Subgroups — C₂²×C₃×#478 #10679082

Order 768.

2574 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#478	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×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₃×#102	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₃×#102	C₂	1	1	yes	maximal, characteristic
9	C₂²×#478	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃×C₄	—	2	1	no	—
12	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×#125	—	2	6	no	—
15	C₂×C₃×#125	C₂²	1	12	yes	complemented
16	C₂×C₃×C₄×D₄	—	2	6	no	—
17	C₂×C₃×C₄×D₄	C₂²	1	12	yes	—
18	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
19	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
20	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
24	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#478	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×C₄⋊C₈	C₂²	1	12	yes	complemented
29	C₂×C₃×#102	C₂²	1	12	yes	—
30	C₂×C₃×#102	C₂²	1	12	yes	—
31	C₂²×#125	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
33	C₂×#478	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×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×#102	C₂×C₃	1	1	yes	characteristic
38	C₂²×#102	C₂×C₃	1	1	yes	characteristic
39	C₂⁵×C₃	—	2	1	no	—
40	C₂²×C₃×D₄	—	2	3	no	—
41	C₂²×C₃×D₄	—	2	4	no	—
42	C₂²×C₃×D₄	Q₈	1	8	yes	—
43	C₂²×C₃×D₄	C₂³	1	6	yes	—
44	C₂³×C₃×C₄	—	2	1	no	—
45	C₂³×C₃×C₄	—	2	1	no	—
46	C₂³×C₃×C₄	—	2	1	no	—
47	C₂³×C₃×C₄	—	2	12	no	—
48	C₂³×C₃×C₄	—	4	8	no	—
49	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
50	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
51	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₈	—	2	4	no	—
53	C₂²×C₃×C₈	—	2	4	no	—
54	C₂²×C₃×C₈	C₂³	1	6	yes	—
55	C₂²×C₃×C₈	C₂³	1	6	yes	—
56	C₃×#125	—	2	24	no	—
57	C₃×#125	C₂³	1	16	yes	complemented
58	C₃×C₄×D₄	—	2	24	no	—
59	C₃×C₄×D₄	C₂³	1	16	yes	—
60	C₂×C₃×C₄⋊C₄	—	2	12	no	—
61	C₂×C₃×C₄⋊C₄	—	2	12	no	—
62	C₂×C₃×C₄⋊C₄	—	2	3	no	—
63	C₂×C₃×C₄⋊C₄	—	2	3	no	—
64	C₂×C₃×C₄⋊C₄	—	2	3	no	—
65	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
66	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
67	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
68	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
69	C₂×C₃×C₄²	D₄	1	6	yes	—
70	C₂×C₃×C₄²	C₂³	1	6	yes	—
71	C₃×#107	C₂³	1	16	yes	complemented
72	C₃×#106	C₂³	1	16	yes	complemented
73	C₃×C₄⋊C₈	C₂³	1	16	yes	complemented
74	C₃×#102	C₂³	1	16	yes	—
75	C₃×#102	C₂³	1	16	yes	—
76	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
77	C₂⁴×C₄	—	2	1	no	—
78	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×#125	—	2	6	no	—
81	C₂×#125	C₂²×C₃	1	12	yes	complemented
82	C₂×C₄×D₄	—	2	6	no	—
83	C₂×C₄×D₄	C₂²×C₃	1	12	yes	—
84	C₂²×C₄⋊C₄	—	2	1	no	—
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×C₄²	C₂²×C₃	1	1	yes	characteristic
