Subgroups — C₂×C₃×Q₁₆×D₉ #13153455

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×C₃×Q₈×D₉	C₂	1	2	yes	maximal
2	C₂×C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3674	C₂	1	2	yes	maximal, complemented
4	C₃×Q₁₆×D₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₈×D₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃×Dic₃₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×D₃×Q₁₆	—	3	1	no	maximal
8	C₂×Q₁₆×D₉	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
10	C₃×Q₈×D₉	—	2	4	no	—
11	C₃×Q₈×D₉	C₂²	1	8	yes	—
12	C₂×C₃×C₄×D₉	—	2	2	no	—
13	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₈	—	2	2	no	—
15	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
16	C₃×C₉×Q₁₆	C₂²	1	4	yes	complemented
17	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₃×#3674	C₂²	1	8	yes	complemented
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₈×D₉	C₂²	1	4	yes	—
21	C₃×Dic₃₆	C₂²	1	4	yes	complemented
22	C₂×C₃×D₃×Q₈	—	3	2	no	—
23	C₂×C₃²×Q₁₆	D₃	1	1	yes	characteristic
24	C₂×C₃×#216	—	3	2	no	—
25	C₃×D₃×Q₁₆	—	3	8	no	—
26	C₂×C₃×Dic₁₂	—	3	1	no	—
27	C₂×C₃×D₃×C₈	—	3	1	no	—
28	C₂×Q₈×D₉	C₂×C₃	1	2	yes	—
29	C₂×C₉×Q₁₆	—	2	1	no	—
30	C₂×C₉×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#3674	C₂×C₃	1	2	yes	complemented
32	Q₁₆×D₉	C₂×C₃	1	8	yes	direct product factor
33	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic
34	C₂×Dic₃₆	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×D₉	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×C₃×Dic₉	—	2	2	no	—
41	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
42	C₃×C₄×D₉	—	2	8	no	—
43	C₃×C₄×D₉	D₄	1	2	yes	—
44	C₃×C₄×D₉	C₂³	1	2	yes	—
45	C₃×Dic₁₈	—	2	2	no	—
46	C₃×Dic₁₈	—	2	8	no	—
47	C₃×Dic₁₈	C₂³	1	4	yes	—
48	C₃×C₈×C₉	C₂³	1	2	yes	—
49	C₃×C₉⋊C₈	C₂³	1	2	yes	—
50	C₂²×C₃×Q₁₆	—	9	1	no	—
51	C₂×D₃×Q₁₆	—	3	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
53	C₃×D₃×Q₈	—	3	8	no	—
54	C₃×D₃×Q₈	—	6	4	no	—
55	C₂×C₃×C₄×D₃	—	3	1	no	—
56	C₂×C₃×C₄×D₃	—	6	2	no	—
57	C₂×C₃×Dic₆	—	3	2	no	—
58	C₂×C₃×Dic₆	—	6	2	no	—
59	C₃²×Q₁₆	C₂×D₃	1	4	yes	—
60	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#216	—	3	8	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₃×Dic₁₂	—	3	4	no	—
64	C₃×D₃×C₈	—	3	4	no	—
65	C₂×Q₈×C₉	—	2	2	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
67	Q₈×D₉	—	2	4	no	—
68	Q₈×D₉	C₂²×C₃	1	8	yes	—
69	C₂×C₄×D₉	—	2	2	no	—
70	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
71	C₂×Dic₁₈	—	2	2	no	—
72	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
73	C₉×Q₁₆	—	2	4	no	—
74	C₉×Q₁₆	C₂²×C₃	1	4	yes	complemented
75	C₂×C₈×C₉	—	2	1	no	—
76	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
77	#3674	C₂²×C₃	1	8	yes	complemented
