Subgroups — #13656

Order 256.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2262	C₂	1	1	yes	maximal, characteristic
2	#2178	C₂	1	2	yes	maximal, complemented
3	#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	#1417	C₂	1	2	yes	maximal, complemented
5	#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	#508	C₂²	1	2	yes	—
7	#443	—	2	2	no	—
8	Q₆₄	C₄	1	1	yes	characteristic, complemented
9	SD₆₄	C₄	1	2	yes	complemented
10	D₃₂	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	#368	C₂²	1	2	yes	—
14	#359	—	2	2	no	—
15	C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#135	—	2	2	no	—
17	#131	—	4	2	no	—
18	Q₃₂	C₂×C₄	1	2	yes	—
19	SD₃₂	—	2	2	no	—
20	D₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	#108	—	2	2	no	—
24	#104	—	4	2	no	—
25	C₄×C₈	D₄	1	1	yes	characteristic
26	C₃₂	—	2	1	no	—
27	C₃₂	C₂×C₄	1	1	yes	characteristic
28	C₃₂	C₂×C₄	1	1	yes	characteristic
29	#41	—	4	2	no	—
30	Q₁₆	—	2	2	no	—
31	SD₁₆	—	4	2	no	—
32	D₈	—	2	2	no	—
33	M₁₆	—	4	2	no	—
34	M₁₆	—	8	2	no	—
35	C₂×C₈	—	8	2	no	—
36	C₂×C₈	D₈	1	1	yes	characteristic
37	C₂×C₈	C₂×D₄	1	1	yes	characteristic
38	C₄²	D₈	1	1	yes	characteristic
39	C₁₆	—	2	1	no	—
40	C₁₆	C₂²×C₄	1	1	yes	characteristic
41	C₁₆	#41	1	1	yes	characteristic
42	Q₈	—	4	2	no	—
43	D₄	—	4	2	no	—
44	D₄	—	8	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	8	2	no	—
47	C₂×C₄	C₂×D₈	1	1	yes	characteristic
48	C₈	—	8	2	no	—
49	C₈	D₁₆	1	2	yes	—
50	C₈	C₄×D₄	1	1	yes	characteristic
51	C₈	#135	1	1	yes	characteristic
52	C₂²	—	8	2	no	—
53	C₂²	#508	1	1	yes	characteristic
54	C₄	—	4	1	no	—
55	C₄	—	8	2	no	—
56	C₄	C₄×D₈	1	1	yes	characteristic
57	C₄	C₂×D₁₆	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	16	2	no	—
60	C₂	C₄×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	C₄×D₁₆
prime radical	#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2178	C₂	semidirect
3. #1425	C₂	semidirect
4. #1417	C₂	semidirect
5. #1397	C₂	semidirect
8. Q₆₄	C₄	semidirect
9. SD₆₄	C₄	semidirect
10. D₃₂	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2262	2
#2178	2
#1425	2
#1417	2
#1397	2