Subgroups — #20569

Order 256.

495 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄	C₂	1	6	yes	maximal
4	C₄×C₄⋊C₈	C₂	1	3	yes	maximal
5	#1744	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	C₄	1	2	yes	—
7	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
8	C₈×Q₈	C₄	1	16	yes	complemented
9	C₂×C₄⋊C₈	C₂²	1	6	yes	—
10	C₂×C₄×C₈	C₂²	1	6	yes	—
11	C₂×C₄×C₈	C₂²	1	6	yes	—
12	C₄×C₄⋊C₄	C₄	1	6	yes	—
13	C₄×C₄⋊C₄	C₂²	1	3	yes	—
14	C₄³	C₂²	1	3	yes	—
15	#336	C₂²	1	1	yes	characteristic
16	#336	C₂²	1	3	yes	—
17	#336	C₂²	1	6	yes	—
18	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	2	yes	—
20	C₂²×C₈	C₂³	1	6	yes	—
21	C₄×Q₈	C₈	1	16	yes	complemented
22	C₄×Q₈	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
24	C₂×C₄⋊C₄	C₂×C₄	1	6	yes	—
25	C₂×C₄²	C₂×C₄	1	6	yes	—
26	C₂×C₄²	C₂×C₄	1	6	yes	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₄²	C₂³	1	3	yes	—
29	C₂×C₄²	C₂³	1	3	yes	—
30	C₄⋊C₈	C₂×C₄	1	24	yes	—
31	C₄×C₈	—	4	6	no	—
32	C₄×C₈	C₂×C₄	1	24	yes	—
33	#95	C₂×C₄	1	6	yes	—
34	C₂×Q₈	C₄²	1	4	yes	—
35	C₂×Q₈	C₂×C₈	1	8	yes	—
36	C₂²×C₄	C₂²×C₄	1	3	yes	—
37	C₂²×C₄	C₂²×C₄	1	6	yes	—
38	C₂²×C₄	C₂²×C₄	1	2	yes	—
39	C₂²×C₄	C₂²×C₄	1	3	yes	—
40	C₂²×C₄	C₂⁴	1	1	yes	characteristic
41	C₂×C₈	—	2	6	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	6	no	—
45	C₂×C₈	C₂²×C₄	1	4	yes	—
46	C₂×C₈	C₂²×C₄	1	12	yes	—
47	C₄⋊C₄	C₄²	1	12	yes	—
48	C₄⋊C₄	C₂×C₈	1	24	yes	—
49	C₄²	—	2	3	no	—
50	C₄²	—	2	2	no	—
51	C₄²	—	2	3	no	—
52	C₄²	—	4	6	no	—
53	C₄²	C₂×C₈	1	24	yes	—
54	C₄²	C₂²×C₄	1	12	yes	—
55	C₂³	C₂³×C₄	1	1	yes	characteristic
56	Q₈	C₄×C₈	1	16	yes	complemented
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	6	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	C₂×C₄²	1	6	yes	—
63	C₂×C₄	C₂×C₄²	1	6	yes	—
64	C₂×C₄	C₂²×C₈	1	4	yes	—
65	C₂×C₄	C₂²×C₈	1	12	yes	—
66	C₂×C₄	C₂²×C₈	1	12	yes	—
67	C₂×C₄	C₂³×C₄	1	2	yes	—
68	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
71	C₈	—	2	4	no	—
72	C₈	—	2	12	no	—
73	C₂²	C₂²×C₄²	1	1	yes	characteristic
74	C₂²	#519	1	1	yes	characteristic
75	C₂²	#520	1	1	yes	characteristic
76	C₂²	C₂³×C₈	1	2	yes	—
77	C₂²	#568	1	2	yes	—
78	C₄	—	2	6	no	—
79	C₄	—	2	4	no	—
80	C₄	—	2	12	no	—
81	C₄	C₂×C₄×C₈	1	12	yes	—
82	C₄	#519	1	2	yes	—
83	C₄	#520	1	2	yes	—
84	C₂	#2276	1	1	yes	characteristic
85	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
86	C₂	#2875	1	1	yes	characteristic
87	C₂	#2982	1	2	yes	—
88	C₂	#2988	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈	C₄	semidirect
21. C₄×Q₈	C₈	semidirect
56. Q₈	C₄×C₈	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
C₄²×Q₈	2
C₈×C₄⋊C₄	2
C₄×C₄⋊C₈	2
#1744	2