78	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	C₈×D₉	C₂²×C₃	1	4	yes	—
80	Dic₃₆	C₂²×C₃	1	4	yes	complemented
81	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×D₉	Q₁₆	1	4	yes	direct product factor
83	C₂×C₃×D₉	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	C₃×Dic₉	—	2	4	no	—
88	C₃×Dic₉	C₂×D₄	1	2	yes	—
89	C₂²×C₃×Q₈	—	9	2	no	—
90	C₂×D₃×Q₈	—	3	2	no	—
91	C₂×C₃×Q₁₆	—	2	1	no	—
92	C₂×C₃×Q₁₆	—	9	2	no	—
93	C₂×C₃×Q₁₆	—	9	8	no	—
94	C₂×C₃×Q₁₆	—	9	1	no	—
95	C₂×C₃×Q₁₆	D₉	1	1	yes	characteristic, direct product factor
96	C₂×C₃×Q₁₆	C₃×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₈	—	9	1	no	—
98	C₂×#216	—	3	2	no	—
99	D₃×Q₁₆	—	3	8	no	—
100	C₂×Dic₁₂	—	3	1	no	—
101	C₂×D₃×C₈	—	3	1	no	—
102	C₂²×C₃×D₃	—	3	1	no	—
103	C₃²×Q₈	—	2	2	no	—
104	C₃²×Q₈	C₂²×D₃	1	4	yes	—
105	C₂×C₃²×C₄	—	2	2	no	—
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	—	3	1	no	—
108	C₂×C₃×Dic₃	—	6	2	no	—
109	C₃×C₄×D₃	—	3	2	no	—
110	C₃×C₄×D₃	—	3	2	no	—
111	C₃×C₄×D₃	—	6	8	no	—
112	C₃×Dic₆	—	3	4	no	—
113	C₃×Dic₆	—	6	8	no	—
114	C₃×Dic₆	—	6	2	no	—
115	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
116	C₃²×C₈	C₂²×D₃	1	2	yes	—
117	C₃×C₃⋊C₈	—	3	2	no	—
118	Q₈×C₉	—	2	2	no	—
119	Q₈×C₉	—	2	4	no	—
120	Q₈×C₉	—	4	2	no	—
121	Q₈×C₉	C₂³×C₃	1	4	yes	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	4	2	no	—
125	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₉	—	2	2	no	—
127	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
128	C₄×D₉	—	2	8	no	—
129	C₄×D₉	C₃×D₄	1	2	yes	—
130	C₄×D₉	C₂³×C₃	1	2	yes	—
131	Dic₁₈	—	2	2	no	—
132	Dic₁₈	—	2	8	no	—
133	Dic₁₈	C₂³×C₃	1	4	yes	—
134	C₈×C₉	—	2	2	no	—
135	C₈×C₉	C₂³×C₃	1	2	yes	—
136	C₉⋊C₈	C₂³×C₃	1	2	yes	—
137	C₂²×Q₁₆	—	9	1	no	Sylow
138	C₂×C₃×C₉	C₂×Q₁₆	1	2	yes	complemented
139	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
140	C₃×D₉	C₂×Q₁₆	1	4	yes	direct product factor
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	—	9	8	no	—
143	C₂×C₃×Q₈	—	9	2	no	—
144	C₂×C₃×Q₈	—	18	2	no	—
145	C₂×C₃×Q₈	—	18	4	no	—
146	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
147	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
148	C₂²×C₃×C₄	—	9	1	no	—
149	C₂²×C₃×C₄	—	18	2	no	—
150	D₃×Q₈	—	3	8	no	—
151	D₃×Q₈	—	6	4	no	—
152	C₂×C₄×D₃	—	3	1	no	—
153	C₂×C₄×D₃	—	6	2	no	—
154	C₂×Dic₆	—	3	2	no	—
155	C₂×Dic₆	—	6	2	no	—
156	C₃×Q₁₆	—	2	4	no	—
157	C₃×Q₁₆	—	9	8	no	—
158	C₃×Q₁₆	—	9	4	no	—
159	C₃×Q₁₆	C₂×D₉	1	4	yes	direct product factor
160	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₂×C₃×C₈	—	9	1	no	—
163	C₂×C₃×C₈	—	9	4	no	—
164	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
165	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
166	#216	—	3	8	no	—
167	C₂×C₃⋊C₈	—	3	1	no	—
168	Dic₁₂	—	3	4	no	—
