Subgroups — C₂×#13660 #512783

Order 512.

342 conjugacy classes of subgroups.

The subgroup lattice has 131 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	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₃₂	C₂	1	2	yes	maximal
5	C₂×#1397	C₂	1	2	yes	maximal
6	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
8	C₂²×C₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₃₂	C₂²	1	2	yes	—
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	Q₈×C₁₆	C₄	1	4	yes	—
12	Q₈×C₁₆	C₂²	1	4	yes	—
13	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₆	C₂²	1	2	yes	—
17	C₄⋊C₃₂	C₂²	1	4	yes	—
18	C₄⋊C₃₂	C₂²	1	8	yes	—
19	#1397	C₂²	1	8	yes	—
20	C₄×C₃₂	C₂²	1	4	yes	—
21	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₁₆	C₂³	1	1	yes	characteristic
25	C₈×Q₈	C₈	1	4	yes	—
26	C₈×Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₈	C₈	1	2	yes	—
28	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	—	2	4	no	—
33	C₂×C₃₂	Q₈	1	4	yes	complemented
34	C₂×C₃₂	C₂³	1	2	yes	—
35	C₂×C₃₂	C₂³	1	4	yes	—
36	C₂×C₃₂	C₂³	1	2	yes	—
37	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
38	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
39	C₄×C₁₆	C₂×C₄	1	2	yes	—
40	C₄×C₁₆	C₂×C₄	1	4	yes	—
41	C₄×C₁₆	C₂³	1	2	yes	—
42	C₄×C₁₆	C₂³	1	4	yes	—
43	C₂²×Q₈	C₁₆	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×C₈	1	2	yes	—
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄×Q₈	C₁₆	1	4	yes	—
48	C₄×Q₈	C₂×C₈	1	4	yes	—
49	C₂×C₄⋊C₄	C₁₆	1	2	yes	—
50	C₂×C₄⋊C₄	C₁₆	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₈	1	2	yes	—
52	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	4	no	—
55	C₂×C₁₆	C₂²×C₄	1	2	yes	—
56	C₂×C₁₆	C₂²×C₄	1	4	yes	—
57	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₁₆	C₂×Q₈	1	2	yes	—
59	C₂×C₁₆	#41	1	2	yes	—
60	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₈	C₂×C₈	1	8	yes	—
62	C₄⋊C₈	C₂×C₈	1	4	yes	—
63	C₄×C₈	C₂×C₈	1	4	yes	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₄×C₈	C₂²×C₄	1	2	yes	—
66	C₄×C₈	C₂²×C₄	1	4	yes	—
67	C₃₂	—	2	2	no	—
68	C₃₂	—	2	4	no	—
69	C₃₂	C₂×Q₈	1	4	yes	complemented
70	C₂×Q₈	—	2	4	no	—
71	C₂×Q₈	C₂×C₁₆	1	4	yes	—
72	C₂²×C₄	C₂×C₁₆	1	2	yes	—
73	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
74	C₂²×C₄	C₂×C₁₆	1	2	yes	—
75	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
76	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	C₄×Q₈	1	2	yes	—
80	C₂×C₈	C₂²×C₈	1	2	yes	—
81	C₂×C₈	C₂²×C₈	1	4	yes	—
82	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₈	#131	1	2	yes	—
84	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
85	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
86	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
87	C₄²	C₂×C₁₆	1	4	yes	—
88	C₄²	C₂×C₁₆	1	2	yes	—
89	C₄²	C₂²×C₈	1	4	yes	—
90	C₄²	C₂²×C₈	1	2	yes	—
91	C₁₆	—	2	2	no	—
92	C₁₆	—	2	4	no	—
93	C₁₆	C₄×Q₈	1	2	yes	—
94	C₁₆	C₂²×Q₈	1	1	yes	characteristic
95	C₁₆	C₂×#41	1	1	yes	characteristic
96	C₂³	C₂²×C₁₆	1	1	yes	characteristic
97	Q₈	—	2	8	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	M₆₄	1	4	yes	—
102	C₂×C₄	C₈×Q₈	1	2	yes	—
103	C₂×C₄	C₂²×C₁₆	1	4	yes	—
104	C₂×C₄	C₂²×C₁₆	1	2	yes	—
105	C₂×C₄	C₂²×C₁₆	1	4	yes	—
106	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
107	C₂×C₄	C₂²×C₁₆	1	2	yes	—
108	C₂×C₄	#504	1	2	yes	—
109	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	2	4	no	—
112	C₈	C₈×Q₈	1	2	yes	—
113	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₈	C₂×#131	1	1	yes	characteristic
115	C₂²	Q₈×C₁₆	1	2	yes	—
116	C₂²	C₂×M₆₄	1	2	yes	—
117	C₂²	#2258	1	2	yes	—
118	C₂²	C₂³×C₁₆	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	2	4	no	—
121	C₄	—	2	2	no	—
122	C₄	Q₈×C₁₆	1	2	yes	—
123	C₄	C₂×M₆₄	1	4	yes	—
124	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
125	C₄	C₂×#504	1	1	yes	characteristic
126	C₂	#13660	1	4	yes	direct product factor
127	C₂	C₂×Q₈×C₁₆	1	1	yes	characteristic
128	C₂	C₂²×M₆₄	1	1	yes	characteristic
129	C₂	C₂×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle, prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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	direct (N × H)
33. C₂×C₃₂	Q₈	semidirect
69. C₃₂	C₂×Q₈	semidirect
126. C₂	#13660	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
C₂×C₄⋊C₃₂	2
C₂×#1397	2
C₂×C₄×C₃₂	2

Subgroups — C2×#13660 #512783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13660 #512783