Subgroups — C₂×C₃×#5056 #11873427

Order 1152.

989 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	C₂×C₃²×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#835	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#832	C₂	1	2	yes	maximal
5	C₃×#5056	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#488	—	3	1	no	maximal
8	C₂×#5056	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×D₄	C₂²	1	1	yes	characteristic
11	C₃²×#124	—	2	2	no	—
12	C₃²×#124	C₂²	1	4	yes	—
13	C₂×C₃²×C₂²⋊C₄	—	2	2	no	—
14	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
17	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃×#870	—	2	2	no	—
19	C₃×#870	C₂²	1	4	yes	complemented
20	C₃×#835	C₂²	1	8	yes	complemented
21	C₃×#832	C₂²	1	8	yes	—
22	C₃×C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
23	C₂×C₃×#125	—	3	1	no	—
24	C₂×C₃×#124	—	2	1	no	—
25	C₂×C₃×#124	D₃	1	1	yes	characteristic
26	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
27	C₂×#870	C₂×C₃	1	1	yes	characteristic, complemented
28	C₃×#488	—	3	8	no	—
29	C₂×C₃×#103	—	3	2	no	—
30	C₂×C₃×#102	—	3	2	no	—
31	C₂×C₃×C₄×C₈	—	3	1	no	—
32	C₂×#835	C₂×C₃	1	2	yes	complemented
33	C₂×#832	C₂×C₃	1	2	yes	—
34	#5056	C₂×C₃	1	8	yes	direct product factor
35	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂⁴×C₃²	—	2	1	no	—
37	C₂×C₃²×Q₈	—	2	1	no	—
38	C₂×C₃²×Q₈	—	2	4	no	—
39	C₂×C₃²×Q₈	C₂³	1	2	yes	—
40	C₂×C₃²×D₄	—	2	1	no	—
41	C₂×C₃²×D₄	—	2	4	no	—
42	C₂×C₃²×D₄	C₂³	1	2	yes	—
43	C₂²×C₃²×C₄	—	2	1	no	—
44	C₂²×C₃²×C₄	D₄	1	2	yes	—
45	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×Dic₃	—	2	2	no	—
47	C₃²×C₂²⋊C₄	—	2	8	no	—
48	C₃²×C₄²	D₄	1	2	yes	—
49	C₃²×C₄²	C₂³	1	2	yes	—
50	C₃×C₄⋊Dic₃	—	2	2	no	—
51	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
52	C₃×Dic₃⋊C₄	—	2	8	no	—
53	C₂×C₃×C₃⋊C₈	—	2	4	no	—
54	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
55	C₂×#488	—	3	1	no	Sylow
56	C₂²×C₃×Q₈	—	2	1	no	—
57	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×D₄	—	2	1	no	—
60	C₂²×C₃×D₄	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	—	3	2	no	—
63	C₃×#125	—	3	4	no	—
64	C₃×#125	—	6	2	no	—
65	C₃×#124	—	2	4	no	—
66	C₃×#124	—	2	2	no	—
67	C₃×#124	—	2	2	no	—
68	C₃×#124	—	2	4	no	—
69	C₃×#124	C₂×D₃	1	4	yes	—
70	C₃×#124	C₂²×C₃	1	4	yes	—
71	C₂×C₃×C₄⋊C₄	—	3	2	no	—
72	C₂×C₃×C₄⋊C₄	—	6	2	no	—
73	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
74	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
75	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
76	C₂×C₃×C₄²	—	2	1	no	—
77	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
80	C₂×Dic₃⋊C₄	—	2	2	no	—
81	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
82	#870	—	2	2	no	—
83	#870	C₂²×C₃	1	4	yes	complemented
84	C₃×#103	—	3	8	no	—
85	C₃×#102	—	3	8	no	—
