Subgroups — C₂×D₃×C₉×Q₁₆ #13153459

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₈×C₉	C₂	1	2	yes	maximal
2	C₂×C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#216	C₂	1	2	yes	maximal, complemented
4	D₃×C₉×Q₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×D₃×Q₁₆	C₃	1	1	yes	maximal, characteristic
8	C₂²×C₉×Q₁₆	—	3	1	no	maximal
9	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
10	D₃×Q₈×C₉	—	2	4	no	—
11	D₃×Q₈×C₉	C₂²	1	8	yes	—
12	C₂×C₄×D₃×C₉	—	2	2	no	—
13	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₉×Dic₆	—	2	2	no	—
15	C₂×C₉×Dic₆	C₂²	1	2	yes	—
16	C₃×C₉×Q₁₆	C₂²	1	4	yes	complemented
17	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₉×#216	C₂²	1	8	yes	complemented
19	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₉×Dic₁₂	C₂²	1	4	yes	complemented
21	D₃×C₈×C₉	C₂²	1	4	yes	—
22	C₂×C₃×D₃×Q₈	C₂×C₃	1	2	yes	—
23	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#216	C₂×C₃	1	2	yes	—
25	C₃×D₃×Q₁₆	C₂×C₃	1	8	yes	—
26	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×D₃×C₈	C₂×C₃	1	1	yes	characteristic
28	C₂²×Q₈×C₉	—	3	2	no	—
29	C₂×C₉×Q₁₆	—	2	1	no	—
30	C₂×C₉×Q₁₆	—	3	2	no	—
31	C₂×C₉×Q₁₆	—	3	8	no	—
32	C₂×C₉×Q₁₆	—	3	1	no	—
33	C₂×C₉×Q₁₆	D₃	1	1	yes	characteristic, direct product factor
34	C₂²×C₈×C₉	—	3	1	no	—
35	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
36	C₃×Q₈×C₉	—	2	2	no	—
37	C₃×Q₈×C₉	C₂³	1	4	yes	—
38	C₂×C₃×C₄×C₉	—	2	2	no	—
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₉×Dic₃	—	2	2	no	—
41	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
42	C₄×D₃×C₉	—	2	8	no	—
43	C₄×D₃×C₉	D₄	1	2	yes	—
44	C₄×D₃×C₉	C₂³	1	2	yes	—
45	C₉×Dic₆	—	2	2	no	—
46	C₉×Dic₆	—	2	8	no	—
47	C₉×Dic₆	C₂³	1	4	yes	—
48	C₃×C₈×C₉	C₂³	1	2	yes	—
49	C₉×C₃⋊C₈	C₂³	1	2	yes	—
50	C₂²×C₃×Q₁₆	—	3	1	no	—
51	C₂×D₃×Q₁₆	C₉	1	1	yes	characteristic, direct product factor
52	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
53	C₃×D₃×Q₈	—	2	4	no	—
54	C₃×D₃×Q₈	C₂²×C₃	1	8	yes	—
55	C₂×C₃×C₄×D₃	—	2	2	no	—
56	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×Dic₆	—	2	2	no	—
58	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
59	C₃²×Q₁₆	C₂²×C₃	1	4	yes	—
60	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃×#216	C₂²×C₃	1	8	yes	—
62	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₃×Dic₁₂	C₂²×C₃	1	4	yes	—
64	C₃×D₃×C₈	C₂²×C₃	1	4	yes	—
65	C₂×Q₈×C₉	—	2	2	no	—
66	C₂×Q₈×C₉	—	3	8	no	—
67	C₂×Q₈×C₉	—	3	2	no	—
68	C₂×Q₈×C₉	—	6	2	no	—
69	C₂×Q₈×C₉	—	6	4	no	—
70	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
71	C₂²×C₄×C₉	—	3	1	no	—
72	C₂²×C₄×C₉	—	6	2	no	—
73	C₉×Q₁₆	—	2	4	no	—
74	C₉×Q₁₆	—	3	8	no	—
75	C₉×Q₁₆	—	3	4	no	—
76	C₉×Q₁₆	C₂×D₃	1	4	yes	direct product factor
77	C₂×C₈×C₉	—	2	1	no	—
78	C₂×C₈×C₉	—	3	1	no	—
79	C₂×C₈×C₉	—	3	4	no	—
80	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×D₃×C₉	Q₁₆	1	4	yes	direct product factor
83	C₂×D₃×C₉	C₂×D₄	1	2	yes	—
84	C₃×C₄×C₉	—	2	4	no	—
