Subgroups — #7196

Order 256.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2163	C₂	1	2	yes	maximal
2	C₂×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₄⋊C₁₆	—	2	2	no	—
7	C₂²⋊C₁₆	C₄	1	4	yes	—
8	#346	—	2	2	no	—
9	#322	—	2	2	no	—
10	C₂²×C₈	—	2	1	no	—
11	C₂²×C₈	—	2	1	no	—
12	C₂²×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆	—	2	4	no	—
15	C₄×C₈	—	2	1	no	—
16	C₄×C₈	—	2	1	no	—
17	C₄×C₈	C₈	1	2	yes	—
18	C₄×C₈	C₈	1	2	yes	—
19	C₄×C₈	D₄	1	2	yes	—
20	C₄×C₈	D₄	1	1	yes	characteristic
21	C₄×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₄	C₄²	1	1	yes	characteristic
23	C₂²×C₄	M₁₆	1	1	yes	characteristic
24	C₂²×C₄	M₁₆	1	1	yes	characteristic
25	C₂×C₈	—	2	4	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	4	2	no	—
29	C₂×C₈	—	4	2	no	—
30	C₂×C₈	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈	C₄⋊C₄	1	2	yes	—
32	C₄²	C₂²⋊C₄	1	1	yes	characteristic
33	C₄²	C₄⋊C₄	1	1	yes	characteristic
34	C₄²	C₂×C₈	1	1	yes	characteristic
35	C₄²	C₂×C₈	1	1	yes	characteristic
36	C₁₆	—	4	4	no	—
37	C₂³	#97	1	1	yes	characteristic
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	#95	1	1	yes	characteristic
42	C₂×C₄	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
47	C₈	—	2	4	no	—
48	C₈	—	2	2	no	—
49	C₈	—	2	2	no	—
50	C₈	—	4	2	no	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	1	no	—
53	C₂²	#336	1	1	yes	characteristic
54	C₂²	#347	1	2	yes	—
55	C₄	—	2	1	no	—
56	C₄	—	2	1	no	—
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	#344	1	2	yes	—
60	C₄	#364	1	2	yes	—
61	C₂	—	2	2	no	—
62	C₂	#1281	1	1	yes	characteristic
63	C₂	#1375	1	1	yes	characteristic
64	C₂	#1376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#95
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
#2163	2
C₂×#322	2