Subgroups — D₃×C₉×C₄⋊Q₈ #13153305

Order 1728.

938 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#944	C₂	1	2	yes	maximal, complemented
4	D₃×C₉×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₉×#888	C₂	1	4	yes	maximal, complemented
6	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic
7	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×D₃×C₄⋊Q₈	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×C₄⋊Q₈	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
11	D₃×Q₈×C₉	—	2	8	no	—
12	C₂×C₄×D₃×C₉	C₂²	1	4	yes	—
13	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
15	C₂×C₉×Dic₆	C₂²	1	2	yes	—
16	C₂×C₉×Dic₆	C₂²	1	4	yes	—
17	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
18	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
20	C₉×Dic₃⋊C₄	C₂²	1	8	yes	—
21	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
22	C₄×C₉×Dic₃	C₂²	1	2	yes	—
23	C₂×C₃×D₃×Q₈	C₂×C₃	1	2	yes	—
24	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
25	C₃×#944	C₂×C₃	1	2	yes	—
26	C₃×D₃×C₄⋊C₄	C₂×C₃	1	4	yes	—
27	C₃×#888	C₂×C₃	1	4	yes	—
28	C₃×C₄²×D₃	C₂×C₃	1	1	yes	characteristic
29	C₃×#869	C₂×C₃	1	1	yes	characteristic
30	C₂²×Q₈×C₉	—	3	2	no	—
31	C₉×C₄⋊Q₈	—	2	1	no	—
32	C₉×C₄⋊Q₈	—	3	2	no	—
33	C₉×C₄⋊Q₈	—	3	4	no	—
34	C₉×C₄⋊Q₈	—	3	1	no	—
35	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic, direct product factor
36	C₂×C₉×C₄⋊C₄	—	3	4	no	—
37	C₂×C₄²×C₉	—	3	1	no	—
38	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
39	C₃×Q₈×C₉	—	2	4	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₉×Dic₃	C₂³	1	4	yes	—
44	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
45	C₂×C₉×Dic₃	C₂³	1	2	yes	—
46	C₄×D₃×C₉	—	2	8	no	—
47	C₄×D₃×C₉	D₄	1	4	yes	—
48	C₄×D₃×C₉	Q₈	1	8	yes	complemented
49	C₉×Dic₆	—	2	4	no	—
50	C₉×Dic₆	—	2	8	no	—
51	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
52	D₃×C₄⋊Q₈	C₉	1	1	yes	characteristic, direct product factor
53	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
54	C₃×D₃×Q₈	—	2	8	no	—
55	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
56	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
58	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
59	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
60	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
61	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
63	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
64	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
65	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
66	C₂×Q₈×C₉	—	2	2	no	—
67	C₂×Q₈×C₉	—	3	4	no	—
68	C₂×Q₈×C₉	—	3	2	no	—
69	C₂×Q₈×C₉	—	6	8	no	—
70	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
71	C₂²×C₄×C₉	—	3	2	no	—
72	C₂²×C₄×C₉	—	3	1	no	—
73	C₂²×C₄×C₉	—	3	4	no	—
74	C₉×C₄⋊C₄	—	2	4	no	—
75	C₉×C₄⋊C₄	—	3	8	no	—
76	C₉×C₄⋊C₄	—	3	4	no	—
77	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
78	C₄²×C₉	—	2	1	no	—
79	C₄²×C₉	—	3	1	no	—
80	C₄²×C₉	—	3	2	no	—
81	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
83	C₂×D₃×C₉	C₂×D₄	1	2	yes	—
84	C₂×D₃×C₉	C₂×Q₈	1	4	yes	—
85	C₃×C₄×C₉	—	2	4	no	—
86	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
87	C₃×C₄×C₉	C₂×Q₈	1	4	yes	complemented
88	C₉×Dic₃	—	2	4	no	—
89	C₉×Dic₃	C₂×D₄	1	2	yes	—
90	C₉×Dic₃	C₂×Q₈	1	4	yes	complemented
91	C₂²×C₃×Q₈	—	3	2	no	—
92	C₂×D₃×Q₈	C₂×C₉	1	2	yes	—
93	C₃×C₄⋊Q₈	—	2	1	no	—
94	C₃×C₄⋊Q₈	—	3	2	no	—
95	C₃×C₄⋊Q₈	—	3	4	no	—
96	C₃×C₄⋊Q₈	—	3	1	no	—
97	C₃×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
98	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
99	C₂×C₃×C₄⋊C₄	—	3	4	no	—
100	C₂×C₃×C₄²	—	3	1	no	—
101	#944	C₂×C₉	1	2	yes	complemented
102	D₃×C₄⋊C₄	C₂×C₉	1	4	yes	—
103	#888	C₂×C₉	1	4	yes	complemented
104	C₄²×D₃	C₂×C₉	1	1	yes	characteristic
105	#869	C₂×C₉	1	1	yes	characteristic, complemented
