Subgroups — #7444

Order 256.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2246	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#1371	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆	—	2	1	no	—
5	C₂×D₁₆	C₄	1	2	yes	—
6	#493	—	2	1	no	—
7	C₂×C₃₂	—	2	1	no	—
8	C₄×C₁₆	C₂²	1	1	yes	characteristic
9	#335	—	2	1	no	—
10	C₂×D₈	—	2	2	no	—
11	C₂×D₈	—	4	1	no	—
12	#127	—	4	1	no	—
13	D₁₆	—	4	2	no	—
14	D₁₆	—	4	1	no	—
15	D₁₆	—	4	1	no	—
16	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆	D₄	1	1	yes	characteristic
18	C₄⋊C₈	—	4	1	no	—
19	C₄×C₈	D₄	1	1	yes	characteristic
20	C₃₂	—	4	1	no	—
21	C₂×D₄	—	4	2	no	—
22	C₂×D₄	—	8	1	no	—
23	C₂×D₄	—	8	1	no	—
24	D₈	—	4	2	no	—
25	D₈	—	4	2	no	—
26	D₈	—	8	2	no	—
27	C₂×C₈	—	8	1	no	—
28	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈	SD₁₆	1	1	yes	characteristic
30	C₄²	D₈	1	1	yes	characteristic
31	C₁₆	—	2	1	no	—
32	C₁₆	D₈	1	1	yes	characteristic
33	C₁₆	SD₁₆	1	1	yes	characteristic
34	C₂³	—	8	2	no	—
35	D₄	—	8	2	no	—
36	D₄	—	8	2	no	—
37	D₄	—	16	2	no	—
38	D₄	—	16	1	no	—
39	D₄	—	16	1	no	—
40	C₂×C₄	#102	1	1	yes	characteristic
41	C₂×C₄	D₁₆	1	1	yes	characteristic
42	C₂×C₄	SD₃₂	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	16	1	no	—
45	C₈	#100	1	1	yes	characteristic
46	C₈	#102	1	1	yes	characteristic
47	C₂²	—	16	2	no	—
48	C₂²	—	16	2	no	—
49	C₂²	—	16	2	no	—
50	C₂²	#357	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	#331	1	1	yes	characteristic
53	C₄	#361	1	1	yes	characteristic
54	C₄	D₃₂	1	1	yes	characteristic
55	C₄	SD₆₄	1	1	yes	characteristic
56	C₂	—	32	2	no	—
57	C₂	#1361	1	1	yes	characteristic
58	C₂	#1415	1	1	yes	characteristic
59	C₂	#1419	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#357
commutator	C₂×C₁₆	C₂×C₄
Frattini, cosocle	C₄×C₁₆	C₂²
prime radical	#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₃₂	C₂	semidirect
3. #1371	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2246	2
C₄⋊C₃₂	2
#1371	2