Subgroups — C₂²×C₃×#485 #10679110

Order 768.

2358 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#485	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×#485	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊Q₈	—	2	6	no	—
15	C₂×C₃×C₂²⋊Q₈	C₂²	1	12	yes	—
16	C₂×C₃×#117	—	2	6	no	—
17	C₂×C₃×#117	C₂²	1	12	yes	—
18	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
19	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
23	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
24	C₂²×C₃×C₄²	—	2	1	no	—
25	C₃×#485	C₂²	1	64	yes	direct product factor
26	C₂×C₃×#107	C₂²	1	12	yes	complemented
27	C₂×C₃×#106	C₂²	1	12	yes	complemented
28	C₂×C₃×#103	C₂²	1	12	yes	complemented
29	C₂×C₃×#103	C₂²	1	12	yes	complemented
30	C₂×C₃×C₂²⋊C₈	C₂²	1	12	yes	—
31	C₂²×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	C₂²×#117	C₂×C₃	1	1	yes	characteristic
33	C₂×#485	C₂×C₃	1	24	yes	direct product factor
34	C₂²×#107	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×#106	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂²×#103	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×#103	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
39	C₂⁵×C₃	D₄	1	1	yes	characteristic
40	C₂²×C₃×Q₈	—	2	3	no	—
41	C₂²×C₃×Q₈	—	2	4	no	—
42	C₂²×C₃×Q₈	—	2	4	no	—
43	C₂²×C₃×Q₈	C₂³	1	6	yes	—
44	C₂³×C₃×C₄	—	2	4	no	—
45	C₂³×C₃×C₄	—	2	4	no	—
46	C₂³×C₃×C₄	—	2	1	no	—
47	C₂³×C₃×C₄	—	2	1	no	—
48	C₂³×C₃×C₄	—	2	1	no	—
49	C₂³×C₃×C₄	—	2	1	no	—
50	C₂³×C₃×C₄	D₄	1	6	yes	—
51	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
52	C₂³×C₃×C₄	C₂³	1	6	yes	—
53	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₈	—	2	4	no	—
55	C₂²×C₃×C₈	—	2	4	no	—
56	C₂²×C₃×C₈	C₂³	1	6	yes	—
57	C₂²×C₃×C₈	C₂³	1	6	yes	—
58	C₃×C₂²⋊Q₈	—	2	24	no	—
59	C₃×C₂²⋊Q₈	C₂³	1	16	yes	—
60	C₃×#117	—	2	24	no	—
61	C₃×#117	C₂³	1	16	yes	—
62	C₂×C₃×C₄⋊C₄	—	2	12	no	—
63	C₂×C₃×C₄⋊C₄	—	2	3	no	—
64	C₂×C₃×C₄⋊C₄	—	2	3	no	—
65	C₂×C₃×C₄⋊C₄	—	2	3	no	—
66	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
67	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
68	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
69	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
70	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
71	C₂×C₃×C₄²	—	2	12	no	—
72	C₃×#107	C₂³	1	16	yes	complemented
73	C₃×#106	C₂³	1	16	yes	complemented
74	C₃×#103	C₂³	1	16	yes	complemented
75	C₃×#103	C₂³	1	16	yes	complemented
76	C₃×C₂²⋊C₈	C₂³	1	16	yes	—
77	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₂²⋊Q₈	—	2	6	no	—
82	C₂×C₂²⋊Q₈	C₂²×C₃	1	12	yes	—
83	C₂×#117	—	2	6	no	—
84	C₂×#117	C₂²×C₃	1	12	yes	—
85	C₂²×C₄⋊C₄	—	2	1	no	—
86	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂²×C₂²⋊C₄	—	2	1	no	—
90	C₂²×C₂²⋊C₄	—	2	1	no	—
