Subgroups — C₃²×#5045 #13162481

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#926	C₂	1	1	yes	maximal, characteristic
3	C₃²×#835	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#830	C₂	1	1	yes	maximal, characteristic
6	C₃²×#808	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#485	—	3	1	no	maximal
9	C₃×#5045	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₃³×C₄⋊C₄	—	2	1	no	—
13	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃³×C₂²⋊C₄	—	2	1	no	—
15	C₃²×#217	—	2	1	no	—
16	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₃²×C₄×Dic₃	—	2	1	no	—
19	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×C₂²⋊Q₈	—	2	4	no	—
22	C₃²×C₂²⋊Q₈	D₃	1	1	yes	characteristic
23	C₃²×C₂²⋊Q₈	C₂×C₃	1	4	yes	—
24	C₃²×#117	—	3	1	no	—
25	C₃×#926	C₂×C₃	1	4	yes	—
26	C₃²×#107	—	3	1	no	—
27	C₃²×#106	—	3	1	no	—
28	C₃²×#103	—	3	1	no	—
29	C₃²×#103	—	3	1	no	—
30	C₃²×C₂²⋊C₈	—	3	1	no	—
31	C₃×#835	C₂×C₃	1	4	yes	complemented
32	C₃×#835	C₂×C₃	1	4	yes	complemented
33	C₃×#830	C₂×C₃	1	4	yes	—
34	C₃×#808	C₂×C₃	1	4	yes	complemented
35	C₃×#807	C₂×C₃	1	4	yes	complemented
36	C₂³×C₃³	D₄	1	1	yes	characteristic
37	C₃³×Q₈	—	2	1	no	—
38	C₃³×Q₈	—	2	1	no	—
39	C₂×C₃³×C₄	—	2	1	no	—
40	C₂×C₃³×C₄	—	2	1	no	—
41	C₂×C₃³×C₄	—	2	1	no	—
42	C₂×C₃³×C₄	—	2	1	no	—
43	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
44	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃²×Dic₃	—	2	1	no	—
46	C₂×C₃²×Dic₃	—	2	1	no	—
47	C₃²×C₃⋊C₈	—	2	1	no	—
48	C₃²×C₃⋊C₈	—	2	1	no	—
49	C₃×#485	—	3	4	no	—
50	#5045	C₃²	1	1	yes	characteristic, direct product factor
51	C₂×C₃²×Q₈	—	2	4	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
54	C₂²×C₃²×C₄	—	2	4	no	—
55	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₃²×C₈	—	3	1	no	—
58	C₂×C₃²×C₈	—	3	1	no	—
59	C₃²×C₄⋊C₄	—	2	8	no	—
60	C₃²×C₄⋊C₄	—	2	4	no	—
61	C₃²×C₄⋊C₄	—	2	1	no	—
62	C₃²×C₄⋊C₄	—	2	4	no	—
63	C₃²×C₄⋊C₄	—	3	1	no	—
64	C₃²×C₄⋊C₄	—	3	1	no	—
65	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
67	C₃²×C₂²⋊C₄	—	2	8	no	—
68	C₃²×C₂²⋊C₄	—	2	1	no	—
69	C₃²×C₂²⋊C₄	—	2	4	no	—
70	C₃²×C₂²⋊C₄	—	6	1	no	—
71	C₃²×C₄²	—	6	1	no	—
72	C₃×#217	—	2	4	no	—
73	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
74	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
75	C₃×C₄×Dic₃	—	2	4	no	—
76	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
77	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
78	C₂²×C₃³	—	2	1	no	—
79	C₂²×C₃³	—	2	1	no	—
80	C₂²×C₃³	—	2	1	no	—
81	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
82	C₃³×C₄	—	2	1	no	—
83	C₃³×C₄	—	4	1	no	—
84	C₃³×C₄	—	4	1	no	—
85	C₃³×C₄	#41	1	1	yes	characteristic
86	C₃³×C₄	#41	1	1	yes	characteristic
87	C₃²×Dic₃	—	4	1	no	—
88	C₃²×Dic₃	—	4	1	no	—
