Subgroups — #7841

Order 256.

287 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1744	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#322	C₂	1	2	yes	maximal
4	C₂×C₈²	C₂	1	2	yes	maximal
5	C₂×C₄×C₈	C₄	1	4	yes	—
6	C₂×C₄×C₈	C₂²	1	4	yes	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₂²	1	1	yes	characteristic
10	#336	C₄	1	8	yes	—
11	#322	C₄	1	8	yes	complemented
12	C₈²	C₄	1	8	yes	complemented
13	C₂²×C₈	C₂×C₄	1	8	yes	—
14	C₂²×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄²	C₂×C₄	1	4	yes	—
17	C₂×C₄²	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂³	1	1	yes	characteristic
19	C₄×C₈	C₈	1	16	yes	complemented
20	C₄×C₈	C₂×C₄	1	8	yes	—
21	C₄×C₈	C₂×C₄	1	4	yes	—
22	C₄×C₈	C₂×C₄	1	8	yes	—
23	C₄×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄	C₄²	1	4	yes	—
25	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄	C₂²×C₄	1	2	yes	—
27	C₂×C₈	—	2	16	no	—
28	C₂×C₈	C₄²	1	16	yes	—
29	C₂×C₈	C₄²	1	4	yes	—
30	C₂×C₈	C₄²	1	4	yes	—
31	C₂×C₈	C₂×C₈	1	8	yes	—
32	C₂×C₈	C₂×C₈	1	8	yes	—
33	C₄²	—	2	8	no	—
34	C₄²	C₂×C₈	1	8	yes	—
35	C₄²	C₂²×C₄	1	2	yes	—
36	C₄²	C₂²×C₄	1	2	yes	—
37	C₂³	C₂×C₄²	1	1	yes	characteristic
38	C₂×C₄	—	2	8	no	—
39	C₂×C₄	C₄×C₈	1	8	yes	—
40	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄	C₂×C₄²	1	4	yes	—
43	C₂×C₄	C₂²×C₈	1	4	yes	—
44	C₂×C₄	#131	1	8	yes	—
45	C₈	—	2	16	no	—
46	C₈	C₄×C₈	1	8	yes	—
47	C₈	C₄×C₈	1	8	yes	—
48	C₂²	C₄³	1	1	yes	characteristic
49	C₂²	C₂×C₄×C₈	1	2	yes	—
50	C₂²	#405	1	4	yes	—
51	C₄	—	2	8	no	—
52	C₄	C₂×C₄×C₈	1	2	yes	—
53	C₄	C₂×C₄×C₈	1	2	yes	—
54	C₄	#405	1	8	yes	—
55	C₂	#1450	1	4	yes	—
56	C₂	C₄²×C₈	1	1	yes	characteristic
57	C₂	#1726	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₄²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #322	C₄	semidirect
12. C₈²	C₄	semidirect
19. C₄×C₈	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1744	2
C₄²×C₈	2
C₂×#322	2
C₂×C₈²	2