Subgroups — #11410

Order 256.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	C₂	1	2	yes	maximal
2	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
3	#1312	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆	C₄	1	4	yes	—
9	C₂²⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₁₆	C₂²	1	4	yes	—
11	C₈²	C₄	1	1	yes	characteristic
12	C₈²	C₄	1	1	yes	characteristic
13	C₈²	C₂²	1	1	yes	characteristic
14	C₈²	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₈	1	2	yes	—
16	C₂²×C₈	C₈	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₄	1	8	yes	—
20	C₄×C₈	C₈	1	2	yes	—
21	C₄×C₈	C₈	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₄	C₄²	1	1	yes	characteristic
30	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈	—	2	4	no	—
33	C₂×C₈	—	2	4	no	—
34	C₂×C₈	C₁₆	1	8	yes	complemented
35	C₂×C₈	C₄²	1	2	yes	—
36	C₂×C₈	C₂×C₈	1	2	yes	—
37	C₂×C₈	C₂×C₈	1	2	yes	—
38	C₂×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₈	C₂×C₈	1	2	yes	—
40	C₂×C₈	C₂²×C₄	1	2	yes	—
41	C₄²	C₄²	1	1	yes	characteristic
42	C₄²	C₂×C₈	1	1	yes	characteristic
43	C₄²	C₂×C₈	1	1	yes	characteristic
44	C₄²	C₂²×C₄	1	1	yes	characteristic
45	C₁₆	—	2	8	no	—
46	C₂³	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄	C₂×C₁₆	1	4	yes	—
53	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
56	C₈	—	2	2	no	—
57	C₈	—	2	2	no	—
58	C₈	C₂×C₁₆	1	4	yes	—
59	C₈	C₂×C₁₆	1	4	yes	—
60	C₈	M₃₂	1	4	yes	—
61	C₈	#131	1	4	yes	—
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	C₄×C₁₆	1	2	yes	—
65	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	2	1	no	—
68	C₄	C₄×C₁₆	1	2	yes	—
69	C₄	#405	1	2	yes	—
70	C₄	C₂²×C₁₆	1	2	yes	—
71	C₄	C₂×M₃₂	1	2	yes	—
72	C₂	—	2	2	no	—
73	C₂	#1450	1	1	yes	characteristic
74	C₂	C₂×C₄×C₁₆	1	1	yes	characteristic
75	C₂	C₄×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈²	C₂²
commutator	C₂	C₂×C₄×C₁₆
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×C₄×C₈
prime radical	C₂³	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1312	C₂	semidirect
4. C₈×C₁₆	C₂	semidirect
34. C₂×C₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
C₂×C₈²	2
#1312	2
C₈×C₁₆	2