Subgroups — C₃×#10588482 #13141863

Order 1728.

492 conjugacy classes of subgroups.

The subgroup lattice has 368 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	1	yes	maximal, characteristic
2	C₂×C₃×#3663	C₂	1	1	yes	maximal, characteristic
3	C₃×#63454	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63368	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4802	—	3	1	no	maximal
9	#10588482	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×Q₈×D₉	—	2	2	no	—
12	C₃×Q₈×D₉	C₄	1	4	yes	complemented
13	C₂×C₃×C₄×D₉	—	2	1	no	—
14	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₈	—	2	1	no	—
16	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
20	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
21	C₃×C₄×Dic₉	—	2	1	no	—
22	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×#3663	—	2	2	no	—
24	C₂×C₃×D₃×Q₈	—	3	1	no	—
25	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
26	C₃×#892	—	3	1	no	—
27	C₂×Q₈×D₉	C₂×C₃	1	1	yes	characteristic
28	C₃²×#103	D₃	1	1	yes	characteristic
29	C₃×#835	—	3	1	no	—
30	C₃×#816	—	3	1	no	—
31	C₃×#810	—	3	1	no	—
32	C₂×#3663	C₂×C₃	1	1	yes	characteristic
33	#63454	C₂×C₃	1	1	yes	characteristic
34	C₉×#103	—	2	1	no	—
35	C₉×#103	C₂×C₃	1	1	yes	characteristic, complemented
36	#63395	C₂×C₃	1	1	yes	characteristic, complemented
37	#63376	C₂×C₃	1	1	yes	characteristic, complemented
38	#63368	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×D₉	D₄	1	1	yes	characteristic
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₃×Q₈×C₉	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₄×C₉	—	2	1	no	—
43	C₂×C₃×C₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃×Dic₉	—	2	1	no	—
46	C₂×C₃×Dic₉	—	2	1	no	—
47	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
48	C₃×C₄×D₉	—	2	4	no	—
49	C₃×C₄×D₉	C₂×C₄	1	2	yes	—
50	C₃×C₄×D₉	D₄	1	2	yes	—
51	C₃×Dic₁₈	—	2	1	no	—
52	C₃×Dic₁₈	—	2	4	no	—
53	C₃×Dic₁₈	C₂×C₄	1	2	yes	—
54	C₃×C₈×C₉	—	2	1	no	—
55	C₃×C₉⋊C₈	—	2	1	no	—
56	C₃×#419	—	9	1	no	—
57	#4802	—	3	1	no	—
58	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₃×D₃×Q₈	—	3	4	no	—
60	C₃×D₃×Q₈	—	6	2	no	—
61	C₂×C₃×C₄×D₃	—	3	1	no	—
62	C₂×C₃×C₄×D₃	—	6	1	no	—
63	C₂×C₃×Dic₆	—	3	1	no	—
64	C₂×C₃×Dic₆	—	6	1	no	—
65	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
68	C₃×C₄⋊Dic₃	—	3	1	no	—
69	C₃×C₄×Dic₃	—	6	1	no	—
70	C₂×C₃×C₃⋊C₈	—	3	1	no	—
71	C₃×ΓL(1,𝔽₂₅)	—	6	2	no	—
72	C₂×Q₈×C₉	—	2	1	no	—
73	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
74	Q₈×D₉	—	2	2	no	—
75	Q₈×D₉	C₃×C₄	1	4	yes	complemented
76	C₂×C₄×D₉	—	2	1	no	—
77	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
78	C₂×Dic₁₈	—	2	1	no	—
79	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₈×C₉	—	2	1	no	—
81	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
82	C₉×C₄⋊C₄	—	2	1	no	—
83	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	(C₂×D₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
86	C₄×Dic₉	—	2	1	no	—
87	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
88	#3663	—	2	2	no	—
