Subgroups — C₂×C₁₆×#41 #7622649

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#41	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×D₄×C₁₆	C₂	1	6	yes	maximal, complemented
5	C₂×#2162	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄×C₁₆	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#41	C₄	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	6	yes	—
9	C₈×#41	C₄	1	8	yes	—
10	C₈×#41	C₂²	1	8	yes	—
11	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
12	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₄	C₄	1	3	yes	—
14	C₂×C₈×D₄	C₂²	1	3	yes	—
15	C₂×#431	C₄	1	3	yes	—
16	C₂²×C₄×C₈	C₂²	1	3	yes	—
17	Q₈×C₁₆	C₂²	1	16	yes	complemented
18	D₄×C₁₆	C₂²	1	48	yes	complemented
19	#2162	C₂²	1	24	yes	complemented
20	C₂×C₄⋊C₁₆	C₂²	1	6	yes	—
21	C₂×C₂²⋊C₁₆	C₂²	1	6	yes	—
22	C₂×C₄×C₁₆	C₂²	1	3	yes	—
23	C₂×C₄×C₁₆	C₂²	1	6	yes	—
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₁₆	C₂²	1	24	yes	complemented
26	C₂²×#41	C₈	1	1	yes	characteristic
27	C₂³×C₈	C₂×C₄	1	3	yes	—
28	C₂³×C₈	C₂³	1	3	yes	—
29	C₄×#41	C₈	1	8	yes	—
30	C₄×#41	C₂×C₄	1	8	yes	—
31	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
32	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₄	C₈	1	3	yes	—
34	C₂×C₄×D₄	C₂×C₄	1	3	yes	—
35	C₂×#117	C₈	1	3	yes	—
36	C₂²×C₄²	C₂×C₄	1	3	yes	—
37	C₂²×C₁₆	—	2	24	no	—
38	C₂²×C₁₆	C₂³	1	6	yes	—
39	C₂²×C₁₆	C₂³	1	24	yes	—
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	C₈×Q₈	C₂×C₄	1	8	yes	—
42	C₈×Q₈	C₂×C₄	1	4	yes	—
43	C₈×Q₈	C₂³	1	4	yes	—
44	C₈×D₄	C₂×C₄	1	24	yes	—
45	C₈×D₄	C₂×C₄	1	12	yes	—
46	C₈×D₄	C₂³	1	12	yes	—
47	#431	C₂×C₄	1	24	yes	—
48	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
49	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
50	C₂×C₂²⋊C₈	C₂×C₄	1	6	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
52	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
53	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	3	yes	—
56	C₂×C₄×C₈	C₂³	1	12	yes	—
57	C₄⋊C₁₆	C₂³	1	24	yes	—
58	C₂²⋊C₁₆	C₂³	1	24	yes	—
59	C₄×C₁₆	C₂³	1	12	yes	—
60	C₄×C₁₆	C₂³	1	24	yes	—
61	C₄×C₁₆	C₂³	1	4	yes	—
62	C₂×#41	C₁₆	1	16	yes	direct product factor
63	C₂×#41	C₂×C₈	1	8	yes	—
64	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
65	C₂²×D₄	C₂×C₈	1	3	yes	—
66	C₂³×C₄	C₂×C₈	1	3	yes	—
67	C₂³×C₄	C₂×C₈	1	3	yes	—
68	C₂³×C₄	C₂²×C₄	1	3	yes	—
69	C₂²×C₈	—	2	12	no	—
70	C₂²×C₈	—	2	12	no	—
71	C₂²×C₈	C₂²×C₄	1	3	yes	—
