Subgroups — C₂×C₃×#1951 #10639786

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#428	C₂	1	2	yes	maximal
3	C₂²×C₃×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₃×#1951	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#343	C₂	1	2	yes	maximal
7	C₂×#1951	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×#108	C₄	1	8	yes	complemented
14	C₂×C₃×#108	C₂²	1	4	yes	—
15	C₃×#428	—	2	4	no	—
16	C₃×#428	C₂²	1	8	yes	—
17	C₂×C₃×#107	—	2	2	no	—
18	C₂×C₃×#106	—	2	1	no	—
19	C₂×C₃×#106	—	2	1	no	—
20	C₂×C₃×#106	—	2	2	no	—
21	C₂×C₃×#106	—	2	2	no	—
22	C₂×C₃×#106	C₂²	1	4	yes	—
23	C₃×#405	—	2	2	no	—
24	C₃×#405	C₂²	1	4	yes	—
25	C₂×C₃×#97	—	2	1	no	—
26	C₂×C₃×C₄×C₈	—	2	1	no	—
27	C₃×#343	C₂²	1	8	yes	—
28	C₂²×#108	C₂×C₃	1	1	yes	characteristic
29	C₂×#428	C₂×C₃	1	2	yes	—
30	C₂²×#106	C₂×C₃	1	1	yes	characteristic
31	C₂×#405	C₂×C₃	1	1	yes	characteristic
32	#1951	C₂×C₃	1	8	yes	direct product factor
33	C₂×#343	C₂×C₃	1	2	yes	—
34	C₂³×C₃×C₄	—	2	2	no	—
35	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₃×M₁₆	—	2	1	no	—
37	C₂×C₃×M₁₆	—	2	2	no	—
38	C₂×C₃×M₁₆	—	2	4	no	—
39	C₂×C₃×M₁₆	—	2	2	no	—
40	C₂×C₃×M₁₆	C₂×C₄	1	8	yes	—
41	C₂×C₃×M₁₆	Q₈	1	4	yes	—
42	C₂×C₃×M₁₆	C₂³	1	4	yes	—
43	C₂×C₃×M₁₆	C₂³	1	2	yes	—
44	C₂²×C₃×C₈	—	2	1	no	—
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
47	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₃×C₈	D₄	1	2	yes	—
50	C₂²×C₃×C₈	Q₈	1	4	yes	—
51	C₂²×C₃×C₈	C₂³	1	2	yes	—
52	C₃×#117	—	2	1	no	—
53	C₃×#117	—	2	1	no	—
54	C₃×#117	D₄	1	2	yes	—
55	C₃×#117	C₂³	1	2	yes	—
56	C₂×C₃×C₄⋊C₄	—	2	2	no	—
57	C₂×C₃×C₄⋊C₄	—	2	2	no	—
58	C₂×C₃×C₄⋊C₄	—	2	2	no	—
59	C₂×C₃×C₄⋊C₄	—	2	2	no	—
60	C₂×C₃×C₄⋊C₄	—	2	2	no	—
61	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
62	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
63	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
64	C₂×C₃×C₄²	—	2	1	no	—
65	C₃×#108	C₂×C₄	1	16	yes	complemented
66	C₃×#107	—	2	8	no	—
67	C₃×#106	—	2	4	no	—
68	C₃×#106	—	2	4	no	—
69	C₃×#106	—	2	8	no	—
70	C₃×#97	—	2	4	no	—
71	C₃×C₄×C₈	—	2	4	no	—
72	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
73	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×#117	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×#108	C₃×C₄	1	8	yes	complemented
78	C₂×#108	C₂²×C₃	1	4	yes	—
79	#428	—	2	4	no	—
80	#428	C₂²×C₃	1	8	yes	—
81	C₂×#107	—	2	2	no	—
82	C₂×#106	—	2	1	no	—
83	C₂×#106	—	2	1	no	—
84	C₂×#106	—	2	2	no	—
85	C₂×#106	—	2	2	no	—
86	C₂×#106	C₂²×C₃	1	4	yes	—
87	#405	—	2	2	no	—
88	#405	C₂²×C₃	1	4	yes	—
89	C₂×#97	—	2	1	no	—
90	C₂×C₄×C₈	—	2	1	no	—
91	#343	C₂²×C₃	1	8	yes	—
92	C₂⁴×C₃	#41	1	1	yes	characteristic
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	—	4	2	no	—
96	C₂²×C₃×C₄	—	4	2	no	—
97	C₂²×C₃×C₄	—	8	4	no	—
98	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
99	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
104	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
105	C₂²×C₃×C₄	#41	1	2	yes	—
106	C₂²×C₃×C₄	#41	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
108	C₃×M₁₆	—	2	8	no	—
109	C₃×M₁₆	—	2	4	no	—
110	C₃×M₁₆	—	2	4	no	—
111	C₃×M₁₆	—	2	2	no	—
112	C₃×M₁₆	C₂²×C₄	1	8	yes	—
113	C₃×M₁₆	C₂×Q₈	1	4	yes	—
114	C₂×C₃×C₈	—	2	8	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
119	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
120	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	—
121	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	2	4	no	—
127	C₃×C₄⋊C₄	—	2	4	no	—
128	C₃×C₄⋊C₄	—	4	2	no	—
129	C₃×C₄⋊C₄	—	4	2	no	—
130	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
131	C₃×C₂²⋊C₄	—	2	2	no	—
132	C₃×C₂²⋊C₄	—	2	2	no	—
133	C₃×C₄²	—	2	2	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₂³×C₄	—	2	2	no	—
136	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂×M₁₆	—	2	1	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	C₂×C₃×C₄	1	8	yes	—
142	C₂×M₁₆	C₃×Q₈	1	4	yes	—
143	C₂×M₁₆	C₂³×C₃	1	4	yes	—
144	C₂×M₁₆	C₂³×C₃	1	2	yes	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
148	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
149	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
150	C₂²×C₈	C₃×D₄	1	2	yes	—
151	C₂²×C₈	C₃×Q₈	1	4	yes	—
152	C₂²×C₈	C₂³×C₃	1	2	yes	—
153	#117	—	2	1	no	—
154	#117	—	2	1	no	—
155	#117	C₃×D₄	1	2	yes	—
156	#117	C₂³×C₃	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₃×D₄	1	2	yes	—
163	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
164	C₂×C₂²⋊C₄	—	2	1	no	—
165	C₂×C₄²	—	2	1	no	—
166	#108	C₂×C₃×C₄	1	16	yes	complemented
167	#107	—	2	8	no	—
168	#106	—	2	4	no	—
169	#106	—	2	4	no	—
170	#106	—	2	8	no	—
171	#97	—	2	4	no	—
172	C₄×C₈	—	2	4	no	—
173	C₂³×C₃	—	2	4	no	—
174	C₂³×C₃	C₄×Q₈	1	2	yes	—
175	C₂³×C₃	#121	1	1	yes	characteristic
176	C₂³×C₃	#121	1	1	yes	characteristic
177	C₂³×C₃	#125	1	2	yes	—
178	C₂³×C₃	C₂×#41	1	1	yes	characteristic
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	4	4	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	—	8	4	no	—