89	C₃×C₂²⋊Q₈	—	2	4	no	—
90	C₃×C₂²⋊Q₈	C₃×D₃	1	4	yes	—
91	C₃×C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic
92	C₃×#117	—	3	4	no	—
93	#926	C₂×C₃²	1	1	yes	characteristic
94	C₃×#107	—	3	4	no	—
95	C₃×#106	—	3	4	no	—
96	C₃×#103	—	3	4	no	—
97	C₃×#103	—	3	4	no	—
98	C₃×C₂²⋊C₈	—	3	4	no	—
99	#835	C₂×C₃²	1	1	yes	characteristic, complemented
100	#835	C₂×C₃²	1	1	yes	characteristic, complemented
101	#830	C₂×C₃²	1	1	yes	characteristic
102	#808	C₂×C₃²	1	1	yes	characteristic, complemented
103	#807	C₂×C₃²	1	1	yes	characteristic, complemented
104	C₂³×C₃²	—	2	4	no	—
105	C₂³×C₃²	#67	1	1	yes	characteristic
106	C₂³×C₃²	C₃×D₄	1	4	yes	—
107	C₃²×Q₈	—	2	1	no	—
108	C₃²×Q₈	—	2	1	no	—
109	C₃²×Q₈	—	2	4	no	—
110	C₃²×Q₈	—	2	4	no	—
111	C₃²×Q₈	—	4	4	no	—
112	C₃²×Q₈	—	4	4	no	—
113	C₂×C₃²×C₄	—	2	1	no	—
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	—	2	8	no	—
116	C₂×C₃²×C₄	—	2	8	no	—
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	—	2	4	no	—
120	C₂×C₃²×C₄	—	2	4	no	—
121	C₂×C₃²×C₄	—	2	4	no	—
122	C₂×C₃²×C₄	—	2	4	no	—
123	C₂×C₃²×C₄	—	2	4	no	—
124	C₂×C₃²×C₄	—	2	4	no	—
125	C₂×C₃²×C₄	—	4	4	no	—
126	C₂×C₃²×C₄	—	4	4	no	—
127	C₂×C₃²×C₄	—	6	1	no	—
128	C₂×C₃²×C₄	—	6	1	no	—
129	C₂×C₃²×C₄	#67	1	1	yes	characteristic
130	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
131	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
133	C₂×C₃×Dic₃	—	2	4	no	—
134	C₂×C₃×Dic₃	—	2	4	no	—
135	C₃²×C₈	—	6	1	no	—
136	C₃²×C₈	—	6	1	no	—
137	C₃×C₃⋊C₈	—	2	4	no	—
138	C₃×C₃⋊C₈	—	2	4	no	—
139	#485	—	3	1	no	Sylow
140	C₂×C₃³	—	4	1	no	—
141	C₂×C₃³	#123	1	1	yes	characteristic
142	C₂×C₃³	#135	1	1	yes	characteristic
143	C₂×C₃³	#137	1	1	yes	characteristic
144	C₂×C₃×Q₈	—	2	4	no	—
145	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
146	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
147	C₂²×C₃×C₄	—	2	4	no	—
148	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
149	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
150	C₂×C₃×C₈	—	3	4	no	—
151	C₂×C₃×C₈	—	3	4	no	—
152	C₃×C₄⋊C₄	—	2	8	no	—
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	—	2	4	no	—
156	C₃×C₄⋊C₄	—	3	4	no	—
157	C₃×C₄⋊C₄	—	3	4	no	—
158	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
159	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	8	no	—
161	C₃×C₂²⋊C₄	—	2	4	no	—
162	C₃×C₂²⋊C₄	—	2	1	no	—
163	C₃×C₂²⋊C₄	—	6	4	no	—
164	C₃×C₄²	—	6	4	no	—
165	#217	—	2	1	no	—
166	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
167	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
168	C₄×Dic₃	—	2	1	no	—
169	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
170	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
171	C₂²×C₃²	—	2	8	no	—
