Subgroups — #7071

Order 256.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#1367	C₂	1	1	yes	maximal, characteristic
3	#1312	C₂	1	1	yes	maximal, characteristic
4	C₈×Q₈	—	2	1	no	—
5	C₄×Q₁₆	C₄	1	1	yes	characteristic
6	C₄⋊C₁₆	—	2	1	no	—
7	C₄×C₁₆	—	2	1	no	—
8	#335	C₄	1	1	yes	characteristic
9	Q₈⋊C₈	—	2	1	no	—
10	C₈²	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆	C₈	1	1	yes	characteristic
12	C₄×Q₈	—	2	1	no	—
13	C₂×C₁₆	—	2	2	no	—
14	C₂×C₁₆	—	4	1	no	—
15	#107	C₈	1	1	yes	characteristic
16	C₄⋊C₈	—	2	1	no	—
17	C₄⋊C₈	—	4	1	no	—
18	#103	—	2	1	no	—
19	C₄×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×Q₈	—	2	1	no	—
24	Q₁₆	—	2	1	no	—
25	Q₁₆	C₁₆	1	2	yes	complemented
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	—	4	1	no	—
28	C₂×C₈	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈	M₁₆	1	1	yes	characteristic
30	C₂×C₈	D₈	1	1	yes	characteristic
31	C₂×C₈	SD₁₆	1	1	yes	characteristic
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	—	4	1	no	—
34	C₄²	—	4	1	no	—
35	C₄²	C₂²⋊C₄	1	1	yes	characteristic
36	C₁₆	—	4	2	no	—
37	C₁₆	—	8	1	no	—
38	Q₈	—	2	2	no	—
39	Q₈	—	4	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄	#102	1	1	yes	characteristic
44	C₂×C₄	#104	1	1	yes	characteristic
45	C₈	—	2	2	no	—
46	C₈	—	2	1	no	—
47	C₈	—	2	1	no	—
48	C₈	—	2	1	no	—
49	C₈	—	8	1	no	—
50	C₈	C₂×C₁₆	1	1	yes	characteristic
51	C₈	M₃₂	1	1	yes	characteristic
52	C₂²	D₄⋊C₈	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	2	no	—
55	C₄	—	8	1	no	—
56	C₄	C₂²⋊C₁₆	1	1	yes	characteristic
57	C₄	#350	1	1	yes	characteristic
58	C₄	#360	1	1	yes	characteristic
59	C₄	#362	1	1	yes	characteristic
60	C₂	D₄⋊C₁₆	1	1	yes	characteristic
61	C₂	#1334	1	1	yes	characteristic
62	C₂	#1336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#102
commutator	C₈	C₂×C₁₆
Frattini, cosocle	C₈²	C₂²
socle, prime radical	C₂²	D₄⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
25. Q₁₆	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₁₆	2
#1367	2
#1312	2