Subgroups — C₃×#69485 #11790065

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
5	C₃×#4524	C₂	1	2	yes	maximal
6	C₃×#1940	—	3	1	no	maximal
7	#69485	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×C₄⋊C₄	C₄	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃²×#95	C₄	1	1	yes	characteristic
15	C₃²×#95	C₄	1	1	yes	characteristic
16	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
17	C₃×C₄⋊(C₃⋊C₈)	—	2	4	no	—
18	C₃×#803	—	2	2	no	—
19	C₂×C₃×C₄⋊C₈	—	3	2	no	—
20	C₂×C₃×C₄⋊C₈	—	3	1	no	—
21	C₂×C₃×#97	—	3	1	no	—
22	C₃×C₄×C₄⋊C₄	—	2	1	no	—
23	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
24	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
25	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
26	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	—
27	C₂×#803	C₂×C₃	1	1	yes	characteristic
28	C₃×#336	—	3	2	no	—
29	#4524	C₂×C₃	1	2	yes	—
30	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
35	C₃²×C₄⋊C₄	—	2	4	no	—
36	C₃²×C₄²	—	2	2	no	—
37	C₃²×C₄²	—	2	2	no	—
38	C₃²×C₄²	D₄	1	2	yes	—
39	C₃²×C₄²	Q₈	1	2	yes	—
40	C₂×C₃×C₃⋊C₈	—	2	1	no	—
41	C₂×C₃×C₃⋊C₈	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
49	#1940	—	3	1	no	Sylow
50	C₂²×C₃×C₈	—	3	1	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂²×C₃×C₈	—	3	2	no	—
53	C₂×C₃×C₄⋊C₄	—	2	2	no	—
54	C₂×C₃×C₄⋊C₄	Dic₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
56	C₂×C₃×C₄²	—	2	1	no	—
57	C₂×C₃×C₄²	—	2	2	no	—
58	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
63	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₃×C₄⋊C₈	—	6	4	no	—
66	C₃×C₄⋊C₈	—	6	2	no	—
67	C₃×#97	—	6	2	no	—
68	C₃×#95	—	2	1	no	—
69	C₃×#95	—	2	1	no	—
70	C₃×#95	Dic₃	1	1	yes	characteristic
71	C₃×#95	Dic₃	1	1	yes	characteristic
72	C₃×#95	C₃×C₄	1	1	yes	characteristic
73	C₃×#95	C₃×C₄	1	1	yes	characteristic
74	C₄⋊(C₃⋊C₈)	—	2	2	no	—
75	C₄⋊(C₃⋊C₈)	—	2	4	no	—
76	#803	—	2	2	no	—
77	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
86	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
87	C₂×C₃²×C₄	M₁₆	1	4	yes	—
88	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₃²×C₄	#41	1	1	yes	characteristic
93	C₂×C₃²×C₄	#41	1	1	yes	characteristic
94	C₃×C₃⋊C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	4	2	no	—
97	C₂×C₄⋊C₈	—	3	2	no	—
98	C₂×C₄⋊C₈	—	3	1	no	—
99	C₂×#97	—	3	1	no	—
100	C₄×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
101	#336	—	3	2	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
108	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
109	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
112	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
115	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₂×C₃×C₈	—	6	1	no	—
120	C₂×C₃×C₈	—	6	1	no	—
121	C₂×C₃×C₈	—	6	1	no	—
122	C₂×C₃×C₈	—	6	1	no	—
123	C₂×C₃×C₈	—	6	2	no	—
124	C₂×C₃×C₈	—	6	2	no	—
125	C₂×C₃×C₈	—	6	2	no	—
126	C₃×C₄⋊C₄	—	2	8	no	—
127	C₃×C₄⋊C₄	—	2	4	no	—
128	C₃×C₄⋊C₄	—	2	4	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	—	2	2	no	—
133	C₃×C₄²	—	2	2	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	4	2	no	—
136	C₃×C₄²	—	4	2	no	—
137	C₃×C₄²	Dic₆	1	2	yes	—
138	C₃×C₄²	D₁₂	1	2	yes	—
139	C₃×C₄²	C₃×D₄	1	2	yes	—
140	C₃×C₄²	C₃×Q₈	1	2	yes	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂×C₃⋊C₈	—	2	1	no	—
143	C₂×C₃⋊C₈	—	2	1	no	—
144	C₂×C₃⋊C₈	—	2	1	no	—
145	C₂×C₃⋊C₈	—	2	2	no	—
146	C₂×C₃⋊C₈	—	2	2	no	—
147	C₂×C₃⋊C₈	—	2	2	no	—
148	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
149	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
150	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
153	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
154	C₂²×C₃²	#131	1	2	yes	—
155	C₃²×C₄	—	2	2	no	—
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	—	2	4	no	—
158	C₃²×C₄	—	4	2	no	—
159	C₃²×C₄	#121	1	1	yes	characteristic
160	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
161	C₃²×C₄	#125	1	1	yes	characteristic
162	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₈	—	3	1	no	—
164	C₂²×C₈	—	3	1	no	—
165	C₂²×C₈	—	3	2	no	—
166	C₂×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
167	C₂×C₄²	C₂×C₃×D₃	1	2	yes	—
168	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
169	C₄⋊C₈	—	6	4	no	—
170	C₄⋊C₈	—	6	2	no	—
171	#97	—	6	2	no	—
172	#95	C₃×Dic₃	1	1	yes	characteristic
173	#95	C₃×Dic₃	1	1	yes	characteristic
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
176	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	4	2	no	—
207	C₂×C₃×C₄	#208	1	4	yes	—
208	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
209	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
