Subgroups — #8789

Order 256.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1842	C₂	1	1	yes	maximal, characteristic
2	#1831	C₂	1	1	yes	maximal, characteristic
3	#1744	C₂	1	1	yes	maximal, characteristic
4	#1277	C₂	1	4	yes	maximal
5	C₂×C₄⋊C₈	—	2	2	no	—
6	C₂×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	2	yes	—
9	C₄³	C₂²	1	1	yes	characteristic
10	#336	—	2	2	no	—
11	#336	C₄	1	2	yes	—
12	#336	C₄	1	2	yes	—
13	C₂²×C₈	—	2	2	no	—
14	C₂²×C₈	—	2	2	no	—
15	C₂²×C₈	C₂×C₄	1	4	yes	—
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²	D₄	1	2	yes	—
19	C₂×C₄²	Q₈	1	2	yes	—
20	C₂×C₄²	C₂³	1	1	yes	characteristic
21	C₄⋊C₈	—	2	4	no	—
22	C₄⋊C₈	—	2	4	no	—
23	C₄⋊C₈	—	4	4	no	—
24	C₄×C₈	—	2	2	no	—
25	C₄×C₈	—	2	2	no	—
26	C₂²×C₄	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈	—	2	4	no	—
32	C₂×C₈	—	2	4	no	—
33	C₂×C₈	—	2	4	no	—
34	C₂×C₈	—	4	2	no	—
35	C₂×C₈	—	4	2	no	—
36	C₂×C₈	—	4	4	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	—	4	2	no	—
39	C₄²	—	2	4	no	—
40	C₄²	—	2	2	no	—
41	C₄²	—	2	2	no	—
42	C₄²	—	2	2	no	—
43	C₄²	—	2	2	no	—
44	C₄²	#41	1	2	yes	—
45	C₄²	#41	1	2	yes	—
46	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	#108	1	4	yes	—
54	C₂×C₄	#117	1	2	yes	—
55	C₂×C₄	#124	1	1	yes	characteristic
56	C₂×C₄	#124	1	1	yes	characteristic
57	C₂×C₄	#125	1	2	yes	—
58	C₂×C₄	#131	1	2	yes	—
59	C₂×C₄	#131	1	2	yes	—
60	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄	#137	1	2	yes	—
62	C₈	—	4	4	no	—
63	C₈	—	8	2	no	—
64	C₈	—	8	2	no	—
65	C₂²	#382	1	1	yes	characteristic
66	C₂²	#424	1	1	yes	characteristic
67	C₂²	#424	1	1	yes	characteristic
68	C₂²	#428	1	2	yes	—
69	C₂²	C₂×#108	1	2	yes	—
70	C₄	—	2	4	no	—
71	C₄	—	2	2	no	—
72	C₄	—	4	2	no	—
73	C₄	#433	1	4	yes	—
74	C₄	#489	1	2	yes	—
75	C₄	#490	1	1	yes	characteristic
76	C₄	#491	1	1	yes	characteristic
77	C₂	#1573	1	2	yes	—
78	C₂	#1574	1	2	yes	—
79	C₂	#1832	1	1	yes	characteristic
80	C₂	#1836	1	1	yes	characteristic
81	C₂	#1837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#1842	2
#1831	2
#1744	2
#1277	2