Subgroups — #7144

Order 256.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2163	C₂	1	1	yes	maximal, characteristic
2	#2162	C₂	1	1	yes	maximal, characteristic
3	#1746	C₂	1	1	yes	maximal, characteristic
4	C₂×#97	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	#376	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆	—	2	1	no	—
8	C₄⋊C₁₆	—	2	1	no	—
9	C₂²⋊C₁₆	—	2	1	no	—
10	C₂²⋊C₁₆	—	2	1	no	—
11	#346	C₄	1	2	yes	complemented
12	C₄×C₁₆	C₄	1	2	yes	complemented
13	#336	—	2	1	no	—
14	#336	—	2	1	no	—
15	C₂²×C₈	—	2	1	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄²	—	2	1	no	—
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	#97	—	2	1	no	—
23	#97	C₈	1	2	yes	—
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₄×C₈	Q₈	1	1	yes	characteristic
28	#95	—	2	1	no	—
29	#95	C₈	1	2	yes	—
30	C₂²×C₄	—	2	2	no	—
31	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄	M₁₆	1	1	yes	characteristic
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	—	4	1	no	—
36	C₂×C₈	—	4	1	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₄²	—	4	1	no	—
44	C₄²	—	4	1	no	—
45	C₄²	C₄²	1	1	yes	characteristic
46	C₄²	C₄⋊C₄	1	1	yes	characteristic
47	C₄²	C₂×C₈	1	1	yes	characteristic
48	C₄²	M₁₆	1	1	yes	characteristic
49	C₁₆	—	4	2	no	—
50	C₁₆	—	4	2	no	—
51	C₂³	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	#95	1	1	yes	characteristic
60	C₂×C₄	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄	#97	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	2	1	no	—
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₈	—	4	2	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	#336	1	1	yes	characteristic
73	C₂²	#350	1	2	yes	—
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	8	2	no	—
80	C₄	H(3,ℤ/4)	1	1	yes	characteristic
81	C₄	H(3,ℤ/4)	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#1274	1	1	yes	characteristic
84	C₂	#1378	1	1	yes	characteristic
85	C₂	#1379	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#95
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #346	C₄	semidirect
12. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2163	2
#2162	2
#1746	2