Subgroups — C₂×D₃×C₄⋊C₁₆ #10948160

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×C₄⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₄⋊C₁₆	—	3	1	no	maximal, Sylow
8	C₂²×D₃×C₈	C₄	1	1	yes	characteristic
9	C₂²×D₃×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₃	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
17	C₂×D₃×C₁₆	—	2	8	no	—
18	C₂×D₃×C₁₆	C₂²	1	8	yes	—
19	C₄×D₃×C₈	C₄	1	4	yes	—
20	C₄×D₃×C₈	C₂²	1	4	yes	—
21	C₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	Dic₃⋊C₁₆	C₂²	1	8	yes	complemented
23	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
24	C₂³×C₁₆	—	3	2	no	—
25	C₂²×C₄×C₈	—	3	1	no	—
26	C₂×C₄⋊C₁₆	—	3	2	no	—
27	C₂×C₄⋊C₁₆	—	3	1	no	—
28	C₂×C₄⋊C₁₆	—	3	8	no	—
29	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic, direct product factor
30	C₂²×C₄×D₃	C₈	1	1	yes	characteristic
31	C₂²×C₄×D₃	C₈	1	1	yes	characteristic
32	C₂²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₃	C₈	1	1	yes	characteristic
37	C₂×C₄×Dic₃	C₈	1	1	yes	characteristic
38	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×D₃×C₈	—	2	4	no	—
42	C₂×D₃×C₈	C₂×C₄	1	4	yes	—
43	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
44	C₂×D₃×C₈	D₄	1	4	yes	—
45	C₂×D₃×C₈	Q₈	1	4	yes	—
46	C₂×D₃×C₈	C₂³	1	2	yes	—
47	C₄²×D₃	C₈	1	4	yes	—
48	C₄²×D₃	C₂×C₄	1	4	yes	—
49	C₂×C₃×C₁₆	—	2	4	no	—
50	C₂×C₃×C₁₆	C₂³	1	4	yes	—
51	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
52	C₃×C₄×C₈	C₂³	1	2	yes	—
53	C₈×Dic₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₃	C₂×C₄	1	4	yes	—
55	C₈×Dic₃	C₂³	1	2	yes	—
56	C₂×C₃⋊C₁₆	—	2	4	no	—
57	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
58	C₄×C₃⋊C₈	C₂×C₄	1	4	yes	—
59	D₃×C₁₆	—	2	16	no	—
60	C₂³×C₈	—	3	1	no	—
61	C₂³×C₈	—	3	1	no	—
62	C₂²×C₄²	—	3	1	no	—
63	C₂²×C₁₆	—	3	2	no	—
64	C₂²×C₁₆	—	3	8	no	—
65	C₂²×C₁₆	—	6	8	no	—
66	C₂²×C₁₆	C₂×D₃	1	2	yes	—
67	C₂×C₄×C₈	—	3	1	no	—
68	C₂×C₄×C₈	—	3	1	no	—
69	C₂×C₄×C₈	—	3	1	no	—
70	C₂×C₄×C₈	—	3	4	no	—
71	C₂×C₄×C₈	—	3	4	no	—
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₄⋊C₁₆	—	3	8	no	—
74	C₄⋊C₁₆	—	3	4	no	—
75	C₄⋊C₁₆	C₂×D₃	1	4	yes	direct product factor
76	C₂³×D₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₃	—	2	4	no	—
84	C₂×C₄×D₃	C₁₆	1	8	yes	complemented
85	C₂×C₄×D₃	C₄⋊C₄	1	4	yes	—
86	C₂×C₄×D₃	C₂×C₈	1	4	yes	—
87	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₃	C₂×C₈	1	4	yes	—
89	C₂×C₄×D₃	M₁₆	1	4	yes	—
90	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
96	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
97	C₃×C₄²	C₂×C₈	1	2	yes	—
98	C₃×C₄²	C₂²×C₄	1	2	yes	—
99	C₄×Dic₃	C₂×C₈	1	2	yes	—
100	C₄×Dic₃	C₂×C₈	1	4	yes	—
101	C₄×Dic₃	C₂×C₈	1	4	yes	—
102	C₄×Dic₃	C₂²×C₄	1	2	yes	—
103	C₂×C₃⋊C₈	—	2	2	no	—
104	C₂×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
105	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
107	D₃×C₈	—	2	8	no	—
108	D₃×C₈	C₄⋊C₄	1	8	yes	—
109	D₃×C₈	C₂×D₄	1	4	yes	—
110	D₃×C₈	C₂×Q₈	1	4	yes	—
111	C₃×C₁₆	—	2	4	no	—
112	C₃⋊C₁₆	—	2	4	no	—
113	C₂³×C₄	—	3	1	no	—
114	C₂³×C₄	—	3	1	no	—
115	C₂³×C₄	—	3	1	no	—
