Subgroups — #8787

Order 256.

181 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1842	C₂	1	2	yes	maximal
2	#1744	C₂	1	1	yes	maximal, characteristic
3	#1277	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈	—	2	2	no	—
5	C₂×C₄⋊C₈	—	2	2	no	—
6	C₂×C₄⋊C₈	C₂²	1	4	yes	—
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₄³	C₂²	1	1	yes	characteristic
9	#336	C₄	1	4	yes	—
10	C₂²×C₈	—	2	4	no	—
11	C₂²×C₈	C₂×C₄	1	4	yes	—
12	C₂×C₄²	C₂×C₄	1	2	yes	—
13	C₂×C₄²	D₄	1	1	yes	characteristic
14	C₂×C₄²	D₄	1	1	yes	characteristic
15	C₂×C₄²	Q₈	1	2	yes	—
16	C₂×C₄²	C₂³	1	1	yes	characteristic
17	C₄⋊C₈	—	2	8	no	—
18	C₄⋊C₈	—	4	4	no	—
19	C₄⋊C₈	—	4	4	no	—
20	C₄×C₈	—	2	4	no	—
21	C₂²×C₄	C₄⋊C₄	1	2	yes	—
22	C₂²×C₄	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈	—	2	4	no	—
27	C₂×C₈	—	2	8	no	—
28	C₂×C₈	—	4	4	no	—
29	C₂×C₈	—	4	4	no	—
30	C₂×C₈	—	4	4	no	—
31	C₄²	—	2	4	no	—
32	C₄²	—	4	2	no	—
33	C₄²	D₈	1	2	yes	—
34	C₄²	SD₁₆	1	4	yes	—
35	C₄²	Q₁₆	1	2	yes	—
36	C₄²	#41	1	4	yes	—
37	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	#106	1	4	yes	—
44	C₂×C₄	#107	1	4	yes	—
45	C₂×C₄	#117	1	2	yes	—
46	C₂×C₄	#124	1	2	yes	—
47	C₂×C₄	#125	1	2	yes	—
48	C₂×C₄	#131	1	4	yes	—
49	C₂×C₄	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
52	C₈	—	4	4	no	—
53	C₈	—	8	4	no	—
54	C₂²	#382	1	1	yes	characteristic
55	C₂²	#424	1	2	yes	—
56	C₂²	C₂×#106	1	1	yes	characteristic
57	C₂²	C₂×#107	1	1	yes	characteristic
58	C₂²	#430	1	2	yes	—
59	C₄	—	2	4	no	—
60	C₄	—	2	4	no	—
61	C₄	—	2	2	no	—
62	C₄	#433	1	4	yes	—
63	C₄	#486	1	2	yes	—
64	C₄	#487	1	2	yes	—
65	C₂	#1570	1	2	yes	—
66	C₂	#1571	1	2	yes	—
67	C₂	#1832	1	1	yes	characteristic
68	C₂	#1834	1	1	yes	characteristic
69	C₂	#1838	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
#1744	2
#1277	2