72	C₂²×C₈	C₂²×C₄	1	12	yes	—
73	C₂²×C₈	C₂²×C₄	1	3	yes	—
74	C₂²×C₈	C₂²×C₄	1	6	yes	—
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂⁴	1	6	yes	—
77	C₂²×C₈	C₂⁴	1	1	yes	characteristic
78	C₄×Q₈	C₂×C₈	1	8	yes	—
79	C₄×Q₈	C₂×C₈	1	4	yes	—
80	C₄×Q₈	C₂²×C₄	1	4	yes	—
81	C₄×D₄	C₂×C₈	1	24	yes	—
82	C₄×D₄	C₂×C₈	1	12	yes	—
83	C₄×D₄	C₂²×C₄	1	12	yes	—
84	#117	C₂×C₈	1	24	yes	—
85	C₂×C₄⋊C₄	C₂×C₈	1	3	yes	—
86	C₂×C₄⋊C₄	C₂×C₈	1	3	yes	—
87	C₂×C₂²⋊C₄	C₂×C₈	1	6	yes	—
88	C₂×C₄²	C₂×C₈	1	3	yes	—
89	C₂×C₄²	C₂×C₈	1	3	yes	—
90	C₂×C₄²	C₂×C₈	1	12	yes	—
91	C₂×C₄²	C₂²×C₄	1	3	yes	—
92	C₂×C₄²	C₂²×C₄	1	12	yes	—
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₁₆	—	2	12	no	—
95	C₂×C₁₆	—	2	48	no	—
96	C₂×C₁₆	#41	1	8	yes	direct product factor
97	C₂×C₁₆	C₂⁴	1	12	yes	—
98	C₂×C₁₆	C₂⁴	1	4	yes	—
99	C₄⋊C₈	C₂²×C₄	1	12	yes	—
100	C₄⋊C₈	C₂²×C₄	1	12	yes	—
101	C₂²⋊C₈	C₂²×C₄	1	24	yes	—
102	C₄×C₈	C₂²×C₄	1	2	yes	—
103	C₄×C₈	C₂²×C₄	1	12	yes	—
104	C₄×C₈	C₂²×C₄	1	12	yes	—
105	C₄×C₈	C₂²×C₄	1	6	yes	—
106	C₄×C₈	C₂⁴	1	2	yes	—
107	C₄×C₈	C₂⁴	1	6	yes	—
108	C₂⁴	C₂²×C₈	1	3	yes	—
109	#41	C₂×C₁₆	1	64	yes	direct product factor
110	C₂×Q₈	C₂×C₁₆	1	8	yes	complemented
111	C₂×Q₈	C₂²×C₈	1	4	yes	—
112	C₂×D₄	C₂×C₁₆	1	24	yes	complemented
113	C₂×D₄	C₂²×C₈	1	12	yes	—
114	C₂²×C₄	—	2	6	no	—
115	C₂²×C₄	—	2	6	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	C₂×C₁₆	1	24	yes	complemented
118	C₂²×C₄	C₂²×C₈	1	3	yes	—
119	C₂²×C₄	C₂²×C₈	1	6	yes	—
120	C₂²×C₄	C₂²×C₈	1	3	yes	—
121	C₂²×C₄	C₂²×C₈	1	12	yes	—
122	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₈	1	3	yes	—
124	C₂²×C₄	C₂²×C₈	1	12	yes	—
125	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₄	1	6	yes	—
127	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₈	—	2	6	no	—
129	C₂×C₈	—	2	6	no	—
130	C₂×C₈	—	2	12	no	—
131	C₂×C₈	—	2	24	no	—
132	C₂×C₈	—	2	12	no	—
133	C₂×C₈	#131	1	4	yes	—
134	C₂×C₈	C₂³×C₄	1	6	yes	—
135	C₂×C₈	C₂³×C₄	1	6	yes	—
136	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₈	C₂³×C₄	1	2	yes	—
138	C₂×C₈	C₂×#41	1	4	yes	—
139	C₂×C₈	C₂⁵	1	1	yes	characteristic
140	C₄⋊C₄	C₂²×C₈	1	12	yes	—
141	C₄⋊C₄	C₂²×C₈	1	12	yes	—
142	C₂²⋊C₄	C₂²×C₈	1	24	yes	—
143	C₄²	C₂²×C₈	1	2	yes	—
144	C₄²	C₂²×C₈	1	12	yes	—
145	C₄²	C₂²×C₈	1	12	yes	—
146	C₄²	C₂²×C₈	1	6	yes	—
147	C₄²	C₂³×C₄	1	2	yes	—
148	C₄²	C₂³×C₄	1	6	yes	—
149	C₁₆	—	2	12	no	—
150	C₁₆	C₂×#41	1	8	yes	direct product factor
151	C₂³	—	2	6	no	—
152	C₂³	—	2	6	no	—
153	C₂³	C₂²×C₁₆	1	12	yes	—
