Subgroups — #7068

Order 256.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₈	C₂	1	1	yes	maximal, characteristic
2	#1370	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₈	C₄	1	1	yes	characteristic
5	C₈×D₄	—	2	1	no	—
6	C₄⋊C₁₆	—	2	1	no	—
7	C₄×C₁₆	—	2	1	no	—
8	#335	C₄	1	1	yes	characteristic
9	D₄⋊C₈	—	2	1	no	—
10	C₈²	C₂²	1	1	yes	characteristic
11	C₂×D₈	C₈	1	1	yes	characteristic
12	C₂²×C₈	—	4	1	no	—
13	C₄×D₄	—	2	1	no	—
14	C₂×C₁₆	—	2	2	no	—
15	C₂×C₁₆	—	4	1	no	—
16	#107	C₈	1	1	yes	characteristic
17	C₄⋊C₈	—	2	1	no	—
18	#102	—	2	1	no	—
19	C₂²⋊C₈	—	4	1	no	—
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈	D₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×D₄	—	2	1	no	—
24	C₂²×C₄	—	4	1	no	—
25	D₈	—	2	1	no	—
26	D₈	C₁₆	1	2	yes	complemented
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	4	1	no	—
29	C₂×C₈	—	8	1	no	—
30	C₂×C₈	—	8	1	no	—
31	C₂×C₈	C₂×C₈	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	1	yes	characteristic
35	C₄⋊C₄	—	2	1	no	—
36	C₂²⋊C₄	—	4	1	no	—
37	C₄²	C₂²⋊C₄	1	1	yes	characteristic
38	C₁₆	—	2	4	no	—
39	C₁₆	—	8	1	no	—
40	C₂³	—	4	1	no	—
41	D₄	—	2	2	no	—
42	D₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	8	1	no	—
45	C₂×C₄	—	8	1	no	—
46	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄	#102	1	1	yes	characteristic
48	C₂×C₄	#104	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	2	1	no	—
51	C₈	—	2	1	no	—
52	C₈	—	8	1	no	—
53	C₈	C₂×C₁₆	1	1	yes	characteristic
54	C₈	M₃₂	1	1	yes	characteristic
55	C₈	D₁₆	1	1	yes	characteristic
56	C₈	SD₃₂	1	1	yes	characteristic
57	C₂²	—	4	2	no	—
58	C₂²	—	8	1	no	—
59	C₂²	—	8	1	no	—
60	C₂²	D₄⋊C₈	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	8	1	no	—
63	C₄	C₂²⋊C₁₆	1	1	yes	characteristic
64	C₄	#350	1	1	yes	characteristic
65	C₄	#357	1	1	yes	characteristic
66	C₄	#359	1	1	yes	characteristic
67	C₂	—	8	2	no	—
68	C₂	D₄⋊C₁₆	1	1	yes	characteristic
69	C₂	#1331	1	1	yes	characteristic
70	C₂	#1335	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₈
commutator	C₈	C₂×C₁₆
Frattini, cosocle	C₈²	C₂²
socle	C₂²	D₄⋊C₈
prime radical	C₂×D₈	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1370	C₂	semidirect
3. C₈×C₁₆	C₂	semidirect
26. D₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₈	2
#1370	2
C₈×C₁₆	2