89	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×D₉	—	2	1	no	—
91	C₂×C₃×D₉	—	2	1	no	—
92	C₂×C₃×D₉	C₂²⋊C₄	1	2	yes	—
93	C₃×C₄×C₉	—	2	2	no	—
94	C₃×C₄×C₉	—	4	1	no	—
95	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
96	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₃×Dic₉	—	2	2	no	—
98	C₃×Dic₉	—	4	1	no	—
99	C₃×Dic₉	C₂²⋊C₄	1	2	yes	—
100	C₂²×C₃×Q₈	—	9	1	no	—
101	C₂×D₃×Q₈	—	3	1	no	—
102	C₂×C₃×M₁₆	—	9	1	no	—
103	C₃×#117	—	9	1	no	—
104	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
105	#892	—	3	1	no	—
106	C₃×#103	—	2	1	no	—
107	C₃×#103	—	9	1	no	—
108	C₃×#103	—	9	1	no	—
109	C₃×#103	—	9	1	no	—
110	C₃×#103	D₉	1	1	yes	characteristic, complemented
111	C₃×#103	C₃×D₃	1	1	yes	characteristic
112	#835	—	3	1	no	—
113	#816	—	3	1	no	—
114	#810	—	3	1	no	—
115	C₂²×C₃×D₃	—	3	1	no	—
116	C₃²×Q₈	—	2	1	no	—
117	C₃²×Q₈	C₄×D₃	1	2	yes	—
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	—	2	1	no	—
120	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₃×Dic₃	—	3	1	no	—
122	C₂×C₃×Dic₃	—	6	1	no	—
123	C₂×C₃×Dic₃	—	6	1	no	—
124	C₃×C₄×D₃	—	3	2	no	—
125	C₃×C₄×D₃	—	3	2	no	—
126	C₃×C₄×D₃	—	6	4	no	—
127	C₃×Dic₆	—	3	2	no	—
128	C₃×Dic₆	—	6	4	no	—
129	C₃×Dic₆	—	6	1	no	—
130	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
131	C₃²×C₈	—	2	1	no	—
132	C₃×C₃⋊C₈	—	6	1	no	—
133	Q₈×C₉	—	2	1	no	—
134	Q₈×C₉	—	2	2	no	—
135	Q₈×C₉	—	4	1	no	—
136	Q₈×C₉	C₂×C₃×C₄	1	2	yes	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	—	2	1	no	—
139	C₂×C₄×C₉	—	2	1	no	—
140	C₂×C₄×C₉	—	4	1	no	—
141	C₂×C₄×C₉	—	4	1	no	—
142	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
143	C₂×Dic₉	—	2	1	no	—
144	C₂×Dic₉	—	2	1	no	—
145	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
146	C₄×D₉	—	2	4	no	—
147	C₄×D₉	C₂×C₃×C₄	1	2	yes	—
148	C₄×D₉	C₃×D₄	1	2	yes	—
149	Dic₁₈	—	2	1	no	—
150	Dic₁₈	—	2	4	no	—
151	Dic₁₈	C₂×C₃×C₄	1	2	yes	—
152	C₈×C₉	—	2	1	no	—
153	C₈×C₉	—	4	1	no	—
154	C₉⋊C₈	—	2	1	no	—
155	#419	—	9	1	no	Sylow
156	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₉	#137	1	1	yes	characteristic
158	C₂×C₃×C₉	#137	1	1	yes	characteristic
159	C₃×D₉	—	2	2	no	—
160	C₂×C₃×Q₈	—	2	1	no	—
161	C₂×C₃×Q₈	—	9	4	no	—
162	C₂×C₃×Q₈	—	9	1	no	—
163	C₂×C₃×Q₈	—	18	1	no	—
164	C₂×C₃×Q₈	—	18	2	no	—
165	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
166	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
167	C₂²×C₃×C₄	—	9	1	no	—
168	C₂²×C₃×C₄	—	18	1	no	—
169	D₃×Q₈	—	3	4	no	—
170	D₃×Q₈	—	6	2	no	—
171	C₂×C₄×D₃	—	3	1	no	—
172	C₂×C₄×D₃	—	6	1	no	—
173	C₂×Dic₆	—	3	1	no	—
174	C₂×Dic₆	—	6	1	no	—
175	C₃×M₁₆	—	18	2	no	—
176	C₂×C₃×C₈	—	2	1	no	—
177	C₂×C₃×C₈	—	9	1	no	—
178	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
179	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
180	C₃×C₄⋊C₄	—	2	1	no	—
181	C₃×C₄⋊C₄	—	9	1	no	—
182	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
183	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
184	C₃×C₂²⋊C₄	—	18	1	no	—
185	C₃×C₄²	—	18	1	no	—
186	(C₂×D₃)⋊C₄	—	6	1	no	—
187	C₄⋊Dic₃	—	3	1	no	—