116	C₂²×C₈	—	3	1	no	—
117	C₂²×C₈	—	3	2	no	—
118	C₂²×C₈	—	3	4	no	—
119	C₂²×C₈	—	3	4	no	—
120	C₂²×C₈	—	3	2	no	—
121	C₂²×C₈	—	3	1	no	—
122	C₂²×C₈	—	3	4	no	—
123	C₂²×C₈	—	6	4	no	—
124	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₄²	—	3	1	no	—
127	C₂×C₄²	—	3	1	no	—
128	C₂×C₄²	—	3	1	no	—
129	C₂×C₄²	—	3	4	no	—
130	C₂×C₄²	—	3	4	no	—
131	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	—	3	4	no	—
134	C₂×C₁₆	—	6	16	no	—
135	C₂×C₁₆	—	6	4	no	—
136	C₂×C₁₆	C₂²×D₃	1	4	yes	—
137	C₄×C₈	—	3	4	no	—
138	C₄×C₈	—	3	2	no	—
139	C₄×C₈	—	3	2	no	—
140	C₄×C₈	—	3	4	no	—
141	C₄×C₈	C₄×D₃	1	2	yes	—
142	C₄×C₈	C₂²×D₃	1	2	yes	—
143	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
144	C₂²×D₃	C₄⋊C₈	1	4	yes	—
145	C₂²×D₃	C₂×C₁₆	1	4	yes	—
146	C₂²×D₃	M₃₂	1	4	yes	—
147	C₂²×D₃	C₂²×C₈	1	2	yes	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	complemented
150	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
152	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
154	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
155	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
156	C₂×Dic₃	—	2	2	no	—
157	C₂×Dic₃	C₄⋊C₈	1	4	yes	—
158	C₂×Dic₃	C₂×C₁₆	1	4	yes	complemented
159	C₂×Dic₃	C₂²×C₈	1	2	yes	—
160	C₂×Dic₃	C₂²×C₈	1	2	yes	—
161	C₂×Dic₃	C₂²×C₈	1	2	yes	—
162	C₄×D₃	—	2	8	no	—
163	C₄×D₃	C₄⋊C₈	1	8	yes	—
164	C₄×D₃	C₂×C₁₆	1	16	yes	complemented
165	C₄×D₃	C₂×C₄⋊C₄	1	4	yes	—
166	C₄×D₃	C₂×M₁₆	1	4	yes	—
167	C₃×C₈	—	2	2	no	—
168	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
169	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
170	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
171	C₃⋊C₈	—	2	2	no	—
172	C₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
173	C₂⁴	—	3	1	no	—
174	C₂²×C₄	—	3	1	no	—
175	C₂²×C₄	—	3	1	no	—
176	C₂²×C₄	—	3	8	no	—
177	C₂²×C₄	—	3	4	no	—
178	C₂²×C₄	—	3	4	no	—
179	C₂²×C₄	—	3	2	no	—
180	C₂²×C₄	—	3	4	no	—
181	C₂²×C₄	—	3	2	no	—
182	C₂²×C₄	—	3	1	no	—
183	C₂²×C₄	—	3	4	no	—
184	C₂²×C₄	—	6	4	no	—
185	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
186	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	3	4	no	—
190	C₂×C₈	—	3	2	no	—
191	C₂×C₈	—	3	8	no	—
192	C₂×C₈	—	3	4	no	—
193	C₂×C₈	—	3	4	no	—
194	C₂×C₈	—	3	2	no	—
195	C₂×C₈	—	6	8	no	—
196	C₂×C₈	—	6	2	no	—
197	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₈	D₃×D₄	1	2	yes	—
200	C₂×C₈	D₃×Q₈	1	2	yes	—
201	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
202	C₄²	—	3	4	no	—
203	C₄²	—	3	4	no	—
204	C₄²	—	3	2	no	—
205	C₄²	—	3	2	no	—
206	C₄²	D₃×C₈	1	2	yes	—
207	C₄²	C₂×C₄×D₃	1	2	yes	—
208	C₁₆	—	2	4	no	—
209	C₁₆	—	6	4	no	—
210	C₂²×C₃	C₂×C₄⋊C₈	1	2	yes	—
211	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
212	C₂²×C₃	C₂×M₃₂	1	2	yes	—
213	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
214	C₂×D₃	C₄⋊C₁₆	1	16	yes	direct product factor
215	C₂×D₃	C₂×C₄⋊C₈	1	4	yes	—
216	C₂×D₃	C₂²×C₁₆	1	4	yes	—
217	C₂×D₃	C₂×M₃₂	1	4	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	C₂×C₄⋊C₈	1	2	yes	—
220	C₃×C₄	C₂²×C₁₆	1	4	yes	complemented
