Subgroups — C₃×#10588709 #13142778

Order 1728.

430 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3675	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
4	C₃×#63395	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63369	C₂	1	1	yes	maximal, characteristic
6	C₃×#63366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5026	—	3	1	no	maximal
9	#10588709	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₉	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	—	2	1	no	—
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	—	2	1	no	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₃×#3675	C₄	1	4	yes	complemented
18	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
19	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
23	C₂×C₃×#215	—	3	1	no	—
24	C₃×C₄×D₁₂	—	3	1	no	—
25	C₃×#835	—	3	1	no	—
26	C₃×#809	—	3	1	no	—
27	C₃×#807	—	3	1	no	—
28	C₃×#803	—	3	1	no	—
29	C₄×Q₈×C₉	—	2	1	no	—
30	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#3675	C₂×C₃	1	1	yes	characteristic
32	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic
33	#63395	C₂×C₃	1	1	yes	characteristic, complemented
34	#63369	C₂×C₃	1	1	yes	characteristic
35	#63366	C₂×C₃	1	1	yes	characteristic, complemented
36	#63362	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×C₃×D₉	—	2	1	no	—
38	C₃×Q₈×C₉	—	2	1	no	—
39	C₃×Q₈×C₉	C₂×C₄	1	2	yes	complemented
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	—	2	1	no	—
42	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃×Dic₉	—	2	1	no	—
46	C₃×D₃₆	—	2	1	no	—
47	C₃×D₃₆	C₂×C₄	1	2	yes	—
48	C₃×C₄×D₉	—	4	2	no	—
49	C₃×C₉⋊C₈	—	2	1	no	—
50	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	9	1	no	—
52	#5026	—	3	1	no	—
53	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×D₁₂	—	3	1	no	—
55	C₂×C₃×C₄×D₃	—	6	1	no	—
56	C₃²×C₄⋊C₄	—	2	1	no	—
57	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄²	—	2	1	no	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃×#215	—	3	4	no	—
61	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
62	C₃×C₄⋊Dic₃	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₂×C₃×C₃⋊C₈	—	3	1	no	—
65	C₂×Q₈×C₉	—	2	1	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×D₉	—	2	1	no	—
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	2	1	no	—
72	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₄²×C₉	—	2	2	no	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	—	2	1	no	—
76	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
77	#3675	C₃×C₄	1	4	yes	complemented
78	(C₂×D₉)⋊C₄	—	2	1	no	—
79	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×D₉	—	2	2	no	—
84	C₂×C₃×D₉	—	4	1	no	—
85	C₂×C₃×D₉	—	4	1	no	—
86	C₃×C₄×C₉	—	2	2	no	—
87	C₃×C₄×C₉	—	2	1	no	—
88	C₃×C₄×C₉	—	2	1	no	—
89	C₃×C₄×C₉	—	4	1	no	—
90	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
91	C₃×C₄×C₉	#41	1	1	yes	characteristic
92	C₃×Dic₉	—	4	1	no	—
93	C₂×C₃×SD₁₆	—	9	1	no	—
94	C₃×C₄×Q₈	—	2	1	no	—
95	C₃×C₄×Q₈	D₉	1	1	yes	characteristic, complemented
96	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
97	C₃×C₄×D₄	—	9	1	no	—
98	C₂×#215	—	3	1	no	—
99	C₄×D₁₂	—	3	1	no	—
100	C₃×#107	—	9	1	no	—
101	C₃×#103	—	9	1	no	—
102	C₃×#102	—	9	1	no	—
103	C₃×#97	—	9	1	no	—
104	#835	—	3	1	no	—
105	#809	—	3	1	no	—
106	#807	—	3	1	no	—
107	#803	—	3	1	no	—
108	C₂²×C₃×D₃	—	6	1	no	—
109	C₃²×Q₈	—	2	1	no	—
110	C₃²×Q₈	C₄×D₃	1	2	yes	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	—	2	1	no	—
113	C₂×C₃²×C₄	#67	1	1	yes	characteristic
114	C₂×C₃²×C₄	#67	1	1	yes	characteristic
115	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₃×Dic₃	—	6	1	no	—
117	C₃×D₁₂	—	3	2	no	—
118	C₃×D₁₂	—	6	1	no	—