91	C₂²×C₄²	—	2	1	no	—
92	#485	C₂²×C₃	1	64	yes	direct product factor
93	C₂×#107	C₂²×C₃	1	12	yes	complemented
94	C₂×#106	C₂²×C₃	1	12	yes	complemented
95	C₂×#103	C₂²×C₃	1	12	yes	complemented
96	C₂×#103	C₂²×C₃	1	12	yes	complemented
97	C₂×C₂²⋊C₈	C₂²×C₃	1	12	yes	—
98	C₂⁴×C₃	—	2	4	no	—
99	C₂⁴×C₃	—	2	4	no	—
100	C₂⁴×C₃	—	2	4	no	—
101	C₂⁴×C₃	C₂×D₄	1	6	yes	—
102	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×Q₈	—	2	6	no	—
104	C₂×C₃×Q₈	—	2	12	no	—
105	C₂×C₃×Q₈	—	2	12	no	—
106	C₂×C₃×Q₈	—	4	12	no	—
107	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	—	2	24	no	—
110	C₂²×C₃×C₄	—	2	12	no	—
111	C₂²×C₃×C₄	—	2	12	no	—
112	C₂²×C₃×C₄	—	2	6	no	—
113	C₂²×C₃×C₄	—	2	6	no	—
114	C₂²×C₃×C₄	—	2	6	no	—
115	C₂²×C₃×C₄	—	2	6	no	—
116	C₂²×C₃×C₄	—	4	4	no	—
117	C₂²×C₃×C₄	—	4	4	no	—
118	C₂²×C₃×C₄	—	4	4	no	—
119	C₂²×C₃×C₄	—	4	4	no	—
120	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
121	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
122	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
123	C₂²×C₃×C₄	#41	1	4	yes	—
124	C₂²×C₃×C₄	#41	1	4	yes	—
125	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
126	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
127	C₂×C₃×C₈	—	2	12	no	—
128	C₂×C₃×C₈	—	2	12	no	—
129	C₂×C₃×C₈	C₂⁴	1	4	yes	—
130	C₂×C₃×C₈	C₂⁴	1	4	yes	—
131	C₃×C₄⋊C₄	—	2	16	no	—
132	C₃×C₄⋊C₄	—	2	6	no	—
133	C₃×C₄⋊C₄	—	2	6	no	—
134	C₃×C₄⋊C₄	—	2	6	no	—
135	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
136	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
137	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
138	C₃×C₂²⋊C₄	—	2	16	no	—
139	C₃×C₂²⋊C₄	—	2	16	no	—
140	C₃×C₄²	—	2	16	no	—
141	C₂⁵	C₃×D₄	1	1	yes	characteristic
142	C₂²×Q₈	—	2	3	no	—
143	C₂²×Q₈	—	2	4	no	—
144	C₂²×Q₈	—	2	4	no	—
145	C₂²×Q₈	C₂³×C₃	1	6	yes	—
146	C₂³×C₄	—	2	4	no	—
147	C₂³×C₄	—	2	4	no	—
148	C₂³×C₄	—	2	1	no	—
149	C₂³×C₄	—	2	1	no	—
150	C₂³×C₄	—	2	1	no	—
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
153	C₂³×C₄	C₃×D₄	1	6	yes	—
154	C₂³×C₄	C₂³×C₃	1	6	yes	—
155	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	4	no	—
158	C₂²×C₈	C₂³×C₃	1	6	yes	—
159	C₂²×C₈	C₂³×C₃	1	6	yes	—
160	C₂²⋊Q₈	—	2	24	no	—
161	C₂²⋊Q₈	C₂³×C₃	1	16	yes	—
162	#117	—	2	24	no	—
163	#117	C₂³×C₃	1	16	yes	—
164	C₂×C₄⋊C₄	—	2	12	no	—
165	C₂×C₄⋊C₄	—	2	3	no	—
166	C₂×C₄⋊C₄	—	2	3	no	—
167	C₂×C₄⋊C₄	—	2	3	no	—
168	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
169	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
170	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
171	C₂×C₂²⋊C₄	—	2	12	no	—
172	C₂×C₂²⋊C₄	—	2	12	no	—
173	C₂×C₄²	—	2	12	no	—
174	#107	C₂³×C₃	1	16	yes	complemented
175	#106	C₂³×C₃	1	16	yes	complemented
176	#103	C₂³×C₃	1	16	yes	complemented
