Subgroups — #7471

Order 256.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#366	C₂	1	2	yes	maximal
3	C₂²×C₁₆	C₂²	1	1	yes	characteristic
4	C₂×#107	—	2	2	no	—
5	C₂×C₃₂	—	2	2	no	—
6	C₂×C₃₂	C₄	1	2	yes	complemented
7	#366	—	2	2	no	—
8	#366	C₄	1	4	yes	complemented
9	C₂²×C₈	D₄	1	1	yes	characteristic
10	C₂×C₄⋊C₄	—	4	2	no	—
11	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆	C₂×C₄	1	2	yes	—
13	C₂×C₁₆	D₄	1	2	yes	—
14	C₂×C₁₆	Q₈	1	1	yes	characteristic
15	#107	—	2	4	no	—
16	#107	—	4	2	no	—
17	C₃₂	—	2	2	no	—
18	C₂²×C₄	—	8	2	no	—
19	C₂²×C₄	D₈	1	1	yes	characteristic
20	C₂×C₈	C₂²⋊C₄	1	2	yes	—
21	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₈	SD₁₆	1	2	yes	—
23	C₂×C₈	Q₁₆	1	1	yes	characteristic
24	C₄⋊C₄	—	4	4	no	—
25	C₄⋊C₄	—	8	2	no	—
26	C₁₆	C₄²	1	1	yes	characteristic
27	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆	C₄⋊C₄	1	2	yes	—
29	C₂³	D₁₆	1	1	yes	characteristic
30	C₂×C₄	—	8	4	no	—
31	C₂×C₄	—	16	2	no	—
32	C₂×C₄	—	16	2	no	—
33	C₂×C₄	#102	1	2	yes	—
34	C₂×C₄	#107	1	1	yes	characteristic
35	C₂×C₄	SD₃₂	1	2	yes	—
36	C₂×C₄	Q₃₂	1	1	yes	characteristic
37	C₈	#95	1	1	yes	characteristic
38	C₈	#103	1	2	yes	—
39	C₈	#106	1	1	yes	characteristic
40	C₂²	#357	1	2	yes	—
41	C₂²	#366	1	1	yes	characteristic
42	C₂²	D₃₂	1	1	yes	characteristic
43	C₂²	SD₆₄	1	2	yes	—
44	C₂²	Q₆₄	1	1	yes	characteristic
45	C₄	—	16	4	no	—
46	C₄	#340	1	1	yes	characteristic
47	C₄	#358	1	2	yes	—
48	C₄	#367	1	1	yes	characteristic
49	C₂	#1380	1	1	yes	characteristic
50	C₂	#1415	1	2	yes	—
51	C₂	#1416	1	2	yes	—
52	C₂	#1423	1	1	yes	characteristic
53	C₂	#1424	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	D₁₆
commutator	C₁₆	C₄²
Frattini, cosocle	C₂²×C₁₆	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
6. C₂×C₃₂	C₄	semidirect
8. #366	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₂	2
C₂×#366	2