Subgroups — #13701

Order 256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#2258	C₂	1	1	yes	maximal, characteristic
3	#2222	C₂	1	2	yes	maximal
4	#1425	C₂	1	1	yes	maximal, characteristic, complemented
5	#1420	C₂	1	2	yes	maximal, complemented
6	C₂×Q₃₂	C₂²	1	2	yes	—
7	#504	C₂²	1	1	yes	characteristic
8	#473	—	2	2	no	—
9	Q₆₄	—	2	1	no	—
10	Q₆₄	—	2	1	no	—
11	M₆₄	—	2	1	no	—
12	M₆₄	C₂²	1	1	yes	characteristic
13	C₂×C₃₂	—	2	1	no	—
14	C₂×C₃₂	C₂²	1	1	yes	characteristic
15	#368	C₂²	1	2	yes	—
16	#362	—	2	2	no	—
17	#137	—	4	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	#131	D₄	1	1	yes	characteristic
20	Q₃₂	—	2	2	no	—
21	Q₃₂	—	2	2	no	—
22	M₃₂	—	2	1	no	—
23	M₃₂	D₄	1	1	yes	characteristic
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	C₂³	1	1	yes	characteristic
26	#108	—	2	2	no	—
27	#101	—	4	2	no	—
28	C₃₂	—	2	1	no	—
29	C₃₂	D₄	1	1	yes	characteristic
30	C₃₂	D₄	1	1	yes	characteristic
31	#41	D₈	1	1	yes	characteristic
32	C₂×Q₈	—	4	2	no	—
33	Q₁₆	—	4	2	no	—
34	Q₁₆	—	4	2	no	—
35	Q₁₆	—	8	2	no	—
36	SD₁₆	—	8	2	no	—
37	M₁₆	—	2	1	no	—
38	M₁₆	—	4	2	no	—
39	M₁₆	D₈	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₂×D₄	1	1	yes	characteristic
42	C₁₆	—	2	1	no	—
43	C₁₆	C₂×D₄	1	1	yes	characteristic
44	C₁₆	#41	1	1	yes	characteristic
45	Q₈	—	8	2	no	—
46	Q₈	—	8	2	no	—
47	Q₈	D₁₆	1	1	yes	characteristic
48	D₄	—	2	1	no	—
49	D₄	D₁₆	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	8	2	no	—
52	C₂×C₄	C₂×D₈	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	2	no	—
55	C₈	#121	1	1	yes	characteristic
56	C₈	#135	1	1	yes	characteristic
57	C₂²	—	2	1	no	—
58	C₂²	#508	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	16	2	no	—
61	C₄	#466	1	1	yes	characteristic
62	C₄	C₂×D₁₆	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	#2215	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #504 ◁ 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
1. C₂×Q₆₄	C₂	semidirect
4. #1425	C₂	semidirect
5. #1420	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₆₄	2
#2258	2
#2222	2
#1425	2
#1420	2