Subgroups — #33497

Order 256.

243 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	8	yes	maximal, complemented
3	#2163	C₂	1	4	yes	maximal
4	#2162	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₈×Q₈	C₄	1	4	yes	—
7	C₈×Q₈	C₂²	1	4	yes	—
8	C₂×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	C₄⋊C₁₆	C₂²	1	4	yes	—
13	C₄⋊C₁₆	C₂²	1	8	yes	—
14	C₂²⋊C₁₆	C₂²	1	4	yes	—
15	#346	C₂²	1	8	yes	—
16	C₄×C₁₆	C₂²	1	4	yes	—
17	C₂²×Q₈	C₈	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	C₄×Q₈	C₈	1	4	yes	—
22	C₄×Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄	C₈	1	2	yes	—
24	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
25	C₂×C₄²	C₂×C₄	1	2	yes	—
26	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆	C₂³	1	8	yes	—
28	C₄⋊C₈	C₂×C₄	1	8	yes	—
29	C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₄×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₈	C₂×C₄	1	4	yes	—
32	C₄×C₈	C₂³	1	2	yes	—
33	C₄×C₈	C₂³	1	4	yes	—
34	C₂×Q₈	—	2	4	no	—
35	C₂×Q₈	C₂×C₈	1	4	yes	—
36	C₂²×C₄	C₂×C₈	1	2	yes	—
37	C₂²×C₄	C₂×C₈	1	2	yes	—
38	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂²×C₄	1	2	yes	—
46	C₂×C₈	C₂²×C₄	1	4	yes	—
47	C₂×C₈	C₂⁴	1	1	yes	characteristic
48	C₄⋊C₄	C₂×C₈	1	8	yes	—
49	C₄⋊C₄	C₂×C₈	1	4	yes	—
50	C₄²	C₂×C₈	1	4	yes	—
51	C₄²	C₂×C₈	1	2	yes	—
52	C₄²	C₂²×C₄	1	4	yes	—
53	C₄²	C₂²×C₄	1	2	yes	—
54	C₁₆	—	2	8	no	—
55	C₂³	C₂²×C₈	1	1	yes	characteristic
56	Q₈	—	2	8	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	M₃₂	1	4	yes	—
63	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄	C₂²×C₈	1	4	yes	—
65	C₂×C₄	C₂²×C₈	1	2	yes	—
66	C₂×C₄	C₂²×C₈	1	4	yes	—
67	C₂×C₄	C₂²×C₈	1	2	yes	—
68	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	2	2	no	—
71	C₈	—	2	4	no	—
72	C₈	2₋¹⁺⁴	1	2	yes	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	C₂×M₃₂	1	2	yes	—
76	C₂²	C₂³×C₈	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	2	4	no	—
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	C₂×M₃₂	1	4	yes	—
82	C₄	#519	1	1	yes	characteristic
83	C₄	#568	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#2982	1	1	yes	characteristic
86	C₂	C₂²×M₃₂	1	1	yes	characteristic
87	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2183	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈	2
#2183	2
#2163	2
#2162	2