183	C₂×C₃×C₄	—	8	4	no	—
184	C₂×C₃×C₄	—	8	4	no	—
185	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
186	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
188	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
190	C₂×C₃×C₄	#121	1	1	yes	characteristic
191	C₂×C₃×C₄	#121	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
193	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
194	C₂×C₃×C₄	#125	1	2	yes	—
195	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
196	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
197	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
203	C₃×C₈	—	2	8	no	—
204	C₃×C₈	—	2	4	no	—
205	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
206	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
207	C₂⁴	C₃×#41	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	8	4	no	—
213	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
214	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
219	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
220	C₂²×C₄	C₃×#41	1	2	yes	—
221	C₂²×C₄	C₃×#41	1	1	yes	characteristic
222	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
223	M₁₆	—	2	8	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	C₂²×C₃×C₄	1	8	yes	—
228	M₁₆	C₂×C₃×Q₈	1	4	yes	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
234	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
235	C₂×C₈	C₃×C₄⋊C₄	1	8	yes	—
236	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
237	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
238	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
239	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₂²×C₃	—	2	4	no	—
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	#389	1	2	yes	—
255	C₂²×C₃	#389	1	2	yes	—
256	C₂²×C₃	#471	1	4	yes	—
257	C₂²×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
258	C₂²×C₃	C₂×#121	1	1	yes	characteristic
259	C₂²×C₃	C₂×#125	1	1	yes	characteristic
260	C₃×C₄	—	4	4	no	—
261	C₃×C₄	—	8	4	no	—
262	C₃×C₄	#389	1	2	yes	—
263	C₃×C₄	#389	1	2	yes	—
264	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
265	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
266	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
267	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
268	C₂³	—	2	4	no	—
269	C₂³	C₃×C₄×Q₈	1	2	yes	—
270	C₂³	C₃×#121	1	1	yes	characteristic
271	C₂³	C₃×#121	1	1	yes	characteristic
272	C₂³	C₃×#125	1	2	yes	—
273	C₂³	C₂×C₃×#41	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
281	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
282	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
283	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
284	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
285	C₂×C₄	C₃×#121	1	1	yes	characteristic
286	C₂×C₄	C₃×#121	1	1	yes	characteristic
287	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
289	C₂×C₄	C₃×#125	1	2	yes	—
290	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
291	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
292	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
297	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
298	C₈	—	2	8	no	—
299	C₈	—	2	4	no	—
300	C₈	C₂×C₃×C₄⋊C₄	1	4	yes	—
301	C₈	C₂×C₃×C₄⋊C₄	1	4	yes	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	#1951	1	4	yes	direct product factor
306	C₂×C₃	C₂×#389	1	1	yes	characteristic
307	C₂×C₃	C₂×#471	1	2	yes	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	C₃×#389	1	2	yes	—
313	C₂²	C₃×#389	1	2	yes	—
314	C₂²	C₃×#471	1	4	yes	—
315	C₂²	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
316	C₂²	C₂×C₃×#121	1	1	yes	characteristic
317	C₂²	C₂×C₃×#125	1	1	yes	characteristic
318	C₄	—	4	4	no	—
319	C₄	—	8	4	no	—
320	C₄	C₃×#389	1	2	yes	—
321	C₄	C₃×#389	1	2	yes	—
322	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
323	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
324	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
325	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
326	C₃	C₂×#1951	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	C₃×#1951	1	4	yes	direct product factor
331	C₂	C₂×C₃×#389	1	1	yes	characteristic
332	C₂	C₂×C₃×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	#121
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂⁴×C₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃×#1951	C₂	direct (N × H)
7. C₂×#1951	C₃	direct (N × H)
13. C₂×C₃×#108	C₄	semidirect
32. #1951	C₂×C₃	direct (N × H)
65. C₃×#108	C₂×C₄	semidirect
77. C₂×#108	C₃×C₄	semidirect
166. #108	C₂×C₃×C₄	semidirect
305. C₂×C₃	#1951	direct (N × H)
326. C₃	C₂×#1951	direct (N × H)
330. C₂	C₃×#1951	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#108	2
C₂×C₃×#428	2
C₂²×C₃×#106	2
C₂×C₃×#405	2
C₃×#1951	2
C₂×C₃×#343	2
C₂×#1951	3

Subgroups — C2×C3×#1951 #10639786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1951 #10639786