Subgroups — C₂×C₃×#1945 #10639768

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 314 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×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#424	C₂	1	2	yes	maximal
4	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×#1945	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#339	C₂	1	2	yes	maximal
7	C₂×#1945	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#117	C₂²	1	2	yes	—
12	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×#108	—	2	4	no	—
14	C₂×C₃×#108	C₂²	1	4	yes	—
15	C₃×#427	C₄	1	4	yes	complemented
16	C₃×#427	C₂²	1	4	yes	—
17	C₂×C₃×#107	C₄	1	2	yes	complemented
18	C₂×C₃×#106	C₄	1	2	yes	complemented
19	C₃×#424	—	2	4	no	—
20	C₃×#424	C₂²	1	8	yes	—
21	C₂×C₃×C₄⋊C₈	—	2	4	no	—
22	C₂×C₃×C₄×C₈	—	2	4	no	—
23	C₂×C₃×C₄×C₈	C₂²	1	4	yes	—
24	C₃×#339	C₂²	1	8	yes	—
25	C₂²×#108	C₂×C₃	1	1	yes	characteristic
26	C₂×#427	C₂×C₃	1	1	yes	characteristic
27	C₂×#424	C₂×C₃	1	2	yes	—
28	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
29	#1945	C₂×C₃	1	8	yes	direct product factor
30	C₂×#339	C₂×C₃	1	2	yes	—
31	C₂³×C₃×C₄	—	2	1	no	—
32	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×M₁₆	—	2	2	no	—
34	C₂×C₃×M₁₆	—	2	4	no	—
35	C₂×C₃×M₁₆	—	2	4	no	—
36	C₂×C₃×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₃×C₈	—	2	2	no	—
38	C₂²×C₃×C₈	—	2	2	no	—
39	C₂²×C₃×C₈	—	2	2	no	—
40	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₃×C₈	D₄	1	2	yes	—
44	C₂²×C₃×C₈	D₄	1	2	yes	—
45	C₂²×C₃×C₈	D₄	1	4	yes	—
46	C₂²×C₃×C₈	Q₈	1	4	yes	—
47	C₂²×C₃×C₈	C₂³	1	2	yes	—
48	C₂²×C₃×C₈	C₂³	1	2	yes	—
49	C₃×#117	—	2	2	no	—
50	C₃×#117	—	2	2	no	—
51	C₃×#117	C₂×C₄	1	4	yes	—
52	C₃×#117	C₂³	1	4	yes	—
53	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
54	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
55	C₂×C₃×C₄²	—	2	2	no	—
56	C₂×C₃×C₄²	—	2	2	no	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	Q₈	1	4	yes	—
59	C₂×C₃×C₄²	C₂³	1	2	yes	—
60	C₃×#108	—	2	8	no	—
61	C₃×#107	C₂×C₄	1	8	yes	complemented
62	C₃×#106	C₂×C₄	1	8	yes	complemented
63	C₃×C₄⋊C₈	—	2	16	no	—
64	C₃×C₄×C₈	—	2	8	no	—
65	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
66	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×#117	C₂²×C₃	1	2	yes	—
69	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₂×#108	—	2	4	no	—
71	C₂×#108	C₂²×C₃	1	4	yes	—
72	#427	C₃×C₄	1	4	yes	complemented
73	#427	C₂²×C₃	1	4	yes	—
74	C₂×#107	C₃×C₄	1	2	yes	complemented
75	C₂×#106	C₃×C₄	1	2	yes	complemented
76	#424	—	2	4	no	—
77	#424	C₂²×C₃	1	8	yes	—
78	C₂×C₄⋊C₈	—	2	4	no	—
79	C₂×C₄×C₈	—	2	4	no	—
80	C₂×C₄×C₈	C₂²×C₃	1	4	yes	—
81	#339	C₂²×C₃	1	8	yes	—
82	C₂⁴×C₃	#41	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	8	no	—
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	4	no	—
87	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
94	C₂²×C₃×C₄	#41	1	1	yes	characteristic
95	C₂²×C₃×C₄	#41	1	2	yes	—
96	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
97	C₃×M₁₆	—	2	4	no	—
98	C₃×M₁₆	—	2	4	no	—
99	C₃×M₁₆	—	4	4	no	—
100	C₂×C₃×C₈	—	2	4	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	4	no	—
105	C₂×C₃×C₈	—	4	4	no	—
106	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
107	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
108	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	—
109	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
112	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
113	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
114	C₃×C₄⋊C₄	—	2	4	no	—
115	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
116	C₃×C₂²⋊C₄	—	2	4	no	—
117	C₃×C₂²⋊C₄	—	2	4	no	—
118	C₃×C₄²	—	2	4	no	—
119	C₃×C₄²	—	2	4	no	—
120	C₃×C₄²	—	2	2	no	—
121	C₃×C₄²	—	2	4	no	—
122	C₃×C₄²	C₂×Q₈	1	4	yes	—
123	C₂³×C₄	—	2	1	no	—
124	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂³×C₃	1	4	yes	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
135	C₂²×C₈	C₃×D₄	1	2	yes	—
136	C₂²×C₈	C₃×D₄	1	2	yes	—
137	C₂²×C₈	C₃×D₄	1	4	yes	—
138	C₂²×C₈	C₃×Q₈	1	4	yes	—
139	C₂²×C₈	C₂³×C₃	1	2	yes	—
140	C₂²×C₈	C₂³×C₃	1	2	yes	—
141	#117	—	2	2	no	—
142	#117	—	2	2	no	—
143	#117	C₂×C₃×C₄	1	4	yes	—
144	#117	C₂³×C₃	1	4	yes	—
145	C₂×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₄²	—	2	2	no	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	C₃×Q₈	1	4	yes	—
151	C₂×C₄²	C₂³×C₃	1	2	yes	—
152	#108	—	2	8	no	—
153	#107	C₂×C₃×C₄	1	8	yes	complemented
154	#106	C₂×C₃×C₄	1	8	yes	complemented
155	C₄⋊C₈	—	2	16	no	—
156	C₄×C₈	—	2	8	no	—
157	C₂³×C₃	—	2	4	no	—
158	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
159	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
160	C₂³×C₃	C₂²⋊Q₈	1	2	yes	—
161	C₂³×C₃	#125	1	2	yes	—
