Subgroups — #7155

Order 256.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	2	yes	maximal
3	C₂²×C₁₆	C₄	1	2	yes	—
4	#431	—	2	2	no	—
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₄×C₁₆	—	2	2	no	—
7	#335	C₄	1	4	yes	—
8	#334	—	2	2	no	—
9	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈	Q₈	1	1	yes	characteristic
11	C₂×C₄²	D₄	1	1	yes	characteristic
12	C₂×C₁₆	—	2	2	no	—
13	C₂×C₁₆	—	2	2	no	—
14	C₂×C₁₆	C₈	1	4	yes	complemented
15	C₄⋊C₈	—	2	4	no	—
16	C₂²⋊C₈	—	4	2	no	—
17	C₄×C₈	—	4	2	no	—
18	C₄×C₈	C₂×C₄	1	2	yes	—
19	C₄×C₈	D₄	1	2	yes	—
20	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄	SD₁₆	1	2	yes	—
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	—	4	4	no	—
24	C₂×C₈	C₄²	1	1	yes	characteristic
25	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈	C₄⋊C₄	1	2	yes	—
27	C₂×C₈	C₂×C₈	1	2	yes	—
28	C₂×C₈	M₁₆	1	2	yes	—
29	C₄²	C₂²⋊C₄	1	2	yes	—
30	C₄²	D₈	1	1	yes	characteristic
31	C₄²	Q₁₆	1	1	yes	characteristic
32	C₁₆	—	2	4	no	—
33	C₂³	#106	1	1	yes	characteristic
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	#95	1	1	yes	characteristic
36	C₂×C₄	#102	1	2	yes	—
37	C₂×C₄	#103	1	2	yes	—
38	C₂×C₄	C₄⋊C₈	1	2	yes	—
39	C₂×C₄	#107	1	1	yes	characteristic
40	C₂×C₄	#108	1	2	yes	—
41	C₈	—	2	2	no	—
42	C₈	—	8	4	no	—
43	C₈	C₄×C₈	1	1	yes	characteristic
44	C₈	#97	1	1	yes	characteristic
45	C₈	C₂²⋊C₈	1	2	yes	—
46	C₂²	—	2	2	no	—
47	C₂²	#334	1	2	yes	—
48	C₂²	#340	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	#335	1	2	yes	—
51	C₄	#336	1	1	yes	characteristic
52	C₄	#341	1	1	yes	characteristic
53	C₄	#359	1	4	yes	—
54	C₂	—	2	2	no	—
55	C₂	#1277	1	1	yes	characteristic
56	C₂	#1382	1	2	yes	—

	subgroup	G/H
center	C₄²	D₈
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#340
prime radical	C₂³	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. C₂×C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆	2
#1564	2