169	D₃×C₈	—	3	4	no	—
170	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
171	C₂×C₃×D₃	—	3	4	no	—
172	C₂×C₃×D₃	—	3	2	no	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
175	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
176	C₃×Dic₃	—	3	2	no	—
177	C₃×Dic₃	—	6	4	no	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×D₉	C₃×Q₁₆	1	4	yes	direct product factor
181	C₂×D₉	C₂×C₃×D₄	1	2	yes	—
182	C₄×C₉	—	2	4	no	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	4	4	no	—
186	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
187	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
188	Dic₉	—	2	4	no	—
189	Dic₉	C₂×C₃×D₄	1	2	yes	—
190	C₂²×Q₈	—	9	2	no	—
191	C₂×Q₁₆	—	9	2	no	—
192	C₂×Q₁₆	—	9	8	no	—
193	C₂×Q₁₆	—	9	1	no	—
194	C₂×Q₁₆	C₃×D₉	1	1	yes	characteristic, direct product factor
195	C₂²×C₈	—	9	1	no	—
196	C₃×C₉	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₃	—	9	1	no	—
198	C₂²×D₃	—	3	1	no	—
199	C₃×Q₈	—	2	2	no	—
200	C₃×Q₈	—	2	2	no	—
201	C₃×Q₈	—	2	4	no	—
202	C₃×Q₈	—	4	2	no	—
203	C₃×Q₈	—	9	4	no	—
204	C₃×Q₈	—	18	8	no	—
205	C₃×Q₈	—	18	2	no	—
206	C₃×Q₈	C₂²×D₉	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	—	4	2	no	—
212	C₂×C₃×C₄	—	9	1	no	—
213	C₂×C₃×C₄	—	9	2	no	—
214	C₂×C₃×C₄	—	9	2	no	—
215	C₂×C₃×C₄	—	18	8	no	—
216	C₂×C₃×C₄	—	18	2	no	—
217	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
219	C₂×Dic₃	—	3	1	no	—
220	C₂×Dic₃	—	6	2	no	—
221	C₄×D₃	—	3	2	no	—
222	C₄×D₃	—	3	2	no	—
223	C₄×D₃	—	6	8	no	—
224	Dic₆	—	3	4	no	—
225	Dic₆	—	6	8	no	—
226	Dic₆	—	6	2	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	9	2	no	—
229	C₃×C₈	C₂²×D₉	1	2	yes	—
230	C₃×C₈	C₂²×C₃×D₃	1	2	yes	—
231	C₃⋊C₈	—	3	2	no	—
232	C₂×C₃²	D₃×Q₁₆	1	2	yes	—
233	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
234	C₃×D₃	—	3	4	no	—
235	C₂×C₉	—	2	2	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	C₂×C₃×Q₁₆	1	2	yes	complemented
238	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
239	D₉	C₂×C₃×Q₁₆	1	4	yes	direct product factor
240	C₂×Q₈	—	9	8	no	—
241	C₂×Q₈	—	9	2	no	—
242	C₂×Q₈	—	18	2	no	—
243	C₂×Q₈	—	18	4	no	—
244	C₂×Q₈	C₂×C₃×D₉	1	2	yes	—
245	C₂²×C₄	—	9	1	no	—
246	C₂²×C₄	—	18	2	no	—
247	Q₁₆	—	9	8	no	—
248	Q₁₆	—	9	4	no	—
249	Q₁₆	C₂×C₃×D₉	1	4	yes	direct product factor
250	C₂×C₈	—	9	1	no	—
251	C₂×C₈	—	9	4	no	—
252	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	9	2	no	—
255	C₂²×C₃	—	9	4	no	—
256	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
257	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
258	C₂×D₃	—	3	4	no	—
259	C₂×D₃	—	3	2	no	—
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	—	4	4	no	—
265	C₃×C₄	—	9	2	no	—
266	C₃×C₄	—	18	4	no	—
267	C₃×C₄	D₄×D₉	1	1	yes	characteristic
