Subgroups — #7143

Order 256.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2163	C₂	1	2	yes	maximal
2	#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
5	C₄³	C₄	1	1	yes	characteristic
6	C₄⋊C₁₆	—	2	2	no	—
7	C₂²⋊C₁₆	—	2	2	no	—
8	#346	C₄	1	4	yes	complemented
9	#336	—	2	2	no	—
10	C₂²×C₈	—	2	1	no	—
11	C₂²×C₈	D₄	1	2	yes	—
12	C₂×C₄²	—	2	1	no	—
13	C₂×C₄²	—	2	1	no	—
14	C₂×C₄²	C₈	1	2	yes	—
15	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆	—	2	4	no	—
17	C₄×C₈	—	2	1	no	—
18	C₄×C₈	C₈	1	2	yes	—
19	C₄×C₈	C₂×C₄	1	2	yes	—
20	C₄×C₈	D₄	1	1	yes	characteristic
21	C₄×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₄	—	2	2	no	—
23	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄	M₁₆	1	1	yes	characteristic
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	4	2	no	—
28	C₂×C₈	—	4	2	no	—
29	C₂×C₈	C₂²⋊C₄	1	2	yes	—
30	C₂×C₈	C₄⋊C₄	1	2	yes	—
31	C₄²	—	2	4	no	—
32	C₄²	—	4	2	no	—
33	C₄²	—	4	2	no	—
34	C₄²	C₄²	1	1	yes	characteristic
35	C₄²	C₄⋊C₄	1	1	yes	characteristic
36	C₄²	C₂×C₈	1	1	yes	characteristic
37	C₄²	M₁₆	1	1	yes	characteristic
38	C₁₆	—	4	4	no	—
39	C₂³	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	—	4	2	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	#95	1	1	yes	characteristic
47	C₂×C₄	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄	#97	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₈	—	2	2	no	—
54	C₈	—	4	2	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	#336	1	1	yes	characteristic
58	C₂²	#349	1	2	yes	—
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	2	1	no	—
63	C₄	—	4	4	no	—
64	C₄	H(3,ℤ/4)	1	2	yes	—
65	C₄	#364	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#1274	1	1	yes	characteristic
68	C₂	#1376	1	1	yes	characteristic
69	C₂	#1377	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#95
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
8. #346	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2163	2
#1744	2