119	C₃×C₄×D₃	—	12	2	no	—
120	C₂²×D₉	—	2	1	no	—
121	C₃×C₃⋊C₈	—	3	2	no	—
122	C₃×C₃⋊C₈	—	6	1	no	—
123	Q₈×C₉	—	2	2	no	—
124	Q₈×C₉	—	2	1	no	—
125	Q₈×C₉	—	2	2	no	—
126	Q₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
127	C₂×C₄×C₉	—	2	2	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	2	2	no	—
130	C₂×C₄×C₉	—	2	1	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	2	1	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
135	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
137	C₂×Dic₉	—	2	1	no	—
138	D₃₆	—	2	1	no	—
139	D₃₆	C₂×C₃×C₄	1	2	yes	—
140	C₄×D₉	—	4	2	no	—
141	C₉⋊C₈	—	2	1	no	—
142	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	complemented
143	#440	—	9	1	no	Sylow
144	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₃×C₉	#137	1	1	yes	characteristic
147	C₃×D₉	—	4	2	no	—
148	C₂×C₃×Q₈	—	2	1	no	—
149	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
150	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂×C₃×D₄	—	9	1	no	—
152	C₂²×C₃×C₄	—	18	1	no	—
153	C₂×D₁₂	—	3	1	no	—
154	C₂×C₄×D₃	—	6	1	no	—
155	C₃×SD₁₆	—	9	4	no	—
156	C₂×C₃×C₈	—	9	1	no	—
157	C₂×C₃×C₈	—	9	1	no	—
158	C₃×C₄⋊C₄	—	2	2	no	—
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₄⋊C₄	—	2	1	no	—
161	C₃×C₄⋊C₄	—	2	1	no	—
162	C₃×C₄⋊C₄	—	9	1	no	—
163	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
164	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
165	C₃×C₂²⋊C₄	—	18	1	no	—
166	C₃×C₄²	—	2	2	no	—
167	C₃×C₄²	—	2	1	no	—
168	C₃×C₄²	—	2	1	no	—
169	C₃×C₄²	—	2	1	no	—
170	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
171	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
172	#215	—	3	4	no	—
173	(C₂×D₃)⋊C₄	—	6	1	no	—
174	C₄⋊Dic₃	—	3	1	no	—
175	C₂×C₃⋊C₈	—	3	1	no	—
176	C₂×C₃⋊C₈	—	3	1	no	—
177	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
178	C₂×C₃×D₃	—	6	2	no	—
179	C₂×C₃×D₃	—	12	1	no	—
180	C₂×C₃×D₃	—	12	1	no	—
181	C₃²×C₄	—	2	2	no	—
182	C₃²×C₄	—	2	1	no	—
183	C₃²×C₄	—	2	1	no	—
184	C₃²×C₄	—	4	1	no	—
185	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
186	C₃²×C₄	#235	1	1	yes	characteristic
187	C₃×Dic₃	—	12	1	no	—
188	C₂²×C₉	—	2	1	no	—
189	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×D₉	—	2	2	no	—
191	C₂×D₉	—	4	1	no	—
192	C₂×D₉	—	4	1	no	—
193	C₄×C₉	—	2	2	no	—
194	C₄×C₉	—	2	4	no	—
195	C₄×C₉	—	2	2	no	—
196	C₄×C₉	—	2	1	no	—
197	C₄×C₉	—	2	1	no	—
198	C₄×C₉	—	2	1	no	—
199	C₄×C₉	—	2	1	no	—
200	C₄×C₉	—	4	1	no	—
201	C₄×C₉	—	4	2	no	—
202	C₄×C₉	—	4	1	no	—
203	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
204	C₄×C₉	C₃×#41	1	1	yes	characteristic
205	Dic₉	—	4	1	no	—
206	C₂×SD₁₆	—	9	1	no	—
207	C₄×Q₈	C₃×D₉	1	1	yes	characteristic, complemented
208	C₄×D₄	—	9	1	no	—
209	#107	—	9	1	no	—
210	#103	—	9	1	no	—
211	#102	—	9	1	no	—
212	#97	—	9	1	no	—
213	C₃×C₉	#440	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₃	—	18	1	no	—
215	C₂²×D₃	—	6	1	no	—
216	C₃×Q₈	—	2	2	no	—
217	C₃×Q₈	—	2	1	no	—
218	C₃×Q₈	—	2	1	no	—
219	C₃×Q₈	—	2	2	no	—
220	C₃×Q₈	C₄×D₉	1	2	yes	complemented
221	C₃×Q₈	C₃×C₄×D₃	1	2	yes	—
222	C₃×D₄	—	9	2	no	—
223	C₃×D₄	—	18	1	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	—	18	1	no	—
234	C₂×C₃×C₄	—	36	2	no	—
235	C₂×C₃×C₄	#611	1	1	yes	characteristic
236	C₂×C₃×C₄	#611	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
238	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
239	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
241	C₂×Dic₃	—	6	1	no	—
242	D₁₂	—	3	2	no	—
243	D₁₂	—	6	1	no	—
244	C₄×D₃	—	12	2	no	—
245	C₃×C₈	—	9	2	no	—
246	C₃×C₈	—	18	1	no	—
247	C₃⋊C₈	—	3	2	no	—
248	C₃⋊C₈	—	6	1	no	—
249	C₂×C₃²	C₄×#67	1	1	yes	characteristic
250	C₂×C₃²	#942	1	1	yes	characteristic