85	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
87	C₉×Dic₃	—	2	4	no	—
88	C₉×Dic₃	C₂×D₄	1	2	yes	—
89	C₂²×C₃×Q₈	—	3	2	no	—
90	C₂×D₃×Q₈	C₂×C₉	1	2	yes	—
91	C₂×C₃×Q₁₆	—	2	1	no	—
92	C₂×C₃×Q₁₆	—	3	2	no	—
93	C₂×C₃×Q₁₆	—	3	8	no	—
94	C₂×C₃×Q₁₆	—	3	1	no	—
95	C₂×C₃×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
96	C₂×C₃×Q₁₆	C₃×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₈	—	3	1	no	—
98	C₂×#216	C₂×C₉	1	2	yes	complemented
99	D₃×Q₁₆	C₂×C₉	1	8	yes	direct product factor
100	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
101	C₂×D₃×C₈	C₂×C₉	1	1	yes	characteristic
102	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
103	C₃²×Q₈	—	2	2	no	—
104	C₃²×Q₈	C₂³×C₃	1	4	yes	—
105	C₂×C₃²×C₄	—	2	2	no	—
106	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	—	2	2	no	—
108	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
109	C₃×C₄×D₃	—	2	8	no	—
110	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
111	C₃×C₄×D₃	C₂³×C₃	1	2	yes	—
112	C₃×Dic₆	—	2	2	no	—
113	C₃×Dic₆	—	2	8	no	—
114	C₃×Dic₆	C₂³×C₃	1	4	yes	—
115	C₂³×C₉	—	3	1	no	—
116	C₃²×C₈	C₂³×C₃	1	2	yes	—
117	C₃×C₃⋊C₈	C₂³×C₃	1	2	yes	—
118	Q₈×C₉	—	2	2	no	—
119	Q₈×C₉	—	2	4	no	—
120	Q₈×C₉	—	3	4	no	—
121	Q₈×C₉	—	4	2	no	—
122	Q₈×C₉	—	6	8	no	—
123	Q₈×C₉	—	6	2	no	—
124	Q₈×C₉	C₂²×D₃	1	4	yes	—
125	C₂×C₄×C₉	—	2	2	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	3	1	no	—
128	C₂×C₄×C₉	—	3	2	no	—
129	C₂×C₄×C₉	—	3	2	no	—
130	C₂×C₄×C₉	—	4	2	no	—
131	C₂×C₄×C₉	—	6	8	no	—
132	C₂×C₄×C₉	—	6	2	no	—
133	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
134	C₈×C₉	—	2	2	no	—
135	C₈×C₉	—	3	2	no	—
136	C₈×C₉	C₂²×D₃	1	2	yes	—
137	C₂²×Q₁₆	—	3	1	no	Sylow
138	C₂×C₃×C₉	C₂×Q₁₆	1	2	yes	complemented
139	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
140	D₃×C₉	C₂×Q₁₆	1	4	yes	direct product factor
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	—	3	8	no	—
143	C₂×C₃×Q₈	—	3	2	no	—
144	C₂×C₃×Q₈	—	6	2	no	—
145	C₂×C₃×Q₈	—	6	4	no	—
146	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
147	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
148	C₂²×C₃×C₄	—	3	1	no	—
149	C₂²×C₃×C₄	—	6	2	no	—
150	D₃×Q₈	—	2	4	no	—
151	D₃×Q₈	C₂²×C₉	1	8	yes	—
152	C₂×C₄×D₃	—	2	2	no	—
153	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
154	C₂×Dic₆	—	2	2	no	—
155	C₂×Dic₆	C₂²×C₉	1	2	yes	—
156	C₃×Q₁₆	—	2	4	no	—
157	C₃×Q₁₆	—	3	8	no	—
158	C₃×Q₁₆	—	3	4	no	—
159	C₃×Q₁₆	C₂²×C₉	1	4	yes	complemented
160	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₂×C₃×C₈	—	3	1	no	—
163	C₂×C₃×C₈	—	3	4	no	—
164	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
165	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
166	#216	C₂²×C₉	1	8	yes	complemented
167	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
168	Dic₁₂	C₂²×C₉	1	4	yes	complemented
169	D₃×C₈	C₂²×C₉	1	4	yes	—
170	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₃×D₃	C₃×Q₁₆	1	4	yes	—