162	C₂³×C₃	C₂×#41	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	16	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	4	8	no	—
168	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
173	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
174	C₂×C₃×C₄	#121	1	2	yes	—
175	C₂×C₃×C₄	#121	1	2	yes	—
176	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
177	C₂×C₃×C₄	#125	1	2	yes	—
178	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
179	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
180	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
186	C₃×C₈	—	2	4	no	—
187	C₃×C₈	—	4	4	no	—
188	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
189	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
190	C₂⁴	C₃×#41	1	1	yes	characteristic
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
201	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
202	C₂²×C₄	C₃×#41	1	1	yes	characteristic
203	C₂²×C₄	C₃×#41	1	2	yes	—
204	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
215	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
216	C₂×C₈	C₃×C₄⋊C₄	1	8	yes	—
217	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
218	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
219	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
220	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
221	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	4	no	—
230	C₄²	C₂×C₃×Q₈	1	4	yes	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	4	no	—
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	#389	1	2	yes	—
236	C₂²×C₃	#389	1	2	yes	—
237	C₂²×C₃	#443	1	4	yes	—
238	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂²×C₃	C₂×#125	1	1	yes	characteristic
241	C₃×C₄	—	2	8	no	—
242	C₃×C₄	—	4	8	no	—
243	C₃×C₄	#389	1	2	yes	—
244	C₃×C₄	#389	1	2	yes	—
245	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
246	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
247	C₃×C₄	C₂×#121	1	1	yes	characteristic
248	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
249	C₂³	—	2	4	no	—
250	C₂³	C₃×C₄×D₄	1	1	yes	characteristic
251	C₂³	C₃×C₄×D₄	1	1	yes	characteristic
252	C₂³	C₃×C₂²⋊Q₈	1	2	yes	—
253	C₂³	C₃×#125	1	2	yes	—
254	C₂³	C₂×C₃×#41	1	1	yes	characteristic
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
261	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
262	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
263	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
264	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
265	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
266	C₂×C₄	C₃×#121	1	2	yes	—
267	C₂×C₄	C₃×#121	1	2	yes	—
268	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
269	C₂×C₄	C₃×#125	1	2	yes	—
270	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
271	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
272	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	4	4	no	—
280	C₈	C₂×C₃×C₄⋊C₄	1	4	yes	—
281	C₈	C₂×C₃×C₄⋊C₄	1	4	yes	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	#1945	1	4	yes	direct product factor
286	C₂×C₃	C₂×#389	1	1	yes	characteristic
287	C₂×C₃	C₂×#443	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	C₃×#389	1	2	yes	—
293	C₂²	C₃×#389	1	2	yes	—
294	C₂²	C₃×#443	1	4	yes	—
295	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
296	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
297	C₂²	C₂×C₃×#125	1	1	yes	characteristic
298	C₄	—	2	8	no	—
299	C₄	—	4	8	no	—
300	C₄	C₃×#389	1	2	yes	—
301	C₄	C₃×#389	1	2	yes	—
302	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
303	C₄	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
304	C₄	C₂×C₃×#121	1	1	yes	characteristic
305	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
306	C₃	C₂×#1945	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	C₃×#1945	1	4	yes	direct product factor
311	C₂	C₂×C₃×#389	1	1	yes	characteristic
312	C₂	C₂×C₃×#443	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₄×D₄
prime radical	C₂⁴×C₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃×#1945	C₂	direct (N × H)
7. C₂×#1945	C₃	direct (N × H)
15. C₃×#427	C₄	semidirect
17. C₂×C₃×#107	C₄	semidirect
18. C₂×C₃×#106	C₄	semidirect
29. #1945	C₂×C₃	direct (N × H)
61. C₃×#107	C₂×C₄	semidirect
62. C₃×#106	C₂×C₄	semidirect
72. #427	C₃×C₄	semidirect
74. C₂×#107	C₃×C₄	semidirect
75. C₂×#106	C₃×C₄	semidirect
153. #107	C₂×C₃×C₄	semidirect
154. #106	C₂×C₃×C₄	semidirect
285. C₂×C₃	#1945	direct (N × H)
306. C₃	C₂×#1945	direct (N × H)
310. C₂	C₃×#1945	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×#427	2
C₂×C₃×#424	2
C₂²×C₃×C₄×C₈	2
C₃×#1945	2
C₂×C₃×#339	2
C₂×#1945	3

Subgroups — C2×C3×#1945 #10639768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1945 #10639768