Subgroups — #12590

Order 256.

207 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆	C₂²	1	4	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₄	1	1	yes	characteristic
10	C₄×C₁₆	C₄	1	8	yes	complemented
11	C₂²×C₈	C₂×C₄	1	2	yes	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²	C₈	1	2	yes	—
15	C₂×C₄²	C₈	1	2	yes	—
16	C₂×C₄²	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆	—	2	8	no	—
19	C₂×C₁₆	C₂×C₄	1	8	yes	—
20	C₄×C₈	—	2	4	no	—
21	C₄×C₈	—	2	4	no	—
22	C₄×C₈	C₈	1	4	yes	—
23	C₄×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄	C₂×C₈	1	4	yes	—
25	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	C₄²	1	2	yes	—
31	C₂×C₈	C₂×C₈	1	4	yes	—
32	C₂×C₈	M₁₆	1	4	yes	—
33	C₂×C₈	C₂²×C₄	1	2	yes	—
34	C₂×C₈	#41	1	4	yes	—
35	C₄²	—	2	4	no	—
36	C₄²	—	2	2	no	—
37	C₄²	—	2	2	no	—
38	C₄²	C₁₆	1	8	yes	complemented
39	C₄²	C₄²	1	2	yes	—
40	C₄²	C₂×C₈	1	2	yes	—
41	C₁₆	—	2	8	no	—
42	C₂³	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₄×C₈	1	2	yes	—
46	C₂×C₄	C₄×C₈	1	2	yes	—
47	C₂×C₄	C₂×C₁₆	1	8	yes	—
48	C₂×C₄	C₂×C₁₆	1	4	yes	—
49	C₂×C₄	M₃₂	1	4	yes	—
50	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄	#117	1	2	yes	—
52	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄	C₂×M₁₆	1	2	yes	—
54	C₈	—	2	4	no	—
55	C₈	#97	1	4	yes	—
56	C₈	#117	1	4	yes	—
57	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
58	C₂²	#431	1	2	yes	—
59	C₂²	C₂²×C₁₆	1	2	yes	—
60	C₂²	C₂×M₃₂	1	2	yes	—
61	C₄	—	2	8	no	—
62	C₄	C₄×C₁₆	1	4	yes	—
63	C₄	#346	1	4	yes	—
64	C₄	#376	1	1	yes	characteristic
65	C₄	C₂×#97	1	1	yes	characteristic
66	C₄	#431	1	2	yes	—
67	C₂	#1744	1	1	yes	characteristic
68	C₂	C₂×C₄×C₁₆	1	1	yes	characteristic
69	C₂	C₂×#346	1	1	yes	characteristic
70	C₂	#2162	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³
socle, prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆	C₄	semidirect
38. C₄²	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆	2
C₄²×C₈	2
#1374	2