Subgroups — #7293

Order 256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2163	C₂	1	1	yes	maximal, characteristic
2	#1568	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#432	—	2	1	no	—
5	#431	—	2	1	no	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₄⋊C₁₆	—	2	1	no	—
8	C₂²⋊C₁₆	C₄	1	2	yes	—
9	#346	—	2	1	no	—
10	#335	—	2	1	no	—
11	#335	—	2	1	no	—
12	#334	C₄	1	2	yes	—
13	#334	C₄	1	2	yes	—
14	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈	Q₈	1	1	yes	characteristic
16	C₂×C₄²	D₄	1	1	yes	characteristic
17	C₂×C₁₆	—	2	2	no	—
18	C₄⋊C₈	—	2	2	no	—
19	C₄⋊C₈	—	2	2	no	—
20	C₂²⋊C₈	—	4	1	no	—
21	C₂²⋊C₈	—	4	1	no	—
22	#97	—	4	1	no	—
23	C₄×C₈	—	4	1	no	—
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄	D₈	1	1	yes	characteristic
30	C₂²×C₄	Q₁₆	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	4	1	no	—
33	C₂×C₈	—	4	2	no	—
34	C₂×C₈	—	4	2	no	—
35	C₂×C₈	C₄²	1	1	yes	characteristic
36	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₄²	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²	SD₁₆	1	1	yes	characteristic
42	C₄²	SD₁₆	1	1	yes	characteristic
43	C₁₆	—	4	2	no	—
44	C₂³	#107	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	#95	1	1	yes	characteristic
50	C₂×C₄	#102	1	1	yes	characteristic
51	C₂×C₄	#102	1	1	yes	characteristic
52	C₂×C₄	#103	1	1	yes	characteristic
53	C₂×C₄	#103	1	1	yes	characteristic
54	C₂×C₄	#106	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	—	2	1	no	—
57	C₈	—	2	1	no	—
58	C₈	—	8	2	no	—
59	C₈	—	8	2	no	—
60	C₈	#108	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	#340	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	#341	1	1	yes	characteristic
71	C₄	#344	1	1	yes	characteristic
72	C₂	—	4	1	no	—
73	C₂	#1301	1	1	yes	characteristic
74	C₂	#1384	1	1	yes	characteristic
75	C₂	#1388	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#340
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²
prime radical	C₂³	#107
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
#1568	2
#1564	2