Q₈	C₃×A₄	direct (N × H)
53. C₂²×C₃	C₃²×Q₈	semidirect
54. A₄	C₃²×Q₈	direct (N × H)
59. C₃²	Q₈×A₄	direct (N × H)
63. Q₈	C₃²×A₄	direct (N × H)
72. C₂²	C₃³×Q₈	semidirect
76. C₃	C₃×Q₈×A₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄×A₄	2
C₂²×C₃²×Q₈	3
C₃×Q₈×A₄	3
C₃³×Q₈	4

Subgroups — C32×Q8×A4 #11724637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Q₈×A₄ #11724637