210	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
211	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
212	C₂×C₃×C₄	C₃×M₁₆	1	4	yes	—
213	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
215	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	#237	1	1	yes	characteristic
217	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
218	C₂×C₃×C₄	#239	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
224	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
225	C₃×C₈	—	6	2	no	—
226	C₃×C₈	—	6	2	no	—
227	C₃×C₈	—	12	2	no	—
228	C₃⋊C₈	—	2	2	no	—
229	C₃⋊C₈	—	2	2	no	—
230	C₃⋊C₈	—	4	2	no	—
231	C₂×C₃²	#389	1	1	yes	characteristic
232	C₂×C₃²	#423	1	1	yes	characteristic
233	C₂×C₃²	#424	1	1	yes	characteristic
234	C₂×C₃²	#435	1	2	yes	—
235	C₂×C₃²	#446	1	2	yes	—
236	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
237	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
238	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
241	C₂×C₈	—	3	2	no	—
242	C₂×C₈	—	3	2	no	—
243	C₂×C₈	—	6	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	—	6	1	no	—
247	C₂×C₈	—	6	2	no	—
248	C₂×C₈	—	6	2	no	—
249	C₂×C₈	—	6	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₃×Dic₆	1	2	yes	—
254	C₄²	C₃×D₁₂	1	2	yes	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	C₂×#208	1	2	yes	—
261	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
262	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
263	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
264	C₂²×C₃	#948	1	2	yes	—
265	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
266	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
267	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
268	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
269	C₂²×C₃	C₃×#131	1	2	yes	—
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	4	4	no	—
283	C₃×C₄	—	4	4	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	#888	1	1	yes	characteristic
287	C₃×C₄	#890	1	1	yes	characteristic
288	C₃×C₄	#895	1	1	yes	characteristic
289	C₃×C₄	#897	1	1	yes	characteristic
290	C₃×C₄	C₃×#121	1	1	yes	characteristic
291	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
292	C₃×C₄	C₃×#125	1	1	yes	characteristic
293	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
294	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
295	C₂³	C₂²×C₃×Dic₃	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	C₃×#208	1	4	yes	—
304	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
305	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
306	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
307	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
308	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
309	C₂×C₄	C₃×#237	1	1	yes	characteristic
310	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
311	C₂×C₄	C₃×#239	1	1	yes	characteristic
312	C₈	—	6	2	no	—
313	C₈	—	6	2	no	—
314	C₈	—	12	2	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	#4924	1	1	yes	characteristic
321	C₂×C₃	#4984	1	1	yes	characteristic
322	C₂×C₃	#4999	1	1	yes	characteristic
323	C₂×C₃	#5011	1	2	yes	—
324	C₂×C₃	#5024	1	2	yes	—
325	C₂×C₃	C₃×#389	1	1	yes	characteristic
326	C₂×C₃	C₃×#423	1	1	yes	characteristic
327	C₂×C₃	C₃×#424	1	1	yes	characteristic
328	C₂×C₃	C₃×#435	1	2	yes	—
329	C₂×C₃	C₃×#446	1	2	yes	—
330	C₂²	C₂×C₃×#208	1	2	yes	—
331	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
332	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
333	C₂²	C₃×Q₈×Dic₃	1	1	yes	characteristic
334	C₂²	C₃×#948	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	4	2	no	—
339	C₄	C₃×#888	1	1	yes	characteristic
340	C₄	C₃×#890	1	1	yes	characteristic
341	C₄	C₃×#895	1	1	yes	characteristic
342	C₄	C₃×#897	1	1	yes	characteristic
343	C₃	—	2	1	no	—
344	C₃	#69485	1	1	yes	characteristic, direct product factor
345	C₃	C₃×#1940	1	1	yes	characteristic, complemented
346	C₂	C₃×#4924	1	1	yes	characteristic
347	C₂	C₃×#4984	1	1	yes	characteristic
348	C₂	C₃×#4999	1	1	yes	characteristic
349	C₂	C₃×#5011	1	2	yes	—
350	C₂	C₃×#5024	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄×C₄⋊C₄	D₃
hypocenter	C₃	C₃×#1940
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #69485	C₃	direct (N × H)
294. C₃²	#1940	semidirect
344. C₃	#69485	direct (N × H)
345. C₃	C₃×#1940	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₄⋊C₄	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×C₃×#803	2
C₃×#4524	2
C₃×#1940	3
#69485	3

Subgroups — C3×#69485 #11790065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69485 #11790065