Subgroups — C₃×#70456 #11791422

Order 1152.

717 conjugacy classes of subgroups.

The subgroup lattice has 326 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	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#816	C₂	1	4	yes	maximal
4	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₃×#1981	—	3	1	no	maximal
6	#70456	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×C₄×Dic₃	—	2	1	no	—
14	C₃×#869	—	2	2	no	—
15	C₃×#869	C₄	1	4	yes	complemented
16	C₃²×C₄⋊C₈	—	2	2	no	—
17	C₃×C₃⋊#95	—	2	1	no	—
18	C₃×#816	—	2	8	no	—
19	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
20	C₂×#869	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₈	—	2	1	no	—
22	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#103	—	3	4	no	—
25	C₃×C₄×C₄⋊C₄	—	3	1	no	—
26	C₂×#816	C₂×C₃	1	4	yes	—
27	C₄×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
28	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	2	no	—
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	C₂×C₃²×C₈	—	2	2	no	—
38	C₂×C₃²×C₈	—	2	4	no	—
39	C₃²×C₄²	C₂×C₄	1	2	yes	—
40	C₃²×C₄²	D₄	1	2	yes	—
41	C₃×C₄⋊Dic₃	—	2	4	no	—
42	C₃×C₄⋊Dic₃	—	2	4	no	—
43	C₃×C₄⋊Dic₃	—	2	2	no	—
44	C₃×C₄⋊Dic₃	D₄	1	4	yes	—
45	C₃×C₄×Dic₃	—	4	2	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	—	2	2	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
51	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
52	C₃×C₄⋊Q₈	—	3	4	no	—
53	C₃×C₄⋊Q₈	—	6	2	no	—
54	C₂×C₃×C₄⋊C₄	—	3	2	no	—
55	C₂×C₃×C₄⋊C₄	—	6	1	no	—
56	C₂×C₃×C₄⋊C₄	—	6	1	no	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	—	6	1	no	—
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	2	1	no	—
62	C₂×C₄⋊Dic₃	—	2	1	no	—
63	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
64	C₂×C₄×Dic₃	—	2	1	no	—
65	#869	—	2	2	no	—
66	#869	C₃×C₄	1	4	yes	complemented
67	C₃×C₄⋊C₈	—	2	4	no	—
68	C₃×C₄⋊C₈	—	2	2	no	—
69	C₃×C₄⋊C₈	—	2	2	no	—
70	C₃×#103	—	6	8	no	—
71	C₃×#95	—	6	1	no	—
72	C₃⋊#95	—	2	1	no	—
73	#816	—	2	8	no	—
74	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
79	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
80	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	2	yes	—
84	C₂×C₃×Dic₃	—	2	4	no	—
85	C₂×C₃×Dic₃	—	2	4	no	—
86	C₂×C₃×Dic₃	—	4	4	no	—
87	C₂×C₃×Dic₃	—	4	2	no	—
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₃×Dic₆	—	2	8	no	—
90	C₃×Dic₆	—	4	2	no	—
91	C₃×Dic₆	—	4	2	no	—
92	C₃²×C₈	—	4	2	no	—
93	C₂×C₄⋊Q₈	—	3	1	no	—
94	C₂×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
95	C₂×#103	—	3	4	no	—
96	C₄×C₄⋊C₄	—	3	1	no	—
97	C₂×C₃×Q₈	—	3	4	no	—
98	C₂×C₃×Q₈	—	6	2	no	—
99	C₂×C₃×Q₈	—	6	4	no	—
100	C₂×C₃×Q₈	—	6	4	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	6	2	no	—
105	C₂²×C₃×C₄	—	6	2	no	—
106	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
107	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×Dic₃	—	2	2	no	—
113	C₂²×Dic₃	—	2	2	no	—
114	C₂×Dic₆	—	2	4	no	—
115	C₂×Dic₆	—	2	4	no	—
116	C₂×Dic₆	—	2	2	no	—
117	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	4	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	4	4	no	—
124	C₃×C₄⋊C₄	—	3	4	no	—
