Subgroups — #33498

Order 256.

247 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	6	yes	maximal, complemented
3	Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
4	#2163	C₂	1	3	yes	maximal
5	#2162	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₈×Q₈	C₄	1	3	yes	—
8	C₈×Q₈	C₄	1	1	yes	characteristic
9	C₈×Q₈	C₂²	1	1	yes	characteristic
10	C₈×Q₈	C₂²	1	3	yes	—
11	C₂×C₄⋊C₈	C₄	1	3	yes	—
12	C₂×C₄×C₈	C₂²	1	3	yes	—
13	C₄⋊C₁₆	C₂²	1	6	yes	—
14	C₄⋊C₁₆	C₂²	1	6	yes	—
15	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²⋊C₁₆	C₂²	1	3	yes	—
17	#346	C₂²	1	6	yes	—
18	C₄×C₁₆	C₂²	1	6	yes	—
19	C₂²×Q₈	C₈	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	C₄×Q₈	C₈	1	3	yes	—
23	C₄×Q₈	C₈	1	1	yes	characteristic
24	C₄×Q₈	C₂×C₄	1	3	yes	—
25	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄	C₈	1	3	yes	—
27	C₂×C₄²	C₂×C₄	1	3	yes	—
28	C₂×C₁₆	C₂³	1	2	yes	—
29	C₂×C₁₆	C₂³	1	6	yes	—
30	C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₄×C₈	C₂×C₄	1	3	yes	—
34	C₄×C₈	C₂×C₄	1	3	yes	—
35	C₄×C₈	C₂³	1	3	yes	—
36	C₄×C₈	C₂³	1	3	yes	—
37	C₂×Q₈	—	2	4	no	—
38	C₂×Q₈	C₂×C₈	1	3	yes	—
39	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄	C₂×C₈	1	3	yes	—
41	C₂²×C₄	C₂×C₈	1	3	yes	—
42	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	4	3	no	—
45	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈	C₂²×C₄	1	3	yes	—
47	C₂×C₈	C₂²×C₄	1	3	yes	—
48	C₂×C₈	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄	C₂×C₈	1	6	yes	—
50	C₄⋊C₄	C₂×C₈	1	3	yes	—
51	C₄⋊C₄	C₂×C₈	1	3	yes	—
52	C₄²	C₂×C₈	1	3	yes	—
53	C₄²	C₂×C₈	1	3	yes	—
54	C₄²	C₂²×C₄	1	3	yes	—
55	C₄²	C₂²×C₄	1	3	yes	—
56	C₁₆	—	2	2	no	—
57	C₁₆	—	2	6	no	—
58	C₂³	C₂²×C₈	1	1	yes	characteristic
59	Q₈	—	2	6	no	—
60	Q₈	M₃₂	1	4	yes	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	6	no	—
65	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄	C₂²×C₈	1	3	yes	—
67	C₂×C₄	C₂²×C₈	1	3	yes	—
68	C₂×C₄	C₂²×C₈	1	3	yes	—
69	C₂×C₄	C₂²×C₈	1	3	yes	—
70	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	3	no	—
73	C₈	—	2	3	no	—
74	C₈	2₋¹⁺⁴	1	2	yes	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	#504	1	2	yes	—
78	C₂²	C₂³×C₈	1	1	yes	characteristic
79	C₄	—	2	3	no	—
80	C₄	—	2	3	no	—
81	C₄	—	2	3	no	—
82	C₄	—	2	1	no	—
83	C₄	C₂×M₃₂	1	6	yes	—
84	C₄	#519	1	1	yes	characteristic
85	C₄	#568	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	#2982	1	1	yes	characteristic
88	C₂	C₂²×M₃₂	1	1	yes	characteristic
89	C₂	C₂×#504	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
3. Q₈×C₁₆	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
Q₈×C₁₆	2
#2163	2
#2162	2