172	C₂²×C₃²	—	2	1	no	—
173	C₂²×C₃²	—	2	1	no	—
174	C₂²×C₃²	—	2	1	no	—
175	C₂²×C₃²	—	2	4	no	—
176	C₂²×C₃²	—	2	4	no	—
177	C₂²×C₃²	—	2	4	no	—
178	C₂²×C₃²	—	2	4	no	—
179	C₂²×C₃²	—	4	4	no	—
180	C₂²×C₃²	—	4	4	no	—
181	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
182	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
183	C₃²×C₄	—	2	1	no	—
184	C₃²×C₄	—	2	4	no	—
185	C₃²×C₄	—	2	4	no	—
186	C₃²×C₄	—	2	4	no	—
187	C₃²×C₄	—	4	4	no	—
188	C₃²×C₄	—	4	8	no	—
189	C₃²×C₄	—	4	8	no	—
190	C₃²×C₄	—	4	1	no	—
191	C₃²×C₄	—	4	1	no	—
192	C₃²×C₄	—	4	4	no	—
193	C₃²×C₄	—	4	4	no	—
194	C₃²×C₄	—	12	1	no	—
195	C₃²×C₄	—	12	1	no	—
196	C₃²×C₄	#237	1	1	yes	characteristic
197	C₃²×C₄	#237	1	1	yes	characteristic
198	C₃²×C₄	C₃×#41	1	4	yes	—
199	C₃²×C₄	C₃×#41	1	4	yes	—
200	C₃×Dic₃	—	4	4	no	—
201	C₃×Dic₃	—	4	4	no	—
202	C₂²⋊Q₈	C₃²×D₃	1	1	yes	characteristic
203	#117	—	3	1	no	—
204	#107	—	3	1	no	—
205	#106	—	3	1	no	—
206	#103	—	3	1	no	—
207	#103	—	3	1	no	—
208	C₂²⋊C₈	—	3	1	no	—
209	C₃³	#485	1	1	yes	characteristic, complemented, Sylow
210	C₂³×C₃	—	2	4	no	—
211	C₂³×C₃	C₃×#67	1	4	yes	—
212	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
213	C₃×Q₈	—	2	4	no	—
214	C₃×Q₈	—	2	4	no	—
215	C₃×Q₈	—	2	1	no	—
216	C₃×Q₈	—	2	1	no	—
217	C₃×Q₈	—	4	4	no	—
218	C₃×Q₈	—	4	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	8	no	—
224	C₂×C₃×C₄	—	2	8	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	4	no	—
230	C₂×C₃×C₄	—	2	4	no	—
231	C₂×C₃×C₄	—	4	4	no	—
232	C₂×C₃×C₄	—	4	4	no	—
233	C₂×C₃×C₄	—	6	4	no	—
234	C₂×C₃×C₄	—	6	4	no	—
235	C₂×C₃×C₄	C₃×#67	1	4	yes	—
236	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
238	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
239	C₂×Dic₃	—	2	1	no	—
240	C₂×Dic₃	—	2	1	no	—
241	C₃×C₈	—	6	4	no	—
242	C₃×C₈	—	6	4	no	—
243	C₃⋊C₈	—	2	1	no	—
244	C₃⋊C₈	—	2	1	no	—
245	C₂×C₃²	—	2	4	no	—
246	C₂×C₃²	—	2	4	no	—
247	C₂×C₃²	—	2	4	no	—
248	C₂×C₃²	—	4	8	no	—
249	C₂×C₃²	—	4	1	no	—
250	C₂×C₃²	—	4	4	no	—
251	C₂×C₃²	#935	1	1	yes	characteristic
252	C₂×C₃²	#942	1	1	yes	characteristic
253	C₂×C₃²	#950	1	1	yes	characteristic
254	C₂×C₃²	C₃×#123	1	4	yes	—
255	C₂×C₃²	C₃×#135	1	4	yes	—
256	C₂×C₃²	C₃×#137	1	4	yes	—
257	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
259	C₂×C₈	—	3	1	no	—
260	C₂×C₈	—	3	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	—	3	1	no	—
264	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	6	1	no	—
267	C₄²	—	6	1	no	—
268	C₂²×C₃	—	2	8	no	—
269	C₂²×C₃	—	2	4	no	—
270	C₂²×C₃	—	2	4	no	—
271	C₂²×C₃	—	2	4	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	4	no	—
276	C₂²×C₃	—	4	4	no	—
277	C₂²×C₃	—	4	4	no	—
278	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
