Subgroups — C₂⁴×C₃×Q₁₆ #10681686

Order 768.

11166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×Q₈	C₂	1	2	yes	maximal
2	C₂³×C₃×Q₁₆	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×Q₈	—	2	30	no	—
6	C₂³×C₃×Q₈	C₂²	1	60	yes	—
7	C₂⁴×C₃×C₄	—	2	2	no	—
8	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Q₁₆	C₂²	1	560	yes	direct product factor
10	C₂³×C₃×C₈	C₂²	1	30	yes	—
11	C₂⁴×Q₈	C₂×C₃	1	2	yes	—
12	C₂³×Q₁₆	C₂×C₃	1	60	yes	direct product factor
13	C₂⁴×C₈	C₂×C₃	1	1	yes	characteristic
14	C₂⁵×C₃	D₄	1	1	yes	characteristic
15	C₂²×C₃×Q₈	—	2	420	no	—
16	C₂²×C₃×Q₈	C₂³	1	280	yes	—
17	C₂³×C₃×C₄	—	2	60	no	—
18	C₂³×C₃×C₄	D₄	1	15	yes	—
19	C₂³×C₃×C₄	C₂³	1	15	yes	—
20	C₂×C₃×Q₁₆	C₂³	1	960	yes	direct product factor
21	C₂²×C₃×C₈	C₂³	1	140	yes	—
22	C₂³×Q₈	—	2	30	no	—
23	C₂³×Q₈	C₂²×C₃	1	60	yes	—
24	C₂⁴×C₄	—	2	2	no	—
25	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
26	C₂²×Q₁₆	C₂²×C₃	1	560	yes	direct product factor
27	C₂³×C₈	C₂²×C₃	1	30	yes	—
28	C₂⁴×C₃	Q₁₆	1	16	yes	direct product factor
29	C₂⁴×C₃	C₂×D₄	1	15	yes	—
30	C₂×C₃×Q₈	—	2	840	no	—
31	C₂×C₃×Q₈	C₂⁴	1	240	yes	—
32	C₂²×C₃×C₄	—	2	280	no	—
33	C₂²×C₃×C₄	C₂×D₄	1	105	yes	—
34	C₂²×C₃×C₄	C₂⁴	1	35	yes	—
35	C₃×Q₁₆	C₂⁴	1	256	yes	direct product factor
36	C₂×C₃×C₈	C₂⁴	1	120	yes	—
37	C₂⁵	C₃×D₄	1	1	yes	characteristic
38	C₂²×Q₈	—	2	420	no	—
39	C₂²×Q₈	C₂³×C₃	1	280	yes	—
40	C₂³×C₄	—	2	60	no	—
41	C₂³×C₄	C₃×D₄	1	15	yes	—
42	C₂³×C₄	C₂³×C₃	1	15	yes	—
43	C₂×Q₁₆	C₂³×C₃	1	960	yes	direct product factor
44	C₂²×C₈	C₂³×C₃	1	140	yes	—
45	C₂³×C₃	C₂×Q₁₆	1	120	yes	direct product factor
46	C₂³×C₃	C₂²×D₄	1	35	yes	—
47	C₃×Q₈	—	2	240	no	—
48	C₃×Q₈	C₂⁵	1	32	yes	—
49	C₂×C₃×C₄	—	2	240	no	—
50	C₂×C₃×C₄	C₂²×D₄	1	105	yes	—
51	C₂×C₃×C₄	C₂⁵	1	15	yes	—
52	C₃×C₈	C₂⁵	1	16	yes	—
53	C₂⁴	C₃×Q₁₆	1	16	yes	direct product factor
54	C₂⁴	C₂×C₃×D₄	1	15	yes	—
55	C₂×Q₈	—	2	840	no	—
56	C₂×Q₈	C₂⁴×C₃	1	240	yes	—
57	C₂²×C₄	—	2	280	no	—
58	C₂²×C₄	C₂×C₃×D₄	1	105	yes	—
59	C₂²×C₄	C₂⁴×C₃	1	35	yes	—
60	Q₁₆	C₂⁴×C₃	1	256	yes	direct product factor
61	C₂×C₈	C₂⁴×C₃	1	120	yes	—
62	C₂²×C₃	C₂²×Q₁₆	1	140	yes	direct product factor
63	C₂²×C₃	C₂³×D₄	1	15	yes	—
64	C₃×C₄	—	2	32	no	—
65	C₃×C₄	C₂³×D₄	1	15	yes	—
66	C₃×C₄	C₂⁶	1	1	yes	characteristic
67	C₂³	C₂×C₃×Q₁₆	1	120	yes	direct product factor
68	C₂³	C₂²×C₃×D₄	1	35	yes	—
69	Q₈	—	2	240	no	—
70	Q₈	C₂⁵×C₃	1	32	yes	—
71	C₂×C₄	—	2	240	no	—
72	C₂×C₄	C₂²×C₃×D₄	1	105	yes	—
73	C₂×C₄	C₂⁵×C₃	1	15	yes	—
74	C₈	C₂⁵×C₃	1	16	yes	—
75	C₂×C₃	C₂³×Q₁₆	1	30	yes	direct product factor
76	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
77	C₂²	C₂²×C₃×Q₁₆	1	140	yes	direct product factor
78	C₂²	C₂³×C₃×D₄	1	15	yes	—
79	C₄	—	2	32	no	—
80	C₄	C₂³×C₃×D₄	1	15	yes	—
81	C₄	C₂⁶×C₃	1	1	yes	characteristic
82	C₃	C₂⁴×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
83	C₂	C₂³×C₃×Q₁₆	1	30	yes	direct product factor
84	C₂	C₂⁴×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Q₁₆	C₂	direct (N × H)
4. C₂⁴×Q₁₆	C₃	direct (N × H)
9. C₂²×C₃×Q₁₆	C₂²	direct (N × H)
12. C₂³×Q₁₆	C₂×C₃	direct (N × H)
20. C₂×C₃×Q₁₆	C₂³	direct (N × H)
26. C₂²×Q₁₆	C₂²×C₃	direct (N × H)
28. C₂⁴×C₃	Q₁₆	direct (N × H)
35. C₃×Q₁₆	C₂⁴	direct (N × H)
43. C₂×Q₁₆	C₂³×C₃	direct (N × H)
45. C₂³×C₃	C₂×Q₁₆	direct (N × H)
53. C₂⁴	C₃×Q₁₆	direct (N × H)
60. Q₁₆	C₂⁴×C₃	direct (N × H)
62. C₂²×C₃	C₂²×Q₁₆	direct (N × H)
67. C₂³	C₂×C₃×Q₁₆	direct (N × H)
75. C₂×C₃	C₂³×Q₁₆	direct (N × H)
77. C₂²	C₂²×C₃×Q₁₆	direct (N × H)
82. C₃	C₂⁴×Q₁₆	direct (N × H)
83. C₂	C₂³×C₃×Q₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×Q₈	2
C₂³×C₃×Q₁₆	2
C₂⁴×C₃×C₈	2
C₂⁴×Q₁₆	3

Subgroups — C24×C3×Q16 #10681686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×Q₁₆ #10681686