Subgroups — #103×C₃⋊D₉ #13143974

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720582	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720539	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720527	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃⋊D₃×#103	—	3	1	no	maximal
9	D₉×#103	—	3	3	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	Q₈×C₃⋊D₉	—	2	2	no	—
12	Q₈×C₃⋊D₉	C₄	1	4	yes	complemented
13	C₂×C₄×C₃⋊D₉	—	2	1	no	—
14	C₂×C₄×C₃⋊D₉	—	2	1	no	—
15	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
16	C₂×#6226	—	2	1	no	—
17	C₂×#6226	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
19	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
21	#88534	—	2	1	no	—
22	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
23	C₈×C₃⋊D₉	—	2	2	no	—
24	C₂×Q₈×C₃⋊D₃	—	3	1	no	—
25	C₂×C₈×C₃⋊D₃	—	3	1	no	—
26	C₄⋊C₄×C₃⋊D₃	—	3	1	no	—
27	C₂×Q₈×D₉	—	3	3	no	—
28	C₃²×#103	D₃	1	1	yes	characteristic
29	#63662	—	3	1	no	—
30	#63643	—	3	1	no	—
31	#63637	—	3	1	no	—
32	C₂×C₈×D₉	—	3	3	no	—
33	C₄⋊C₄×D₉	—	3	3	no	—
34	C₉×#103	D₃	1	3	yes	complemented
35	#63395	—	3	3	no	—
36	#63376	—	3	3	no	—
37	#63368	—	3	3	no	—
38	C₂²×C₃⋊D₉	D₄	1	1	yes	characteristic
39	C₃×Q₈×C₉	—	2	1	no	—
40	C₃×Q₈×C₉	C₂×C₄	1	2	yes	complemented
41	C₂×C₃×C₄×C₉	—	2	1	no	—
42	C₂×C₃×C₄×C₉	—	2	1	no	—
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₃⋊Dic₉	—	2	1	no	—
45	C₂×C₃⋊Dic₉	—	2	1	no	—
46	C₂×C₃⋊Dic₉	D₄	1	1	yes	characteristic
47	C₄×C₃⋊D₉	—	2	4	no	—
48	C₄×C₃⋊D₉	—	4	2	no	—
49	C₄×C₃⋊D₉	C₂×C₄	1	2	yes	—
50	C₄×C₃⋊D₉	D₄	1	2	yes	—
51	#6226	—	2	1	no	—
52	#6226	—	2	4	no	—
53	#6226	C₂×C₄	1	2	yes	complemented
54	C₃×C₈×C₉	—	2	1	no	—
55	C₃⋊(C₉⋊C₈)	—	2	1	no	—
56	D₃×#103	—	9	3	no	—
57	D₃×#103	—	9	1	no	—
58	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
59	Q₈×C₃⋊D₃	—	3	4	no	—
60	Q₈×C₃⋊D₃	—	6	2	no	—
61	C₂×C₄×C₃⋊D₃	—	3	1	no	—
62	C₂×C₄×C₃⋊D₃	—	6	1	no	—
63	C₂×C₄×C₃⋊D₃	—	6	1	no	—
64	C₂×#634	—	3	1	no	—
65	C₂×#634	—	6	1	no	—
66	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
67	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
68	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
69	#3750	—	6	1	no	—
70	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
71	C₈×C₃⋊D₃	—	6	2	no	—
72	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
73	Q₈×D₉	—	3	12	no	—
74	Q₈×D₉	—	6	6	no	—
75	C₂×C₄×D₉	—	3	3	no	—
76	C₂×C₄×D₉	—	6	3	no	—
77	C₂×C₄×D₉	—	6	3	no	—
78	C₂×Dic₁₈	—	3	3	no	—
79	C₂×Dic₁₈	—	6	3	no	—
80	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
81	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
82	C₄⋊Dic₉	—	3	3	no	—
83	Dic₉⋊C₄	—	6	3	no	—
84	C₂×C₉⋊C₈	—	3	3	no	—
85	C₈×D₉	—	6	6	no	—
