Subgroups — C₂×C₃×#3147 #10679239

Order 768.

3366 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	C₂²×C₃×#135	C₂	1	4	yes	maximal, complemented
2	C₂×C₃×#532	C₂	1	2	yes	maximal, complemented
3	C₃×#3147	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#495	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#492	C₂	1	2	yes	maximal, complemented
8	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#3147	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×#41	C₂²	1	4	yes	—
11	C₂×C₃×#135	—	2	32	no	—
12	C₂×C₃×#135	C₂²	1	32	yes	complemented
13	C₂²×C₃×Q₁₆	C₂²	1	4	yes	—
14	C₂²×C₃×SD₁₆	C₂²	1	8	yes	—
15	C₂²×C₃×D₈	C₂²	1	4	yes	—
16	C₂³×C₃×C₈	C₂²	1	2	yes	—
17	C₃×#532	—	2	8	no	—
18	C₃×#532	C₂²	1	16	yes	complemented
19	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₃×#127	C₂²	1	2	yes	—
21	C₂×C₃×#124	C₂²	1	4	yes	—
22	C₂×C₃×#121	—	2	4	no	—
23	C₂×C₃×C₄×D₄	—	2	4	no	—
24	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#495	C₂²	1	32	yes	complemented
26	C₃×#494	C₂²	1	8	yes	complemented
27	C₃×#493	C₂²	1	8	yes	complemented
28	C₃×#492	C₂²	1	16	yes	complemented
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
32	C₂×C₃×C₄×C₈	C₂²	1	8	yes	complemented
33	C₂²×#135	C₂×C₃	1	4	yes	complemented
34	C₂×#532	C₂×C₃	1	2	yes	complemented
35	#3147	C₂×C₃	1	16	yes	direct product factor
36	C₂×#495	C₂×C₃	1	4	yes	complemented
37	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×#493	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#492	C₂×C₃	1	2	yes	complemented
40	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃×#41	—	2	8	no	—
42	C₂×C₃×#41	—	2	32	no	—
43	C₂×C₃×#41	C₂³	1	16	yes	—
44	C₂²×C₃×Q₈	C₂³	1	4	yes	—
45	C₂²×C₃×D₄	—	2	4	no	—
46	C₂²×C₃×D₄	—	2	4	no	—
47	C₂²×C₃×D₄	C₂³	1	4	yes	—
48	C₂³×C₃×C₄	—	2	4	no	—
49	C₂³×C₃×C₄	D₄	1	2	yes	—
50	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₃×#135	—	2	128	no	—
52	C₂×C₃×Q₁₆	—	2	16	no	—
53	C₂×C₃×Q₁₆	C₂³	1	16	yes	—
54	C₂×C₃×SD₁₆	—	2	32	no	—
55	C₂×C₃×SD₁₆	C₂³	1	32	yes	—
56	C₂×C₃×D₈	—	2	16	no	—
57	C₂×C₃×D₈	C₂³	1	16	yes	—
58	C₂²×C₃×C₈	D₄	1	16	yes	complemented
59	C₂²×C₃×C₈	C₂³	1	2	yes	—
60	C₂²×C₃×C₈	C₂³	1	2	yes	—
61	C₂²×C₃×C₈	C₂³	1	8	yes	—
62	C₃×C₄⋊Q₈	—	2	4	no	—
63	C₃×C₄⋊Q₈	C₂³	1	8	yes	—
64	C₃×#127	—	2	4	no	—
65	C₃×#127	C₂³	1	8	yes	—
66	C₃×#124	—	2	8	no	—
67	C₃×#124	C₂³	1	16	yes	—
68	C₃×#121	—	2	32	no	—
69	C₃×C₄×D₄	—	2	32	no	—
70	C₂×C₃×C₄⋊C₄	—	2	4	no	—
71	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
72	C₂×C₃×C₄²	D₄	1	4	yes	—
73	C₂×C₃×C₄²	D₄	1	2	yes	—
74	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂³	1	4	yes	—
77	C₃×C₄×C₈	C₂³	1	4	yes	—
78	C₃×C₄×C₈	C₂³	1	4	yes	—
79	C₃×C₄×C₈	C₂³	1	8	yes	—
80	C₂²×#41	C₂²×C₃	1	4	yes	—
81	C₂×#135	—	2	32	no	—
82	C₂×#135	C₂²×C₃	1	32	yes	complemented
83	C₂²×Q₁₆	C₂²×C₃	1	4	yes	—
84	C₂²×SD₁₆	C₂²×C₃	1	8	yes	—
85	C₂²×D₈	C₂²×C₃	1	4	yes	—
86	C₂³×C₈	C₂²×C₃	1	2	yes	—
87	#532	—	2	8	no	—
