Subgroups — C₃×#69486 #11791425

Order 1152.

745 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#810	C₂	1	4	yes	maximal
4	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₃×#1981	—	3	1	no	maximal
6	#69486	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
8	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	—	2	1	no	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
12	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×#869	—	2	2	no	—
15	C₃×#869	C₄	1	4	yes	complemented
16	C₃²×#95	—	2	1	no	—
17	C₃×#810	—	2	8	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
19	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
20	C₂×#869	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₈	—	3	1	no	—
22	C₂×C₃×#103	—	3	4	no	—
23	C₃×C₄×C₄⋊C₄	—	2	1	no	—
24	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
25	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₂×#810	C₂×C₃	1	4	yes	—
27	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
28	C₂²×C₃²×C₄	—	2	2	no	—
29	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	—	2	2	no	—
33	C₂×C₃×Dic₆	—	2	4	no	—
34	C₂×C₃×Dic₆	—	2	4	no	—
35	C₂×C₃×Dic₆	—	2	2	no	—
36	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
37	C₃²×C₄⋊C₄	—	2	2	no	—
38	C₃²×C₄⋊C₄	D₄	1	4	yes	—
39	C₃²×C₄²	—	4	2	no	—
40	C₃²×C₄²	C₂×C₄	1	2	yes	—
41	C₃²×C₄²	D₄	1	2	yes	—
42	C₃×C₄⋊Dic₃	—	2	4	no	—
43	C₃×C₄⋊Dic₃	—	2	4	no	—
44	C₂×C₃×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×C₃⋊C₈	—	2	4	no	—
46	#1981	—	3	1	no	Sylow
47	C₂²×C₃×Q₈	—	3	2	no	—
48	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
49	C₂²×C₃×C₈	—	3	2	no	—
50	C₃×C₄⋊Q₈	—	3	4	no	—
51	C₃×C₄⋊Q₈	—	6	2	no	—
52	C₂×C₃×C₄⋊C₄	—	2	2	no	—
53	C₂×C₃×C₄⋊C₄	—	6	1	no	—
54	C₂×C₃×C₄⋊C₄	—	6	1	no	—
55	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
56	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₃×C₄²	—	2	2	no	—
58	C₂×C₃×C₄²	—	2	1	no	—
59	C₂×C₃×C₄²	—	2	1	no	—
60	C₂×C₃×C₄²	—	2	1	no	—
61	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄⋊Dic₃	—	2	1	no	—
64	C₂×C₄⋊Dic₃	—	2	1	no	—
65	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
66	#869	—	2	2	no	—
67	#869	C₃×C₄	1	4	yes	complemented
68	C₃×C₄⋊C₈	—	6	2	no	—
69	C₃×#103	—	6	8	no	—
70	C₃×#95	—	2	2	no	—
71	C₃×#95	—	2	1	no	—
72	C₃×#95	—	2	1	no	—
73	#810	—	2	8	no	—
74	C₄⋊(C₃⋊C₈)	—	2	2	no	—
75	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	4	2	no	—
79	C₂×C₃²×C₄	—	4	2	no	—
80	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
83	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
84	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	2	yes	—
88	C₂×C₃×Dic₃	—	2	4	no	—
89	C₂×C₃×Dic₃	—	4	4	no	—
90	C₃×Dic₆	—	2	8	no	—
91	C₃×Dic₆	—	4	2	no	—
92	C₃×Dic₆	—	4	2	no	—
93	C₃×C₃⋊C₈	—	4	2	no	—
94	C₂×C₄⋊Q₈	—	3	1	no	—
95	C₂×C₄⋊C₈	—	3	1	no	—
96	C₂×#103	—	3	4	no	—
97	C₄×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
98	C₂×C₃×Q₈	—	3	4	no	—
99	C₂×C₃×Q₈	—	6	2	no	—
100	C₂×C₃×Q₈	—	6	4	no	—
101	C₂×C₃×Q₈	—	6	4	no	—
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	6	2	no	—
109	C₂²×C₃×C₄	#67	1	1	yes	characteristic
110	C₂²×C₃×C₄	#67	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂²×Dic₃	—	2	2	no	—
116	C₂×Dic₆	—	2	4	no	—
117	C₂×Dic₆	—	2	4	no	—
118	C₂×Dic₆	—	2	2	no	—
119	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
