Subgroups — #13730

Order 256.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄	C₂	1	2	yes	maximal, complemented
2	C₂×SD₆₄	C₂	1	2	yes	maximal
3	#2249	C₂	1	1	yes	maximal, characteristic
4	#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂	—	2	1	no	—
7	C₂×Q₃₂	C₂²	1	2	yes	—
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×SD₃₂	—	2	1	no	—
10	C₂×D₁₆	C₂²	1	1	yes	characteristic
11	#495	—	2	1	no	—
12	#494	—	2	1	no	—
13	Q₆₄	—	2	4	no	—
14	SD₆₄	—	2	4	no	—
15	C₂×C₃₂	C₂²	1	2	yes	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	1	no	—
19	C₂×Q₁₆	—	4	1	no	—
20	C₂×SD₁₆	—	4	1	no	—
21	C₂×D₈	—	2	1	no	—
22	C₄⋊Q₈	—	4	1	no	—
23	#124	—	4	1	no	—
24	Q₃₂	—	2	2	no	—
25	Q₃₂	—	2	2	no	—
26	Q₃₂	—	2	2	no	—
27	Q₃₂	—	4	2	no	—
28	SD₃₂	—	4	2	no	—
29	D₁₆	—	2	2	no	—
30	C₂×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₆	C₂³	1	1	yes	characteristic
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₃₂	D₄	1	4	yes	—
34	C₂×Q₈	—	4	1	no	—
35	C₂×Q₈	—	4	2	no	—
36	C₂×D₄	—	4	1	no	—
37	Q₁₆	—	4	2	no	—
38	Q₁₆	—	4	1	no	—
39	Q₁₆	—	4	1	no	—
40	Q₁₆	—	4	2	no	—
41	Q₁₆	—	8	2	no	—
42	SD₁₆	—	8	2	no	—
43	D₈	—	4	1	no	—
44	D₈	—	4	1	no	—
45	C₂×C₈	D₈	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄	—	8	2	no	—
48	C₂²⋊C₄	—	8	2	no	—
49	C₄²	D₈	1	1	yes	characteristic
50	C₁₆	D₈	1	2	yes	—
51	C₁₆	C₂×D₄	1	2	yes	—
52	C₂³	—	8	1	no	—
53	Q₈	—	8	1	no	—
54	Q₈	—	8	2	no	—
55	Q₈	—	8	1	no	—
56	Q₈	—	8	2	no	—
57	D₄	—	8	1	no	—
58	D₄	—	8	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	D₁₆	1	2	yes	—
62	C₂×C₄	C₂×D₈	1	1	yes	characteristic
63	C₈	D₁₆	1	2	yes	—
64	C₈	#127	1	1	yes	characteristic
65	C₈	C₂×D₈	1	1	yes	characteristic
66	C₂²	—	16	1	no	—
67	C₂²	—	16	2	no	—
68	C₂²	C₂×D₁₆	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	16	1	no	—
71	C₄	—	16	2	no	—
72	C₄	#493	1	1	yes	characteristic
73	C₄	C₂×D₁₆	1	1	yes	characteristic
74	C₂	—	32	1	no	—
75	C₂	#2246	1	1	yes	characteristic
76	C₂	#2264	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×Q₆₄	C₂	semidirect
4. #2247	C₂	semidirect
5. #1397	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₆₄	2
C₂×SD₆₄	2
#2249	2
#2247	2
#1397	2