Subgroups — #13689

Order 256.

108 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	C₂×D₃₂	C₂	1	1	yes	maximal, characteristic
3	#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#1416	C₂	1	1	yes	maximal, characteristic, complemented
7	#1415	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×SD₃₂	—	2	1	no	—
10	C₂×D₁₆	C₂²	1	1	yes	characteristic
11	C₂×D₁₆	C₂²	1	1	yes	characteristic
12	#495	—	2	1	no	—
13	C₄×D₈	—	2	1	no	—
14	SD₆₄	—	2	2	no	—
15	D₃₂	—	2	2	no	—
16	C₂×C₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₃₂	C₂²	1	1	yes	characteristic
18	#366	C₂²	1	1	yes	characteristic
19	#357	—	2	1	no	—
20	C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆	—	2	1	no	—
22	C₂×SD₁₆	—	4	1	no	—
23	C₂×D₈	—	2	1	no	—
24	C₂×D₈	—	2	1	no	—
25	#124	—	4	1	no	—
26	C₄×D₄	—	4	1	no	—
27	Q₃₂	—	2	1	no	—
28	Q₃₂	—	2	1	no	—
29	SD₃₂	—	4	2	no	—
30	D₁₆	—	2	1	no	—
31	D₁₆	—	2	1	no	—
32	D₁₆	—	2	1	no	—
33	D₁₆	D₄	1	2	yes	—
34	C₂×C₁₆	D₄	1	1	yes	characteristic
35	C₂×C₁₆	C₂³	1	1	yes	characteristic
36	#107	—	2	1	no	—
37	#102	—	4	1	no	—
38	C₄×C₈	D₄	1	1	yes	characteristic
39	C₃₂	—	2	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	4	1	no	—
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	—	4	1	no	—
44	C₂²×C₄	—	8	1	no	—
45	Q₁₆	—	4	1	no	—
46	Q₁₆	—	4	1	no	—
47	SD₁₆	—	8	2	no	—
48	D₈	—	2	2	no	—
49	D₈	—	4	1	no	—
50	D₈	—	4	1	no	—
51	D₈	—	4	1	no	—
52	C₂×C₈	D₈	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄	—	4	1	no	—
55	C₂²⋊C₄	—	8	1	no	—
56	C₂²⋊C₄	—	8	1	no	—
57	C₂²⋊C₄	—	8	1	no	—
58	C₄²	D₈	1	1	yes	characteristic
59	C₁₆	—	2	1	no	—
60	C₁₆	C₂×D₄	1	1	yes	characteristic
61	C₁₆	#41	1	1	yes	characteristic
62	C₂³	—	8	1	no	—
63	C₂³	—	8	1	no	—
64	Q₈	—	8	1	no	—
65	Q₈	—	8	1	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	1	no	—
68	D₄	—	8	1	no	—
69	D₄	—	8	1	no	—
70	C₂×C₄	—	8	1	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	—	16	2	no	—
73	C₂×C₄	D₁₆	1	1	yes	characteristic
74	C₂×C₄	D₁₆	1	1	yes	characteristic
75	C₂×C₄	C₂×D₈	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	#121	1	1	yes	characteristic
78	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²	—	8	2	no	—
80	C₂²	—	16	1	no	—
81	C₂²	—	16	1	no	—
82	C₂²	—	16	1	no	—
83	C₂²	—	16	1	no	—
84	C₂²	—	16	1	no	—
85	C₂²	C₂×D₁₆	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	16	1	no	—
89	C₄	—	16	1	no	—
90	C₄	#459	1	1	yes	characteristic
91	C₄	#509	1	1	yes	characteristic
92	C₂	—	16	2	no	—
93	C₂	—	32	1	no	—
94	C₂	#2206	1	1	yes	characteristic
95	C₂	#2262	1	1	yes	characteristic
96	C₂	#2263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₃₂	C₂
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
1. C₂×SD₆₄	C₂	semidirect
3. #2249	C₂	semidirect
4. C₄×D₁₆	C₂	semidirect
5. C₄⋊C₃₂	C₂	semidirect
6. #1416	C₂	semidirect
7. #1415	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
C₂×D₃₂	2
#2249	2
C₄×D₁₆	2
C₄⋊C₃₂	2
#1416	2
#1415	2