Subgroups — C₂×C₃²×M₁₆ #64179

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₃²×M₁₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×M₁₆	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₄	C₄	1	2	yes	—
8	C₂×C₃²×C₄	C₂²	1	2	yes	—
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₃²×C₈	C₂²	1	4	yes	complemented
11	C₂²×C₃×C₄	C₂×C₃	1	4	yes	—
12	C₃×M₁₆	C₂×C₃	1	16	yes	direct product factor
13	C₂×C₃×C₈	C₂×C₃	1	8	yes	complemented
14	C₂²×C₃²	—	2	2	no	—
15	C₂²×C₃²	C₂×C₄	1	2	yes	—
16	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
17	C₃²×C₄	C₂×C₄	1	2	yes	—
18	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₂×M₁₆	C₃²	1	1	yes	characteristic, direct product factor, Sylow
21	C₂³×C₃	C₃×C₄	1	4	yes	—
22	C₂×C₃×C₄	C₃×C₄	1	8	yes	—
23	C₂×C₃×C₄	C₃×C₄	1	4	yes	—
24	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
25	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
26	C₃×C₈	C₂²×C₃	1	16	yes	complemented
27	C₂×C₃²	—	2	2	no	—
28	C₂×C₃²	M₁₆	1	2	yes	direct product factor
29	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic
31	M₁₆	C₂×C₃²	1	4	yes	direct product factor
32	C₂×C₈	C₂×C₃²	1	2	yes	complemented
33	C₂²×C₃	—	2	8	no	—
34	C₂²×C₃	C₂×C₃×C₄	1	8	yes	—
35	C₂²×C₃	C₂×C₃×C₄	1	4	yes	—
36	C₃×C₄	C₂×C₃×C₄	1	8	yes	—
37	C₃×C₄	C₂×C₃×C₄	1	4	yes	—
38	C₃×C₄	C₂³×C₃	1	4	yes	—
39	C₃²	C₂×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
40	C₂³	C₃²×C₄	1	1	yes	characteristic
41	C₂×C₄	C₃²×C₄	1	2	yes	—
42	C₂×C₄	C₃²×C₄	1	1	yes	characteristic
43	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₃²	1	2	yes	—
45	C₈	C₂²×C₃²	1	4	yes	complemented
46	C₂×C₃	—	2	8	no	—
47	C₂×C₃	C₃×M₁₆	1	8	yes	direct product factor
48	C₂×C₃	C₂²×C₃×C₄	1	4	yes	—
49	C₂²	—	2	2	no	—
50	C₂²	C₂×C₃²×C₄	1	2	yes	—
51	C₂²	C₂×C₃²×C₄	1	1	yes	characteristic
52	C₄	C₂×C₃²×C₄	1	1	yes	characteristic
53	C₄	C₂×C₃²×C₄	1	2	yes	—
54	C₄	C₂³×C₃²	1	1	yes	characteristic
55	C₃	C₂×C₃×M₁₆	1	4	yes	direct product factor
56	C₂	—	2	2	no	—
57	C₂	C₃²×M₁₆	1	2	yes	direct product factor
58	C₂	C₂²×C₃²×C₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃²×M₁₆	C₂	direct (N × H)
3. C₂×C₃²×C₈	C₂	semidirect
4. C₂×C₃×M₁₆	C₃	direct (N × H)
10. C₃²×C₈	C₂²	semidirect
12. C₃×M₁₆	C₂×C₃	direct (N × H)
13. C₂×C₃×C₈	C₂×C₃	semidirect
20. C₂×M₁₆	C₃²	direct (N × H)
26. C₃×C₈	C₂²×C₃	semidirect
28. C₂×C₃²	M₁₆	direct (N × H)
31. M₁₆	C₂×C₃²	direct (N × H)
32. C₂×C₈	C₂×C₃²	semidirect
39. C₃²	C₂×M₁₆	direct (N × H)
45. C₈	C₂²×C₃²	semidirect
47. C₂×C₃	C₃×M₁₆	direct (N × H)
55. C₃	C₂×C₃×M₁₆	direct (N × H)
57. 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
C₂²×C₃²×C₄	2
C₃²×M₁₆	2
C₂×C₃²×C₈	2
C₂×C₃×M₁₆	3

Subgroups — C2×C32×M16 #64179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×M₁₆ #64179