Subgroups — C₂³×C₃×M₁₆ #87454

Order 384.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×M₁₆	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₃×C₈	C₂	1	2	yes	maximal, complemented
4	C₂³×M₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁵×C₃	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₄	1	14	yes	—
8	C₂³×C₃×C₄	C₂²	1	14	yes	—
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	112	yes	direct product factor
11	C₂²×C₃×C₈	C₂²	1	28	yes	complemented
12	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂²×M₁₆	C₂×C₃	1	28	yes	direct product factor
14	C₂³×C₈	C₂×C₃	1	2	yes	complemented
15	C₂⁴×C₃	—	2	8	no	—
16	C₂⁴×C₃	C₂×C₄	1	14	yes	—
17	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄	C₂×C₄	1	84	yes	—
19	C₂²×C₃×C₄	C₂×C₄	1	14	yes	—
20	C₂²×C₃×C₄	C₂×C₄	1	7	yes	—
21	C₂²×C₃×C₄	C₂³	1	7	yes	—
22	C₂²×C₃×C₄	C₂³	1	28	yes	—
23	C₃×M₁₆	C₂³	1	64	yes	direct product factor
24	C₂×C₃×C₈	C₂³	1	56	yes	complemented
25	C₂⁵	C₃×C₄	1	1	yes	characteristic
26	C₂³×C₄	C₃×C₄	1	14	yes	—
27	C₂³×C₄	C₃×C₄	1	1	yes	characteristic
28	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
29	C₂³×C₄	C₂²×C₃	1	14	yes	—
30	C₂×M₁₆	C₂²×C₃	1	112	yes	direct product factor
31	C₂²×C₈	C₂²×C₃	1	28	yes	complemented
32	C₂³×C₃	—	2	56	no	—
33	C₂³×C₃	M₁₆	1	8	yes	direct product factor
34	C₂³×C₃	C₂²×C₄	1	28	yes	—
35	C₂³×C₃	C₂²×C₄	1	7	yes	—
36	C₂×C₃×C₄	C₂²×C₄	1	56	yes	—
37	C₂×C₃×C₄	C₂²×C₄	1	28	yes	—
38	C₂×C₃×C₄	C₂²×C₄	1	21	yes	—
39	C₂×C₃×C₄	C₂⁴	1	7	yes	—
40	C₂×C₃×C₄	C₂⁴	1	8	yes	—
41	C₃×C₈	C₂⁴	1	16	yes	complemented
42	C₂⁴	—	2	8	no	—
43	C₂⁴	C₂×C₃×C₄	1	14	yes	—
44	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
45	C₂²×C₄	C₂×C₃×C₄	1	84	yes	—
46	C₂²×C₄	C₂×C₃×C₄	1	7	yes	—
47	C₂²×C₄	C₂×C₃×C₄	1	14	yes	—
48	C₂²×C₄	C₂³×C₃	1	7	yes	—
49	C₂²×C₄	C₂³×C₃	1	28	yes	—
50	M₁₆	C₂³×C₃	1	64	yes	direct product factor
51	C₂×C₈	C₂³×C₃	1	56	yes	complemented
52	C₂²×C₃	—	2	56	no	—
53	C₂²×C₃	C₂×M₁₆	1	28	yes	direct product factor
54	C₂²×C₃	C₂³×C₄	1	8	yes	—
55	C₂²×C₃	C₂³×C₄	1	7	yes	—
56	C₃×C₄	C₂³×C₄	1	7	yes	—
57	C₃×C₄	C₂³×C₄	1	8	yes	—
58	C₃×C₄	C₂⁵	1	1	yes	characteristic
59	C₂³	—	2	56	no	—
60	C₂³	C₃×M₁₆	1	8	yes	direct product factor
61	C₂³	C₂²×C₃×C₄	1	28	yes	—
62	C₂³	C₂²×C₃×C₄	1	7	yes	—
63	C₂×C₄	C₂²×C₃×C₄	1	56	yes	—
64	C₂×C₄	C₂²×C₃×C₄	1	21	yes	—
65	C₂×C₄	C₂²×C₃×C₄	1	28	yes	—
66	C₂×C₄	C₂⁴×C₃	1	7	yes	—
67	C₂×C₄	C₂⁴×C₃	1	8	yes	—
68	C₈	C₂⁴×C₃	1	16	yes	complemented
69	C₂×C₃	—	2	8	no	—
70	C₂×C₃	C₂²×M₁₆	1	14	yes	direct product factor
71	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
72	C₂²	—	2	56	no	—
73	C₂²	C₂×C₃×M₁₆	1	28	yes	direct product factor
74	C₂²	C₂³×C₃×C₄	1	8	yes	—
75	C₂²	C₂³×C₃×C₄	1	7	yes	—
76	C₄	C₂³×C₃×C₄	1	7	yes	—
77	C₄	C₂³×C₃×C₄	1	8	yes	—
78	C₄	C₂⁵×C₃	1	1	yes	characteristic
79	C₃	C₂³×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	—	2	8	no	—
81	C₂	C₂²×C₃×M₁₆	1	14	yes	direct product factor
82	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	C₂²
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₄	C₂⁵×C₃
socle	C₂⁴×C₃	C₂×C₄
prime radical	C₂⁵×C₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×M₁₆	C₂	direct (N × H)
3. C₂³×C₃×C₈	C₂	semidirect
4. C₂³×M₁₆	C₃	direct (N × H)
10. C₂×C₃×M₁₆	C₂²	direct (N × H)
11. C₂²×C₃×C₈	C₂²	semidirect
13. C₂²×M₁₆	C₂×C₃	direct (N × H)
14. C₂³×C₈	C₂×C₃	semidirect
23. C₃×M₁₆	C₂³	direct (N × H)
24. C₂×C₃×C₈	C₂³	semidirect
30. C₂×M₁₆	C₂²×C₃	direct (N × H)
31. C₂²×C₈	C₂²×C₃	semidirect
33. C₂³×C₃	M₁₆	direct (N × H)
41. C₃×C₈	C₂⁴	semidirect
50. M₁₆	C₂³×C₃	direct (N × H)
51. C₂×C₈	C₂³×C₃	semidirect
53. C₂²×C₃	C₂×M₁₆	direct (N × H)
60. C₂³	C₃×M₁₆	direct (N × H)
68. C₈	C₂⁴×C₃	semidirect
70. C₂×C₃	C₂²×M₁₆	direct (N × H)
73. C₂²	C₂×C₃×M₁₆	direct (N × H)
79. C₃	C₂³×M₁₆	direct (N × H)
81. C₂	C₂²×C₃×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
C₂⁴×C₃×C₄	2
C₂²×C₃×M₁₆	2
C₂³×C₃×C₈	2
C₂³×M₁₆	3

Subgroups — C23×C3×M16 #87454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×M₁₆ #87454