Subgroups — #7161

Order 256.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	2	yes	maximal
2	C₂×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂	—	2	2	no	—
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	#364	—	2	2	no	—
8	#364	C₄	1	4	yes	—
9	#346	—	2	2	no	—
10	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈	D₄	1	1	yes	characteristic
12	C₂×C₄²	Q₈	1	1	yes	characteristic
13	M₃₂	—	2	4	no	—
14	M₃₂	—	4	2	no	—
15	C₂×C₁₆	—	2	1	no	—
16	C₂×C₁₆	—	2	1	no	—
17	C₂×C₁₆	—	2	1	no	—
18	C₂×C₁₆	—	2	1	no	—
19	C₂×C₁₆	—	4	2	no	—
20	C₂×C₁₆	C₈	1	2	yes	—
21	C₂×C₁₆	C₈	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈	D₄	1	2	yes	—
24	C₂²×C₄	—	2	1	no	—
25	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈	C₄²	1	1	yes	characteristic
27	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈	C₂²⋊C₄	1	2	yes	—
29	C₂×C₈	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈	M₁₆	1	1	yes	characteristic
32	C₂×C₈	M₁₆	1	1	yes	characteristic
33	C₂×C₈	D₈	1	1	yes	characteristic
34	C₂×C₈	SD₁₆	1	2	yes	—
35	C₂×C₈	Q₁₆	1	1	yes	characteristic
36	C₄²	—	2	1	no	—
37	C₄²	C₄⋊C₄	1	2	yes	—
38	C₁₆	—	2	2	no	—
39	C₁₆	—	2	2	no	—
40	C₁₆	—	4	4	no	—
41	C₂³	#108	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	#95	1	1	yes	characteristic
45	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄	#106	1	1	yes	characteristic
48	C₂×C₄	#107	1	1	yes	characteristic
49	C₂×C₄	#108	1	1	yes	characteristic
50	C₈	C₄×C₈	1	1	yes	characteristic
51	C₈	#97	1	1	yes	characteristic
52	C₈	C₂²⋊C₈	1	1	yes	characteristic
53	C₈	C₂²⋊C₈	1	1	yes	characteristic
54	C₈	#102	1	2	yes	—
55	C₈	#103	1	2	yes	—
56	C₂²	—	2	2	no	—
57	C₂²	#334	1	1	yes	characteristic
58	C₂²	#335	1	1	yes	characteristic
59	C₂²	#341	1	1	yes	characteristic
60	C₄	—	4	2	no	—
61	C₄	#334	1	1	yes	characteristic
62	C₄	#335	1	1	yes	characteristic
63	C₄	#336	1	1	yes	characteristic
64	C₄	#340	1	1	yes	characteristic
65	C₂	—	2	1	no	—
66	C₂	—	2	1	no	—
67	C₂	#1277	1	1	yes	characteristic
68	C₂	#1369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#106
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#334
prime radical	C₂³	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#364	2
C₂×#346	2