Subgroups — #13708

Order 256.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#1424	C₂	1	1	yes	maximal, characteristic
4	#1423	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#1416	C₂	1	1	yes	maximal, characteristic
7	#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	#499	—	2	1	no	—
10	C₄×Q₁₆	—	2	1	no	—
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	#367	—	2	1	no	—
14	#366	C₂²	1	1	yes	characteristic
15	#366	C₂²	1	1	yes	characteristic
16	#366	C₂²	1	1	yes	characteristic
17	#358	—	2	1	no	—
18	C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆	—	2	1	no	—
20	#125	—	4	1	no	—
21	C₄×Q₈	—	4	1	no	—
22	Q₃₂	—	2	1	no	—
23	Q₃₂	Q₈	1	2	yes	—
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₆	C₂³	1	1	yes	characteristic
26	#107	—	2	1	no	—
27	#107	—	2	1	no	—
28	#107	—	2	1	no	—
29	#106	—	4	1	no	—
30	#103	—	4	1	no	—
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₃₂	—	2	1	no	—
33	C₃₂	—	2	1	no	—
34	C₂×Q₈	—	4	1	no	—
35	Q₁₆	—	2	2	no	—
36	Q₁₆	—	4	1	no	—
37	C₂×C₈	D₈	1	1	yes	characteristic
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄	—	4	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	4	1	no	—
42	C₄⋊C₄	—	8	1	no	—
43	C₄⋊C₄	—	8	1	no	—
44	C₄⋊C₄	—	8	1	no	—
45	C₄²	—	8	1	no	—
46	C₄²	D₈	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₁₆	#41	1	1	yes	characteristic
50	Q₈	—	4	2	no	—
51	Q₈	—	8	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	8	1	no	—
54	C₂×C₄	—	8	1	no	—
55	C₂×C₄	—	8	1	no	—
56	C₂×C₄	D₁₆	1	1	yes	characteristic
57	C₂×C₄	D₁₆	1	1	yes	characteristic
58	C₂×C₄	C₂×D₈	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈	#137	1	1	yes	characteristic
62	C₂²	C₂×D₁₆	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	16	1	no	—
67	C₄	—	16	1	no	—
68	C₄	—	16	1	no	—
69	C₄	#474	1	1	yes	characteristic
70	C₄	#510	1	1	yes	characteristic
71	C₂	#2224	1	1	yes	characteristic
72	C₂	#2262	1	1	yes	characteristic
73	C₂	#2264	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

Maximal subgroups

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

subgroup	index
#2253	2
C₄×Q₃₂	2
#1424	2
#1423	2
C₄⋊C₃₂	2
#1416	2