Subgroups — #13698

Order 256.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
3	#2216	C₂	1	2	yes	maximal
4	#1424	C₂	1	1	yes	maximal, characteristic, complemented
5	#1416	C₂	1	2	yes	maximal, complemented
6	C₂×Q₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	#467	—	2	2	no	—
9	Q₆₄	—	2	2	no	—
10	M₆₄	—	2	2	no	—
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	#366	C₂²	1	2	yes	—
14	#358	—	2	2	no	—
15	C₂×Q₁₆	—	2	2	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²⋊Q₈	—	4	2	no	—
18	Q₃₂	—	2	2	no	—
19	Q₃₂	—	2	2	no	—
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	—	2	1	no	—
22	C₂×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₆	C₂³	1	1	yes	characteristic
24	#107	—	2	2	no	—
25	#103	—	4	2	no	—
26	C₃₂	—	2	1	no	—
27	C₃₂	D₄	1	2	yes	—
28	C₂×Q₈	—	4	2	no	—
29	C₂²×C₄	D₈	1	1	yes	characteristic
30	Q₁₆	—	4	2	no	—
31	Q₁₆	—	4	2	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	D₈	1	1	yes	characteristic
35	C₂×C₈	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄	—	4	2	no	—
37	C₄⋊C₄	—	8	2	no	—
38	C₂²⋊C₄	—	8	2	no	—
39	C₁₆	—	2	1	no	—
40	C₁₆	C₂×D₄	1	1	yes	characteristic
41	C₁₆	#41	1	1	yes	characteristic
42	C₂³	D₁₆	1	1	yes	characteristic
43	Q₈	—	8	2	no	—
44	Q₈	—	8	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	8	2	no	—
48	C₂×C₄	—	8	2	no	—
49	C₂×C₄	D₁₆	1	1	yes	characteristic
50	C₂×C₄	C₂×D₈	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	#121	1	1	yes	characteristic
53	C₈	#135	1	1	yes	characteristic
54	C₂²	—	2	1	no	—
55	C₂²	—	2	2	no	—
56	C₂²	C₂×D₁₆	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	16	2	no	—
59	C₄	—	16	2	no	—
60	C₄	#466	1	1	yes	characteristic
61	C₄	#508	1	1	yes	characteristic
62	C₂	—	4	1	no	—
63	C₂	#2215	1	1	yes	characteristic
64	C₂	#2264	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂³	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×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
1. C₂×Q₆₄	C₂	semidirect
4. #1424	C₂	semidirect
5. #1416	C₂	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
C₂×M₆₄	2
#2216	2
#1424	2
#1416	2