86	C₃×C₄×C₈	—	3	4	no	—
87	#835	C₂²×C₃	1	8	yes	complemented
88	#832	C₂²×C₃	1	8	yes	—
89	C₄×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
90	C₂³×C₃²	—	2	2	no	—
91	C₂³×C₃²	—	4	2	no	—
92	C₂³×C₃²	—	4	4	no	—
93	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
94	C₃²×Q₈	—	2	4	no	—
95	C₃²×Q₈	—	4	2	no	—
96	C₃²×D₄	—	2	4	no	—
97	C₃²×D₄	—	4	2	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	—	4	2	no	—
101	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
102	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	#41	1	4	yes	—
104	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	2	4	no	—
106	C₂×C₃×Dic₃	—	4	4	no	—
107	C₃×C₃⋊C₈	—	2	4	no	—
108	C₂×#125	—	3	1	no	—
109	C₂×#124	C₃×D₃	1	1	yes	characteristic
110	#488	—	3	8	no	—
111	C₂×#103	—	3	2	no	—
112	C₂×#102	—	3	2	no	—
113	C₂×C₄×C₈	—	3	1	no	—
114	C₂⁴×C₃	—	2	2	no	—
115	C₂⁴×C₃	—	2	1	no	—
116	C₂⁴×C₃	—	2	1	no	—
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	—	2	1	no	—
119	C₂×C₃×Q₈	—	2	1	no	—
120	C₂×C₃×Q₈	—	2	4	no	—
121	C₂×C₃×Q₈	—	2	4	no	—
122	C₂×C₃×Q₈	—	2	2	no	—
123	C₂×C₃×Q₈	—	4	4	no	—
124	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
125	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
126	C₂×C₃×D₄	—	2	2	no	—
127	C₂×C₃×D₄	—	2	4	no	—
128	C₂×C₃×D₄	—	2	1	no	—
129	C₂×C₃×D₄	—	2	1	no	—
130	C₂×C₃×D₄	—	2	4	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	4	4	no	—
133	C₂×C₃×D₄	C₂²×D₃	1	2	yes	—
134	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	—	2	2	no	—
139	C₂²×C₃×C₄	—	2	1	no	—
140	C₂²×C₃×C₄	—	6	2	no	—
141	C₂²×C₃×C₄	#67	1	2	yes	—
142	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
143	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂²×Dic₃	—	2	2	no	—
146	C₂×C₃×C₈	—	3	4	no	—
147	C₂×C₃×C₈	—	6	4	no	—
148	C₃×C₄⋊C₄	—	3	4	no	—
149	C₃×C₄⋊C₄	—	6	8	no	—
150	C₃×C₄⋊C₄	—	6	2	no	—
151	C₃×C₂²⋊C₄	—	2	16	no	—
152	C₃×C₂²⋊C₄	—	2	8	no	—
153	C₃×C₂²⋊C₄	—	2	8	no	—
154	C₃×C₄²	—	2	2	no	—
155	C₃×C₄²	—	2	2	no	—
156	C₃×C₄²	#67	1	2	yes	—
157	C₃×C₄²	C₃×D₄	1	2	yes	—
158	C₃×C₄²	C₂²×D₃	1	2	yes	—
159	C₃×C₄²	C₂³×C₃	1	2	yes	—
160	C₄⋊Dic₃	—	2	2	no	—
161	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
162	Dic₃⋊C₄	—	2	8	no	—
163	C₂×C₃⋊C₈	—	2	4	no	—
164	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
165	C₂²×C₃²	—	4	2	no	—
166	C₂²×C₃²	—	4	4	no	—
167	C₂²×C₃²	—	8	4	no	—
168	C₂²×C₃²	#124	1	2	yes	—
169	C₂²×C₃²	#135	1	4	yes	—
170	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
171	C₃²×C₄	—	2	4	no	—
172	C₃²×C₄	—	4	2	no	—
173	C₃²×C₄	#124	1	2	yes	—
174	C₃²×C₄	C₂×#41	1	2	yes	—
175	C₃×Dic₃	—	4	4	no	—
176	C₂²×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
177	C₂²×D₄	C₂×C₃×D₃	1	1	yes	characteristic
178	C₂²×C₈	—	3	2	no	—
179	#125	—	3	4	no	—
180	#125	—	6	2	no	—
181	#124	—	2	2	no	—
