Subgroups — C₂×C₁₆×D₈ #483484

Order 512.

602 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₈	C₂	1	1	yes	maximal, characteristic
3	C₁₆×D₈	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₁₆	—	2	2	no	—
8	C₂×C₄×D₈	C₄	1	1	yes	characteristic
9	C₂×C₈×D₄	C₂²	1	2	yes	—
10	D₄×C₁₆	—	2	4	no	—
11	D₄×C₁₆	C₂²	1	8	yes	complemented
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₁₆	—	2	2	no	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₈×D₈	C₄	1	4	yes	—
17	C₈×D₈	C₂²	1	4	yes	—
18	C₂×#335	C₄	1	1	yes	characteristic
19	C₂×D₄⋊C₈	C₄	1	2	yes	—
20	C₂×C₈²	C₂²	1	1	yes	characteristic
21	#1370	C₂²	1	4	yes	complemented
22	D₄⋊C₁₆	C₂²	1	8	yes	complemented
23	C₈×C₁₆	C₂²	1	4	yes	complemented
24	C₂²×D₈	C₈	1	1	yes	characteristic
25	C₂³×C₈	—	2	2	no	—
26	C₂×C₄×D₄	C₂×C₄	1	2	yes	—
27	C₂²×C₁₆	—	2	1	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	C₂²×C₁₆	—	2	8	no	—
30	C₂²×C₁₆	—	4	8	no	—
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	D₄	1	1	yes	characteristic
33	C₄×D₈	C₈	1	4	yes	—
34	C₄×D₈	C₂×C₄	1	4	yes	—
35	C₈×D₄	—	2	2	no	—
36	C₈×D₄	—	2	2	no	—
37	C₈×D₄	C₂×C₄	1	4	yes	—
38	C₈×D₄	C₂³	1	4	yes	—
39	C₂×#107	C₈	1	1	yes	characteristic
40	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×#102	C₈	1	2	yes	—
42	C₂×C₂²⋊C₈	—	2	2	no	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₄⋊C₁₆	—	2	2	no	—
47	C₄⋊C₁₆	C₂³	1	4	yes	—
48	C₂²⋊C₁₆	—	2	8	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	D₄	1	2	yes	—
51	C₄×C₁₆	C₂³	1	2	yes	—
52	C₄×C₁₆	C₂³	1	2	yes	—
53	#335	C₂×C₄	1	4	yes	—
54	D₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₈²	C₂×C₄	1	2	yes	—
56	C₈²	C₂³	1	2	yes	—
57	C₂²×D₄	C₂×C₈	1	2	yes	—
58	C₂³×C₄	—	2	2	no	—
59	C₂×D₈	C₁₆	1	8	yes	direct product factor
60	C₂×D₈	C₂×C₈	1	4	yes	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	2	no	—
63	C₂²×C₈	—	2	4	no	—
64	C₂²×C₈	—	2	4	no	—
65	C₂²×C₈	—	4	4	no	—
66	C₂²×C₈	—	4	4	no	—
67	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	#41	1	1	yes	characteristic
71	C₄×D₄	—	2	2	no	—
72	C₄×D₄	—	2	2	no	—
73	C₄×D₄	C₂×C₈	1	4	yes	—
74	C₄×D₄	C₂²×C₄	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₂×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	—	4	4	no	—
83	C₂×C₁₆	—	4	16	no	—
84	C₂×C₁₆	D₈	1	4	yes	direct product factor
85	C₂×C₁₆	C₂×D₄	1	2	yes	—
86	C₂×C₁₆	C₂×D₄	1	2	yes	—
87	#107	C₂×C₈	1	4	yes	—
88	C₄⋊C₈	—	2	2	no	—
89	C₄⋊C₈	C₂²×C₄	1	4	yes	—
90	#102	C₂×C₈	1	8	yes	—
91	C₂²⋊C₈	—	2	4	no	—
92	C₂²⋊C₈	—	2	4	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	C₂×C₈	1	2	yes	—
95	C₄×C₈	C₂²×C₄	1	2	yes	—
96	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₄×C₈	C₂²×C₄	1	2	yes	—
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	#41	1	1	yes	characteristic
100	C₄×C₈	C₂⁴	1	1	yes	characteristic
101	C₂⁴	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	4	no	—
104	C₂×D₄	C₂×C₁₆	1	8	yes	complemented
105	C₂×D₄	C₂²×C₈	1	4	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	—	4	4	no	—
111	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
113	C₂²×C₄	#131	1	1	yes	characteristic