91	#478	C₂²×C₃	1	64	yes	direct product factor
92	C₂×#107	C₂²×C₃	1	12	yes	complemented
93	C₂×#106	C₂²×C₃	1	12	yes	complemented
94	C₂×C₄⋊C₈	C₂²×C₃	1	12	yes	complemented
95	C₂×#102	C₂²×C₃	1	12	yes	—
96	C₂×#102	C₂²×C₃	1	12	yes	—
97	C₂⁴×C₃	—	2	8	no	—
98	C₂⁴×C₃	—	2	6	no	—
99	C₂⁴×C₃	—	4	4	no	—
100	C₂⁴×C₃	—	4	4	no	—
101	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃×D₄	—	2	24	no	—
103	C₂×C₃×D₄	—	2	6	no	—
104	C₂×C₃×D₄	—	2	12	no	—
105	C₂×C₃×D₄	C₂×Q₈	1	24	yes	—
106	C₂×C₃×D₄	C₂⁴	1	4	yes	—
107	C₂²×C₃×C₄	—	2	4	no	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	—	2	6	no	—
110	C₂²×C₃×C₄	—	2	6	no	—
111	C₂²×C₃×C₄	—	2	6	no	—
112	C₂²×C₃×C₄	—	2	16	no	—
113	C₂²×C₃×C₄	—	4	48	no	—
114	C₂²×C₃×C₄	—	4	4	no	—
115	C₂²×C₃×C₄	—	4	4	no	—
116	C₂²×C₃×C₄	—	4	4	no	—
117	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
118	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
119	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
120	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
121	C₂²×C₃×C₄	#41	1	4	yes	—
122	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
123	C₂×C₃×C₈	—	2	12	no	—
124	C₂×C₃×C₈	—	2	12	no	—
125	C₂×C₃×C₈	C₂⁴	1	4	yes	—
126	C₂×C₃×C₈	C₂⁴	1	4	yes	—
127	C₃×C₄⋊C₄	—	2	16	no	—
128	C₃×C₄⋊C₄	—	2	16	no	—
129	C₃×C₄⋊C₄	—	2	6	no	—
130	C₃×C₄⋊C₄	—	2	6	no	—
131	C₃×C₄⋊C₄	—	2	6	no	—
132	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
133	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
134	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
135	C₃×C₂²⋊C₄	—	2	16	no	—
136	C₃×C₄²	C₂×D₄	1	12	yes	—
137	C₃×C₄²	C₂⁴	1	4	yes	—
138	C₂⁵	—	2	1	no	—
139	C₂²×D₄	—	2	3	no	—
140	C₂²×D₄	—	2	4	no	—
141	C₂²×D₄	C₃×Q₈	1	8	yes	—
142	C₂²×D₄	C₂³×C₃	1	6	yes	—
143	C₂³×C₄	—	2	1	no	—
144	C₂³×C₄	—	2	1	no	—
145	C₂³×C₄	—	2	1	no	—
146	C₂³×C₄	—	2	12	no	—
147	C₂³×C₄	—	4	8	no	—
148	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
149	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
150	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂³×C₃	1	6	yes	—
154	C₂²×C₈	C₂³×C₃	1	6	yes	—
155	#125	—	2	24	no	—
156	#125	C₂³×C₃	1	16	yes	complemented
157	C₄×D₄	—	2	24	no	—
158	C₄×D₄	C₂³×C₃	1	16	yes	—
159	C₂×C₄⋊C₄	—	2	12	no	—
160	C₂×C₄⋊C₄	—	2	12	no	—
161	C₂×C₄⋊C₄	—	2	3	no	—
162	C₂×C₄⋊C₄	—	2	3	no	—
163	C₂×C₄⋊C₄	—	2	3	no	—
164	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
165	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
166	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
167	C₂×C₂²⋊C₄	—	2	12	no	—
168	C₂×C₄²	C₃×D₄	1	6	yes	—
169	C₂×C₄²	C₂³×C₃	1	6	yes	—
170	#107	C₂³×C₃	1	16	yes	complemented
171	#106	C₂³×C₃	1	16	yes	complemented
172	C₄⋊C₈	C₂³×C₃	1	16	yes	complemented
173	#102	C₂³×C₃	1	16	yes	—
174	#102	C₂³×C₃	1	16	yes	—
175	C₂³×C₃	—	2	24	no	—
176	C₂³×C₃	—	2	4	no	—
177	C₂³×C₃	—	4	12	no	—
