Subgroups — #8816

Order 256.

211 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1845	C₂	1	2	yes	maximal
2	#1834	C₂	1	4	yes	maximal
3	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₂×#107	—	2	4	no	—
5	C₂×C₄×C₈	C₂²	1	2	yes	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	#384	C₂²	1	4	yes	—
8	#340	—	2	8	no	—
9	C₈²	C₄	1	4	yes	complemented
10	C₂²×C₈	D₄	1	4	yes	—
11	C₂²×C₈	D₄	1	2	yes	—
12	C₂×C₄⋊C₄	—	2	8	no	—
13	C₂×C₄⋊C₄	—	2	4	no	—
14	C₂×C₄²	C₂³	1	1	yes	characteristic
15	#107	—	4	8	no	—
16	C₄×C₈	C₂×C₄	1	4	yes	—
17	C₄×C₈	C₂×C₄	1	2	yes	—
18	C₄×C₈	Q₈	1	4	yes	—
19	C₄×C₈	Q₈	1	2	yes	—
20	C₂²×C₄	—	4	4	no	—
21	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄	C₂×D₄	1	2	yes	—
23	C₂×C₈	—	2	8	no	—
24	C₂×C₈	C₄⋊C₄	1	8	yes	—
25	C₂×C₈	C₄⋊C₄	1	4	yes	—
26	C₂×C₈	D₈	1	4	yes	—
27	C₂×C₈	Q₁₆	1	4	yes	—
28	C₄⋊C₄	—	4	8	no	—
29	C₄⋊C₄	—	4	16	no	—
30	C₄²	C₂²×C₄	1	1	yes	characteristic
31	C₄²	C₂×Q₈	1	1	yes	characteristic
32	C₄²	C₂×Q₈	1	2	yes	—
33	C₂³	#127	1	1	yes	characteristic
34	C₂×C₄	—	8	4	no	—
35	C₂×C₄	—	8	8	no	—
36	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
38	C₂×C₄	C₄⋊Q₈	1	2	yes	—
39	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₄	C₂×D₈	1	2	yes	—
41	C₂×C₄	C₂×Q₁₆	1	2	yes	—
42	C₂×C₄	#135	1	8	yes	—
43	C₈	—	2	8	no	—
44	C₈	#107	1	8	yes	—
45	C₂²	#384	1	1	yes	characteristic
46	C₂²	#493	1	1	yes	characteristic
47	C₂²	#494	1	1	yes	characteristic
48	C₂²	#495	1	4	yes	—
49	C₄	—	16	4	no	—
50	C₄	C₂×#107	1	2	yes	—
51	C₄	#427	1	4	yes	—
52	C₄	#499	1	4	yes	—
53	C₄	#500	1	2	yes	—
54	C₂	#1709	1	2	yes	—
55	C₂	#1710	1	2	yes	—
56	C₂	#1845	1	1	yes	characteristic
57	C₂	#1846	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
9. C₈²	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1845	2
#1834	2
C₂×C₈²	2