Subgroups — #12083

Order 256.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	#1591	C₂	1	2	yes	maximal
3	#1371	C₂	1	1	yes	maximal, characteristic
4	#1332	C₂	1	2	yes	maximal
5	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂	C₄	1	1	yes	characteristic
7	#446	—	2	2	no	—
8	C₄×Q₁₆	C₂²	1	2	yes	—
9	#366	C₄	1	1	yes	characteristic
10	#358	C₄	1	2	yes	—
11	C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#335	C₂²	1	2	yes	—
14	Q₈⋊C₈	—	2	2	no	—
15	C₈²	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆	C₂×C₄	1	2	yes	—
17	C₄×Q₈	—	2	2	no	—
18	Q₃₂	—	2	2	no	—
19	C₂×C₁₆	—	2	1	no	—
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	#107	C₂×C₄	1	2	yes	—
23	C₄⋊C₈	—	2	2	no	—
24	C₄⋊C₈	—	4	2	no	—
25	#103	—	2	2	no	—
26	#97	—	4	2	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂×Q₈	—	2	2	no	—
31	Q₁₆	—	2	2	no	—
32	Q₁₆	—	2	2	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	4	2	no	—
35	C₂×C₈	D₈	1	1	yes	characteristic
36	C₂×C₈	D₈	1	1	yes	characteristic
37	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈	#41	1	1	yes	characteristic
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	4	2	no	—
41	C₄²	—	4	2	no	—
42	C₄²	C₂×D₄	1	1	yes	characteristic
43	C₁₆	—	2	2	no	—
44	C₁₆	—	2	1	no	—
45	C₁₆	M₁₆	1	2	yes	—
46	Q₈	—	4	2	no	—
47	Q₈	—	4	2	no	—
48	C₂×C₄	—	4	2	no	—
49	C₂×C₄	—	4	2	no	—
50	C₂×C₄	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄	#135	1	1	yes	characteristic
53	C₈	—	2	2	no	—
54	C₈	—	2	1	no	—
55	C₈	—	2	1	no	—
56	C₈	—	8	2	no	—
57	C₈	Q₃₂	1	2	yes	complemented
58	C₈	C₂×M₁₆	1	1	yes	characteristic
59	C₈	#131	1	1	yes	characteristic
60	C₂²	C₄×D₈	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	8	2	no	—
63	C₄	—	8	2	no	—
64	C₄	#436	1	1	yes	characteristic
65	C₄	#443	1	1	yes	characteristic
66	C₄	C₂×Q₃₂	1	1	yes	characteristic
67	C₄	#508	1	1	yes	characteristic
68	C₂	#1589	1	1	yes	characteristic
69	C₂	C₄×Q₃₂	1	1	yes	characteristic
70	C₂	#2178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂²	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ 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
57. C₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₃₂	2
#1591	2
#1371	2
#1332	2
C₈×C₁₆	2