106	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
107	C₃²×Q₈	—	2	4	no	—
108	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
109	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
111	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
112	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
114	C₃×C₄×D₃	—	2	8	no	—
115	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
116	C₃×C₄×D₃	C₃×Q₈	1	8	yes	—
117	C₃×Dic₆	—	2	4	no	—
118	C₃×Dic₆	—	2	8	no	—
119	C₂³×C₉	—	3	1	no	—
120	Q₈×C₉	—	2	4	no	—
121	Q₈×C₉	—	4	4	no	—
122	Q₈×C₉	—	6	8	no	—
123	Q₈×C₉	—	6	4	no	—
124	C₂×C₄×C₉	—	2	2	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	4	no	—
127	C₂×C₄×C₉	—	3	2	no	—
128	C₂×C₄×C₉	—	3	1	no	—
129	C₂×C₄×C₉	—	3	4	no	—
130	C₂×C₄×C₉	—	3	8	no	—
131	C₂×C₄×C₉	—	3	4	no	—
132	C₂×C₄×C₉	—	6	8	no	—
133	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
134	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
136	C₂×C₄⋊Q₈	—	3	1	no	Sylow
137	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₃×C₉	C₂²×Q₈	1	2	yes	—
139	D₃×C₉	C₄⋊Q₈	1	4	yes	direct product factor
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	—	3	4	no	—
142	C₂×C₃×Q₈	—	3	2	no	—
143	C₂×C₃×Q₈	—	6	8	no	—
144	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
145	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
146	C₂²×C₃×C₄	—	3	2	no	—
147	C₂²×C₃×C₄	—	3	1	no	—
148	C₂²×C₃×C₄	—	3	4	no	—
149	D₃×Q₈	—	2	8	no	—
150	C₂×C₄×D₃	C₂²×C₉	1	4	yes	—
151	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
152	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
153	C₂×Dic₆	C₂²×C₉	1	2	yes	—
154	C₂×Dic₆	C₂²×C₉	1	4	yes	—
155	C₃×C₄⋊C₄	—	2	4	no	—
156	C₃×C₄⋊C₄	—	3	8	no	—
157	C₃×C₄⋊C₄	—	3	4	no	—
158	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
159	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
160	C₃×C₄²	—	2	1	no	—
161	C₃×C₄²	—	3	1	no	—
162	C₃×C₄²	—	3	2	no	—
163	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
164	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
165	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
166	Dic₃⋊C₄	C₂²×C₉	1	8	yes	—
167	C₄×Dic₃	C₂²×C₉	1	2	yes	—
168	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
169	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
170	C₂×C₃×D₃	C₂×C₃×D₄	1	2	yes	—
171	C₂×C₃×D₃	C₂×C₃×Q₈	1	4	yes	—
172	C₃²×C₄	—	2	4	no	—
173	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
175	C₃×Dic₃	—	2	4	no	—
176	C₃×Dic₃	C₂×C₃×D₄	1	2	yes	—
177	C₃×Dic₃	C₂×C₃×Q₈	1	4	yes	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	3	2	no	—
180	C₂²×C₉	—	3	4	no	—
181	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
182	C₄×C₉	—	2	4	no	—
183	C₄×C₉	—	2	2	no	—
184	C₄×C₉	—	2	4	no	—
185	C₄×C₉	—	3	2	no	—
186	C₄×C₉	—	3	4	no	—
187	C₄×C₉	—	4	4	no	—
188	C₄×C₉	—	6	4	no	—
189	C₄×C₉	D₃×D₄	1	2	yes	—
190	C₄×C₉	D₃×Q₈	1	4	yes	complemented
191	C₂²×Q₈	—	3	2	no	—
192	C₄⋊Q₈	—	3	2	no	—
193	C₄⋊Q₈	—	3	4	no	—
194	C₄⋊Q₈	—	3	1	no	—
195	C₄⋊Q₈	D₃×C₉	1	1	yes	characteristic, direct product factor
196	C₂×C₄⋊C₄	—	3	4	no	—
197	C₂×C₄²	—	3	1	no	—
198	C₃×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	3	1	no	—
200	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
201	C₃×Q₈	—	2	4	no	—
202	C₃×Q₈	—	2	4	no	—
203	C₃×Q₈	—	4	4	no	—
204	C₃×Q₈	—	6	8	no	—
205	C₃×Q₈	—	6	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	—	3	2	no	—
210	C₂×C₃×C₄	—	3	1	no	—
211	C₂×C₃×C₄	—	3	4	no	—
212	C₂×C₃×C₄	—	3	8	no	—
213	C₂×C₃×C₄	—	3	4	no	—
214	C₂×C₃×C₄	—	6	8	no	—
215	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
216	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
218	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
219	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
221	C₂×Dic₃	C₂³×C₉	1	4	yes	—
222	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