120	C₂×C₃×C₈	—	6	4	no	—
121	C₂×C₃×C₈	—	6	2	no	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	4	no	—
126	C₃×C₄⋊C₄	—	6	4	no	—
127	C₃×C₄⋊C₄	—	6	4	no	—
128	C₃×C₄⋊C₄	D₁₂	1	4	yes	—
129	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	—	4	4	no	—
133	C₃×C₄²	—	4	2	no	—
134	C₃×C₄²	—	4	2	no	—
135	C₃×C₄²	C₄×D₃	1	2	yes	—
136	C₃×C₄²	D₁₂	1	2	yes	—
137	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
138	C₃×C₄²	C₃×D₄	1	2	yes	—
139	C₄⋊Dic₃	—	2	4	no	—
140	C₄⋊Dic₃	—	2	4	no	—
141	C₂×C₃⋊C₈	—	2	2	no	—
142	C₂×C₃⋊C₈	—	2	4	no	—
143	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₃²	#121	1	2	yes	—
145	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
146	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
147	C₂²×C₃²	#137	1	2	yes	—
148	C₃²×C₄	—	2	2	no	—
149	C₃²×C₄	—	4	4	no	—
150	C₃²×C₄	#103	1	4	yes	complemented
151	C₃²×C₄	C₄×D₄	1	2	yes	—
152	C₃²×C₄	#124	1	1	yes	characteristic
153	C₃²×C₄	#127	1	1	yes	characteristic
154	C₃×Dic₃	—	4	4	no	—
155	C₂²×Q₈	—	3	2	no	—
156	C₂²×C₈	—	3	2	no	—
157	C₄⋊Q₈	—	3	4	no	—
158	C₄⋊Q₈	—	6	2	no	—
159	C₂×C₄⋊C₄	—	6	1	no	—
160	C₂×C₄⋊C₄	—	6	1	no	—
161	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
164	C₄⋊C₈	—	6	2	no	—
165	#103	—	6	8	no	—
166	#95	—	2	1	no	—
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	C₂×#67	1	1	yes	characteristic
169	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
170	C₃×Q₈	—	6	8	no	—
171	C₃×Q₈	—	12	2	no	—
172	C₃×Q₈	—	12	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	8	no	—
176	C₂×C₃×C₄	—	2	4	no	—
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	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	4	no	—
188	C₂×C₃×C₄	—	4	4	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	6	4	no	—
194	C₂×C₃×C₄	—	12	4	no	—
195	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
196	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
197	C₂×C₃×C₄	#214	1	2	yes	—
198	C₂×C₃×C₄	#216	1	2	yes	—
199	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
200	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
201	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
202	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
205	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₃×C₄	#235	1	2	yes	—
207	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×#41	1	2	yes	—
211	C₂×Dic₃	—	2	4	no	—
212	C₂×Dic₃	—	4	4	no	—
213	Dic₆	—	2	8	no	—
214	Dic₆	—	4	2	no	—
215	Dic₆	—	4	2	no	—
216	C₃×C₈	—	12	2	no	—
217	C₃⋊C₈	—	4	2	no	—
218	C₂×C₃²	#390	1	1	yes	characteristic
219	C₂×C₃²	C₂×#103	1	1	yes	characteristic
220	C₂×C₃²	#419	1	1	yes	characteristic
221	C₂×C₃²	#461	1	2	yes	—
222	C₂×C₃²	#462	1	2	yes	—
223	C₂×Q₈	—	3	4	no	—
224	C₂×Q₈	—	6	2	no	—
225	C₂×Q₈	—	6	4	no	—
226	C₂×Q₈	—	6	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	6	2	no	—
229	C₂²×C₄	C₃×#67	1	1	yes	characteristic
230	C₂²×C₄	C₃×#67	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
232	C₂×C₈	—	6	4	no	—
233	C₂×C₈	—	6	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	6	4	no	—
236	C₄⋊C₄	—	6	4	no	—
237	C₄⋊C₄	C₃×D₁₂	1	4	yes	—
238	C₄²	—	4	2	no	—
239	C₄²	C₃×C₄×D₃	1	2	yes	—
240	C₄²	C₃×D₁₂	1	2	yes	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
247	C₂²×C₃	#930	1	2	yes	—
248	C₂²×C₃	C₂×#214	1	1	yes	characteristic