177	#103	C₂³×C₃	1	16	yes	complemented
178	C₂²⋊C₈	C₂³×C₃	1	16	yes	—
179	C₂³×C₃	—	2	12	no	—
180	C₂³×C₃	—	2	12	no	—
181	C₂³×C₃	—	2	12	no	—
182	C₂³×C₃	—	4	16	no	—
183	C₂³×C₃	#123	1	4	yes	—
184	C₂³×C₃	#135	1	4	yes	—
185	C₂³×C₃	#137	1	4	yes	—
186	C₂³×C₃	C₂²×D₄	1	4	yes	—
187	C₂³×C₃	C₂²×D₄	1	3	yes	—
188	C₃×Q₈	—	2	4	no	—
189	C₃×Q₈	—	2	4	no	—
190	C₃×Q₈	—	4	12	no	—
191	C₂×C₃×C₄	—	2	12	no	—
192	C₂×C₃×C₄	—	2	24	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	4	12	no	—
200	C₂×C₃×C₄	—	4	12	no	—
201	C₂×C₃×C₄	—	4	12	no	—
202	C₂×C₃×C₄	—	4	12	no	—
203	C₂×C₃×C₄	#123	1	12	yes	—
204	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
205	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
206	C₂×C₃×C₄	C₂×#41	1	6	yes	—
207	C₂×C₃×C₄	C₂×#41	1	6	yes	—
208	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
209	C₃×C₈	—	2	4	no	—
210	C₃×C₈	—	2	4	no	—
211	C₂⁴	—	2	4	no	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	—	2	4	no	—
214	C₂⁴	C₂×C₃×D₄	1	6	yes	—
215	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×Q₈	—	2	6	no	—
217	C₂×Q₈	—	2	12	no	—
218	C₂×Q₈	—	2	12	no	—
219	C₂×Q₈	—	4	12	no	—
220	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	12	no	—
225	C₂²×C₄	—	2	6	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	2	6	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
234	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
235	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
236	C₂²×C₄	C₃×#41	1	4	yes	—
237	C₂²×C₄	C₃×#41	1	4	yes	—
238	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
240	C₂×C₈	—	2	12	no	—
241	C₂×C₈	—	2	12	no	—
242	C₂×C₈	C₂⁴×C₃	1	4	yes	—
243	C₂×C₈	C₂⁴×C₃	1	4	yes	—
244	C₄⋊C₄	—	2	16	no	—
245	C₄⋊C₄	—	2	6	no	—
246	C₄⋊C₄	—	2	6	no	—
247	C₄⋊C₄	—	2	6	no	—
248	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
249	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
250	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
251	C₂²⋊C₄	—	2	16	no	—
252	C₂²⋊C₄	—	2	16	no	—
253	C₄²	—	2	16	no	—
254	C₂²×C₃	—	2	4	no	—
255	C₂²×C₃	—	2	4	no	—
256	C₂²×C₃	—	2	4	no	—
257	C₂²×C₃	—	4	24	no	—
258	C₂²×C₃	#485	1	16	yes	direct product factor
259	C₂²×C₃	C₂×#123	1	6	yes	—
260	C₂²×C₃	C₂×#135	1	6	yes	—
261	C₂²×C₃	C₂×#137	1	6	yes	—
262	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	4	4	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	4	4	no	—
267	C₃×C₄	—	4	4	no	—
268	C₃×C₄	C₂×#123	1	6	yes	—
269	C₃×C₄	C₂²×#41	1	1	yes	characteristic
270	C₃×C₄	C₂²×#41	1	1	yes	characteristic
271	C₂³	—	2	12	no	—
272	C₂³	—	2	12	no	—
273	C₂³	—	2	12	no	—
274	C₂³	—	4	16	no	—
275	C₂³	C₃×#123	1	4	yes	—
276	C₂³	C₃×#135	1	4	yes	—
277	C₂³	C₃×#137	1	4	yes	—
278	C₂³	C₂²×C₃×D₄	1	4	yes	—
279	C₂³	C₂²×C₃×D₄	1	3	yes	—
280	Q₈	—	2	4	no	—
281	Q₈	—	2	4	no	—
