Subgroups — C₂×C₁₆×C₄⋊C₄ #493113

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₆	C₂	1	2	yes	maximal
3	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₁₆×C₄⋊C₄	C₂	1	8	yes	maximal, direct product factor
6	C₂×#1374	C₂	1	2	yes	maximal
7	C₂³×C₁₆	C₂²	1	2	yes	—
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂²×C₄×C₈	C₂²	1	2	yes	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₁₆	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
16	C₂×C₄×C₁₆	—	2	8	no	—
17	C₂×C₄×C₁₆	C₄	1	8	yes	complemented
18	C₂×C₄×C₁₆	C₂²	1	8	yes	—
19	C₂×C₄×C₁₆	C₂²	1	4	yes	—
20	C₈×C₄⋊C₄	C₄	1	4	yes	—
21	C₈×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂×#336	C₄	1	2	yes	—
23	#1374	C₂²	1	8	yes	—
24	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₈	C₂×C₄	1	2	yes	—
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄⋊C₄	C₈	1	2	yes	—
28	C₂²×C₄²	C₂×C₄	1	2	yes	—
29	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₆	—	2	4	no	—
31	C₂²×C₁₆	—	2	2	no	—
32	C₂²×C₁₆	—	2	2	no	—
33	C₂²×C₁₆	—	2	4	no	—
34	C₂²×C₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₁₆	C₂×C₄	1	8	yes	—
36	C₂²×C₁₆	C₂×C₄	1	4	yes	—
37	C₂²×C₁₆	D₄	1	4	yes	—
38	C₂²×C₁₆	Q₈	1	4	yes	—
39	C₂²×C₁₆	C₂³	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	4	yes	—
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	C₂×C₄⋊C₈	C₈	1	8	yes	—
43	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₂×C₄×C₈	—	2	4	no	—
45	C₂×C₄×C₈	—	2	4	no	—
46	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
50	C₂×C₄×C₈	C₂³	1	4	yes	—
51	C₂×C₄×C₈	C₂³	1	2	yes	—
52	C₄×C₄⋊C₄	C₈	1	4	yes	—
53	C₄×C₄⋊C₄	C₂×C₄	1	4	yes	—
54	C₂×#95	C₈	1	2	yes	—
55	C₄⋊C₁₆	C₂×C₄	1	16	yes	complemented
56	C₄×C₁₆	—	2	16	no	—
57	C₄×C₁₆	C₂×C₄	1	16	yes	complemented
58	#336	C₂×C₄	1	8	yes	—
59	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
60	C₂³×C₄	C₂×C₈	1	4	yes	—
61	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
62	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₈	—	2	4	no	—
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	—	2	2	no	—
66	C₂²×C₈	—	2	4	no	—
67	C₂²×C₈	C₄²	1	4	yes	—
68	C₂²×C₈	C₂×C₈	1	8	yes	—
69	C₂²×C₈	C₂²×C₄	1	2	yes	—
70	C₂²×C₈	C₂²×C₄	1	4	yes	—
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄	1	2	yes	—
73	C₂²×C₈	C₂²×C₄	1	2	yes	—
74	C₂²×C₈	C₂×D₄	1	2	yes	—
75	C₂²×C₈	C₂×Q₈	1	2	yes	—
76	C₂²×C₈	#41	1	2	yes	—
77	C₂²×C₈	#41	1	2	yes	—
78	C₂²×C₈	C₂⁴	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₁₆	1	16	yes	direct product factor
80	C₂×C₄⋊C₄	C₂×C₈	1	8	yes	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	—	2	4	no	—
83	C₂×C₄²	C₄²	1	4	yes	—
84	C₂×C₄²	C₂×C₈	1	4	yes	—
85	C₂×C₄²	C₂×C₈	1	2	yes	—
86	C₂×C₄²	C₂×C₈	1	4	yes	—
87	C₂×C₄²	C₂²×C₄	1	4	yes	—
88	C₂×C₄²	C₂²×C₄	1	2	yes	—
89	C₂×C₁₆	—	2	8	no	—
90	C₂×C₁₆	—	2	16	no	—
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	4	no	—
95	C₂×C₁₆	C₄⋊C₄	1	16	yes	direct product factor
96	C₂×C₁₆	C₂²×C₄	1	4	yes	—
97	C₂×C₁₆	C₂²×C₄	1	8	yes	—
98	C₂×C₁₆	C₂²×C₄	1	4	yes	—
99	C₂×C₁₆	C₂×D₄	1	4	yes	—
100	C₂×C₁₆	C₂×Q₈	1	4	yes	—
101	C₄⋊C₈	C₂×C₈	1	16	yes	—
102	C₄×C₈	—	2	8	no	—
103	C₄×C₈	—	2	4	no	—
104	C₄×C₈	—	2	4	no	—
105	C₄×C₈	C₄²	1	8	yes	—
106	C₄×C₈	C₂²×C₄	1	4	yes	—
107	C₄×C₈	C₂²×C₄	1	4	yes	—
108	#95	C₂×C₈	1	8	yes	—
109	C₂⁴	C₂²×C₈	1	1	yes	characteristic
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	C₄×C₈	1	4	yes	—
114	C₂²×C₄	C₄×C₈	1	4	yes	—
115	C₂²×C₄	C₂×C₁₆	1	16	yes	—
116	C₂²×C₄	C₂×C₁₆	1	8	yes	—
117	C₂²×C₄	C₂×C₄²	1	2	yes	—
118	C₂²×C₄	C₄×D₄	1	2	yes	—
119	C₂²×C₄	C₄×Q₈	1	2	yes	—
120	C₂²×C₄	C₂²×C₈	1	2	yes	—