88	#532	C₂²×C₃	1	16	yes	complemented
89	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
90	C₂×#127	C₂²×C₃	1	2	yes	—
91	C₂×#124	C₂²×C₃	1	4	yes	—
92	C₂×#121	—	2	4	no	—
93	C₂×C₄×D₄	—	2	4	no	—
94	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
95	#495	C₂²×C₃	1	32	yes	complemented
96	#494	C₂²×C₃	1	8	yes	complemented
97	#493	C₂²×C₃	1	8	yes	complemented
98	#492	C₂²×C₃	1	16	yes	complemented
99	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₃	1	8	yes	complemented
103	C₂⁴×C₃	—	2	4	no	—
104	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
105	C₃×#41	—	2	64	no	—
106	C₃×#41	—	4	32	no	—
107	C₂×C₃×Q₈	—	2	4	no	—
108	C₂×C₃×Q₈	—	2	16	no	—
109	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
110	C₂×C₃×D₄	—	2	16	no	—
111	C₂×C₃×D₄	—	2	16	no	—
112	C₂×C₃×D₄	—	2	16	no	—
113	C₂×C₃×D₄	—	2	4	no	—
114	C₂×C₃×D₄	—	4	16	no	—
115	C₂×C₃×D₄	—	4	16	no	—
116	C₂×C₃×D₄	C₂⁴	1	8	yes	—
117	C₂²×C₃×C₄	—	2	8	no	—
118	C₂²×C₃×C₄	—	2	4	no	—
119	C₂²×C₃×C₄	—	2	16	no	—
120	C₂²×C₃×C₄	—	4	16	no	—
121	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
122	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
123	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
124	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
125	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
126	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
129	C₃×Q₁₆	—	2	32	no	—
130	C₃×SD₁₆	—	2	64	no	—
131	C₃×D₈	—	2	32	no	—
132	C₂×C₃×C₈	C₂×D₄	1	8	yes	—
133	C₂×C₃×C₈	C₂×D₄	1	8	yes	—
134	C₂×C₃×C₈	C₂×D₄	1	32	yes	complemented
135	C₂×C₃×C₈	C₂⁴	1	4	yes	—
136	C₂×C₃×C₈	C₂⁴	1	4	yes	—
137	C₃×C₄⋊C₄	—	2	16	no	—
138	C₃×C₂²⋊C₄	—	2	32	no	—
139	C₃×C₄²	C₂×D₄	1	8	yes	—
140	C₃×C₄²	C₂×D₄	1	2	yes	—
141	C₃×C₄²	C₂×D₄	1	2	yes	—
142	C₃×C₄²	C₂⁴	1	2	yes	—
143	C₃×C₄²	C₂⁴	1	2	yes	—
144	C₂×#41	—	2	8	no	—
145	C₂×#41	—	2	32	no	—
146	C₂×#41	C₂³×C₃	1	16	yes	—
147	C₂²×Q₈	C₂³×C₃	1	4	yes	—
148	C₂²×D₄	—	2	4	no	—
149	C₂²×D₄	—	2	4	no	—
150	C₂²×D₄	C₂³×C₃	1	4	yes	—
151	C₂³×C₄	—	2	4	no	—
152	C₂³×C₄	C₃×D₄	1	2	yes	—
153	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
154	#135	—	2	128	no	—
155	C₂×Q₁₆	—	2	16	no	—
156	C₂×Q₁₆	C₂³×C₃	1	16	yes	—
157	C₂×SD₁₆	—	2	32	no	—
158	C₂×SD₁₆	C₂³×C₃	1	32	yes	—
159	C₂×D₈	—	2	16	no	—
160	C₂×D₈	C₂³×C₃	1	16	yes	—
161	C₂²×C₈	C₃×D₄	1	16	yes	complemented
162	C₂²×C₈	C₂³×C₃	1	2	yes	—
163	C₂²×C₈	C₂³×C₃	1	2	yes	—
164	C₂²×C₈	C₂³×C₃	1	8	yes	—
165	C₄⋊Q₈	—	2	4	no	—
166	C₄⋊Q₈	C₂³×C₃	1	8	yes	—
167	#127	—	2	4	no	—
168	#127	C₂³×C₃	1	8	yes	—
169	#124	—	2	8	no	—
170	#124	C₂³×C₃	1	16	yes	—
171	#121	—	2	32	no	—
172	C₄×D₄	—	2	32	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	—	2	8	no	—
175	C₂×C₄²	C₃×D₄	1	2	yes	—
176	C₂×C₄²	C₃×D₄	1	4	yes	—
177	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₃	1	4	yes	—
180	C₄×C₈	C₂³×C₃	1	4	yes	—
181	C₄×C₈	C₂³×C₃	1	4	yes	—
182	C₄×C₈	C₂³×C₃	1	8	yes	—
183	C₂³×C₃	—	2	4	no	—
184	C₂³×C₃	—	2	8	no	—
185	C₂³×C₃	—	4	8	no	—