172	C₂×C₃×D₃	C₂×C₃×D₄	1	2	yes	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
176	C₃×Dic₃	—	2	4	no	—
177	C₃×Dic₃	C₂×C₃×D₄	1	2	yes	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	3	4	no	—
180	C₂²×C₉	—	3	2	no	—
181	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
182	C₄×C₉	—	2	4	no	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	3	2	no	—
186	C₄×C₉	—	4	4	no	—
187	C₄×C₉	—	6	4	no	—
188	C₄×C₉	D₃×D₄	1	1	yes	characteristic
189	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
190	C₂²×Q₈	—	3	2	no	—
191	C₂×Q₁₆	—	3	2	no	—
192	C₂×Q₁₆	—	3	8	no	—
193	C₂×Q₁₆	—	3	1	no	—
194	C₂×Q₁₆	D₃×C₉	1	1	yes	characteristic, direct product factor
195	C₂²×C₈	—	3	1	no	—
196	C₃×C₉	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₃	—	3	1	no	—
198	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
199	C₃×Q₈	—	2	2	no	—
200	C₃×Q₈	—	2	2	no	—
201	C₃×Q₈	—	2	4	no	—
202	C₃×Q₈	—	3	4	no	—
203	C₃×Q₈	—	4	2	no	—
204	C₃×Q₈	—	6	8	no	—
205	C₃×Q₈	—	6	2	no	—
206	C₃×Q₈	C₂³×C₉	1	4	yes	—
207	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	3	1	no	—
212	C₂×C₃×C₄	—	3	2	no	—
213	C₂×C₃×C₄	—	3	2	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	—	6	8	no	—
216	C₂×C₃×C₄	—	6	2	no	—
217	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
219	C₂×Dic₃	—	2	2	no	—
220	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
221	C₄×D₃	—	2	8	no	—
222	C₄×D₃	D₄×C₉	1	2	yes	—
223	C₄×D₃	C₂³×C₉	1	2	yes	—
224	Dic₆	—	2	2	no	—
225	Dic₆	—	2	8	no	—
226	Dic₆	C₂³×C₉	1	4	yes	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	3	2	no	—
229	C₃×C₈	C₂³×C₉	1	2	yes	—
230	C₃×C₈	C₂²×C₃×D₃	1	2	yes	—
231	C₃⋊C₈	C₂³×C₉	1	2	yes	—
232	C₂×C₃²	C₂×C₃×Q₁₆	1	2	yes	—
233	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₃×D₃	C₂×C₃×Q₁₆	1	4	yes	—
235	C₂×C₉	—	2	2	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	3	4	no	—
238	C₂×C₉	D₃×Q₁₆	1	2	yes	direct product factor
239	C₂×C₉	C₂×D₃×D₄	1	1	yes	characteristic
240	C₂×Q₈	—	3	8	no	—
241	C₂×Q₈	—	3	2	no	—
242	C₂×Q₈	—	6	2	no	—
243	C₂×Q₈	—	6	4	no	—
244	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
245	C₂²×C₄	—	3	1	no	—
246	C₂²×C₄	—	6	2	no	—
247	Q₁₆	—	3	8	no	—
248	Q₁₆	—	3	4	no	—
249	Q₁₆	C₂×D₃×C₉	1	4	yes	direct product factor
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	—	3	4	no	—
252	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	3	4	no	—
255	C₂²×C₃	—	3	2	no	—
256	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
257	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
258	C₂×D₃	C₉×Q₁₆	1	4	yes	direct product factor
259	C₂×D₃	C₂×D₄×C₉	1	2	yes	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	3	2	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	6	4	no	—
267	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
268	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
269	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
270	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