178	C₂³×C₃	—	4	12	no	—
179	C₂³×C₃	—	4	32	no	—
180	C₂³×C₃	C₂²⋊Q₈	1	4	yes	—
181	C₂³×C₃	#135	1	4	yes	—
182	C₂³×C₃	AGL(1,ℤ/8)	1	4	yes	—
183	C₂³×C₃	C₂²×D₄	1	3	yes	—
184	C₃×D₄	—	2	24	no	—
185	C₃×D₄	—	2	4	no	—
186	C₃×D₄	C₂²×Q₈	1	8	yes	—
187	C₂×C₃×C₄	—	2	12	no	—
188	C₂×C₃×C₄	—	2	12	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	4	32	no	—
193	C₂×C₃×C₄	—	4	12	no	—
194	C₂×C₃×C₄	—	4	12	no	—
195	C₂×C₃×C₄	—	4	12	no	—
196	C₂×C₃×C₄	C₂²⋊Q₈	1	12	yes	—
197	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
198	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
199	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
200	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
201	C₂×C₃×C₄	C₂×#41	1	6	yes	—
202	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
203	C₃×C₈	—	2	4	no	—
204	C₃×C₈	—	2	4	no	—
205	C₂⁴	—	2	8	no	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	—	4	4	no	—
209	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×D₄	—	2	24	no	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	2	12	no	—
213	C₂×D₄	C₂×C₃×Q₈	1	24	yes	—
214	C₂×D₄	C₂⁴×C₃	1	4	yes	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	6	no	—
218	C₂²×C₄	—	2	6	no	—
219	C₂²×C₄	—	2	6	no	—
220	C₂²×C₄	—	2	16	no	—
221	C₂²×C₄	—	4	48	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
226	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
227	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
228	C₂²×C₄	C₂×C₃×Q₈	1	4	yes	—
229	C₂²×C₄	C₃×#41	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
231	C₂×C₈	—	2	12	no	—
232	C₂×C₈	—	2	12	no	—
233	C₂×C₈	C₂⁴×C₃	1	4	yes	—
234	C₂×C₈	C₂⁴×C₃	1	4	yes	—
235	C₄⋊C₄	—	2	16	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₄⋊C₄	—	2	6	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	—	2	6	no	—
240	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
241	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
242	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
243	C₂²⋊C₄	—	2	16	no	—
244	C₄²	C₂×C₃×D₄	1	12	yes	—
245	C₄²	C₂⁴×C₃	1	4	yes	—
246	C₂²×C₃	—	2	8	no	—
247	C₂²×C₃	—	4	48	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	#478	1	16	yes	direct product factor
251	C₂²×C₃	C₂×C₂²⋊Q₈	1	6	yes	—
252	C₂²×C₃	C₂×#135	1	6	yes	—
253	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
254	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	—	4	4	no	—
259	C₃×C₄	—	4	4	no	—
260	C₃×C₄	C₂×C₂²⋊Q₈	1	6	yes	—
261	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
262	C₃×C₄	C₂²×#41	1	1	yes	characteristic
263	C₂³	—	2	4	no	—
264	C₂³	—	2	24	no	—
265	C₂³	—	4	12	no	—
266	C₂³	—	4	12	no	—
267	C₂³	—	4	32	no	—
268	C₂³	C₃×C₂²⋊Q₈	1	4	yes	—
269	C₂³	C₃×#135	1	4	yes	—
270	C₂³	C₃×AGL(1,ℤ/8)	1	4	yes	—
271	C₂³	C₂²×C₃×D₄	1	3	yes	—
272	D₄	—	2	4	no	—
273	D₄	—	2	24	no	—
274	D₄	C₂²×C₃×Q₈	1	8	yes	—
