Subgroups — #13706

Order 256.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#1423	C₂	1	2	yes	maximal
4	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	#1416	C₂	1	2	yes	maximal
6	C₂×Q₃₂	C₂²	1	1	yes	characteristic
7	#500	—	2	1	no	—
8	C₄×Q₁₆	—	2	1	no	—
9	C₂×C₃₂	C₂²	1	2	yes	—
10	#366	—	2	1	no	—
11	#366	C₂²	1	1	yes	characteristic
12	#366	C₂²	1	2	yes	—
13	#358	—	2	1	no	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆	—	2	1	no	—
16	C₄⋊Q₈	—	4	1	no	—
17	C₄×Q₈	—	4	1	no	—
18	Q₃₂	—	2	1	no	—
19	Q₃₂	Q₈	1	2	yes	complemented
20	C₂×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂³	1	1	yes	characteristic
22	#107	—	2	1	no	—
23	#107	—	2	2	no	—
24	#107	—	4	1	no	—
25	#103	—	4	1	no	—
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₃₂	—	2	2	no	—
28	C₂×Q₈	—	4	1	no	—
29	C₂×Q₈	—	8	1	no	—
30	Q₁₆	—	2	2	no	—
31	Q₁₆	—	4	1	no	—
32	C₂×C₈	D₈	1	1	yes	characteristic
33	C₂×C₈	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄	—	4	1	no	—
35	C₄⋊C₄	—	4	2	no	—
36	C₄⋊C₄	—	8	1	no	—
37	C₄⋊C₄	—	8	1	no	—
38	C₄²	—	8	1	no	—
39	C₄²	D₈	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	C₂×Q₈	1	1	yes	characteristic
42	C₁₆	#41	1	1	yes	characteristic
43	Q₈	—	4	2	no	—
44	Q₈	—	8	1	no	—
45	Q₈	—	16	2	no	—
46	C₂×C₄	—	8	1	no	—
47	C₂×C₄	—	8	1	no	—
48	C₂×C₄	—	8	2	no	—
49	C₂×C₄	D₁₆	1	1	yes	characteristic
50	C₂×C₄	D₁₆	1	1	yes	characteristic
51	C₂×C₄	C₂×D₈	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	C₂²⋊Q₈	1	1	yes	characteristic
54	C₈	#137	1	1	yes	characteristic
55	C₂²	C₂×D₁₆	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	8	2	no	—
58	C₄	—	16	1	no	—
59	C₄	—	16	2	no	—
60	C₄	Q₆₄	1	2	yes	—
61	C₄	#474	1	1	yes	characteristic
62	C₄	#510	1	1	yes	characteristic
63	C₂	#2224	1	1	yes	characteristic
64	C₂	C₂×Q₆₄	1	1	yes	characteristic
65	C₂	#2263	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. Q₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
C₄×Q₃₂	2
#1423	2
C₄⋊C₃₂	2
#1416	2