121	C₂²×C₄	C₂²×C₈	1	8	yes	—
122	C₂²×C₄	C₂²×C₈	1	2	yes	—
123	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
124	C₂²×C₄	C₂²×C₈	1	2	yes	—
125	C₂²×C₄	#131	1	2	yes	—
126	C₂²×C₄	#131	1	2	yes	—
127	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	2	8	no	—
133	C₂×C₈	—	2	16	no	—
134	C₂×C₈	C₂×C₄²	1	4	yes	—
135	C₂×C₈	C₂×C₄²	1	2	yes	—
136	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₈	#117	1	4	yes	—
138	C₂×C₈	C₄×D₄	1	4	yes	—
139	C₂×C₈	C₄×D₄	1	2	yes	—
140	C₂×C₈	C₄×Q₈	1	4	yes	—
141	C₂×C₈	C₄×Q₈	1	2	yes	—
142	C₂×C₈	C₂²×C₈	1	8	yes	—
143	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₈	C₂×#41	1	1	yes	characteristic
148	C₂×C₈	C₂×#41	1	1	yes	characteristic
149	C₄⋊C₄	C₂×C₁₆	1	32	yes	direct product factor
150	C₄²	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	8	no	—
153	C₄²	C₄×C₈	1	8	yes	—
154	C₄²	C₂×C₄²	1	4	yes	—
155	C₄²	C₂²×C₈	1	4	yes	—
156	C₁₆	—	2	4	no	—
157	C₁₆	—	2	8	no	—
158	C₁₆	C₂×C₄⋊C₄	1	8	yes	direct product factor
159	C₂³	C₂×C₄×C₈	1	2	yes	—
160	C₂³	C₈×D₄	1	2	yes	—
161	C₂³	C₈×Q₈	1	2	yes	—
162	C₂³	C₂²×C₁₆	1	4	yes	—
163	C₂³	#504	1	4	yes	—
164	C₂³	C₂³×C₈	1	1	yes	characteristic
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	32	no	—
170	C₂×C₄	C₄×C₁₆	1	16	yes	complemented
171	C₂×C₄	C₄×C₄⋊C₄	1	4	yes	—
172	C₂×C₄	C₂×C₄×C₈	1	4	yes	—
173	C₂×C₄	C₂×C₄×C₈	1	4	yes	—
174	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
175	C₂×C₄	#405	1	4	yes	—
176	C₂×C₄	C₈×D₄	1	4	yes	—
177	C₂×C₄	C₈×D₄	1	2	yes	—
178	C₂×C₄	C₈×Q₈	1	4	yes	—
179	C₂×C₄	C₈×Q₈	1	2	yes	—
180	C₂×C₄	C₂²×C₁₆	1	16	yes	—
181	C₂×C₄	C₂²×C₁₆	1	8	yes	—
182	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
186	C₂×C₄	C₂×#131	1	1	yes	characteristic
187	C₂×C₄	C₂×#131	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₈	—	2	8	no	—
190	C₈	C₄×C₄⋊C₄	1	4	yes	—
191	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₈	C₂×#117	1	1	yes	characteristic
193	C₈	C₂×C₄×D₄	1	1	yes	characteristic
194	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
195	C₂²	C₈×C₄⋊C₄	1	4	yes	—
196	C₂²	C₂×C₄×C₁₆	1	4	yes	—
197	C₂²	#2108	1	4	yes	—
198	C₂²	D₄×C₁₆	1	8	yes	—
199	C₂²	Q₈×C₁₆	1	8	yes	—
200	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
201	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
202	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
203	C₂²	C₂³×C₁₆	1	2	yes	—
204	C₂²	C₂×#504	1	2	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	2	16	no	—
207	C₄	C₈×C₄⋊C₄	1	4	yes	—
208	C₄	C₂×C₄×C₁₆	1	8	yes	complemented
209	C₄	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
210	C₄	C₂×#405	1	1	yes	characteristic
211	C₄	C₂×C₈×D₄	1	1	yes	characteristic
212	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
213	C₂	C₁₆×C₄⋊C₄	1	8	yes	direct product factor
214	C₂	C₂×C₈×C₄⋊C₄	1	1	yes	characteristic
215	C₂	C₂²×C₄×C₁₆	1	1	yes	characteristic
216	C₂	C₂×#2108	1	1	yes	characteristic
217	C₂	C₂×D₄×C₁₆	1	2	yes	—
218	C₂	C₂×Q₈×C₁₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₁₆×C₄⋊C₄	C₂	direct (N × H)
14. C₂×C₄⋊C₁₆	C₄	semidirect
17. C₂×C₄×C₁₆	C₄	semidirect
55. C₄⋊C₁₆	C₂×C₄	semidirect
57. C₄×C₁₆	C₂×C₄	semidirect
79. C₂×C₄⋊C₄	C₁₆	direct (N × H)
95. C₂×C₁₆	C₄⋊C₄	direct (N × H)
149. C₄⋊C₄	C₂×C₁₆	direct (N × H)
158. C₁₆	C₂×C₄⋊C₄	direct (N × H)
170. C₂×C₄	C₄×C₁₆	semidirect
208. C₄	C₂×C₄×C₁₆	semidirect
213. C₂	C₁₆×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₁₆	2
C₂²×C₄×C₁₆	2
C₂×C₈×C₄⋊C₄	2
C₁₆×C₄⋊C₄	2
C₂×#1374	2

Subgroups — C2×C16×C4⋊C4 #493113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×C₄⋊C₄ #493113