114	D₈	C₂×C₁₆	1	16	yes	direct product factor
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	4	4	no	—
121	C₂×C₈	—	4	4	no	—
122	C₂×C₈	—	4	4	no	—
123	C₂×C₈	—	4	8	no	—
124	C₂×C₈	C₂×C₁₆	1	4	yes	complemented
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	C₄×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₈	1	2	yes	—
128	C₂×C₈	C₂²×C₈	1	2	yes	—
129	C₂×C₈	C₂×D₈	1	2	yes	—
130	C₂×C₈	#135	1	2	yes	—
131	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₈	C₂×#41	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂²×C₈	1	4	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	C₄×D₄	1	1	yes	characteristic
138	C₄²	C₂²×C₈	1	1	yes	characteristic
139	C₄²	#131	1	1	yes	characteristic
140	C₄²	C₂³×C₄	1	1	yes	characteristic
141	C₁₆	—	2	4	no	—
142	C₁₆	—	2	2	no	—
143	C₁₆	—	4	4	no	—
144	C₁₆	C₂×D₈	1	4	yes	direct product factor
145	C₂³	—	2	4	no	—
146	C₂³	—	2	8	no	—
147	C₂³	—	4	4	no	—
148	C₂³	—	4	4	no	—
149	C₂³	C₈×D₄	1	1	yes	characteristic
150	D₄	—	2	4	no	—
151	D₄	—	2	8	no	—
152	D₄	C₂²×C₁₆	1	8	yes	complemented
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	C₈×D₄	1	1	yes	characteristic
160	C₂×C₄	C₈×D₄	1	1	yes	characteristic
161	C₂×C₄	C₈×D₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₈	1	2	yes	—
163	C₂×C₄	#443	1	2	yes	—
164	C₂×C₄	C₂²×C₁₆	1	2	yes	—
165	C₂×C₄	#504	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
168	C₂×C₄	C₂×#131	1	1	yes	characteristic
169	C₈	—	2	4	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	C₄×D₈	1	2	yes	—
175	C₈	C₂²×C₁₆	1	4	yes	complemented
176	C₈	C₂²×D₈	1	1	yes	characteristic
177	C₈	C₂×#135	1	1	yes	characteristic
178	C₂²	—	2	8	no	—
179	C₂²	—	4	16	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₈×D₈	1	2	yes	—
183	C₂²	D₄×C₁₆	1	2	yes	—
184	C₂²	#2170	1	2	yes	—
185	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	4	no	—
189	C₄	C₈×D₈	1	2	yes	—
190	C₄	D₄×C₁₆	1	2	yes	—
191	C₄	C₂×C₄×D₈	1	1	yes	characteristic
192	C₄	C₂×#443	1	1	yes	characteristic
193	C₄	C₂³×C₁₆	1	1	yes	characteristic
194	C₄	C₂×#504	1	1	yes	characteristic
195	C₂	—	4	8	no	—
196	C₂	C₁₆×D₈	1	4	yes	direct product factor
197	C₂	C₂×C₈×D₈	1	1	yes	characteristic
198	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic
199	C₂	C₂×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₈×D₄
prime radical	C₂²×D₈	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×D₄×C₁₆	C₂	semidirect
3. C₁₆×D₈	C₂	direct (N × H)
4. C₂×#1370	C₂	semidirect
5. C₂×D₄⋊C₁₆	C₂	semidirect
6. C₂×C₈×C₁₆	C₂	semidirect
11. D₄×C₁₆	C₂²	semidirect
21. #1370	C₂²	semidirect
22. D₄⋊C₁₆	C₂²	semidirect
23. C₈×C₁₆	C₂²	semidirect
59. C₂×D₈	C₁₆	direct (N × H)
84. C₂×C₁₆	D₈	direct (N × H)
104. C₂×D₄	C₂×C₁₆	semidirect
114. D₈	C₂×C₁₆	direct (N × H)
124. C₂×C₈	C₂×C₁₆	semidirect
144. C₁₆	C₂×D₈	direct (N × H)
152. D₄	C₂²×C₁₆	semidirect
175. C₈	C₂²×C₁₆	semidirect
196. C₂	C₁₆×D₈	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×D₄×C₁₆	2
C₂×C₈×D₈	2
C₁₆×D₈	2
C₂×#1370	2
C₂×D₄⋊C₁₆	2
C₂×C₈×C₁₆	2

Subgroups — C2×C16×D8 #483484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×D₈ #483484