279	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
280	C₃×C₄	—	2	4	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	4	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	4	4	no	—
285	C₃×C₄	—	4	8	no	—
286	C₃×C₄	—	4	8	no	—
287	C₃×C₄	—	4	4	no	—
288	C₃×C₄	—	4	4	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	12	4	no	—
292	C₃×C₄	—	12	4	no	—
293	C₃×C₄	C₃×#237	1	4	yes	—
294	C₃×C₄	C₃×#237	1	4	yes	—
295	C₃×C₄	C₃²×#41	1	1	yes	characteristic
296	C₃×C₄	C₃²×#41	1	1	yes	characteristic
297	Dic₃	—	4	1	no	—
298	Dic₃	—	4	1	no	—
299	C₃²	—	2	4	no	—
300	C₃²	#5045	1	1	yes	characteristic, direct product factor
301	C₃²	C₃×#485	1	4	yes	complemented
302	C₂³	C₃²×#67	1	1	yes	characteristic
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	6	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	C₃²×#67	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
313	C₈	—	6	1	no	—
314	C₈	—	6	1	no	—
315	C₂×C₃	—	2	4	no	—
316	C₂×C₃	—	2	4	no	—
317	C₂×C₃	—	2	4	no	—
318	C₂×C₃	—	4	8	no	—
319	C₂×C₃	—	4	1	no	—
320	C₂×C₃	—	4	4	no	—
321	C₂×C₃	C₃×#935	1	4	yes	—
322	C₂×C₃	C₃×#942	1	4	yes	—
323	C₂×C₃	C₃×#950	1	4	yes	—
324	C₂×C₃	C₃²×#123	1	1	yes	characteristic
325	C₂×C₃	C₃²×#135	1	1	yes	characteristic
326	C₂×C₃	C₃²×#137	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	12	1	no	—
335	C₄	—	12	1	no	—
336	C₄	C₃²×#237	1	1	yes	characteristic
337	C₄	C₃²×#237	1	1	yes	characteristic
338	C₃	—	2	4	no	—
339	C₃	C₃×#5045	1	4	yes	direct product factor
340	C₃	C₃²×#485	1	1	yes	characteristic, complemented
341	C₂	—	4	1	no	—
342	C₂	C₃²×#935	1	1	yes	characteristic
343	C₂	C₃²×#942	1	1	yes	characteristic
344	C₂	C₃²×#950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₂²⋊Q₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂³×C₃³	D₄
hypercenter	C₃²×C₂²⋊Q₈	D₃
hypocenter	C₃	C₃²×#485
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₃²×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₃²×#835	C₂	semidirect
4. C₃²×#835	C₂	semidirect
6. C₃²×#808	C₂	semidirect
7. C₃²×#807	C₂	semidirect
9. C₃×#5045	C₃	direct (N × H)
31. C₃×#835	C₂×C₃	semidirect
32. C₃×#835	C₂×C₃	semidirect
34. C₃×#808	C₂×C₃	semidirect
35. C₃×#807	C₂×C₃	semidirect
50. #5045	C₃²	direct (N × H)
99. #835	C₂×C₃²	semidirect
100. #835	C₂×C₃²	semidirect
102. #808	C₂×C₃²	semidirect
103. #807	C₂×C₃²	semidirect
209. C₃³	#485	semidirect
300. C₃²	#5045	direct (N × H)
301. C₃²	C₃×#485	semidirect
339. C₃	C₃×#5045	direct (N × H)
340. C₃	C₃²×#485	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₂²⋊Q₈	2
C₃²×#926	2
C₃²×#835	2
C₃²×#830	2
C₃²×#808	2
C₃²×#807	2
C₃²×#485	3
C₃×#5045	3

Subgroups — C32×#5045 #13162481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5045 #13162481