Subgroups — #13688

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄	C₂	1	2	yes	maximal
2	#2247	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	#1416	C₂	1	2	yes	maximal
6	C₂×Q₃₂	—	2	1	no	—
7	C₂×Q₃₂	C₂²	1	2	yes	—
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	#494	—	2	1	no	—
10	C₄×Q₁₆	—	2	1	no	—
11	Q₆₄	—	2	4	no	—
12	C₂×C₃₂	C₂²	1	2	yes	—
13	#366	C₂²	1	1	yes	characteristic
14	#358	—	2	1	no	—
15	C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆	—	2	1	no	—
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	4	1	no	—
19	C₄⋊Q₈	—	4	1	no	—
20	C₄×Q₈	—	4	1	no	—
21	Q₃₂	—	2	2	no	—
22	Q₃₂	—	2	1	no	—
23	Q₃₂	—	2	2	no	—
24	Q₃₂	—	4	2	no	—
25	Q₃₂	D₄	1	2	yes	—
26	C₂×C₁₆	D₄	1	1	yes	characteristic
27	C₂×C₁₆	C₂³	1	1	yes	characteristic
28	#107	—	2	1	no	—
29	#103	—	4	1	no	—
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₃₂	—	2	2	no	—
32	C₂×Q₈	—	4	1	no	—
33	C₂×Q₈	—	4	2	no	—
34	Q₁₆	—	2	2	no	—
35	Q₁₆	—	4	2	no	—
36	Q₁₆	—	4	1	no	—
37	Q₁₆	—	4	2	no	—
38	Q₁₆	—	8	2	no	—
39	C₂×C₈	D₈	1	1	yes	characteristic
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄	—	4	1	no	—
42	C₄⋊C₄	—	8	1	no	—
43	C₄⋊C₄	—	8	1	no	—
44	C₄⋊C₄	—	8	1	no	—
45	C₄²	—	8	1	no	—
46	C₄²	D₈	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₂×D₄	1	1	yes	characteristic
49	C₁₆	#41	1	1	yes	characteristic
50	Q₈	—	4	2	no	—
51	Q₈	—	8	1	no	—
52	Q₈	—	8	2	no	—
53	Q₈	—	8	2	no	—
54	C₂×C₄	—	8	1	no	—
55	C₂×C₄	—	8	1	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	D₁₆	1	1	yes	characteristic
58	C₂×C₄	D₁₆	1	1	yes	characteristic
59	C₂×C₄	C₂×D₈	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	#121	1	1	yes	characteristic
62	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²	C₂×D₁₆	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	16	2	no	—
67	C₄	—	16	1	no	—
68	C₄	Q₆₄	1	2	yes	complemented
69	C₄	#459	1	1	yes	characteristic
70	C₄	#509	1	1	yes	characteristic
71	C₂	#2206	1	1	yes	characteristic
72	C₂	C₂×Q₆₄	1	1	yes	characteristic
73	C₂	#2264	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
68. C₄	Q₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₆₄	2
#2247	2
C₄×Q₃₂	2
C₄⋊C₃₂	2
#1416	2