Subgroups — #12059

Order 256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₆	C₂	1	2	yes	maximal
2	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#1367	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈	C₂²	1	2	yes	—
7	C₄×Q₁₆	C₄	1	1	yes	characteristic
8	C₄⋊C₁₆	—	2	2	no	—
9	C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₁₆	—	2	2	no	—
11	C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#335	C₄	1	1	yes	characteristic
14	Q₈⋊C₈	C₄	1	2	yes	—
15	C₈²	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆	C₈	1	1	yes	characteristic
17	C₄×Q₈	C₂×C₄	1	2	yes	—
18	C₂×C₁₆	—	2	2	no	—
19	C₂×C₁₆	—	2	1	no	—
20	C₂×C₁₆	D₄	1	2	yes	—
21	#107	C₈	1	1	yes	characteristic
22	C₄⋊C₈	—	2	2	no	—
23	C₄⋊C₈	C₂×C₄	1	2	yes	—
24	#103	C₈	1	2	yes	—
25	C₄×C₈	—	2	2	no	—
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂×Q₈	C₂×C₈	1	2	yes	—
30	Q₁₆	C₁₆	1	4	yes	complemented
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈	#41	1	1	yes	characteristic
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	C₂×C₈	1	2	yes	—
39	C₄²	—	2	2	no	—
40	C₄²	C₂²×C₄	1	1	yes	characteristic
41	C₁₆	—	2	2	no	—
42	C₁₆	—	4	1	no	—
43	C₁₆	—	4	2	no	—
44	Q₈	—	2	2	no	—
45	Q₈	C₂×C₁₆	1	4	yes	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄	#131	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	—	4	1	no	—
54	C₈	—	4	2	no	—
55	C₈	C₂×C₁₆	1	2	yes	—
56	C₈	#137	1	2	yes	—
57	C₂²	C₈×D₄	1	1	yes	characteristic
58	C₄	—	2	4	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	#441	1	1	yes	characteristic
62	C₄	#444	1	1	yes	characteristic
63	C₄	C₂²×C₁₆	1	1	yes	characteristic
64	C₄	#504	1	1	yes	characteristic
65	C₂	#1579	1	1	yes	characteristic
66	C₂	D₄×C₁₆	1	1	yes	characteristic
67	C₂	#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂²	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
30. Q₁₆	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₁₆	2
C₈×Q₁₆	2
#1367	2
Q₈⋊C₁₆	2
#1312	2