Subgroups — Q₈×M₆₄ #512787

Order 512.

336 conjugacy classes of subgroups.

The subgroup lattice has 123 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13660	C₂	1	6	yes	maximal, complemented
3	Q₈×C₃₂	C₂	1	2	yes	maximal, complemented
4	#13627	C₂	1	3	yes	maximal
5	C₄×M₆₄	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	3	yes	—
8	C₂×M₆₄	C₂²	1	1	yes	characteristic
9	Q₈×C₁₆	C₄	1	3	yes	—
10	Q₈×C₁₆	C₄	1	1	yes	characteristic
11	Q₈×C₁₆	C₂²	1	1	yes	characteristic
12	Q₈×C₁₆	C₂²	1	3	yes	—
13	C₂×C₄⋊C₁₆	C₄	1	3	yes	—
14	C₂×C₄×C₁₆	C₂²	1	3	yes	—
15	C₄⋊C₃₂	C₂²	1	6	yes	—
16	C₄⋊C₃₂	C₂²	1	6	yes	—
17	#1397	C₂²	1	6	yes	—
18	C₄×C₃₂	C₂²	1	6	yes	—
19	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	3	yes	—
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	C₈×Q₈	C₈	1	3	yes	—
23	C₈×Q₈	C₈	1	1	yes	characteristic
24	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₈×Q₈	C₂×C₄	1	3	yes	—
26	C₂×C₄⋊C₈	C₈	1	3	yes	—
27	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
28	M₆₄	—	2	6	no	—
29	M₆₄	Q₈	1	4	yes	direct product factor
30	C₂×C₃₂	C₂³	1	6	yes	—
31	C₂×C₃₂	C₂³	1	2	yes	—
32	C₄⋊C₁₆	C₂×C₄	1	3	yes	—
33	C₄⋊C₁₆	C₂×C₄	1	3	yes	—
34	C₄⋊C₁₆	C₂×C₄	1	6	yes	—
35	C₄×C₁₆	C₂×C₄	1	3	yes	—
36	C₄×C₁₆	C₂×C₄	1	3	yes	—
37	C₄×C₁₆	C₂³	1	3	yes	—
38	C₄×C₁₆	C₂³	1	3	yes	—
39	C₂²×Q₈	C₁₆	1	1	yes	characteristic
40	C₂²×C₈	C₂×C₈	1	3	yes	—
41	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄×Q₈	C₁₆	1	3	yes	—
43	C₄×Q₈	C₁₆	1	1	yes	characteristic
44	C₄×Q₈	C₂×C₈	1	3	yes	—
45	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄⋊C₄	C₁₆	1	3	yes	—
47	C₂×C₄²	C₂×C₈	1	3	yes	—
48	C₂×C₁₆	—	2	6	no	—
49	C₂×C₁₆	C₂²×C₄	1	3	yes	—
50	C₂×C₁₆	C₂²×C₄	1	3	yes	—
51	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	C₂×Q₈	1	2	yes	—
53	C₂×C₁₆	#41	1	2	yes	—
54	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₈	C₂×C₈	1	6	yes	—
56	C₄⋊C₈	C₂×C₈	1	3	yes	—
57	C₄⋊C₈	C₂×C₈	1	3	yes	—
58	C₄×C₈	C₂×C₈	1	3	yes	—
59	C₄×C₈	C₂×C₈	1	3	yes	—
60	C₄×C₈	C₂²×C₄	1	3	yes	—
61	C₄×C₈	C₂²×C₄	1	3	yes	—
62	C₃₂	—	2	6	no	—
63	C₃₂	C₂×Q₈	1	4	yes	complemented
64	C₂×Q₈	—	2	4	no	—
65	C₂×Q₈	C₂×C₁₆	1	3	yes	—
66	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
67	C₂²×C₄	C₂×C₁₆	1	3	yes	—
68	C₂²×C₄	C₂×C₁₆	1	3	yes	—
69	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₈	—	2	6	no	—
71	C₂×C₈	C₄×Q₈	1	2	yes	—
72	C₂×C₈	C₂²×C₈	1	3	yes	—
73	C₂×C₈	C₂²×C₈	1	3	yes	—
74	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₈	#131	1	2	yes	—
76	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
77	C₄⋊C₄	C₂×C₁₆	1	6	yes	—
78	C₄⋊C₄	C₂×C₁₆	1	3	yes	—
79	C₄⋊C₄	C₂×C₁₆	1	3	yes	—
80	C₄²	C₂×C₁₆	1	3	yes	—
81	C₄²	C₂×C₁₆	1	3	yes	—
82	C₄²	C₂²×C₈	1	3	yes	—
83	C₄²	C₂²×C₈	1	3	yes	—
84	C₁₆	—	2	3	no	—
85	C₁₆	—	2	3	no	—
86	C₁₆	C₄×Q₈	1	2	yes	—
87	C₁₆	C₂²×Q₈	1	1	yes	characteristic
88	C₁₆	C₂×#41	1	1	yes	characteristic
89	C₂³	C₂²×C₁₆	1	1	yes	characteristic
90	Q₈	—	2	6	no	—
91	Q₈	M₆₄	1	4	yes	direct product factor
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	4	3	no	—
94	C₂×C₄	C₈×Q₈	1	2	yes	—
95	C₂×C₄	C₂²×C₁₆	1	3	yes	—
96	C₂×C₄	C₂²×C₁₆	1	3	yes	—
97	C₂×C₄	C₂²×C₁₆	1	3	yes	—
98	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
99	C₂×C₄	C₂²×C₁₆	1	3	yes	—
100	C₂×C₄	#504	1	2	yes	—
101	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
102	C₈	—	2	3	no	—
103	C₈	—	2	3	no	—
104	C₈	C₈×Q₈	1	2	yes	—
105	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
106	C₈	C₂×#131	1	1	yes	characteristic
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	Q₈×C₁₆	1	2	yes	—
110	C₂²	C₂³×C₁₆	1	1	yes	characteristic
111	C₄	—	2	3	no	—
112	C₄	—	2	3	no	—
113	C₄	—	2	3	no	—
114	C₄	Q₈×C₁₆	1	2	yes	—
115	C₄	C₂×M₆₄	1	6	yes	—
116	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
117	C₄	C₂×#504	1	1	yes	characteristic
118	C₂	—	2	2	no	—
119	C₂	C₂×Q₈×C₁₆	1	1	yes	characteristic
120	C₂	C₂²×M₆₄	1	1	yes	characteristic
121	C₂	#33974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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
2. #13660	C₂	semidirect
3. Q₈×C₃₂	C₂	semidirect
29. M₆₄	Q₈	direct (N × H)
63. C₃₂	C₂×Q₈	semidirect
91. Q₈	M₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₆	2
#13660	2
Q₈×C₃₂	2
#13627	2
C₄×M₆₄	2

Subgroups — Q8×M64 #512787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×M₆₄ #512787