Subgroups — C₂³×#106 #60101

Order 256.

1807 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂²×#106	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄	—	2	2	no	—
5	C₂⁴×C₄	C₂²	1	1	yes	characteristic
6	C₂³×C₈	C₄	1	16	yes	complemented
7	C₂³×C₈	C₂²	1	14	yes	—
8	C₂²×C₄⋊C₄	—	2	14	no	—
9	C₂²×C₄⋊C₄	C₂²	1	28	yes	—
10	C₂×#106	C₂²	1	112	yes	direct product factor
11	C₂⁵	D₄	1	1	yes	characteristic
12	C₂³×C₄	—	2	28	no	—
13	C₂³×C₄	—	4	16	no	—
14	C₂³×C₄	C₂×C₄	1	8	yes	—
15	C₂³×C₄	D₄	1	7	yes	—
16	C₂³×C₄	Q₈	1	8	yes	—
17	C₂³×C₄	C₂³	1	7	yes	—
18	C₂²×C₈	C₂×C₄	1	112	yes	complemented
19	C₂²×C₈	C₂³	1	28	yes	—
20	C₂×C₄⋊C₄	—	2	84	no	—
21	C₂×C₄⋊C₄	C₂³	1	56	yes	—
22	#106	C₂³	1	64	yes	direct product factor
23	C₂⁴	C₄⋊C₄	1	8	yes	—
24	C₂⁴	SD₁₆	1	16	yes	—
25	C₂⁴	C₂×D₄	1	7	yes	—
26	C₂²×C₄	—	2	56	no	—
27	C₂²×C₄	—	4	112	no	—
28	C₂²×C₄	C₄⋊C₄	1	56	yes	—
29	C₂²×C₄	C₂²×C₄	1	28	yes	—
30	C₂²×C₄	C₂×D₄	1	21	yes	—
31	C₂²×C₄	C₂×Q₈	1	28	yes	—
32	C₂²×C₄	C₂⁴	1	7	yes	—
33	C₂×C₈	C₂²×C₄	1	112	yes	complemented
34	C₂×C₈	C₂⁴	1	8	yes	—
35	C₄⋊C₄	—	2	56	no	—
36	C₄⋊C₄	C₂⁴	1	16	yes	—
37	C₂³	#106	1	64	yes	direct product factor
38	C₂³	C₂×C₄⋊C₄	1	28	yes	—
39	C₂³	C₂×SD₁₆	1	56	yes	—
40	C₂³	C₂²×D₄	1	7	yes	—
41	C₂×C₄	—	2	16	no	—
42	C₂×C₄	—	4	112	no	—
43	C₂×C₄	C₂×C₄⋊C₄	1	84	yes	—
44	C₂×C₄	C₂³×C₄	1	14	yes	—
45	C₂×C₄	C₂²×D₄	1	7	yes	—
46	C₂×C₄	C₂²×Q₈	1	14	yes	—
47	C₂×C₄	C₂⁵	1	1	yes	characteristic
48	C₈	C₂³×C₄	1	16	yes	complemented
49	C₂²	C₂×#106	1	112	yes	direct product factor
50	C₂²	C₂²×C₄⋊C₄	1	14	yes	—
51	C₂²	C₂²×SD₁₆	1	28	yes	—
52	C₂²	C₂³×D₄	1	1	yes	characteristic
53	C₄	—	4	16	no	—
54	C₄	C₂²×C₄⋊C₄	1	14	yes	—
55	C₄	C₂⁴×C₄	1	1	yes	characteristic
56	C₄	C₂³×Q₈	1	1	yes	characteristic
57	C₂	C₂²×#106	1	28	yes	direct product factor
58	C₂	C₂³×C₄⋊C₄	1	1	yes	characteristic
59	C₂	C₂³×SD₁₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂²×#106	C₂	direct (N × H)
6. C₂³×C₈	C₄	semidirect
10. C₂×#106	C₂²	direct (N × H)
18. C₂²×C₈	C₂×C₄	semidirect
22. #106	C₂³	direct (N × H)
33. C₂×C₈	C₂²×C₄	semidirect
37. C₂³	#106	direct (N × H)
48. C₈	C₂³×C₄	semidirect
49. C₂²	C₂×#106	direct (N × H)
57. C₂	C₂²×#106	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×C₈	2
C₂³×C₄⋊C₄	2
C₂²×#106	2

Subgroups — C23×#106 #60101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#106 #60101