186	C₂³×C₃	—	4	16	no	—
187	C₂³×C₃	#127	1	4	yes	—
188	C₂³×C₃	C₂²×D₄	1	2	yes	—
189	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
190	C₃×Q₈	—	2	16	no	—
191	C₃×Q₈	—	4	8	no	—
192	C₃×D₄	—	2	16	no	—
193	C₃×D₄	—	4	32	no	—
194	C₃×D₄	—	4	8	no	—
195	C₃×D₄	—	4	32	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	16	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	4	8	no	—
200	C₂×C₃×C₄	—	4	32	no	—
201	C₂×C₃×C₄	#127	1	4	yes	—
202	C₂×C₃×C₄	#127	1	8	yes	—
203	C₂×C₃×C₄	#135	1	8	yes	complemented
204	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
205	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
208	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
209	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
210	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
211	C₃×C₈	C₂²×D₄	1	8	yes	—
212	C₃×C₈	C₂²×D₄	1	8	yes	—
213	C₂⁴	—	2	4	no	—
214	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
215	#41	—	2	64	no	—
216	#41	—	4	32	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	16	no	—
219	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
220	C₂×D₄	—	2	16	no	—
221	C₂×D₄	—	2	16	no	—
222	C₂×D₄	—	2	16	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	16	no	—
225	C₂×D₄	—	4	16	no	—
226	C₂×D₄	C₂⁴×C₃	1	8	yes	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	16	no	—
230	C₂²×C₄	—	4	16	no	—
231	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
233	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
235	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
236	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
238	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
239	Q₁₆	—	2	32	no	—
240	SD₁₆	—	2	64	no	—
241	D₈	—	2	32	no	—
242	C₂×C₈	C₂×C₃×D₄	1	8	yes	—
243	C₂×C₈	C₂×C₃×D₄	1	8	yes	—
244	C₂×C₈	C₂×C₃×D₄	1	32	yes	complemented
245	C₂×C₈	C₂⁴×C₃	1	4	yes	—
246	C₂×C₈	C₂⁴×C₃	1	4	yes	—
247	C₄⋊C₄	—	2	16	no	—
248	C₂²⋊C₄	—	2	32	no	—
249	C₄²	C₂×C₃×D₄	1	8	yes	—
250	C₄²	C₂×C₃×D₄	1	2	yes	—
251	C₄²	C₂×C₃×D₄	1	2	yes	—
252	C₄²	C₂⁴×C₃	1	2	yes	—
253	C₄²	C₂⁴×C₃	1	2	yes	—
254	C₂²×C₃	—	2	8	no	—
255	C₂²×C₃	—	2	4	no	—
256	C₂²×C₃	—	4	8	no	—
257	C₂²×C₃	—	4	16	no	—
258	C₂²×C₃	—	8	16	no	—
259	C₂²×C₃	C₂×#127	1	4	yes	—
260	C₂²×C₃	C₂×#127	1	2	yes	—
261	C₂²×C₃	C₂×#135	1	4	yes	—
262	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	4	8	no	—
265	C₃×C₄	C₂×#127	1	4	yes	—
266	C₃×C₄	C₂×#127	1	2	yes	—
267	C₃×C₄	C₂×#135	1	8	yes	complemented
268	C₃×C₄	C₂³×D₄	1	2	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	4	16	no	—
272	C₂³	—	4	8	no	—
273	C₂³	C₃×#127	1	4	yes	—
274	C₂³	C₂²×C₃×D₄	1	2	yes	—
275	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
276	Q₈	—	2	16	no	—
277	Q₈	—	4	8	no	—
278	D₄	—	2	16	no	—
279	D₄	—	4	32	no	—
280	D₄	—	4	32	no	—
281	D₄	—	4	8	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	32	no	—
287	C₂×C₄	C₃×#127	1	4	yes	—
288	C₂×C₄	C₃×#127	1	8	yes	—