86	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃⋊D₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃⋊D₉	SD₁₆	1	2	yes	—
89	C₂×C₃⋊D₉	Q₁₆	1	2	yes	—
90	C₃×C₄×C₉	—	2	2	no	—
91	C₃×C₄×C₉	—	4	1	no	—
92	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₃⋊Dic₉	—	2	2	no	—
95	C₃⋊Dic₉	—	4	1	no	—
96	C₃⋊Dic₉	C₂²⋊C₄	1	2	yes	—
97	C₂×D₃×Q₈	—	9	3	no	—
98	C₂×D₃×Q₈	—	9	1	no	—
99	C₂×D₃×C₈	—	9	3	no	—
100	C₂×D₃×C₈	—	9	1	no	—
101	D₃×C₄⋊C₄	—	9	3	no	—
102	D₃×C₄⋊C₄	—	9	1	no	—
103	C₃×#103	D₉	1	3	yes	complemented
104	C₃×#103	C₃⋊D₃	1	1	yes	characteristic
105	#835	—	9	3	no	—
106	#835	—	9	1	no	—
107	#816	—	9	3	no	—
108	#816	—	9	1	no	—
109	#810	—	9	3	no	—
110	#810	—	9	1	no	—
111	C₂²×C₃⋊D₃	—	3	1	no	—
112	C₃²×Q₈	—	2	1	no	—
113	C₃²×Q₈	C₄×D₃	1	2	yes	—
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	—	2	1	no	—
116	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₃⋊Dic₃	—	3	1	no	—
118	C₂×C₃⋊Dic₃	—	6	1	no	—
119	C₂×C₃⋊Dic₃	—	6	1	no	—
120	C₄×C₃⋊D₃	—	3	2	no	—
121	C₄×C₃⋊D₃	—	3	2	no	—
122	C₄×C₃⋊D₃	—	6	4	no	—
123	C₄×C₃⋊D₃	—	12	2	no	—
124	#634	—	3	2	no	—
125	#634	—	6	4	no	—
126	#634	—	6	1	no	—
127	C₂²×D₉	—	3	3	no	—
128	C₃²×C₈	—	2	1	no	—
129	C₃⋊(C₃⋊C₈)	—	6	1	no	—
130	Q₈×C₉	—	2	3	no	—
131	Q₈×C₉	C₄×D₃	1	6	yes	complemented
132	C₂×C₄×C₉	—	2	3	no	—
133	C₂×C₄×C₉	—	2	3	no	—
134	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
135	C₂×Dic₉	—	3	3	no	—
136	C₂×Dic₉	—	6	3	no	—
137	C₂×Dic₉	—	6	3	no	—
138	C₄×D₉	—	3	6	no	—
139	C₄×D₉	—	3	6	no	—
140	C₄×D₉	—	6	12	no	—
141	C₄×D₉	—	12	6	no	—
142	Dic₁₈	—	3	6	no	—
143	Dic₁₈	—	6	12	no	—
144	Dic₁₈	—	6	3	no	—
145	C₈×C₉	—	2	3	no	—
146	C₉⋊C₈	—	6	3	no	—
147	C₂×#103	—	27	1	no	Sylow
148	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
151	C₃⋊D₉	#103	1	4	yes	direct product factor
152	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
153	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
154	D₃×Q₈	—	9	12	no	—
155	D₃×Q₈	—	9	4	no	—
156	D₃×Q₈	—	18	2	no	—
157	D₃×Q₈	—	18	6	no	—
158	C₂×C₄×D₃	—	9	3	no	—
159	C₂×C₄×D₃	—	9	1	no	—
160	C₂×C₄×D₃	—	18	1	no	—
161	C₂×C₄×D₃	—	18	3	no	—
162	C₂×C₄×D₃	—	18	1	no	—
163	C₂×C₄×D₃	—	18	3	no	—
164	C₂×Dic₆	—	9	3	no	—
165	C₂×Dic₆	—	9	1	no	—
166	C₂×Dic₆	—	18	1	no	—
167	C₂×Dic₆	—	18	3	no	—
168	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
169	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
170	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
171	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
172	C₄⋊Dic₃	—	9	3	no	—
173	C₄⋊Dic₃	—	9	1	no	—
174	Dic₃⋊C₄	—	18	1	no	—
175	Dic₃⋊C₄	—	18	3	no	—
176	C₂×C₃⋊C₈	—	9	3	no	—
177	C₂×C₃⋊C₈	—	9	1	no	—
178	D₃×C₈	—	18	2	no	—
179	D₃×C₈	—	18	6	no	—
180	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
181	C₂×C₃⋊D₃	—	3	2	no	—
182	C₂×C₃⋊D₃	—	3	2	no	—
