Subgroups — #7142

Order 256.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	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	C₄×C₁₆	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	2	no	—
32	C₄²	—	4	2	no	—
33	C₄²	—	4	2	no	—
34	C₄²	C₁₆	1	4	yes	complemented
35	C₄²	C₄²	1	1	yes	characteristic
36	C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₄²	C₂×C₈	1	1	yes	characteristic
38	C₄²	M₁₆	1	1	yes	characteristic
39	C₁₆	—	4	4	no	—
40	C₂³	C₂²⋊C₈	1	1	yes	characteristic
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	C₂×C₄	C₂×C₁₆	1	2	yes	—
53	C₂×C₄	M₃₂	1	2	yes	—
54	C₈	—	2	2	no	—
55	C₈	—	2	2	no	—
56	C₈	—	4	2	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	#336	1	1	yes	characteristic
60	C₂²	C₂²⋊C₁₆	1	2	yes	—
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	C₄×C₁₆	1	1	yes	characteristic
66	C₄	#346	1	1	yes	characteristic
67	C₄	#347	1	2	yes	—
68	C₄	C₄⋊C₁₆	1	2	yes	—
69	C₂	—	2	2	no	—
70	C₂	#1274	1	1	yes	characteristic
71	C₂	#1374	1	1	yes	characteristic
72	C₂	#1375	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄²
commutator	C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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. C₄×C₁₆	C₄	semidirect
34. C₄²	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
#1744	2