275	C₂×C₄	—	2	12	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	32	no	—
281	C₂×C₄	—	4	12	no	—
282	C₂×C₄	—	4	12	no	—
283	C₂×C₄	—	4	12	no	—
284	C₂×C₄	C₃×C₂²⋊Q₈	1	12	yes	—
285	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
286	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
287	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
288	C₂×C₄	C₂²×C₃×Q₈	1	6	yes	—
289	C₂×C₄	C₂×C₃×#41	1	6	yes	—
290	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₂×C₃	—	4	8	no	—
294	C₂×C₃	C₂×#478	1	12	yes	direct product factor
295	C₂×C₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
296	C₂×C₃	C₂²×#135	1	1	yes	characteristic
297	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂²	—	2	8	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	48	no	—
302	C₂²	C₃×#478	1	16	yes	direct product factor
303	C₂²	C₂×C₃×C₂²⋊Q₈	1	6	yes	—
304	C₂²	C₂×C₃×#135	1	6	yes	—
305	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	6	yes	—
306	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
307	C₄	—	2	4	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	4	no	—
312	C₄	C₂×C₃×C₂²⋊Q₈	1	6	yes	—
313	C₄	C₂³×C₃×Q₈	1	1	yes	characteristic
314	C₄	C₂²×C₃×#41	1	1	yes	characteristic
315	C₃	C₂²×#478	1	1	yes	characteristic, direct product factor, Sylow
316	C₂	—	4	8	no	—
317	C₂	C₂×C₃×#478	1	12	yes	direct product factor
318	C₂	C₂²×C₃×C₂²⋊Q₈	1	1	yes	characteristic
319	C₂	C₂²×C₃×#135	1	1	yes	characteristic
320	C₂	C₂²×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#125	C₂	semidirect
3. C₂×C₃×#478	C₂	direct (N × H)
4. C₂²×C₃×#107	C₂	semidirect
5. C₂²×C₃×#106	C₂	semidirect
6. C₂²×C₃×C₄⋊C₈	C₂	semidirect
9. C₂²×#478	C₃	direct (N × H)
15. C₂×C₃×#125	C₂²	semidirect
25. C₃×#478	C₂²	direct (N × H)
26. C₂×C₃×#107	C₂²	semidirect
27. C₂×C₃×#106	C₂²	semidirect
28. C₂×C₃×C₄⋊C₈	C₂²	semidirect
31. C₂²×#125	C₂×C₃	semidirect
33. C₂×#478	C₂×C₃	direct (N × H)
34. C₂²×#107	C₂×C₃	semidirect
35. C₂²×#106	C₂×C₃	semidirect
36. C₂²×C₄⋊C₈	C₂×C₃	semidirect
57. C₃×#125	C₂³	semidirect
71. C₃×#107	C₂³	semidirect
72. C₃×#106	C₂³	semidirect
73. C₃×C₄⋊C₈	C₂³	semidirect
81. C₂×#125	C₂²×C₃	semidirect
91. #478	C₂²×C₃	direct (N × H)
92. C₂×#107	C₂²×C₃	semidirect
93. C₂×#106	C₂²×C₃	semidirect
94. C₂×C₄⋊C₈	C₂²×C₃	semidirect
156. #125	C₂³×C₃	semidirect
170. #107	C₂³×C₃	semidirect
171. #106	C₂³×C₃	semidirect
172. C₄⋊C₈	C₂³×C₃	semidirect
250. C₂²×C₃	#478	direct (N × H)
294. C₂×C₃	C₂×#478	direct (N × H)
302. C₂²	C₃×#478	direct (N × H)
315. C₃	C₂²×#478	direct (N × H)
317. C₂	C₂×C₃×#478	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×#125	2
C₂²×C₃×C₄×D₄	2
C₂×C₃×#478	2
C₂²×C₃×#107	2
C₂²×C₃×#106	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×#102	2
C₂²×#478	3

Subgroups — C22×C3×#478 #10679082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#478 #10679082