223	C₂×Dic₃	C₂³×C₉	1	2	yes	—
224	C₄×D₃	—	2	8	no	—
225	C₄×D₃	D₄×C₉	1	4	yes	—
226	C₄×D₃	Q₈×C₉	1	8	yes	complemented
227	Dic₆	—	2	4	no	—
228	Dic₆	—	2	8	no	—
229	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
230	C₂×C₃²	C₂²×C₃×Q₈	1	2	yes	—
231	C₃×D₃	C₃×C₄⋊Q₈	1	4	yes	—
232	C₂×C₉	—	2	1	no	—
233	C₂×C₉	—	2	2	no	—
234	C₂×C₉	—	3	4	no	—
235	C₂×C₉	C₂×D₃×D₄	1	1	yes	characteristic
236	C₂×C₉	C₂×D₃×Q₈	1	2	yes	—
237	C₂×Q₈	—	3	4	no	—
238	C₂×Q₈	—	3	2	no	—
239	C₂×Q₈	—	6	8	no	—
240	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
241	C₂²×C₄	—	3	2	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	4	no	—
244	C₄⋊C₄	—	3	8	no	—
245	C₄⋊C₄	—	3	4	no	—
246	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
247	C₄²	—	3	1	no	—
248	C₄²	—	3	2	no	—
249	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	3	2	no	—
252	C₂²×C₃	—	3	4	no	—
253	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
254	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
255	C₂×D₃	C₂×D₄×C₉	1	2	yes	—
256	C₂×D₃	C₂×Q₈×C₉	1	4	yes	—
257	C₃×C₄	—	2	4	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	3	2	no	—
262	C₃×C₄	—	3	4	no	—
263	C₃×C₄	—	4	4	no	—
264	C₃×C₄	—	6	4	no	—
265	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
266	C₃×C₄	C₂×Q₈×C₉	1	4	yes	complemented
267	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
268	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
269	Dic₃	—	2	4	no	—
270	Dic₃	C₂×D₄×C₉	1	2	yes	—
271	Dic₃	C₂×Q₈×C₉	1	4	yes	complemented
272	C₃²	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
273	C₉	—	2	1	no	—
274	C₉	D₃×C₄⋊Q₈	1	1	yes	characteristic, direct product factor
275	C₂³	—	3	1	no	—
276	Q₈	—	2	4	no	—
277	Q₈	—	6	8	no	—
278	Q₈	—	6	4	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	3	4	no	—
282	C₂×C₄	—	3	8	no	—
283	C₂×C₄	—	3	4	no	—
284	C₂×C₄	—	6	8	no	—
285	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
286	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	3	4	no	—
291	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
292	C₂×C₃	C₂²×Q₈×C₉	1	2	yes	—
293	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
294	C₂×C₃	C₂×C₃×D₃×Q₈	1	2	yes	—
295	D₃	C₉×C₄⋊Q₈	1	4	yes	direct product factor
296	C₂²	—	3	2	no	—
297	C₂²	—	3	4	no	—
298	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
299	C₄	—	2	4	no	—
300	C₄	—	3	2	no	—
301	C₄	—	3	4	no	—
302	C₄	—	6	4	no	—
303	C₄	D₃×D₄×C₉	1	2	yes	—
304	C₄	D₃×Q₈×C₉	1	4	yes	complemented
305	C₃	—	2	1	no	—
306	C₃	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic, complemented
307	C₃	C₃×D₃×C₄⋊Q₈	1	1	yes	characteristic
308	C₂	—	3	4	no	—
309	C₂	C₂×D₃×D₄×C₉	1	1	yes	characteristic
310	C₂	C₂×D₃×Q₈×C₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂²×C₃×D₃	C₂³×C₃
hypercenter	C₉×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×C₉×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. C₉×#944	C₂	semidirect
5. C₉×#888	C₂	semidirect
7. C₉×#869	C₂	semidirect
35. C₉×C₄⋊Q₈	D₃	direct (N × H)
48. C₄×D₃×C₉	Q₈	semidirect
52. D₃×C₄⋊Q₈	C₉	direct (N × H)
87. C₃×C₄×C₉	C₂×Q₈	semidirect
90. C₉×Dic₃	C₂×Q₈	semidirect
97. C₃×C₄⋊Q₈	C₂×C₉	semidirect
101. #944	C₂×C₉	semidirect
103. #888	C₂×C₉	semidirect
105. #869	C₂×C₉	semidirect
139. D₃×C₉	C₄⋊Q₈	direct (N × H)
190. C₄×C₉	D₃×Q₈	semidirect
195. C₄⋊Q₈	D₃×C₉	direct (N × H)
198. C₃×C₉	C₂×C₄⋊Q₈	semidirect
226. C₄×D₃	Q₈×C₉	semidirect
266. C₃×C₄	C₂×Q₈×C₉	semidirect
271. Dic₃	C₂×Q₈×C₉	semidirect
274. C₉	D₃×C₄⋊Q₈	direct (N × H)
295. D₃	C₉×C₄⋊Q₈	direct (N × H)
304. C₄	D₃×Q₈×C₉	semidirect
306. C₃	C₂×C₉×C₄⋊Q₈	semidirect

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
C₃×C₉×C₄⋊Q₈	2
C₉×#944	2
D₃×C₉×C₄⋊C₄	2
C₉×#888	2
C₄²×D₃×C₉	2
C₉×#869	2
C₃×D₃×C₄⋊Q₈	3
C₂×C₉×C₄⋊Q₈	3

Subgroups — D3×C9×C4⋊Q8 #13153305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×C₄⋊Q₈ #13153305