Subgroups — #13724

Order 256.

87 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	1	yes	maximal, characteristic
2	#2247	C₂	1	1	yes	maximal, characteristic
3	#1416	C₂	1	4	yes	maximal
4	#1397	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂	—	2	1	no	—
6	C₂×Q₃₂	C₂²	1	2	yes	—
7	#499	—	2	1	no	—
8	#494	—	2	1	no	—
9	C₂×C₃₂	C₂²	1	2	yes	—
10	#367	—	2	1	no	—
11	#366	C₂²	1	2	yes	—
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆	—	2	2	no	—
14	C₂×Q₁₆	—	4	1	no	—
15	C₄⋊Q₈	—	4	1	no	—
16	#125	—	4	1	no	—
17	Q₃₂	—	2	4	no	—
18	Q₃₂	—	4	2	no	—
19	C₂×C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₁₆	C₂³	1	1	yes	characteristic
21	#107	—	2	2	no	—
22	#106	—	4	1	no	—
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₃₂	—	2	2	no	—
25	C₂×Q₈	—	4	2	no	—
26	Q₁₆	—	4	2	no	—
27	Q₁₆	—	4	2	no	—
28	Q₁₆	—	8	2	no	—
29	C₂×C₈	D₈	1	1	yes	characteristic
30	C₂×C₈	C₂×D₄	1	1	yes	characteristic
31	C₄⋊C₄	—	4	2	no	—
32	C₄⋊C₄	—	8	2	no	—
33	C₄⋊C₄	—	8	2	no	—
34	C₄²	D₈	1	1	yes	characteristic
35	C₁₆	SD₁₆	1	2	yes	—
36	C₁₆	#41	1	2	yes	—
37	Q₈	—	8	2	no	—
38	Q₈	—	8	2	no	—
39	C₂×C₄	—	8	2	no	—
40	C₂×C₄	—	8	2	no	—
41	C₂×C₄	D₁₆	1	2	yes	—
42	C₂×C₄	C₂×D₈	1	1	yes	characteristic
43	C₈	SD₃₂	1	2	yes	—
44	C₈	#124	1	1	yes	characteristic
45	C₈	C₂×SD₁₆	1	1	yes	characteristic
46	C₂²	C₂×D₁₆	1	1	yes	characteristic
47	C₄	—	2	2	no	—
48	C₄	—	16	2	no	—
49	C₄	—	16	2	no	—
50	C₄	#486	1	1	yes	characteristic
51	C₄	C₂×SD₃₂	1	1	yes	characteristic
52	C₂	#2240	1	1	yes	characteristic
53	C₂	#2264	1	2	yes	—

	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

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