Subgroups — #33496

Order 256.

263 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	Q₈×C₁₆	C₂	1	8	yes	maximal, complemented
3	#2162	C₂	1	6	yes	maximal
4	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
5	C₈×Q₈	C₄	1	4	yes	—
6	C₈×Q₈	C₂²	1	4	yes	—
7	C₂×C₄⋊C₈	C₄	1	3	yes	—
8	C₂×C₄×C₈	C₂²	1	3	yes	—
9	C₄⋊C₁₆	C₂²	1	12	yes	—
10	C₂²⋊C₁₆	C₂²	1	4	yes	—
11	C₄×C₁₆	C₂²	1	12	yes	—
12	C₂²×Q₈	C₈	1	1	yes	characteristic
13	C₂²×C₈	C₂×C₄	1	3	yes	—
14	C₂²×C₈	C₂³	1	1	yes	characteristic
15	C₄×Q₈	C₈	1	4	yes	—
16	C₄×Q₈	C₂×C₄	1	4	yes	—
17	C₂×C₄⋊C₄	C₈	1	3	yes	—
18	C₂×C₄²	C₂×C₄	1	3	yes	—
19	C₂×C₁₆	C₂³	1	8	yes	—
20	C₄⋊C₈	C₂×C₄	1	12	yes	—
21	C₄×C₈	C₂×C₄	1	6	yes	—
22	C₄×C₈	C₂³	1	6	yes	—
23	C₂×Q₈	C₁₆	1	8	yes	complemented
24	C₂×Q₈	C₂×C₈	1	4	yes	—
25	C₂²×C₄	C₂×C₈	1	3	yes	—
26	C₂²×C₄	C₂×C₈	1	3	yes	—
27	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	4	3	no	—
30	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈	C₂²×C₄	1	6	yes	—
32	C₂×C₈	C₂⁴	1	1	yes	characteristic
33	C₄⋊C₄	C₂×C₈	1	12	yes	—
34	C₄²	C₂×C₈	1	6	yes	—
35	C₄²	C₂²×C₄	1	6	yes	—
36	C₁₆	—	2	8	no	—
37	C₂³	C₂²×C₈	1	1	yes	characteristic
38	Q₈	C₂×C₁₆	1	16	yes	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	4	3	no	—
42	C₂×C₄	C₂×C₁₆	1	12	yes	—
43	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₈	1	6	yes	—
45	C₂×C₄	C₂²×C₈	1	6	yes	—
46	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
47	C₈	—	2	1	no	—
48	C₈	—	2	6	no	—
49	C₈	2₋¹⁺⁴	1	2	yes	—
50	C₂²	—	2	1	no	—
51	C₂²	—	2	1	no	—
52	C₂²	C₂²×C₁₆	1	2	yes	—
53	C₂²	C₂³×C₈	1	1	yes	characteristic
54	C₄	—	2	6	no	—
55	C₄	—	2	1	no	—
56	C₄	C₂²×C₁₆	1	12	yes	—
57	C₄	#519	1	1	yes	characteristic
58	C₄	#568	1	1	yes	characteristic
59	C₂	—	2	2	no	—
60	C₂	#2982	1	1	yes	characteristic
61	C₂	C₂³×C₁₆	1	1	yes	characteristic
62	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator	C₂	C₂³×C₁₆
socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂ ▷ 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. Q₈×C₁₆	C₂	semidirect
23. C₂×Q₈	C₁₆	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
Q₈×C₁₆	2
#2162	2