Subgroups — #12440

Order 256.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2247	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1670	C₂	1	2	yes	maximal
4	#1371	C₂	1	1	yes	maximal, characteristic
5	#1362	C₂	1	2	yes	maximal
6	C₂×Q₃₂	—	2	1	no	—
7	C₂×Q₃₂	—	2	1	no	—
8	#494	C₂²	1	2	yes	—
9	#475	—	2	2	no	—
10	C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆	—	2	1	no	—
12	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₆	—	2	1	no	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#335	C₂²	1	2	yes	—
16	#332	—	2	2	no	—
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₄⋊Q₈	—	2	2	no	—
20	C₄×Q₈	D₄	1	1	yes	characteristic
21	Q₃₂	—	4	2	no	—
22	Q₃₂	—	4	2	no	—
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₆	D₄	1	1	yes	characteristic
26	#106	—	4	2	no	—
27	C₄⋊C₈	—	2	1	no	—
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	D₄	1	1	yes	characteristic
30	#103	—	4	2	no	—
31	C₄×C₈	—	2	1	no	—
32	C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂×Q₈	—	4	2	no	—
34	C₂×Q₈	D₈	1	1	yes	characteristic
35	Q₁₆	—	4	2	no	—
36	Q₁₆	—	4	2	no	—
37	Q₁₆	—	8	2	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈	#41	1	1	yes	characteristic
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄⋊C₄	—	4	2	no	—
45	C₄⋊C₄	D₈	1	1	yes	characteristic
46	C₄²	—	2	1	no	—
47	C₄²	C₂×D₄	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	—	4	1	no	—
50	C₁₆	SD₁₆	1	2	yes	—
51	Q₈	—	2	1	no	—
52	Q₈	—	8	2	no	—
53	Q₈	—	8	2	no	—
54	Q₈	Q₃₂	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	8	2	no	—
58	C₂×C₄	#121	1	1	yes	characteristic
59	C₂×C₄	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄	#135	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	4	1	no	—
63	C₈	—	8	2	no	—
64	C₈	C₂×SD₁₆	1	1	yes	characteristic
65	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²	#466	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	16	2	no	—
71	C₄	#460	1	1	yes	characteristic
72	C₄	#472	1	1	yes	characteristic
73	C₄	C₂×Q₃₂	1	1	yes	characteristic
74	C₄	#508	1	1	yes	characteristic
75	C₂	#1668	1	1	yes	characteristic
76	C₂	#2216	1	1	yes	characteristic
77	C₂	#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ 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
54. Q₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2247	2
Q₈×C₁₆	2
#1670	2
#1371	2
#1362	2