Subgroups — #8827

Order 256.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1837	C₂	1	2	yes	maximal
2	#1567	C₂	1	2	yes	maximal
3	#1566	C₂	1	2	yes	maximal
4	C₂×#322	C₂	1	1	yes	maximal, characteristic
5	#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#341	—	2	4	no	—
10	#335	—	2	4	no	—
11	#334	—	2	4	no	—
12	#322	C₄	1	4	yes	—
13	C₂×M₁₆	—	2	4	no	—
14	C₂²×C₈	D₄	1	2	yes	—
15	C₂²×C₈	Q₈	1	2	yes	—
16	C₂²×C₈	Q₈	1	2	yes	—
17	C₂×C₄²	C₂³	1	1	yes	characteristic
18	C₄⋊C₈	—	2	4	no	—
19	C₄⋊C₈	—	2	4	no	—
20	C₄×C₈	C₂×C₄	1	2	yes	—
21	C₄×C₈	C₂×C₄	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈	D₄	1	2	yes	—
24	C₄×C₈	D₄	1	2	yes	—
25	C₄×C₈	Q₈	1	2	yes	—
26	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
29	M₁₆	—	4	8	no	—
30	C₂×C₈	—	2	4	no	—
31	C₂×C₈	—	4	2	no	—
32	C₂×C₈	—	4	2	no	—
33	C₂×C₈	—	4	4	no	—
34	C₂×C₈	C₄⋊C₄	1	4	yes	—
35	C₂×C₈	C₄⋊C₄	1	4	yes	—
36	C₂×C₈	C₄⋊C₄	1	4	yes	—
37	C₄²	C₂²×C₄	1	1	yes	characteristic
38	C₄²	C₂×D₄	1	1	yes	characteristic
39	C₄²	C₂×D₄	1	1	yes	characteristic
40	C₄²	C₂×Q₈	1	1	yes	characteristic
41	C₂³	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄	#127	1	1	yes	characteristic
48	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄	#135	1	4	yes	—
51	C₈	—	2	4	no	—
52	C₈	—	2	4	no	—
53	C₈	—	2	4	no	—
54	C₈	—	8	4	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	#384	1	1	yes	characteristic
58	C₂²	#499	1	2	yes	—
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	#427	1	2	yes	—
62	C₄	#430	1	2	yes	—
63	C₄	#430	1	2	yes	—
64	C₄	#495	1	2	yes	—
65	C₂	—	2	2	no	—
66	C₂	#1846	1	1	yes	characteristic
67	C₂	#1849	1	1	yes	characteristic
68	C₂	#1849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#384
prime radical	C₂³	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ 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
#1837	2
#1567	2
#1566	2
C₂×#322	2