183	C₂×C₃⋊D₃	—	3	2	no	—
184	C₃²×C₄	—	2	2	no	—
185	C₃²×C₄	—	4	1	no	—
186	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
187	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
188	C₃⋊Dic₃	—	3	2	no	—
189	C₃⋊Dic₃	—	6	2	no	—
190	C₃⋊Dic₃	—	12	1	no	—
191	C₂²×C₉	D₃×D₄	1	3	yes	—
192	C₂×D₉	—	3	6	no	—
193	C₂×D₉	—	3	6	no	—
194	C₂×D₉	—	3	6	no	—
195	C₄×C₉	—	2	6	no	—
196	C₄×C₉	—	4	3	no	—
197	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
198	C₄×C₉	D₃×D₄	1	3	yes	—
199	Dic₉	—	3	6	no	—
200	Dic₉	—	6	6	no	—
201	Dic₉	—	12	3	no	—
202	C₂²×Q₈	—	27	1	no	—
203	C₂²×C₈	—	27	1	no	—
204	C₂×C₄⋊C₄	—	27	1	no	—
205	#103	—	27	1	no	—
206	#103	—	27	1	no	—
207	#103	—	27	1	no	—
208	#103	C₃⋊D₉	1	1	yes	characteristic, direct product factor
209	C₃×C₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
210	C₂²×D₃	—	9	3	no	—
211	C₂²×D₃	—	9	1	no	—
212	C₃×Q₈	—	2	3	no	—
213	C₃×Q₈	—	2	1	no	—
214	C₃×Q₈	C₄×D₉	1	6	yes	complemented
215	C₃×Q₈	C₄×C₃⋊D₃	1	2	yes	—
216	C₂×C₃×C₄	—	2	3	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	3	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
221	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
222	C₂×Dic₃	—	9	3	no	—
223	C₂×Dic₃	—	9	1	no	—
224	C₂×Dic₃	—	18	3	no	—
225	C₂×Dic₃	—	18	1	no	—
226	C₂×Dic₃	—	18	3	no	—
227	C₂×Dic₃	—	18	1	no	—
228	C₄×D₃	—	9	6	no	—
229	C₄×D₃	—	9	6	no	—
230	C₄×D₃	—	9	2	no	—
231	C₄×D₃	—	9	2	no	—
232	C₄×D₃	—	18	4	no	—
233	C₄×D₃	—	18	12	no	—
234	C₄×D₃	—	36	2	no	—
235	C₄×D₃	—	36	6	no	—
236	Dic₆	—	9	6	no	—
237	Dic₆	—	9	2	no	—
238	Dic₆	—	18	4	no	—
239	Dic₆	—	18	12	no	—
240	Dic₆	—	18	3	no	—
241	Dic₆	—	18	1	no	—
242	C₃×C₈	—	2	3	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃⋊C₈	—	18	3	no	—
245	C₃⋊C₈	—	18	1	no	—
246	C₂×C₃²	D₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₃²	D₃×SD₁₆	1	1	yes	characteristic
248	C₂×C₃²	D₃×Q₁₆	1	1	yes	characteristic
249	C₃⋊D₃	—	3	4	no	—
250	C₂×C₉	D₃×C₂²⋊C₄	1	3	yes	—
251	C₂×C₉	D₃×SD₁₆	1	3	yes	—
252	C₂×C₉	D₃×Q₁₆	1	3	yes	—
253	D₉	—	3	12	no	—
254	C₂×Q₈	—	27	4	no	—
255	C₂×Q₈	—	27	1	no	—
256	C₂×Q₈	—	54	1	no	—
257	C₂×Q₈	—	54	2	no	—
258	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
259	C₂²×C₄	—	27	1	no	—
260	C₂²×C₄	—	54	1	no	—
261	C₂²×C₄	—	54	1	no	—
262	C₂×C₈	—	27	1	no	—
263	C₂×C₈	—	54	2	no	—
264	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
265	C₄⋊C₄	—	27	1	no	—
266	C₄⋊C₄	—	54	1	no	—
267	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
268	C₂²×C₃	D₄×D₉	1	3	yes	—
269	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
270	C₂×D₃	—	9	2	no	—
271	C₂×D₃	—	9	2	no	—
272	C₂×D₃	—	9	2	no	—
273	C₂×D₃	—	9	6	no	—
274	C₂×D₃	—	9	6	no	—
275	C₂×D₃	—	9	6	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	2	6	no	—
278	C₃×C₄	—	4	1	no	—
