Subgroups — C₄²×M₁₆ #20328

Order 256.

867 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆	C₂	1	6	yes	maximal
2	C₂×C₄³	C₂	1	1	yes	maximal, characteristic
3	C₄×#97	C₂	1	6	yes	maximal, complemented
4	C₄²×C₈	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₄²	C₄	1	4	yes	—
7	C₂²×C₄²	C₂²	1	3	yes	—
8	C₄×M₁₆	C₄	1	48	yes	direct product factor
9	C₂×#97	C₂²	1	12	yes	—
10	C₂×C₄×C₈	C₂²	1	12	yes	—
11	C₄³	C₄	1	4	yes	—
12	C₄³	C₂²	1	3	yes	—
13	C₄³	C₂²	1	1	yes	characteristic
14	C₂³×C₄	C₂×C₄	1	6	yes	—
15	C₂³×C₄	C₂³	1	1	yes	characteristic
16	C₂×M₁₆	C₂×C₄	1	48	yes	—
17	C₂²×C₈	C₂³	1	8	yes	—
18	C₂×C₄²	—	2	16	no	—
19	C₂×C₄²	C₂×C₄	1	12	yes	—
20	C₂×C₄²	C₂×C₄	1	6	yes	—
21	C₂×C₄²	C₂×C₄	1	24	yes	—
22	C₂×C₄²	C₂×C₄	1	4	yes	—
23	C₂×C₄²	C₂³	1	3	yes	—
24	C₂×C₄²	C₂³	1	3	yes	—
25	#97	C₂×C₄	1	48	yes	complemented
26	C₄×C₈	C₂×C₄	1	48	yes	complemented
27	C₂⁴	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄	—	2	24	no	—
29	C₂²×C₄	C₄²	1	24	yes	—
30	C₂²×C₄	C₂²×C₄	1	6	yes	—
31	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄	C₂²×C₄	1	12	yes	—
33	C₂²×C₄	C₂²×C₄	1	6	yes	—
34	C₂²×C₄	C₂⁴	1	1	yes	characteristic
35	M₁₆	C₄²	1	64	yes	direct product factor
36	C₂×C₈	C₂²×C₄	1	48	yes	—
37	C₄²	—	2	24	no	—
38	C₄²	C₄²	1	24	yes	—
39	C₄²	M₁₆	1	16	yes	direct product factor
40	C₄²	C₂²×C₄	1	12	yes	—
41	C₄²	C₂²×C₄	1	12	yes	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×C₄²	1	6	yes	—
44	C₂³	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₄	—	2	12	no	—
46	C₂×C₄	—	2	48	no	—
47	C₂×C₄	C₂×C₄²	1	12	yes	—
48	C₂×C₄	C₂×C₄²	1	6	yes	—
49	C₂×C₄	C₂×C₄²	1	16	yes	—
50	C₂×C₄	C₂×C₄²	1	12	yes	—
51	C₂×C₄	C₂×M₁₆	1	24	yes	—
52	C₂×C₄	C₂³×C₄	1	6	yes	—
53	C₈	C₂×C₄²	1	32	yes	complemented
54	C₂²	—	2	12	no	—
55	C₂²	C₄³	1	4	yes	—
56	C₂²	C₂²×C₄²	1	3	yes	—
57	C₂²	C₂²×M₁₆	1	4	yes	—
58	C₄	—	2	12	no	—
59	C₄	C₄³	1	4	yes	—
60	C₄	C₄×M₁₆	1	24	yes	direct product factor
61	C₄	C₂²×C₄²	1	4	yes	—
62	C₂	—	2	4	no	—
63	C₂	C₂×C₄³	1	1	yes	characteristic
64	C₂	C₂×C₄×M₁₆	1	6	yes	—

	subgroup	G/H
center	C₄³	C₂²
commutator	C₂	C₂×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₄×#97	C₂	semidirect
4. C₄²×C₈	C₂	semidirect
8. C₄×M₁₆	C₄	direct (N × H)
25. #97	C₂×C₄	semidirect
26. C₄×C₈	C₂×C₄	semidirect
35. M₁₆	C₄²	direct (N × H)
39. C₄²	M₁₆	direct (N × H)
53. C₈	C₂×C₄²	semidirect
60. C₄	C₄×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
C₂×C₄×M₁₆	2
C₂×C₄³	2
C₄×#97	2
C₄²×C₈	2

Subgroups — C42×M16 #20328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×M₁₆ #20328