251	C₂×C₃²	#949	1	1	yes	characteristic
252	C₃×D₃	—	12	2	no	—
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
257	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂×C₉	C₃×#137	1	1	yes	characteristic
259	D₉	—	4	2	no	—
260	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
261	C₂×D₄	—	9	1	no	—
262	C₂²×C₄	—	18	1	no	—
263	SD₁₆	—	9	4	no	—
264	C₂×C₈	—	9	1	no	—
265	C₂×C₈	—	9	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	9	1	no	—
268	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	18	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	18	2	no	—
274	C₂²×C₃	—	36	1	no	—
275	C₂²×C₃	—	36	1	no	—
276	C₂²×C₃	C₂×#611	1	1	yes	characteristic
277	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
278	C₂×D₃	—	6	2	no	—
279	C₂×D₃	—	12	1	no	—
280	C₂×D₃	—	12	1	no	—
281	C₃×C₄	—	2	4	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	4	1	no	—
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	—	4	1	no	—
294	C₃×C₄	—	4	1	no	—
295	C₃×C₄	—	36	1	no	—
296	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
297	C₃×C₄	#3697	1	1	yes	characteristic
298	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
299	C₃×C₄	C₃×#235	1	1	yes	characteristic
300	Dic₃	—	12	1	no	—
301	C₃²	#5026	1	1	yes	characteristic
302	C₉	—	2	1	no	—
303	C₉	C₃×#440	1	1	yes	characteristic, complemented
304	C₂³	—	18	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	C₃×C₄×D₉	1	2	yes	complemented
307	D₄	—	9	2	no	—
308	D₄	—	18	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	18	1	no	—
312	C₂×C₄	—	36	2	no	—
313	C₂×C₄	C₃×#611	1	1	yes	characteristic
314	C₂×C₄	C₃×#611	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
316	C₈	—	9	2	no	—
317	C₈	—	18	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	36	2	no	—
322	C₂×C₃	C₄×#611	1	1	yes	characteristic
323	C₂×C₃	#63505	1	1	yes	characteristic
324	C₂×C₃	#63512	1	1	yes	characteristic
325	C₂×C₃	C₃×C₄×#67	1	1	yes	characteristic
326	C₂×C₃	C₃×#942	1	1	yes	characteristic
327	C₂×C₃	C₃×#949	1	1	yes	characteristic
328	D₃	—	12	2	no	—
329	C₂²	—	18	2	no	—
330	C₂²	—	36	1	no	—
331	C₂²	—	36	1	no	—
332	C₂²	C₂×C₃×#611	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	36	1	no	—
338	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
339	C₄	C₃×#3697	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	#10588709	1	1	yes	characteristic, direct product factor
342	C₃	C₃×#5026	1	1	yes	characteristic
343	C₂	—	36	2	no	—
344	C₂	C₃×C₄×#611	1	1	yes	characteristic
345	C₂	C₃×#63505	1	1	yes	characteristic
346	C₂	C₃×#63512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃×D₃₆	C₂²
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄×Q₈	D₉
hypocenter	C₉	C₃×#440
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×Q₈×C₉	C₂	semidirect
4. C₃×#63395	C₂	semidirect
6. C₃×#63366	C₂	semidirect
7. C₃×#63362	C₂	semidirect
9. #10588709	C₃	direct (N × H)
17. C₃×#3675	C₄	semidirect
30. C₄×Q₈×C₉	C₂×C₃	semidirect
33. #63395	C₂×C₃	semidirect
35. #63366	C₂×C₃	semidirect
36. #63362	C₂×C₃	semidirect
39. C₃×Q₈×C₉	C₂×C₄	semidirect
50. C₃×C₉⋊C₈	C₂×C₄	semidirect
77. #3675	C₃×C₄	semidirect
95. C₃×C₄×Q₈	D₉	semidirect
126. Q₈×C₉	C₂×C₃×C₄	semidirect
142. C₉⋊C₈	C₂×C₃×C₄	semidirect
207. C₄×Q₈	C₃×D₉	semidirect
213. C₃×C₉	#440	semidirect
220. C₃×Q₈	C₄×D₉	semidirect
303. C₉	C₃×#440	semidirect
306. Q₈	C₃×C₄×D₉	semidirect
341. C₃	#10588709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₂×C₃×#3675	2
C₃×C₄×D₃₆	2
C₃×#63395	2
C₃×#63369	2
C₃×#63366	2
C₃×#63362	2
C₃×#5026	3
#10588709	3

Subgroups — C3×#10588709 #13142778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588709 #13142778