182	#124	C₂×C₃×D₃	1	4	yes	—
183	C₂×C₄⋊C₄	—	3	2	no	—
184	C₂×C₄⋊C₄	—	6	2	no	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
187	#103	—	3	8	no	—
188	#102	—	3	8	no	—
189	C₄×C₈	—	3	4	no	—
190	C₂³×C₃	—	2	4	no	—
191	C₂³×C₃	—	2	2	no	—
192	C₂³×C₃	—	2	2	no	—
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	4	4	no	—
195	C₂³×C₃	—	4	8	no	—
196	C₂³×C₃	—	4	2	no	—
197	C₂³×C₃	—	4	4	no	—
198	C₂³×C₃	—	4	2	no	—
199	C₂³×C₃	—	4	4	no	—
200	C₂³×C₃	C₂×#67	1	1	yes	characteristic
201	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
202	C₃×Q₈	—	2	4	no	—
203	C₃×Q₈	—	2	4	no	—
204	C₃×Q₈	—	4	4	no	—
205	C₃×Q₈	—	4	4	no	—
206	C₃×Q₈	—	4	2	no	—
207	C₃×Q₈	—	4	2	no	—
208	C₃×D₄	—	2	4	no	—
209	C₃×D₄	—	2	4	no	—
210	C₃×D₄	—	4	4	no	—
211	C₃×D₄	—	4	4	no	—
212	C₃×D₄	—	4	2	no	—
213	C₃×D₄	—	4	2	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	4	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	4	4	no	—
224	C₂×C₃×C₄	—	4	4	no	—
225	C₂×C₃×C₄	—	4	2	no	—
226	C₂×C₃×C₄	—	4	2	no	—
227	C₂×C₃×C₄	—	6	4	no	—
228	C₂×C₃×C₄	—	12	4	no	—
229	C₂×C₃×C₄	#237	1	4	yes	—
230	C₂×C₃×C₄	C₂×#67	1	4	yes	—
231	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
233	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂×C₃×C₄	C₃×#41	1	4	yes	—
235	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
237	C₂×Dic₃	—	2	4	no	—
238	C₂×Dic₃	—	4	4	no	—
239	C₃×C₈	—	6	4	no	—
240	C₃⋊C₈	—	2	4	no	—
241	C₂×C₃²	—	8	2	no	—
242	C₂×C₃²	#488	1	4	yes	complemented
243	C₂×C₃²	C₂×#124	1	1	yes	characteristic
244	C₂×C₃²	C₂×#135	1	2	yes	—
245	C₂⁴	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	C₂²×C₃×D₃	1	2	yes	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	C₂²×C₃×D₃	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	6	2	no	—
254	C₂²×C₄	C₃×#67	1	2	yes	—
255	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
256	C₂×C₈	—	3	4	no	—
257	C₂×C₈	—	6	4	no	—
258	C₄⋊C₄	—	3	4	no	—
259	C₄⋊C₄	—	6	2	no	—
260	C₄⋊C₄	—	6	8	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₄²	C₃×#67	1	2	yes	—
263	C₄²	C₂²×C₃×D₃	1	2	yes	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	2	4	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	4	4	no	—
268	C₂²×C₃	—	4	8	no	—
269	C₂²×C₃	—	4	4	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	4	4	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	8	8	no	—
274	C₂²×C₃	—	8	4	no	—
275	C₂²×C₃	—	8	4	no	—
276	C₂²×C₃	#936	1	2	yes	—
277	C₂²×C₃	#950	1	4	yes	—
278	C₂²×C₃	C₃×#124	1	2	yes	—
279	C₂²×C₃	C₃×#135	1	4	yes	—
280	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
281	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
282	C₃×C₄	—	2	4	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	4	4	no	—
287	C₃×C₄	—	4	4	no	—
288	C₃×C₄	—	4	2	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	12	4	no	—
