Subgroups — #8822

Order 256.

177 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1848	C₂	1	2	yes	maximal
2	#1566	C₂	1	4	yes	maximal
3	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₂×#108	—	2	4	no	—
5	#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#334	—	2	8	no	—
9	C₈²	C₄	1	1	yes	characteristic
10	C₈²	C₄	1	2	yes	—
11	C₈²	C₄	1	1	yes	characteristic
12	C₂×M₁₆	—	2	4	no	—
13	C₂²×C₈	D₄	1	2	yes	—
14	C₂²×C₈	Q₈	1	4	yes	—
15	C₂×C₄²	C₂³	1	1	yes	characteristic
16	#108	—	4	8	no	—
17	C₄⋊C₈	—	2	8	no	—
18	C₄×C₈	C₂×C₄	1	2	yes	—
19	C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈	C₂×C₄	1	2	yes	—
22	C₄×C₈	D₄	1	4	yes	—
23	C₄×C₈	Q₈	1	2	yes	—
24	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄	C₂×Q₈	1	2	yes	—
26	M₁₆	—	4	8	no	—
27	C₂×C₈	—	2	8	no	—
28	C₂×C₈	—	4	4	no	—
29	C₂×C₈	C₄⋊C₄	1	4	yes	—
30	C₂×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₈	C₄⋊C₄	1	2	yes	—
32	C₂×C₈	C₄⋊C₄	1	4	yes	—
33	C₂×C₈	SD₁₆	1	8	yes	—
34	C₄²	C₂²×C₄	1	1	yes	characteristic
35	C₄²	C₂×D₄	1	2	yes	—
36	C₄²	C₂×Q₈	1	1	yes	characteristic
37	C₂³	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×C₄	#127	1	1	yes	characteristic
42	C₂×C₄	C₄⋊Q₈	1	2	yes	—
43	C₂×C₄	C₂×SD₁₆	1	4	yes	—
44	C₈	—	2	4	no	—
45	C₈	—	8	4	no	—
46	C₈	#106	1	4	yes	—
47	C₈	#106	1	4	yes	—
48	C₈	#108	1	8	yes	complemented
49	C₂²	—	2	2	no	—
50	C₂²	#384	1	1	yes	characteristic
51	C₂²	#498	1	2	yes	—
52	C₄	—	2	2	no	—
53	C₄	C₂×#106	1	2	yes	—
54	C₄	C₂×#108	1	4	yes	—
55	C₄	#492	1	2	yes	—
56	C₂	—	2	2	no	—
57	C₂	#1844	1	1	yes	characteristic
58	C₂	#1848	1	2	yes	—

	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

Complements

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

N	H ≅ G/N	type
48. C₈	#108	semidirect

Maximal subgroups

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

subgroup	index
#1848	2
#1566	2
C₂×C₈²	2