154	C₂³	C₂³×C₈	1	6	yes	—
155	C₂³	C₂³×C₈	1	1	yes	characteristic
156	Q₈	C₂²×C₁₆	1	16	yes	complemented
157	D₄	C₂²×C₁₆	1	48	yes	complemented
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	2	24	no	—
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	6	no	—
164	C₂×C₄	C₂²×C₁₆	1	4	yes	—
165	C₂×C₄	C₂²×C₁₆	1	12	yes	—
166	C₂×C₄	C₂²×C₁₆	1	48	yes	complemented
167	C₂×C₄	#504	1	4	yes	—
168	C₂×C₄	C₄×#41	1	2	yes	—
169	C₂×C₄	C₂³×C₈	1	6	yes	—
170	C₂×C₄	C₂³×C₈	1	6	yes	—
171	C₂×C₄	C₂³×C₈	1	2	yes	—
172	C₂×C₄	C₂³×C₈	1	6	yes	—
173	C₂×C₄	C₂³×C₈	1	2	yes	—
174	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
175	C₂×C₄	C₂×#131	1	2	yes	—
176	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
177	C₈	—	2	6	no	—
178	C₈	—	2	6	no	—
179	C₈	C₄×#41	1	2	yes	—
180	C₈	C₂×#131	1	4	yes	—
181	C₈	C₂²×#41	1	2	yes	—
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	—	2	24	no	—
185	C₂²	C₈×#41	1	2	yes	—
186	C₂²	C₂³×C₁₆	1	12	yes	—
187	C₂²	C₂³×C₁₆	1	2	yes	—
188	C₂²	C₂×#504	1	2	yes	—
189	C₂²	C₂⁴×C₈	1	1	yes	characteristic
190	C₄	—	2	6	no	—
191	C₄	—	2	6	no	—
192	C₄	C₈×#41	1	2	yes	—
193	C₄	C₂³×C₁₆	1	4	yes	—
194	C₄	C₂³×C₁₆	1	12	yes	—
195	C₄	C₂×#504	1	4	yes	—
196	C₄	C₂×C₄×#41	1	1	yes	characteristic
197	C₄	C₂²×#131	1	1	yes	characteristic
198	C₂	—	2	12	no	—
199	C₂	C₁₆×#41	1	4	yes	direct product factor
200	C₂	C₂×C₈×#41	1	1	yes	characteristic
201	C₂	C₂⁴×C₁₆	1	1	yes	characteristic
202	C₂	C₂²×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₆	C₂²
commutator	C₂	C₂⁴×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂³	C₂³×C₈
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₁₆×#41	C₂	direct (N × H)
3. C₂×Q₈×C₁₆	C₂	semidirect
4. C₂×D₄×C₁₆	C₂	semidirect
5. C₂×#2162	C₂	semidirect
6. C₂²×C₄×C₁₆	C₂	semidirect
17. Q₈×C₁₆	C₂²	semidirect
18. D₄×C₁₆	C₂²	semidirect
19. #2162	C₂²	semidirect
25. C₂×C₄×C₁₆	C₂²	semidirect
62. C₂×#41	C₁₆	direct (N × H)
96. C₂×C₁₆	#41	direct (N × H)
109. #41	C₂×C₁₆	direct (N × H)
110. C₂×Q₈	C₂×C₁₆	semidirect
112. C₂×D₄	C₂×C₁₆	semidirect
117. C₂²×C₄	C₂×C₁₆	semidirect
150. C₁₆	C₂×#41	direct (N × H)
156. Q₈	C₂²×C₁₆	semidirect
157. D₄	C₂²×C₁₆	semidirect
166. C₂×C₄	C₂²×C₁₆	semidirect
199. C₂	C₁₆×#41	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×C₈×#41	2
C₁₆×#41	2
C₂×Q₈×C₁₆	2
C₂×D₄×C₁₆	2
C₂×#2162	2
C₂²×C₄×C₁₆	2

Subgroups — C2×C16×#41 #7622649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×#41 #7622649