Subgroups — C₂×C₃×D₃×Q₈ #64347

Order 288.

331 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×Q₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₄×D₃	C₂	1	3	yes	maximal
4	C₂×C₃×Dic₆	C₂	1	3	yes	maximal, complemented
5	C₂²×C₃×Q₈	—	3	1	no	maximal
6	C₂×D₃×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×D₃	C₂²	1	1	yes	characteristic
8	C₃²×Q₈	C₂²	1	4	yes	complemented
9	C₂×C₃²×C₄	C₂²	1	3	yes	—
10	C₂×C₃×Dic₃	C₂²	1	3	yes	—
11	C₃×C₄×D₃	C₂²	1	12	yes	—
12	C₃×Dic₆	C₂²	1	12	yes	complemented
13	C₂×C₃×Q₈	—	2	1	no	—
14	C₂×C₃×Q₈	—	3	8	no	—
15	C₂×C₃×Q₈	—	3	3	no	—
16	C₂×C₃×Q₈	D₃	1	1	yes	characteristic, direct product factor
17	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂²×C₃×C₄	—	3	3	no	—
19	D₃×Q₈	C₂×C₃	1	8	yes	direct product factor
20	C₂×C₄×D₃	C₂×C₃	1	3	yes	—
21	C₂×Dic₆	C₂×C₃	1	3	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃×D₃	Q₈	1	4	yes	direct product factor
24	C₂×C₃×D₃	C₂³	1	2	yes	—
25	C₃²×C₄	C₂³	1	6	yes	—
26	C₃×Dic₃	C₂³	1	6	yes	—
27	C₂²×Q₈	—	3	1	no	Sylow
28	C₂³×C₃	—	3	1	no	—
29	C₂²×D₃	C₂²×C₃	1	1	yes	characteristic
30	C₃×Q₈	—	2	4	no	—
31	C₃×Q₈	—	3	12	no	—
32	C₃×Q₈	C₂×D₃	1	4	yes	direct product factor
33	C₃×Q₈	C₂²×C₃	1	4	yes	complemented
34	C₂×C₃×C₄	—	2	3	no	—
35	C₂×C₃×C₄	—	3	3	no	—
36	C₂×C₃×C₄	—	3	12	no	—
37	C₂×C₃×C₄	C₂×D₃	1	3	yes	—
38	C₂×C₃×C₄	C₂²×C₃	1	3	yes	—
39	C₂×Dic₃	C₂²×C₃	1	3	yes	—
40	C₄×D₃	C₂²×C₃	1	12	yes	—
41	Dic₆	C₂²×C₃	1	12	yes	complemented
42	C₂×C₃²	C₂×Q₈	1	2	yes	complemented
43	C₂×C₃²	C₂⁴	1	1	yes	characteristic
44	C₃×D₃	C₂×Q₈	1	4	yes	direct product factor
45	C₂×Q₈	—	3	8	no	—
46	C₂×Q₈	—	3	3	no	—
47	C₂×Q₈	C₃×D₃	1	1	yes	characteristic, direct product factor
48	C₂²×C₄	—	3	3	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	—	3	4	no	—
51	C₂²×C₃	—	3	2	no	—
52	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
53	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
54	C₂×D₃	C₃×Q₈	1	4	yes	direct product factor
55	C₂×D₃	C₂³×C₃	1	2	yes	—
56	C₃×C₄	—	2	6	no	—
57	C₃×C₄	—	3	6	no	—
58	C₃×C₄	C₂²×D₃	1	6	yes	—
59	C₃×C₄	C₂³×C₃	1	6	yes	—
60	Dic₃	C₂³×C₃	1	6	yes	—
61	C₃²	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
62	C₂³	—	3	1	no	—
63	Q₈	—	3	12	no	—
64	Q₈	C₂×C₃×D₃	1	4	yes	direct product factor
65	C₂×C₄	—	3	3	no	—
66	C₂×C₄	—	3	12	no	—
67	C₂×C₄	C₂×C₃×D₃	1	3	yes	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	2	2	no	—
70	C₂×C₃	—	3	4	no	—
71	C₂×C₃	D₃×Q₈	1	2	yes	direct product factor
72	C₂×C₃	C₂×C₃×Q₈	1	2	yes	complemented
73	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
74	C₂×C₃	C₂⁴×C₃	1	1	yes	characteristic
75	D₃	C₂×C₃×Q₈	1	4	yes	direct product factor
76	C₂²	—	3	2	no	—
77	C₂²	—	3	4	no	—
78	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic
79	C₄	—	3	6	no	—
80	C₄	C₂²×C₃×D₃	1	6	yes	—
81	C₃	—	2	1	no	—
82	C₃	C₂×D₃×Q₈	1	1	yes	characteristic, direct product factor
83	C₃	C₂²×C₃×Q₈	1	1	yes	characteristic, complemented
84	C₂	—	3	4	no	—
85	C₂	C₃×D₃×Q₈	1	2	yes	direct product factor
86	C₂	C₂³×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₃
Fitting	C₂×C₃²×Q₈	C₂
socle	C₂²×C₃²	C₂³
prime radical	C₂²×C₃×D₃	C₂²
hypercenter	C₂×C₃×Q₈	D₃
hypocenter	C₃	C₂²×C₃×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×Q₈	C₂	semidirect
2. C₃×D₃×Q₈	C₂	direct (N × H)
4. C₂×C₃×Dic₆	C₂	semidirect
6. C₂×D₃×Q₈	C₃	direct (N × H)
8. C₃²×Q₈	C₂²	semidirect
12. C₃×Dic₆	C₂²	semidirect
16. C₂×C₃×Q₈	D₃	direct (N × H)
17. C₂×C₃×Q₈	C₂×C₃	semidirect
19. D₃×Q₈	C₂×C₃	direct (N × H)
21. C₂×Dic₆	C₂×C₃	semidirect
23. C₂×C₃×D₃	Q₈	direct (N × H)
32. C₃×Q₈	C₂×D₃	direct (N × H)
33. C₃×Q₈	C₂²×C₃	semidirect
41. Dic₆	C₂²×C₃	semidirect
42. C₂×C₃²	C₂×Q₈	semidirect
44. C₃×D₃	C₂×Q₈	direct (N × H)
47. C₂×Q₈	C₃×D₃	direct (N × H)
54. C₂×D₃	C₃×Q₈	direct (N × H)
61. C₃²	C₂²×Q₈	semidirect
64. Q₈	C₂×C₃×D₃	direct (N × H)
71. C₂×C₃	D₃×Q₈	direct (N × H)
72. C₂×C₃	C₂×C₃×Q₈	semidirect
75. D₃	C₂×C₃×Q₈	direct (N × H)
82. C₃	C₂×D₃×Q₈	direct (N × H)
83. C₃	C₂²×C₃×Q₈	semidirect
85. C₂	C₃×D₃×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₈	2
C₃×D₃×Q₈	2
C₂×C₃×C₄×D₃	2
C₂×C₃×Dic₆	2
C₂²×C₃×Q₈	3
C₂×D₃×Q₈	3

Subgroups — C2×C3×D3×Q8 #64347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×Q₈ #64347