249	C₂²×C₃	C₂×#216	1	1	yes	characteristic
250	C₂²×C₃	#951	1	2	yes	—
251	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
252	C₂²×C₃	C₃×#121	1	2	yes	—
253	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
254	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
255	C₂²×C₃	C₃×#137	1	2	yes	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	2	no	—
263	C₃×C₄	—	4	8	no	—
264	C₃×C₄	—	4	4	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	12	4	no	—
267	C₃×C₄	#810	1	4	yes	complemented
268	C₃×C₄	C₃×#103	1	4	yes	complemented
269	C₃×C₄	C₄×D₁₂	1	2	yes	—
270	C₃×C₄	#874	1	1	yes	characteristic
271	C₃×C₄	#875	1	1	yes	characteristic
272	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
273	C₃×C₄	C₃×#124	1	1	yes	characteristic
274	C₃×C₄	C₃×#127	1	1	yes	characteristic
275	Dic₃	—	4	4	no	—
276	C₃²	#1981	1	1	yes	characteristic, complemented, Sylow
277	C₂³	C₂×C₃×#67	1	1	yes	characteristic
278	Q₈	—	6	8	no	—
279	Q₈	—	12	2	no	—
280	Q₈	—	12	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	6	4	no	—
286	C₂×C₄	—	12	4	no	—
287	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
288	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
289	C₂×C₄	C₃×#214	1	2	yes	—
290	C₂×C₄	C₃×#216	1	2	yes	—
291	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
293	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
294	C₂×C₄	C₃×#235	1	2	yes	—
295	C₈	—	12	2	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	2	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	#4939	1	1	yes	characteristic
302	C₂×C₃	C₂×#810	1	1	yes	characteristic
303	C₂×C₃	#5004	1	1	yes	characteristic
304	C₂×C₃	#5019	1	2	yes	—
305	C₂×C₃	#5031	1	2	yes	—
306	C₂×C₃	C₃×#390	1	1	yes	characteristic
307	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
308	C₂×C₃	C₃×#419	1	1	yes	characteristic
309	C₂×C₃	C₃×#461	1	2	yes	—
310	C₂×C₃	C₃×#462	1	2	yes	—
311	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
312	C₂²	C₃×#930	1	2	yes	—
313	C₂²	C₂×C₃×#214	1	1	yes	characteristic
314	C₂²	C₂×C₃×#216	1	1	yes	characteristic
315	C₂²	C₃×#951	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	12	4	no	—
319	C₄	C₃×#810	1	4	yes	complemented
320	C₄	C₃×C₄×D₁₂	1	2	yes	—
321	C₄	C₃×#874	1	1	yes	characteristic
322	C₄	C₃×#875	1	1	yes	characteristic
323	C₃	—	2	1	no	—
324	C₃	#69486	1	1	yes	characteristic, direct product factor
325	C₃	C₃×#1981	1	1	yes	characteristic, complemented
326	C₂	C₃×#4939	1	1	yes	characteristic
327	C₂	C₂×C₃×#810	1	1	yes	characteristic
328	C₂	C₃×#5004	1	1	yes	characteristic
329	C₂	C₃×#5019	1	2	yes	—
330	C₂	C₃×#5031	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄×C₄⋊C₄	D₃
hypocenter	C₃	C₃×#1981
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₃×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #69486	C₃	direct (N × H)
15. C₃×#869	C₄	semidirect
67. #869	C₃×C₄	semidirect
150. C₃²×C₄	#103	semidirect
267. C₃×C₄	#810	semidirect
268. C₃×C₄	C₃×#103	semidirect
276. C₃²	#1981	semidirect
319. C₄	C₃×#810	semidirect
324. C₃	#69486	direct (N × H)
325. C₃	C₃×#1981	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₃²×C₄×C₄⋊C₄	2
C₂×C₃×#810	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₃×#1981	3
#69486	3

Subgroups — C3×#69486 #11791425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69486 #11791425