Subgroups — #12068

Order 256.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2179	C₂	1	1	yes	maximal, characteristic
2	#2171	C₂	1	2	yes	maximal, complemented
3	#1369	C₂	1	1	yes	maximal, characteristic, complemented
4	#1336	C₂	1	2	yes	maximal, complemented
5	#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	#508	C₄	1	1	yes	characteristic
7	#504	—	2	2	no	—
8	#443	C₂²	1	2	yes	—
9	#368	C₄	1	1	yes	characteristic
10	#364	C₂²	1	2	yes	—
11	#359	C₄	1	2	yes	—
12	#350	—	2	2	no	—
13	#346	C₂²	1	1	yes	characteristic
14	#346	C₂²	1	1	yes	characteristic
15	#322	C₂²	1	1	yes	characteristic
16	#135	C₂×C₄	1	2	yes	—
17	#131	—	2	2	no	—
18	Q₃₂	C₈	1	1	yes	characteristic, complemented
19	SD₃₂	C₈	1	2	yes	complemented
20	D₁₆	C₈	1	1	yes	characteristic, complemented
21	M₃₂	—	2	2	no	—
22	M₃₂	—	4	2	no	—
23	C₂×C₁₆	—	4	2	no	—
24	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆	D₄	1	2	yes	—
27	#108	C₂×C₄	1	2	yes	—
28	#104	—	2	2	no	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	C₂³	1	1	yes	characteristic
31	#41	—	2	2	no	—
32	Q₁₆	C₂×C₈	1	2	yes	—
33	SD₁₆	—	2	2	no	—
34	D₈	C₂×C₈	1	2	yes	—
35	M₁₆	—	2	2	no	—
36	M₁₆	—	4	2	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₄²	#41	1	1	yes	characteristic
42	C₁₆	—	2	1	no	—
43	C₁₆	—	2	2	no	—
44	C₁₆	—	4	2	no	—
45	C₁₆	C₂×C₈	1	1	yes	characteristic
46	C₁₆	C₂×C₈	1	1	yes	characteristic
47	Q₈	—	2	2	no	—
48	D₄	—	2	2	no	—
49	D₄	—	4	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	2	no	—
52	C₂×C₄	C₄×D₄	1	1	yes	characteristic
53	C₈	—	4	2	no	—
54	C₈	—	8	1	no	—
55	C₈	C₂²×C₈	1	1	yes	characteristic
56	C₈	#131	1	1	yes	characteristic
57	C₈	AGL(1,ℤ/8)	1	2	yes	—
58	C₂²	—	4	2	no	—
59	C₂²	#444	1	1	yes	characteristic
60	C₄	—	4	1	no	—
61	C₄	—	4	2	no	—
62	C₄	C₈×D₄	1	1	yes	characteristic
63	C₄	#442	1	1	yes	characteristic
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	#1580	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#442
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂	#1580
prime radical	#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2171	C₂	semidirect
3. #1369	C₂	semidirect
4. #1336	C₂	semidirect
5. #1313	C₂	semidirect
18. Q₃₂	C₈	semidirect
19. SD₃₂	C₈	semidirect
20. D₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
#2179	2
#2171	2
#1369	2
#1336	2
#1313	2