279	C₃×C₄	—	4	3	no	—
280	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
281	C₃×C₄	D₄×D₉	1	3	yes	—
282	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
283	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
284	Dic₃	—	9	2	no	—
285	Dic₃	—	9	6	no	—
286	Dic₃	—	18	2	no	—
287	Dic₃	—	18	6	no	—
288	Dic₃	—	36	1	no	—
289	Dic₃	—	36	3	no	—
290	C₃²	D₃×#103	1	1	yes	characteristic
291	C₉	D₃×#103	1	3	yes	complemented
292	C₂³	—	27	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	27	2	no	—
295	Q₈	—	54	4	no	—
296	Q₈	—	54	1	no	—
297	Q₈	C₄×C₃⋊D₉	1	2	yes	complemented
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	27	1	no	—
301	C₂×C₄	—	27	2	no	—
302	C₂×C₄	—	27	2	no	—
303	C₂×C₄	—	54	4	no	—
304	C₂×C₄	—	54	1	no	—
305	C₂×C₄	—	54	1	no	—
306	C₂×C₄	—	108	2	no	—
307	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	54	1	no	—
310	C₂×C₃	C₂²⋊C₄×D₉	1	3	yes	—
311	C₂×C₃	SD₁₆×D₉	1	3	yes	—
312	C₂×C₃	Q₁₆×D₉	1	3	yes	—
313	C₂×C₃	C₂²⋊C₄×C₃⋊D₃	1	1	yes	characteristic
314	C₂×C₃	SD₁₆×C₃⋊D₃	1	1	yes	characteristic
315	C₂×C₃	Q₁₆×C₃⋊D₃	1	1	yes	characteristic
316	D₃	—	9	12	no	—
317	D₃	—	9	4	no	—
318	C₂²	—	27	2	no	—
319	C₂²	—	27	2	no	—
320	C₂²	—	27	2	no	—
321	C₂²	D₄×C₃⋊D₉	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	27	2	no	—
325	C₄	—	54	2	no	—
326	C₄	—	108	1	no	—
327	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
328	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
329	C₃	D₉×#103	1	3	yes	complemented
330	C₃	C₃⋊D₃×#103	1	1	yes	characteristic
331	C₂	—	27	4	no	—
332	C₂	C₂²⋊C₄×C₃⋊D₉	1	1	yes	characteristic
333	C₂	SD₁₆×C₃⋊D₉	1	1	yes	characteristic
334	C₂	Q₁₆×C₃⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₃⋊D₉
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#103	C₂
socle	C₂²×C₃²	D₃×D₄
prime radical	C₂²×C₃⋊D₉	D₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#103	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#103
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
4. C₃×C₉×#103	C₂	semidirect
5. #11720582	C₂	semidirect
6. #11720539	C₂	semidirect
7. #11720527	C₂	semidirect
12. Q₈×C₃⋊D₉	C₄	semidirect
34. C₉×#103	D₃	semidirect
40. C₃×Q₈×C₉	C₂×C₄	semidirect
53. #6226	C₂×C₄	semidirect
103. C₃×#103	D₉	semidirect
131. Q₈×C₉	C₄×D₃	semidirect
151. C₃⋊D₉	#103	direct (N × H)
208. #103	C₃⋊D₉	direct (N × H)
209. C₃×C₉	C₂×#103	semidirect
214. C₃×Q₈	C₄×D₉	semidirect
291. C₉	D₃×#103	semidirect
297. Q₈	C₄×C₃⋊D₉	semidirect
329. C₃	D₉×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₃⋊D₉	2
C₂×C₈×C₃⋊D₉	2
C₄⋊C₄×C₃⋊D₉	2
C₃×C₉×#103	2
#11720582	2
#11720539	2
#11720527	2
C₃⋊D₃×#103	3
D₉×#103	3

Subgroups — #103×C3⋊D9 #13143974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #103×C₃⋊D₉ #13143974