125	C₃×C₄⋊C₄	—	6	4	no	—
126	C₃×C₄⋊C₄	—	6	4	no	—
127	C₃×C₄⋊C₄	—	6	2	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	12	2	no	—
131	C₃×C₄²	C₄×D₃	1	2	yes	—
132	C₃×C₄²	#67	1	2	yes	—
133	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
134	C₃×C₄²	C₃×D₄	1	2	yes	—
135	C₄⋊Dic₃	—	2	4	no	—
136	C₄⋊Dic₃	—	2	4	no	—
137	C₄⋊Dic₃	—	2	2	no	—
138	C₄⋊Dic₃	C₃×D₄	1	4	yes	—
139	C₄×Dic₃	—	4	2	no	—
140	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₃²	#121	1	2	yes	—
142	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
143	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
144	C₂²×C₃²	#137	1	2	yes	—
145	C₃²×C₄	—	2	2	no	—
146	C₃²×C₄	#103	1	4	yes	complemented
147	C₃²×C₄	C₄×D₄	1	2	yes	—
148	C₃²×C₄	#124	1	1	yes	characteristic
149	C₃²×C₄	#127	1	1	yes	characteristic
150	C₃×Dic₃	—	4	4	no	—
151	C₃×Dic₃	—	4	4	no	—
152	C₂²×Q₈	—	3	2	no	—
153	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
154	C₄⋊Q₈	—	3	4	no	—
155	C₄⋊Q₈	—	6	2	no	—
156	C₂×C₄⋊C₄	—	3	2	no	—
157	C₂×C₄⋊C₄	—	6	1	no	—
158	C₂×C₄⋊C₄	—	6	1	no	—
159	C₂×C₄²	—	6	1	no	—
160	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	#103	—	6	8	no	—
163	#95	—	6	1	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
166	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
167	C₃×Q₈	—	6	8	no	—
168	C₃×Q₈	—	12	2	no	—
169	C₃×Q₈	—	12	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	6	4	no	—
182	C₂×C₃×C₄	—	6	4	no	—
183	C₂×C₃×C₄	—	12	4	no	—
184	C₂×C₃×C₄	—	12	2	no	—
185	C₂×C₃×C₄	—	12	2	no	—
186	C₂×C₃×C₄	#204	1	2	yes	—
187	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
188	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
190	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
191	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
192	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
193	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
196	C₂×C₃×C₄	#239	1	2	yes	—
197	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
200	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₃×#41	1	2	yes	—
202	C₂×Dic₃	—	2	4	no	—
203	C₂×Dic₃	—	2	4	no	—
204	C₂×Dic₃	—	4	4	no	—
205	C₂×Dic₃	—	4	2	no	—
206	C₂×Dic₃	—	4	2	no	—
207	Dic₆	—	2	8	no	—
208	Dic₆	—	4	2	no	—
209	Dic₆	—	4	2	no	—
210	C₃×C₈	—	4	4	no	—
211	C₃×C₈	—	4	2	no	—
212	C₃×C₈	—	4	2	no	—
213	C₂×C₃²	#390	1	1	yes	characteristic
214	C₂×C₃²	C₂×#103	1	1	yes	characteristic
215	C₂×C₃²	#419	1	1	yes	characteristic
216	C₂×C₃²	#461	1	2	yes	—
217	C₂×C₃²	#462	1	2	yes	—
218	C₂×Q₈	—	3	4	no	—
219	C₂×Q₈	—	6	2	no	—
220	C₂×Q₈	—	6	4	no	—
221	C₂×Q₈	—	6	4	no	—
222	C₂²×C₄	—	6	2	no	—
223	C₂²×C₄	—	6	2	no	—
224	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
225	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₄⋊C₄	—	3	4	no	—
230	C₄⋊C₄	—	6	4	no	—
231	C₄⋊C₄	—	6	4	no	—
232	C₄⋊C₄	—	6	2	no	—
233	C₄²	—	12	2	no	—
234	C₄²	C₃×C₄×D₃	1	2	yes	—
235	C₄²	C₃×#67	1	2	yes	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	#895	1	2	yes	—
242	C₂²×C₃	C₂×#204	1	1	yes	characteristic
243	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
244	C₂²×C₃	#909	1	2	yes	—
245	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
