Subgroups — C₂²×C₃²×M₁₆ #10596367

Order 576.

894 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×M₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃²×C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×M₁₆	C₃	1	4	yes	maximal, direct product factor
5	C₂⁴×C₃²	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₄	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₄	C₄	1	6	yes	—
8	C₂²×C₃²×C₄	C₂²	1	6	yes	—
9	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₃²×M₁₆	C₂²	1	16	yes	direct product factor
11	C₂×C₃²×C₈	C₂²	1	12	yes	complemented
12	C₂³×C₃×C₄	C₂×C₃	1	4	yes	—
13	C₂×C₃×M₁₆	C₂×C₃	1	48	yes	direct product factor
14	C₂²×C₃×C₈	C₂×C₃	1	8	yes	complemented
15	C₂³×C₃²	—	2	4	no	—
16	C₂³×C₃²	C₂×C₄	1	6	yes	—
17	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₄	C₂×C₄	1	12	yes	—
19	C₂×C₃²×C₄	C₂×C₄	1	6	yes	—
20	C₂×C₃²×C₄	C₂×C₄	1	3	yes	—
21	C₂×C₃²×C₄	C₂³	1	3	yes	—
22	C₂×C₃²×C₄	C₂³	1	4	yes	—
23	C₃²×C₈	C₂³	1	8	yes	complemented
24	C₂²×M₁₆	C₃²	1	1	yes	characteristic, direct product factor, Sylow
25	C₂⁴×C₃	C₃×C₄	1	4	yes	—
26	C₂²×C₃×C₄	C₃×C₄	1	24	yes	—
27	C₂²×C₃×C₄	C₃×C₄	1	4	yes	—
28	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
29	C₂²×C₃×C₄	C₂²×C₃	1	24	yes	—
30	C₃×M₁₆	C₂²×C₃	1	64	yes	direct product factor
31	C₂×C₃×C₈	C₂²×C₃	1	48	yes	complemented
32	C₂²×C₃²	—	2	12	no	—
33	C₂²×C₃²	M₁₆	1	4	yes	direct product factor
34	C₂²×C₃²	C₂²×C₄	1	4	yes	—
35	C₂²×C₃²	C₂²×C₄	1	3	yes	—
36	C₃²×C₄	C₂²×C₄	1	4	yes	—
37	C₃²×C₄	C₂²×C₄	1	3	yes	—
38	C₃²×C₄	C₂⁴	1	1	yes	characteristic
39	C₂³×C₄	C₂×C₃²	1	1	yes	characteristic
40	C₂×M₁₆	C₂×C₃²	1	12	yes	direct product factor
41	C₂²×C₈	C₂×C₃²	1	2	yes	complemented
42	C₂³×C₃	—	2	16	no	—
43	C₂³×C₃	C₂×C₃×C₄	1	24	yes	—
44	C₂³×C₃	C₂×C₃×C₄	1	4	yes	—
45	C₂×C₃×C₄	C₂×C₃×C₄	1	48	yes	—
46	C₂×C₃×C₄	C₂×C₃×C₄	1	24	yes	—
47	C₂×C₃×C₄	C₂×C₃×C₄	1	12	yes	—
48	C₂×C₃×C₄	C₂³×C₃	1	12	yes	—
49	C₂×C₃×C₄	C₂³×C₃	1	16	yes	—
50	C₃×C₈	C₂³×C₃	1	32	yes	complemented
51	C₂×C₃²	—	2	4	no	—
52	C₂×C₃²	C₂×M₁₆	1	6	yes	direct product factor
53	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
54	C₂⁴	C₃²×C₄	1	1	yes	characteristic
55	C₂²×C₄	C₃²×C₄	1	6	yes	—
56	C₂²×C₄	C₃²×C₄	1	1	yes	characteristic
57	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic
58	C₂²×C₄	C₂²×C₃²	1	6	yes	—
