Subgroups — #13702

Order 256.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#2258	C₂	1	1	yes	maximal, characteristic, complemented
3	#2222	C₂	1	1	yes	maximal, characteristic, complemented
4	#2221	C₂	1	1	yes	maximal, characteristic
5	#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	#1420	C₂	1	1	yes	maximal, characteristic, complemented
7	#1419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×D₁₆	C₂²	1	1	yes	characteristic
10	#504	C₂²	1	1	yes	characteristic
11	#473	—	2	1	no	—
12	#472	—	2	1	no	—
13	SD₆₄	—	2	1	no	—
14	SD₆₄	—	2	1	no	—
15	M₆₄	—	2	1	no	—
16	M₆₄	C₂²	1	1	yes	characteristic
17	C₂×C₃₂	—	2	1	no	—
18	C₂×C₃₂	C₂²	1	1	yes	characteristic
19	#368	C₂²	1	1	yes	characteristic
20	#368	C₂²	1	1	yes	characteristic
21	#362	—	2	1	no	—
22	#361	—	2	1	no	—
23	#137	—	4	1	no	—
24	AGL(1,ℤ/8)	—	4	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×D₈	—	2	1	no	—
27	#131	D₄	1	1	yes	characteristic
28	Q₃₂	—	2	1	no	—
29	Q₃₂	—	2	1	no	—
30	D₁₆	—	2	1	no	—
31	D₁₆	—	2	1	no	—
32	M₃₂	—	2	1	no	—
33	M₃₂	D₄	1	1	yes	characteristic
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	C₂³	1	1	yes	characteristic
36	#108	—	2	1	no	—
37	#108	—	2	1	no	—
38	#101	—	4	1	no	—
39	#100	—	4	1	no	—
40	C₃₂	—	2	1	no	—
41	C₃₂	D₄	1	1	yes	characteristic
42	C₃₂	D₄	1	1	yes	characteristic
43	#41	D₈	1	1	yes	characteristic
44	C₂×Q₈	—	4	1	no	—
45	C₂×D₄	—	4	1	no	—
46	Q₁₆	—	4	1	no	—
47	Q₁₆	—	4	1	no	—
48	Q₁₆	—	8	1	no	—
49	SD₁₆	—	8	1	no	—
50	SD₁₆	—	8	1	no	—
51	D₈	—	4	1	no	—
52	D₈	—	4	1	no	—
53	D₈	—	8	1	no	—
54	M₁₆	—	2	1	no	—
55	M₁₆	—	4	1	no	—
56	M₁₆	—	4	1	no	—
57	M₁₆	D₈	1	1	yes	characteristic
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₂×D₄	1	1	yes	characteristic
60	C₁₆	—	2	1	no	—
61	C₁₆	C₂×D₄	1	1	yes	characteristic
62	C₁₆	#41	1	1	yes	characteristic
63	C₂³	—	8	1	no	—
64	Q₈	—	8	1	no	—
65	Q₈	—	8	1	no	—
66	Q₈	D₁₆	1	1	yes	characteristic
67	D₄	—	2	1	no	—
68	D₄	—	8	1	no	—
69	D₄	—	8	1	no	—
70	D₄	D₁₆	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	C₂×D₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	8	1	no	—
76	C₈	—	8	1	no	—
77	C₈	#121	1	1	yes	characteristic
78	C₈	#135	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	—	16	1	no	—
81	C₂²	—	32	1	no	—
82	C₂²	#508	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	16	1	no	—
85	C₄	#466	1	1	yes	characteristic
86	C₄	C₂×D₁₆	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	4	1	no	—
89	C₂	—	32	1	no	—
90	C₂	#2215	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#2221	C₂
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×SD₆₄	C₂	semidirect
2. #2258	C₂	semidirect
3. #2222	C₂	semidirect
5. #1425	C₂	semidirect
6. #1420	C₂	semidirect
7. #1419	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₆₄	2
#2258	2
#2222	2
#2221	2
#1425	2
#1420	2
#1419	2