Subgroups — #12426

Order 256.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2208	C₂	1	2	yes	maximal
2	#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈	C₂	1	1	yes	maximal, characteristic
4	#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	#1362	C₂	1	2	yes	maximal, complemented
6	C₂×SD₃₂	—	2	2	no	—
7	#500	—	2	1	no	—
8	#500	—	2	1	no	—
9	#494	C₂²	1	2	yes	—
10	C₄×D₈	C₂²	1	1	yes	characteristic
11	C₈×D₄	—	2	1	no	—
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	#358	—	2	2	no	—
14	#335	C₂²	1	1	yes	characteristic
15	D₄⋊C₈	—	2	1	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×D₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	—	4	1	no	—
21	C₄⋊Q₈	—	2	2	no	—
22	C₄×D₄	—	2	1	no	—
23	SD₃₂	—	4	4	no	—
24	C₂×C₁₆	—	2	2	no	—
25	#107	—	4	2	no	—
26	#107	—	4	1	no	—
27	#107	—	4	1	no	—
28	#107	D₄	1	1	yes	characteristic
29	C₄⋊C₈	—	2	1	no	—
30	#102	—	2	1	no	—
31	C₂²⋊C₈	—	4	1	no	—
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂×Q₈	—	4	2	no	—
36	C₂×D₄	—	2	1	no	—
37	C₂²×C₄	—	4	1	no	—
38	Q₁₆	—	4	2	no	—
39	Q₁₆	—	4	2	no	—
40	Q₁₆	—	8	2	no	—
41	D₈	—	2	1	no	—
42	D₈	D₈	1	2	yes	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	4	1	no	—
45	C₂×C₈	—	8	2	no	—
46	C₂×C₈	D₈	1	1	yes	characteristic
47	C₂×C₈	D₈	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	#41	1	1	yes	characteristic
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄⋊C₄	—	4	2	no	—
53	C₂²⋊C₄	—	4	1	no	—
54	C₄²	C₂×D₄	1	1	yes	characteristic
55	C₁₆	—	4	2	no	—
56	C₂³	—	4	1	no	—
57	Q₈	—	8	2	no	—
58	Q₈	—	8	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	8	1	no	—
63	C₂×C₄	—	8	1	no	—
64	C₂×C₄	—	8	2	no	—
65	C₂×C₄	#121	1	1	yes	characteristic
66	C₂×C₄	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	—	8	1	no	—
72	C₈	SD₃₂	1	2	yes	complemented
73	C₈	C₂×D₈	1	1	yes	characteristic
74	C₈	#135	1	1	yes	characteristic
75	C₂²	—	4	2	no	—
76	C₂²	—	8	1	no	—
77	C₂²	—	8	1	no	—
78	C₂²	#459	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	8	1	no	—
81	C₄	—	16	2	no	—
82	C₄	#466	1	1	yes	characteristic
83	C₄	#472	1	1	yes	characteristic
84	C₄	C₂×SD₃₂	1	1	yes	characteristic
85	C₄	#510	1	1	yes	characteristic
86	C₂	—	8	2	no	—
87	C₂	#1667	1	1	yes	characteristic
88	C₂	#2208	1	1	yes	characteristic
89	C₂	#2214	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1713	C₂	semidirect
4. #1370	C₂	semidirect
5. #1362	C₂	semidirect
72. C₈	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2208	2
#1713	2
C₈×D₈	2
#1370	2
#1362	2