289	C₂×C₄	C₃×#135	1	8	yes	complemented
290	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
291	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
292	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
293	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
294	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
295	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
296	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
297	C₈	C₂²×C₃×D₄	1	8	yes	—
298	C₈	C₂²×C₃×D₄	1	8	yes	—
299	C₂×C₃	—	2	4	no	—
300	C₂×C₃	—	8	8	no	—
301	C₂×C₃	#3147	1	4	yes	direct product factor
302	C₂×C₃	C₂²×#127	1	1	yes	characteristic
303	C₂×C₃	C₂²×#135	1	2	yes	—
304	C₂²	—	2	8	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	8	no	—
307	C₂²	—	4	16	no	—
308	C₂²	—	8	16	no	—
309	C₂²	C₂×C₃×#127	1	4	yes	—
310	C₂²	C₂×C₃×#127	1	2	yes	—
311	C₂²	C₂×C₃×#135	1	4	yes	—
312	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	4	8	no	—
315	C₄	C₂×C₃×#127	1	4	yes	—
316	C₄	C₂×C₃×#127	1	2	yes	—
317	C₄	C₂×C₃×#135	1	8	yes	complemented
318	C₄	C₂³×C₃×D₄	1	2	yes	—
319	C₃	C₂×#3147	1	1	yes	characteristic, direct product factor, Sylow
320	C₂	—	2	4	no	—
321	C₂	—	8	8	no	—
322	C₂	C₃×#3147	1	4	yes	direct product factor
323	C₂	C₂²×C₃×#127	1	1	yes	characteristic
324	C₂	C₂²×C₃×#135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂³×C₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#135	C₂	semidirect
2. C₂×C₃×#532	C₂	semidirect
3. C₃×#3147	C₂	direct (N × H)
4. C₂×C₃×#495	C₂	semidirect
5. C₂×C₃×#494	C₂	semidirect
6. C₂×C₃×#493	C₂	semidirect
7. C₂×C₃×#492	C₂	semidirect
8. C₂²×C₃×C₄×C₈	C₂	semidirect
9. C₂×#3147	C₃	direct (N × H)
12. C₂×C₃×#135	C₂²	semidirect
18. C₃×#532	C₂²	semidirect
25. C₃×#495	C₂²	semidirect
26. C₃×#494	C₂²	semidirect
27. C₃×#493	C₂²	semidirect
28. C₃×#492	C₂²	semidirect
32. C₂×C₃×C₄×C₈	C₂²	semidirect
33. C₂²×#135	C₂×C₃	semidirect
34. C₂×#532	C₂×C₃	semidirect
35. #3147	C₂×C₃	direct (N × H)
36. C₂×#495	C₂×C₃	semidirect
37. C₂×#494	C₂×C₃	semidirect
38. C₂×#493	C₂×C₃	semidirect
39. C₂×#492	C₂×C₃	semidirect
40. C₂²×C₄×C₈	C₂×C₃	semidirect
58. C₂²×C₃×C₈	D₄	semidirect
82. C₂×#135	C₂²×C₃	semidirect
88. #532	C₂²×C₃	semidirect
95. #495	C₂²×C₃	semidirect
96. #494	C₂²×C₃	semidirect
97. #493	C₂²×C₃	semidirect
98. #492	C₂²×C₃	semidirect
102. C₂×C₄×C₈	C₂²×C₃	semidirect
134. C₂×C₃×C₈	C₂×D₄	semidirect
161. C₂²×C₈	C₃×D₄	semidirect
203. C₂×C₃×C₄	#135	semidirect
244. C₂×C₈	C₂×C₃×D₄	semidirect
267. C₃×C₄	C₂×#135	semidirect
289. C₂×C₄	C₃×#135	semidirect
301. C₂×C₃	#3147	direct (N × H)
317. C₄	C₂×C₃×#135	semidirect
319. C₃	C₂×#3147	direct (N × H)
322. C₂	C₃×#3147	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×#135	2
C₂×C₃×#532	2
C₃×#3147	2
C₂×C₃×#495	2
C₂×C₃×#494	2
C₂×C₃×#493	2
C₂×C₃×#492	2
C₂²×C₃×C₄×C₈	2
C₂×#3147	3

Subgroups — C2×C3×#3147 #10679239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3147 #10679239