291	C₃×C₄	#936	1	2	yes	—
292	C₃×C₄	C₃×#124	1	2	yes	—
293	C₃×C₄	C₂×#237	1	2	yes	—
294	C₃×C₄	C₂×C₃×#41	1	2	yes	—
295	Dic₃	—	4	4	no	—
296	C₃²	C₂×#488	1	1	yes	characteristic, complemented, Sylow
297	C₂³	—	2	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	4	no	—
300	C₂³	C₂×C₃×#67	1	1	yes	characteristic
301	Q₈	—	2	4	no	—
302	Q₈	—	4	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	4	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	6	4	no	—
309	C₂×C₄	—	12	4	no	—
310	C₂×C₄	C₃×#237	1	4	yes	—
311	C₂×C₄	C₂×C₃×#67	1	4	yes	—
312	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
313	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
314	C₈	—	6	4	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	2	4	no	—
318	C₂×C₃	—	8	4	no	—
319	C₂×C₃	—	8	2	no	—
320	C₂×C₃	—	8	2	no	—
321	C₂×C₃	#5056	1	4	yes	direct product factor
322	C₂×C₃	C₃×#488	1	4	yes	complemented
323	C₂×C₃	C₂×#936	1	1	yes	characteristic
324	C₂×C₃	C₂×#950	1	2	yes	—
325	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
326	C₂×C₃	C₂×C₃×#135	1	2	yes	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	8	4	no	—
330	C₂²	C₃×#936	1	2	yes	—
331	C₂²	C₃×#950	1	4	yes	—
332	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	—	12	4	no	—
336	C₄	C₃×#936	1	2	yes	—
337	C₄	C₂×C₃×#237	1	2	yes	—
338	C₃	—	2	1	no	—
339	C₃	C₂×#5056	1	1	yes	characteristic, direct product factor
340	C₃	C₂×C₃×#488	1	1	yes	characteristic, complemented
341	C₂	—	8	2	no	—
342	C₂	C₃×#5056	1	4	yes	direct product factor
343	C₂	C₂×C₃×#936	1	1	yes	characteristic
344	C₂	C₂×C₃×#950	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×#124	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂²×C₃²×D₄	C₂²
hypercenter	C₂×C₃×#124	D₃
hypocenter	C₃	C₂×C₃×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#870	C₂	semidirect
3. C₂×C₃×#835	C₂	semidirect
5. C₃×#5056	C₂	direct (N × H)
6. C₂×C₃×C₄×C₃⋊C₈	C₂	semidirect
8. C₂×#5056	C₃	direct (N × H)
19. C₃×#870	C₂²	semidirect
20. C₃×#835	C₂²	semidirect
22. C₃×C₄×C₃⋊C₈	C₂²	semidirect
27. C₂×#870	C₂×C₃	semidirect
32. C₂×#835	C₂×C₃	semidirect
34. #5056	C₂×C₃	direct (N × H)
35. C₂×C₄×C₃⋊C₈	C₂×C₃	semidirect
83. #870	C₂²×C₃	semidirect
87. #835	C₂²×C₃	semidirect
89. C₄×C₃⋊C₈	C₂²×C₃	semidirect
242. C₂×C₃²	#488	semidirect
296. C₃²	C₂×#488	semidirect
321. C₂×C₃	#5056	direct (N × H)
322. C₂×C₃	C₃×#488	semidirect
339. C₃	C₂×#5056	direct (N × H)
340. C₃	C₂×C₃×#488	semidirect
342. C₂	C₃×#5056	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#124	2
C₂×C₃×#870	2
C₂×C₃×#835	2
C₂×C₃×#832	2
C₃×#5056	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×C₃×#488	3
C₂×#5056	3

Subgroups — C2×C3×#5056 #11873427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5056 #11873427