271	Dic₃	—	2	4	no	—
272	Dic₃	C₂×D₄×C₉	1	2	yes	—
273	C₃²	C₂²×C₃×Q₁₆	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	C₂×D₃×Q₁₆	1	1	yes	characteristic, direct product factor
276	C₂³	—	3	1	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	3	4	no	—
279	Q₈	—	6	8	no	—
280	Q₈	—	6	2	no	—
281	Q₈	C₂²×D₃×C₉	1	4	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	3	1	no	—
284	C₂×C₄	—	3	2	no	—
285	C₂×C₄	—	3	2	no	—
286	C₂×C₄	—	6	8	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
289	C₈	—	3	2	no	—
290	C₈	C₂²×D₃×C₉	1	2	yes	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	3	4	no	—
294	C₂×C₃	C₂×C₉×Q₁₆	1	2	yes	complemented
295	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
296	C₂×C₃	C₃×D₃×Q₁₆	1	2	yes	—
297	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
298	D₃	C₂×C₉×Q₁₆	1	4	yes	direct product factor
299	C₂²	—	3	4	no	—
300	C₂²	—	3	2	no	—
301	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
302	C₄	—	2	4	no	—
303	C₄	—	3	2	no	—
304	C₄	—	6	4	no	—
305	C₄	D₃×D₄×C₉	1	1	yes	characteristic
306	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
307	C₃	—	2	1	no	—
308	C₃	C₂²×C₉×Q₁₆	1	1	yes	characteristic, complemented
309	C₃	C₂×C₃×D₃×Q₁₆	1	1	yes	characteristic
310	C₂	—	3	4	no	—
311	C₂	D₃×C₉×Q₁₆	1	2	yes	direct product factor
312	C₂	C₂×D₃×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₃×C₄	C₂⁴×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×Q₁₆	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂²×C₃×D₃	C₃×D₄
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₂×C₉×Q₁₆	D₃
hypocenter	C₃	C₂²×C₉×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₉×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×Q₁₆	C₂	semidirect
3. C₂×C₉×#216	C₂	semidirect
4. D₃×C₉×Q₁₆	C₂	direct (N × H)
5. C₂×C₉×Dic₁₂	C₂	semidirect
16. C₃×C₉×Q₁₆	C₂²	semidirect
18. C₉×#216	C₂²	semidirect
20. C₉×Dic₁₂	C₂²	semidirect
33. C₂×C₉×Q₁₆	D₃	direct (N × H)
51. C₂×D₃×Q₁₆	C₉	direct (N × H)
76. C₉×Q₁₆	C₂×D₃	direct (N × H)
82. C₂×D₃×C₉	Q₁₆	direct (N × H)
95. C₂×C₃×Q₁₆	C₂×C₉	semidirect
98. C₂×#216	C₂×C₉	semidirect
99. D₃×Q₁₆	C₂×C₉	direct (N × H)
100. C₂×Dic₁₂	C₂×C₉	semidirect
138. C₂×C₃×C₉	C₂×Q₁₆	semidirect
140. D₃×C₉	C₂×Q₁₆	direct (N × H)
159. C₃×Q₁₆	C₂²×C₉	semidirect
166. #216	C₂²×C₉	semidirect
168. Dic₁₂	C₂²×C₉	semidirect
194. C₂×Q₁₆	D₃×C₉	direct (N × H)
196. C₃×C₉	C₂²×Q₁₆	semidirect
238. C₂×C₉	D₃×Q₁₆	direct (N × H)
249. Q₁₆	C₂×D₃×C₉	direct (N × H)
258. C₂×D₃	C₉×Q₁₆	direct (N × H)
275. C₉	C₂×D₃×Q₁₆	direct (N × H)
294. C₂×C₃	C₂×C₉×Q₁₆	semidirect
298. D₃	C₂×C₉×Q₁₆	direct (N × H)
308. C₃	C₂²×C₉×Q₁₆	semidirect
311. C₂	D₃×C₉×Q₁₆	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×D₃×Q₈×C₉	2
C₂×C₃×C₉×Q₁₆	2
C₂×C₉×#216	2
D₃×C₉×Q₁₆	2
C₂×C₉×Dic₁₂	2
C₂×D₃×C₈×C₉	2
C₂×C₃×D₃×Q₁₆	3
C₂²×C₉×Q₁₆	3

Subgroups — C2×D3×C9×Q16 #13153459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₉×Q₁₆ #13153459