246	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²×C₃	C₃×#121	1	2	yes	—
248	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
249	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
250	C₂²×C₃	C₃×#137	1	2	yes	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	4	2	no	—
258	C₃×C₄	—	12	4	no	—
259	C₃×C₄	—	12	4	no	—
260	C₃×C₄	#816	1	4	yes	complemented
261	C₃×C₄	C₃×#103	1	4	yes	complemented
262	C₃×C₄	#893	1	2	yes	—
263	C₃×C₄	#940	1	1	yes	characteristic
264	C₃×C₄	#947	1	1	yes	characteristic
265	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
266	C₃×C₄	C₃×#124	1	1	yes	characteristic
267	C₃×C₄	C₃×#127	1	1	yes	characteristic
268	Dic₃	—	4	4	no	—
269	Dic₃	—	4	4	no	—
270	C₃²	#1981	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
272	Q₈	—	6	8	no	—
273	Q₈	—	12	2	no	—
274	Q₈	—	12	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	6	4	no	—
277	C₂×C₄	—	6	4	no	—
278	C₂×C₄	—	12	4	no	—
279	C₂×C₄	—	12	2	no	—
280	C₂×C₄	—	12	2	no	—
281	C₂×C₄	C₃×#204	1	2	yes	—
282	C₂×C₄	C₃×Dic₁₂	1	2	yes	—
283	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
284	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
285	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
286	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
287	C₂×C₄	C₃×#239	1	2	yes	—
288	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
289	C₈	—	4	2	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	#4841	1	2	yes	—
296	C₂×C₃	#4849	1	2	yes	—
297	C₂×C₃	#4988	1	1	yes	characteristic
298	C₂×C₃	C₂×#816	1	1	yes	characteristic
299	C₂×C₃	#5120	1	1	yes	characteristic
300	C₂×C₃	C₃×#390	1	1	yes	characteristic
301	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
302	C₂×C₃	C₃×#419	1	1	yes	characteristic
303	C₂×C₃	C₃×#461	1	2	yes	—
304	C₂×C₃	C₃×#462	1	2	yes	—
305	C₂²	C₃×#895	1	2	yes	—
306	C₂²	C₂×C₃×#204	1	1	yes	characteristic
307	C₂²	C₂×C₃×Dic₁₂	1	1	yes	characteristic
308	C₂²	C₃×#909	1	2	yes	—
309	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	12	4	no	—
312	C₄	—	12	4	no	—
313	C₄	C₃×#816	1	4	yes	complemented
314	C₄	C₃×#893	1	2	yes	—
315	C₄	C₃×#940	1	1	yes	characteristic
316	C₄	C₃×#947	1	1	yes	characteristic
317	C₃	—	2	1	no	—
318	C₃	#70456	1	1	yes	characteristic, direct product factor
319	C₃	C₃×#1981	1	1	yes	characteristic, complemented
320	C₂	C₃×#4841	1	2	yes	—
321	C₂	C₃×#4849	1	2	yes	—
322	C₂	C₃×#4988	1	1	yes	characteristic
323	C₂	C₂×C₃×#816	1	1	yes	characteristic
324	C₂	C₃×#5120	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₈	D₃
hypocenter	C₃	C₃×#1981
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
6. #70456	C₃	direct (N × H)
15. C₃×#869	C₄	semidirect
66. #869	C₃×C₄	semidirect
146. C₃²×C₄	#103	semidirect
260. C₃×C₄	#816	semidirect
261. C₃×C₄	C₃×#103	semidirect
270. C₃²	#1981	semidirect
313. C₄	C₃×#816	semidirect
318. C₃	#70456	direct (N × H)
319. 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
C₂×C₃²×C₄⋊C₈	2
C₂×C₃×#816	2
C₃×C₄×C₄⋊Dic₃	2
C₃×#1981	3
#70456	3

Subgroups — C3×#70456 #11791422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70456 #11791422