282	Q₈	—	4	12	no	—
283	C₂×C₄	—	2	12	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	24	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	12	no	—
292	C₂×C₄	—	4	12	no	—
293	C₂×C₄	—	4	12	no	—
294	C₂×C₄	—	4	12	no	—
295	C₂×C₄	C₃×#123	1	12	yes	—
296	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
297	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
298	C₂×C₄	C₂×C₃×#41	1	6	yes	—
299	C₂×C₄	C₂×C₃×#41	1	6	yes	—
300	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
301	C₈	—	2	4	no	—
302	C₈	—	2	4	no	—
303	C₂×C₃	—	4	4	no	—
304	C₂×C₃	C₂×#485	1	12	yes	direct product factor
305	C₂×C₃	C₂²×#123	1	1	yes	characteristic
306	C₂×C₃	C₂²×#135	1	1	yes	characteristic
307	C₂×C₃	C₂²×#137	1	1	yes	characteristic
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	24	no	—
312	C₂²	C₃×#485	1	16	yes	direct product factor
313	C₂²	C₂×C₃×#123	1	6	yes	—
314	C₂²	C₂×C₃×#135	1	6	yes	—
315	C₂²	C₂×C₃×#137	1	6	yes	—
316	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	C₂×C₃×#123	1	6	yes	—
323	C₄	C₂²×C₃×#41	1	1	yes	characteristic
324	C₄	C₂²×C₃×#41	1	1	yes	characteristic
325	C₃	C₂²×#485	1	1	yes	characteristic, direct product factor, Sylow
326	C₂	—	4	4	no	—
327	C₂	C₂×C₃×#485	1	12	yes	direct product factor
328	C₂	C₂²×C₃×#123	1	1	yes	characteristic
329	C₂	C₂²×C₃×#135	1	1	yes	characteristic
330	C₂	C₂²×C₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂⁵×C₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#485	C₂	direct (N × H)
4. C₂²×C₃×#107	C₂	semidirect
5. C₂²×C₃×#106	C₂	semidirect
6. C₂²×C₃×#103	C₂	semidirect
7. C₂²×C₃×#103	C₂	semidirect
9. C₂²×#485	C₃	direct (N × H)
25. C₃×#485	C₂²	direct (N × H)
26. C₂×C₃×#107	C₂²	semidirect
27. C₂×C₃×#106	C₂²	semidirect
28. C₂×C₃×#103	C₂²	semidirect
29. C₂×C₃×#103	C₂²	semidirect
33. C₂×#485	C₂×C₃	direct (N × H)
34. C₂²×#107	C₂×C₃	semidirect
35. C₂²×#106	C₂×C₃	semidirect
36. C₂²×#103	C₂×C₃	semidirect
37. C₂²×#103	C₂×C₃	semidirect
72. C₃×#107	C₂³	semidirect
73. C₃×#106	C₂³	semidirect
74. C₃×#103	C₂³	semidirect
75. C₃×#103	C₂³	semidirect
92. #485	C₂²×C₃	direct (N × H)
93. C₂×#107	C₂²×C₃	semidirect
94. C₂×#106	C₂²×C₃	semidirect
95. C₂×#103	C₂²×C₃	semidirect
96. C₂×#103	C₂²×C₃	semidirect
174. #107	C₂³×C₃	semidirect
175. #106	C₂³×C₃	semidirect
176. #103	C₂³×C₃	semidirect
177. #103	C₂³×C₃	semidirect
258. C₂²×C₃	#485	direct (N × H)
304. C₂×C₃	C₂×#485	direct (N × H)
312. C₂²	C₃×#485	direct (N × H)
325. C₃	C₂²×#485	direct (N × H)
327. C₂	C₂×C₃×#485	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×C₂²⋊Q₈	2
C₂²×C₃×#117	2
C₂×C₃×#485	2
C₂²×C₃×#107	2
C₂²×C₃×#106	2
C₂²×C₃×#103	2
C₂²×C₃×C₂²⋊C₈	2
C₂²×#485	3

Subgroups — C22×C3×#485 #10679110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#485 #10679110