Subgroups — #7154

Order 256.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#335	C₂	1	2	yes	maximal
3	C₂²×C₁₆	C₄	1	2	yes	—
4	C₂×C₄⋊C₈	—	2	2	no	—
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₄×C₁₆	—	2	2	no	—
7	#335	—	2	2	no	—
8	#335	C₄	1	4	yes	—
9	C₂²×C₈	—	4	2	no	—
10	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈	Q₈	1	1	yes	characteristic
12	C₂×C₄²	D₄	1	1	yes	characteristic
13	C₂×C₁₆	—	2	2	no	—
14	C₂×C₁₆	—	2	2	no	—
15	C₂×C₁₆	C₈	1	4	yes	complemented
16	C₄⋊C₈	—	2	4	no	—
17	C₄⋊C₈	—	4	2	no	—
18	C₄×C₈	C₂×C₄	1	2	yes	—
19	C₄×C₈	D₄	1	2	yes	—
20	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄	D₈	1	1	yes	characteristic
22	C₂²×C₄	Q₁₆	1	1	yes	characteristic
23	C₂×C₈	—	2	2	no	—
24	C₂×C₈	—	4	4	no	—
25	C₂×C₈	—	8	2	no	—
26	C₂×C₈	—	8	2	no	—
27	C₂×C₈	C₄²	1	1	yes	characteristic
28	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈	C₄⋊C₄	1	2	yes	—
30	C₂×C₈	C₂×C₈	1	2	yes	—
31	C₂×C₈	M₁₆	1	2	yes	—
32	C₄²	C₂²⋊C₄	1	2	yes	—
33	C₄²	SD₁₆	1	2	yes	—
34	C₁₆	—	2	4	no	—
35	C₂³	#107	1	1	yes	characteristic
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	#95	1	1	yes	characteristic
38	C₂×C₄	#102	1	2	yes	—
39	C₂×C₄	#103	1	2	yes	—
40	C₂×C₄	C₄⋊C₈	1	2	yes	—
41	C₂×C₄	#106	1	1	yes	characteristic
42	C₂×C₄	#108	1	2	yes	—
43	C₂×C₄	D₁₆	1	1	yes	characteristic
44	C₂×C₄	SD₃₂	1	2	yes	—
45	C₂×C₄	Q₃₂	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₈	—	8	4	no	—
48	C₈	C₄×C₈	1	1	yes	characteristic
49	C₈	#97	1	1	yes	characteristic
50	C₈	C₂²⋊C₈	1	2	yes	—
51	C₂²	#335	1	2	yes	—
52	C₂²	#340	1	1	yes	characteristic
53	C₂²	#366	1	1	yes	characteristic
54	C₂²	#367	1	1	yes	characteristic
55	C₂²	#368	1	2	yes	—
56	C₄	—	2	2	no	—
57	C₄	#334	1	2	yes	—
58	C₄	#336	1	1	yes	characteristic
59	C₄	#341	1	1	yes	characteristic
60	C₄	#357	1	2	yes	—
61	C₄	#358	1	2	yes	—
62	C₂	#1277	1	1	yes	characteristic
63	C₂	#1371	1	2	yes	—
64	C₂	#1372	1	2	yes	—
65	C₂	#1380	1	1	yes	characteristic
66	C₂	#1381	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle, prime radical	C₂³	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ 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
15. C₂×C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆	2
C₂×#335	2