59	M₁₆	C₂²×C₃²	1	16	yes	direct product factor
60	C₂×C₈	C₂²×C₃²	1	12	yes	complemented
61	C₂²×C₃	—	2	48	no	—
62	C₂²×C₃	C₃×M₁₆	1	16	yes	direct product factor
63	C₂²×C₃	C₂²×C₃×C₄	1	16	yes	—
64	C₂²×C₃	C₂²×C₃×C₄	1	12	yes	—
65	C₃×C₄	C₂²×C₃×C₄	1	12	yes	—
66	C₃×C₄	C₂²×C₃×C₄	1	16	yes	—
67	C₃×C₄	C₂⁴×C₃	1	4	yes	—
68	C₃²	C₂²×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³	—	2	4	no	—
70	C₂³	C₂×C₃²×C₄	1	6	yes	—
71	C₂³	C₂×C₃²×C₄	1	1	yes	characteristic
72	C₂×C₄	C₂×C₃²×C₄	1	12	yes	—
73	C₂×C₄	C₂×C₃²×C₄	1	3	yes	—
74	C₂×C₄	C₂×C₃²×C₄	1	6	yes	—
75	C₂×C₄	C₂³×C₃²	1	3	yes	—
76	C₂×C₄	C₂³×C₃²	1	4	yes	—
77	C₈	C₂³×C₃²	1	8	yes	complemented
78	C₂×C₃	—	2	16	no	—
79	C₂×C₃	C₂×C₃×M₁₆	1	24	yes	direct product factor
80	C₂×C₃	C₂³×C₃×C₄	1	4	yes	—
81	C₂²	—	2	12	no	—
82	C₂²	C₃²×M₁₆	1	4	yes	direct product factor
83	C₂²	C₂²×C₃²×C₄	1	4	yes	—
84	C₂²	C₂²×C₃²×C₄	1	3	yes	—
85	C₄	C₂²×C₃²×C₄	1	3	yes	—
86	C₄	C₂²×C₃²×C₄	1	4	yes	—
87	C₄	C₂⁴×C₃²	1	1	yes	characteristic
88	C₃	C₂²×C₃×M₁₆	1	4	yes	direct product factor
89	C₂	—	2	4	no	—
90	C₂	C₂×C₃²×M₁₆	1	6	yes	direct product factor
91	C₂	C₂³×C₃²×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²×C₄	C₂²
commutator	C₂	C₂³×C₃²×C₄
Frattini, cosocle	C₄	C₂⁴×C₃²
socle	C₂³×C₃²	C₂×C₄
prime radical	C₂⁴×C₃²	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×M₁₆	C₂	direct (N × H)
3. C₂²×C₃²×C₈	C₂	semidirect
4. C₂²×C₃×M₁₆	C₃	direct (N × H)
10. C₃²×M₁₆	C₂²	direct (N × H)
11. C₂×C₃²×C₈	C₂²	semidirect
13. C₂×C₃×M₁₆	C₂×C₃	direct (N × H)
14. C₂²×C₃×C₈	C₂×C₃	semidirect
23. C₃²×C₈	C₂³	semidirect
24. C₂²×M₁₆	C₃²	direct (N × H)
30. C₃×M₁₆	C₂²×C₃	direct (N × H)
31. C₂×C₃×C₈	C₂²×C₃	semidirect
33. C₂²×C₃²	M₁₆	direct (N × H)
40. C₂×M₁₆	C₂×C₃²	direct (N × H)
41. C₂²×C₈	C₂×C₃²	semidirect
50. C₃×C₈	C₂³×C₃	semidirect
52. C₂×C₃²	C₂×M₁₆	direct (N × H)
59. M₁₆	C₂²×C₃²	direct (N × H)
60. C₂×C₈	C₂²×C₃²	semidirect
62. C₂²×C₃	C₃×M₁₆	direct (N × H)
68. C₃²	C₂²×M₁₆	direct (N × H)
77. C₈	C₂³×C₃²	semidirect
79. C₂×C₃	C₂×C₃×M₁₆	direct (N × H)
82. C₂²	C₃²×M₁₆	direct (N × H)
88. C₃	C₂²×C₃×M₁₆	direct (N × H)
90. C₂	C₂×C₃²×M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₄	2
C₂×C₃²×M₁₆	2
C₂²×C₃²×C₈	2
C₂²×C₃×M₁₆	3

Subgroups — C22×C32×M16 #10596367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×M₁₆ #10596367