188	C₄×Dic₃	—	6	1	no	—
189	C₂×C₃⋊C₈	—	3	1	no	—
190	ΓL(1,𝔽₂₅)	—	6	2	no	—
191	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
192	C₂×C₃×D₃	—	3	2	no	—
193	C₂×C₃×D₃	—	6	1	no	—
194	C₂×C₃×D₃	—	6	1	no	—
195	C₃²×C₄	—	2	2	no	—
196	C₃²×C₄	—	4	1	no	—
197	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
199	C₃×Dic₃	—	3	2	no	—
200	C₃×Dic₃	—	6	2	no	—
201	C₃×Dic₃	—	12	1	no	—
202	C₂²×C₉	—	2	1	no	—
203	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×D₉	—	2	1	no	—
205	C₂×D₉	—	2	1	no	—
206	C₂×D₉	C₃×C₂²⋊C₄	1	2	yes	—
207	C₄×C₉	—	2	2	no	—
208	C₄×C₉	—	2	1	no	—
209	C₄×C₉	—	2	1	no	—
210	C₄×C₉	—	4	2	no	—
211	C₄×C₉	—	4	1	no	—
212	C₄×C₉	—	8	1	no	—
213	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
214	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
215	Dic₉	—	2	2	no	—
216	Dic₉	—	4	1	no	—
217	Dic₉	C₃×C₂²⋊C₄	1	2	yes	—
218	C₂²×Q₈	—	9	1	no	—
219	C₂×M₁₆	—	9	1	no	—
220	#117	—	9	1	no	—
221	#103	—	9	1	no	—
222	#103	—	9	1	no	—
223	#103	—	9	1	no	—
224	#103	C₃×D₉	1	1	yes	characteristic, complemented
225	C₃×C₉	#419	1	1	yes	characteristic, complemented, Sylow
226	C₂³×C₃	—	9	1	no	—
227	C₂²×D₃	—	3	1	no	—
228	C₃×Q₈	—	2	1	no	—
229	C₃×Q₈	—	2	1	no	—
230	C₃×Q₈	—	2	2	no	—
231	C₃×Q₈	—	4	1	no	—
232	C₃×Q₈	—	9	2	no	—
233	C₃×Q₈	—	18	4	no	—
234	C₃×Q₈	—	18	1	no	—
235	C₃×Q₈	C₄×D₉	1	2	yes	—
236	C₃×Q₈	C₃×C₄×D₃	1	2	yes	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	2	1	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	4	1	no	—
243	C₂×C₃×C₄	—	4	1	no	—
244	C₂×C₃×C₄	—	9	1	no	—
245	C₂×C₃×C₄	—	9	2	no	—
246	C₂×C₃×C₄	—	9	2	no	—
247	C₂×C₃×C₄	—	18	4	no	—
248	C₂×C₃×C₄	—	18	1	no	—
249	C₂×C₃×C₄	—	18	1	no	—
250	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
252	C₂×Dic₃	—	3	1	no	—
253	C₂×Dic₃	—	6	1	no	—
254	C₂×Dic₃	—	6	1	no	—
255	C₄×D₃	—	3	2	no	—
256	C₄×D₃	—	3	2	no	—
257	C₄×D₃	—	6	4	no	—
258	Dic₆	—	3	2	no	—
259	Dic₆	—	6	4	no	—
260	Dic₆	—	6	1	no	—
261	C₃×C₈	—	2	1	no	—
262	C₃×C₈	—	2	1	no	—
263	C₃×C₈	—	4	1	no	—
264	C₃×C₈	—	18	1	no	—
265	C₃⋊C₈	—	6	1	no	—
266	C₂×C₃²	D₃×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₃²	#915	1	1	yes	characteristic
268	C₂×C₃²	#918	1	1	yes	characteristic
269	C₃×D₃	—	6	2	no	—
270	C₂×C₉	—	2	1	no	—
271	C₂×C₉	—	2	1	no	—
272	C₂×C₉	—	2	1	no	—
273	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₉	C₃×#137	1	1	yes	characteristic
275	C₂×C₉	C₃×#137	1	1	yes	characteristic
276	D₉	—	2	2	no	—
277	C₂×Q₈	—	9	4	no	—
278	C₂×Q₈	—	9	1	no	—
279	C₂×Q₈	—	18	1	no	—
280	C₂×Q₈	—	18	2	no	—
281	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
282	C₂²×C₄	—	9	1	no	—
283	C₂²×C₄	—	18	1	no	—
284	M₁₆	—	18	2	no	—
285	C₂×C₈	—	9	1	no	—
286	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
287	C₄⋊C₄	—	9	1	no	—
288	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
289	C₂²⋊C₄	—	18	1	no	—
290	C₄²	—	18	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	—	9	2	no	—
293	C₂²×C₃	—	18	1	no	—
294	C₂²×C₃	—	18	1	no	—
295	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
296	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