221	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
223	Dic₃	—	2	2	no	—
224	Dic₃	C₂×C₄⋊C₈	1	4	yes	—
225	Dic₃	C₂²×C₁₆	1	4	yes	complemented
226	C₂³	—	3	4	no	—
227	C₂³	—	3	4	no	—
228	C₂³	—	3	2	no	—
229	C₂³	—	3	4	no	—
230	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	3	4	no	—
233	C₂×C₄	—	3	2	no	—
234	C₂×C₄	—	3	2	no	—
235	C₂×C₄	—	3	4	no	—
236	C₂×C₄	—	3	16	no	—
237	C₂×C₄	—	3	4	no	—
238	C₂×C₄	—	3	4	no	—
239	C₂×C₄	—	3	8	no	—
240	C₂×C₄	—	3	2	no	—
241	C₂×C₄	—	6	8	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	D₃×C₁₆	1	4	yes	complemented
244	C₂×C₄	D₃×C₄⋊C₄	1	2	yes	—
245	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
246	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
247	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
248	C₂×C₄	D₃×M₁₆	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	3	4	no	—
252	C₈	—	6	2	no	—
253	C₈	D₃×C₄⋊C₄	1	2	yes	—
254	C₈	C₂×D₃×D₄	1	1	yes	characteristic
255	C₈	C₂×D₃×Q₈	1	1	yes	characteristic
256	C₂×C₃	C₂×C₄⋊C₁₆	1	4	yes	complemented
257	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
258	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
259	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
260	D₃	C₂×C₄⋊C₁₆	1	8	yes	direct product factor
261	C₂²	—	3	4	no	—
262	C₂²	—	3	4	no	—
263	C₂²	—	3	4	no	—
264	C₂²	—	3	16	no	—
265	C₂²	D₃×C₄⋊C₈	1	2	yes	—
266	C₂²	C₂×D₃×C₁₆	1	2	yes	—
267	C₂²	D₃×M₃₂	1	2	yes	—
268	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	3	4	no	—
271	C₄	—	3	4	no	—
272	C₄	—	6	2	no	—
273	C₄	D₃×C₄⋊C₈	1	2	yes	—
274	C₄	C₂×D₃×C₁₆	1	4	yes	complemented
275	C₄	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
276	C₄	C₂×D₃×M₁₆	1	1	yes	characteristic
277	C₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
278	C₂	—	3	8	no	—
279	C₂	D₃×C₄⋊C₁₆	1	4	yes	direct product factor
280	C₂	C₂×D₃×C₄⋊C₈	1	1	yes	characteristic
281	C₂	C₂²×D₃×C₁₆	1	1	yes	characteristic
282	C₂	C₂×D₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂×C₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂³×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄⋊C₁₆	C₂	semidirect
4. C₂×Dic₃⋊C₁₆	C₂	semidirect
5. C₂×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
6. D₃×C₄⋊C₁₆	C₂	direct (N × H)
21. C₃×C₄⋊C₁₆	C₂²	semidirect
22. Dic₃⋊C₁₆	C₂²	semidirect
23. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
29. C₂×C₄⋊C₁₆	D₃	direct (N × H)
75. C₄⋊C₁₆	C₂×D₃	direct (N × H)
84. C₂×C₄×D₃	C₁₆	semidirect
149. C₂×C₃×C₄	C₂×C₁₆	semidirect
158. C₂×Dic₃	C₂×C₁₆	semidirect
164. C₄×D₃	C₂×C₁₆	semidirect
214. C₂×D₃	C₄⋊C₁₆	direct (N × H)
220. C₃×C₄	C₂²×C₁₆	semidirect
225. Dic₃	C₂²×C₁₆	semidirect
243. C₂×C₄	D₃×C₁₆	semidirect
256. C₂×C₃	C₂×C₄⋊C₁₆	semidirect
260. D₃	C₂×C₄⋊C₁₆	direct (N × H)
274. C₄	C₂×D₃×C₁₆	semidirect
277. C₃	C₂²×C₄⋊C₁₆	semidirect
279. C₂	D₃×C₄⋊C₁₆	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×D₃×C₁₆	2
C₂×C₄×D₃×C₈	2
C₂×C₃×C₄⋊C₁₆	2
C₂×Dic₃⋊C₁₆	2
C₂×C₄⋊(C₃⋊C₁₆)	2
D₃×C₄⋊C₁₆	2
C₂²×C₄⋊C₁₆	3

Subgroups — C2×D3×C4⋊C16 #10948160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₄⋊C₁₆ #10948160