Subgroups — #8137

Order 256.

177 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1726	C₂	1	1	yes	maximal, characteristic
2	#1450	C₂	1	2	yes	maximal
3	#1275	C₂	1	2	yes	maximal
4	#1274	C₂	1	2	yes	maximal
5	#431	C₂²	1	4	yes	—
6	C₂×#97	C₄	1	2	yes	—
7	C₂×#97	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#376	C₂²	1	1	yes	characteristic
11	#322	C₄	1	4	yes	complemented
12	C₈²	C₄	1	4	yes	complemented
13	C₂²×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂³	1	1	yes	characteristic
18	C₄⋊C₈	—	2	4	no	—
19	C₂²⋊C₈	—	2	4	no	—
20	#97	—	2	4	no	—
21	#97	—	2	2	no	—
22	C₄×C₈	C₈	1	8	yes	complemented
23	C₄×C₈	C₂×C₄	1	8	yes	—
24	C₄×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₈	C₂×C₄	1	2	yes	—
26	#95	—	2	2	no	—
27	C₂²×C₄	—	2	2	no	—
28	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄	#41	1	2	yes	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	8	no	—
33	C₂×C₈	C₄²	1	1	yes	characteristic
34	C₂×C₈	C₄²	1	2	yes	—
35	C₂×C₈	C₄²	1	1	yes	characteristic
36	C₂×C₈	C₂×C₈	1	4	yes	—
37	C₂×C₈	C₂×C₈	1	4	yes	—
38	C₂×C₈	M₁₆	1	4	yes	—
39	C₄²	—	2	2	no	—
40	C₄²	C₂×C₈	1	4	yes	—
41	C₄²	C₂²×C₄	1	2	yes	—
42	C₄²	#41	1	2	yes	—
43	C₂³	#117	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	—	4	4	no	—
47	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄	#117	1	4	yes	—
49	C₂×C₄	#117	1	1	yes	characteristic
50	C₂×C₄	C₂²×C₈	1	2	yes	—
51	C₂×C₄	C₂×M₁₆	1	2	yes	—
52	C₈	—	2	4	no	—
53	C₈	—	4	8	no	—
54	C₈	C₄×C₈	1	2	yes	—
55	C₈	C₄×C₈	1	2	yes	—
56	C₈	#97	1	4	yes	—
57	C₂²	—	2	2	no	—
58	C₂²	#376	1	1	yes	characteristic
59	C₂²	#431	1	2	yes	—
60	C₄	—	2	4	no	—
61	C₄	—	4	4	no	—
62	C₄	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄	C₂×#97	1	1	yes	characteristic
64	C₄	#431	1	2	yes	—
65	C₂	—	2	2	no	—
66	C₂	#1744	1	1	yes	characteristic
67	C₂	#1747	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#376
prime radical	C₂³	#117
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×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
22. 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
#1726	2
#1450	2
#1275	2
#1274	2