297	C₂×D₃	—	3	2	no	—
298	C₂×D₃	—	6	1	no	—
299	C₂×D₃	—	6	1	no	—
300	C₃×C₄	—	2	2	no	—
301	C₃×C₄	—	2	2	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	4	2	no	—
305	C₃×C₄	—	4	1	no	—
306	C₃×C₄	—	4	1	no	—
307	C₃×C₄	—	8	1	no	—
308	C₃×C₄	—	9	2	no	—
309	C₃×C₄	—	18	2	no	—
310	C₃×C₄	—	36	1	no	—
311	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
312	C₃×C₄	D₄×D₉	1	1	yes	characteristic
313	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
314	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
315	Dic₃	—	3	2	no	—
316	Dic₃	—	6	2	no	—
317	Dic₃	—	12	1	no	—
318	C₃²	#4802	1	1	yes	characteristic
319	C₉	—	2	1	no	—
320	C₉	C₃×#419	1	1	yes	characteristic, complemented
321	C₂³	—	9	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	9	2	no	—
324	Q₈	—	18	4	no	—
325	Q₈	—	18	1	no	—
326	Q₈	C₃×C₄×D₉	1	2	yes	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	9	1	no	—
330	C₂×C₄	—	9	2	no	—
331	C₂×C₄	—	9	2	no	—
332	C₂×C₄	—	18	4	no	—
333	C₂×C₄	—	18	1	no	—
334	C₂×C₄	—	18	1	no	—
335	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	18	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	2	1	no	—
341	C₂×C₃	—	18	2	no	—
342	C₂×C₃	C₂²⋊C₄×D₉	1	1	yes	characteristic
343	C₂×C₃	#63477	1	1	yes	characteristic
344	C₂×C₃	#63480	1	1	yes	characteristic
345	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
346	C₂×C₃	C₃×#915	1	1	yes	characteristic
347	C₂×C₃	C₃×#918	1	1	yes	characteristic
348	D₃	—	6	2	no	—
349	C₂²	—	9	2	no	—
350	C₂²	—	18	1	no	—
351	C₂²	—	18	1	no	—
352	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	9	2	no	—
356	C₄	—	18	2	no	—
357	C₄	—	36	1	no	—
358	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
359	C₄	C₃×D₄×D₉	1	1	yes	characteristic
360	C₃	—	2	1	no	—
361	C₃	#10588482	1	1	yes	characteristic, direct product factor
362	C₃	C₃×#4802	1	1	yes	characteristic
363	C₂	—	18	2	no	—
364	C₂	C₃×C₂²⋊C₄×D₉	1	1	yes	characteristic
365	C₂	C₃×#63477	1	1	yes	characteristic
366	C₂	C₃×#63480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator	C₄×C₉	C₂²×C₃×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×C₄×C₉	C₂³×C₃
hypercenter	C₃×#103	D₉
hypocenter	C₉	C₃×#419
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

Complements

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

N	H ≅ G/N	type
4. C₃×C₉×#103	C₂	semidirect
5. C₃×#63395	C₂	semidirect
6. C₃×#63376	C₂	semidirect
7. C₃×#63368	C₂	semidirect
9. #10588482	C₃	direct (N × H)
12. C₃×Q₈×D₉	C₄	semidirect
35. C₉×#103	C₂×C₃	semidirect
36. #63395	C₂×C₃	semidirect
37. #63376	C₂×C₃	semidirect
38. #63368	C₂×C₃	semidirect
75. Q₈×D₉	C₃×C₄	semidirect
110. C₃×#103	D₉	semidirect
224. #103	C₃×D₉	semidirect
225. C₃×C₉	#419	semidirect
320. C₉	C₃×#419	semidirect
361. C₃	#10588482	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×#3663	2
C₃×#63454	2
C₃×C₉×#103	2
C₃×#63395	2
C₃×#63376	2
C₃×#63368	2
C₃×#4802	3
#10588482	3

Subgroups — C3×#10588482 #13141863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588482 #13141863