268	C₃×C₄	C₂³×D₉	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	—	3	2	no	—
272	Dic₃	—	6	4	no	—
273	C₃²	C₂×D₃×Q₁₆	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	C₂²×C₃×Q₁₆	1	1	yes	characteristic, complemented
276	C₂³	—	9	1	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	9	4	no	—
279	Q₈	—	18	8	no	—
280	Q₈	—	18	2	no	—
281	Q₈	C₂²×C₃×D₉	1	4	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	9	1	no	—
284	C₂×C₄	—	9	2	no	—
285	C₂×C₄	—	9	2	no	—
286	C₂×C₄	—	18	8	no	—
287	C₂×C₄	—	18	2	no	—
288	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
289	C₈	—	9	2	no	—
290	C₈	C₂²×C₃×D₉	1	2	yes	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	9	4	no	—
294	C₂×C₃	Q₁₆×D₉	1	2	yes	direct product factor
295	C₂×C₃	C₂×D₄×D₉	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	—	3	4	no	—
299	C₂²	—	9	4	no	—
300	C₂²	—	9	2	no	—
301	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
302	C₄	—	2	4	no	—
303	C₄	—	9	2	no	—
304	C₄	—	18	4	no	—
305	C₄	C₃×D₄×D₉	1	1	yes	characteristic
306	C₄	C₂³×C₃×D₉	1	1	yes	characteristic
307	C₃	—	2	1	no	—
308	C₃	C₂×Q₁₆×D₉	1	1	yes	characteristic, direct product factor
309	C₃	C₂×C₃×D₃×Q₁₆	1	1	yes	characteristic
310	C₂	—	9	4	no	—
311	C₂	C₃×Q₁₆×D₉	1	2	yes	direct product factor
312	C₂	C₂×C₃×D₄×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×Q₁₆	C₂
socle	C₂²×C₃²	D₃×D₄
prime radical	C₂²×C₃×D₉	D₄
hypercenter	C₂×C₃×Q₁₆	D₉
hypocenter	C₉	C₂²×C₃×Q₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×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×C₃×#3674	C₂	semidirect
4. C₃×Q₁₆×D₉	C₂	direct (N × H)
6. C₂×C₃×Dic₃₆	C₂	semidirect
8. C₂×Q₁₆×D₉	C₃	direct (N × H)
16. C₃×C₉×Q₁₆	C₂²	semidirect
18. C₃×#3674	C₂²	semidirect
21. C₃×Dic₃₆	C₂²	semidirect
30. C₂×C₉×Q₁₆	C₂×C₃	semidirect
31. C₂×#3674	C₂×C₃	semidirect
32. Q₁₆×D₉	C₂×C₃	direct (N × H)
34. C₂×Dic₃₆	C₂×C₃	semidirect
74. C₉×Q₁₆	C₂²×C₃	semidirect
77. #3674	C₂²×C₃	semidirect
80. Dic₃₆	C₂²×C₃	semidirect
82. C₂×C₃×D₉	Q₁₆	direct (N × H)
95. C₂×C₃×Q₁₆	D₉	direct (N × H)
138. C₂×C₃×C₉	C₂×Q₁₆	semidirect
140. C₃×D₉	C₂×Q₁₆	direct (N × H)
159. C₃×Q₁₆	C₂×D₉	direct (N × H)
180. C₂×D₉	C₃×Q₁₆	direct (N × H)
194. C₂×Q₁₆	C₃×D₉	direct (N × H)
196. C₃×C₉	C₂²×Q₁₆	semidirect
237. C₂×C₉	C₂×C₃×Q₁₆	semidirect
239. D₉	C₂×C₃×Q₁₆	direct (N × H)
249. Q₁₆	C₂×C₃×D₉	direct (N × H)
275. C₉	C₂²×C₃×Q₁₆	semidirect
294. C₂×C₃	Q₁₆×D₉	direct (N × H)
308. C₃	C₂×Q₁₆×D₉	direct (N × H)
311. C₂	C₃×Q₁₆×D₉	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×C₃×Q₈×D₉	2
C₂×C₃×C₉×Q₁₆	2
C₂×C₃×#3674	2
C₃×Q₁₆×D₉	2
C₂×C₃×C₈×D₉	2
C₂×C₃×Dic₃₆	2
C₂×C₃×D₃×Q₁₆	3
C₂×Q₁₆×D₉	3

Subgroups — C2×C3×Q16×D9 #13153455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Q₁₆×D₉ #13153455