Subgroups — C₂×C₃×#2211 #10652185

Order 768.

722 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#495	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2211	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#357	C₂	1	1	yes	maximal, characteristic
9	C₂×#2211	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×SD₁₆	—	2	1	no	—
13	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×#124	—	2	1	no	—
15	C₂×C₃×C₄×Q₈	—	2	1	no	—
16	C₂×C₃×Q₃₂	—	2	4	no	—
17	C₂×C₃×Q₃₂	C₂²	1	4	yes	complemented
18	C₂×C₃×SD₃₂	—	2	4	no	—
19	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
20	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
22	C₃×#495	—	2	2	no	—
23	C₃×#495	C₂²	1	4	yes	—
24	C₃×C₄×Q₁₆	—	2	2	no	—
25	C₃×C₄×Q₁₆	C₂²	1	4	yes	complemented
26	C₂×C₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃×#103	—	2	1	no	—
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
30	C₃×#358	C₂²	1	4	yes	complemented
31	C₃×#357	C₂²	1	4	yes	—
32	C₂²×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂²×SD₃₂	C₂×C₃	1	1	yes	characteristic
34	C₂×#495	C₂×C₃	1	1	yes	characteristic
35	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
36	#2211	C₂×C₃	1	8	yes	direct product factor
37	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×#358	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#357	C₂×C₃	1	1	yes	characteristic
40	C₂²×C₃×Q₈	—	2	1	no	—
41	C₂²×C₃×Q₈	—	2	1	no	—
42	C₂²×C₃×D₄	—	2	1	no	—
43	C₂×C₃×Q₁₆	—	2	1	no	—
44	C₂×C₃×Q₁₆	—	2	2	no	—
45	C₂×C₃×Q₁₆	—	2	1	no	—
46	C₂×C₃×Q₁₆	—	2	2	no	—
47	C₂×C₃×Q₁₆	—	2	2	no	—
48	C₂×C₃×Q₁₆	D₄	1	4	yes	—
49	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
50	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
51	C₂×C₃×SD₁₆	—	2	4	no	—
52	C₂×C₃×SD₁₆	—	4	4	no	—
53	C₂×C₃×D₈	—	2	2	no	—
54	C₂×C₃×D₈	—	2	1	no	—
55	C₂×C₃×D₈	—	2	2	no	—
56	C₂×C₃×D₈	C₂³	1	2	yes	—
57	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
59	C₃×#124	—	2	4	no	—
60	C₃×#124	—	4	2	no	—
61	C₃×C₄×Q₈	—	2	4	no	—
62	C₃×C₄×Q₈	—	4	2	no	—
63	C₂×C₃×C₄⋊C₄	—	2	1	no	—
64	C₂×C₃×C₄⋊C₄	—	4	1	no	—
65	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
66	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
67	C₂×C₃×C₄²	—	4	1	no	—
68	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
69	C₃×Q₃₂	—	2	8	no	—
70	C₃×SD₃₂	—	2	8	no	—
71	C₂×C₃×C₁₆	—	2	2	no	—
72	C₂×C₃×C₁₆	—	2	2	no	—
73	C₂×C₃×C₁₆	C₂³	1	2	yes	—
74	C₂×C₃×C₁₆	C₂³	1	2	yes	—
75	C₃×#107	—	2	1	no	—
76	C₃×#107	C₂³	1	2	yes	—
77	C₃×#103	—	2	4	no	—
78	C₃×C₄×C₈	D₄	1	2	yes	—
79	C₃×C₄×C₈	C₂³	1	2	yes	—
80	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂²×SD₁₆	—	2	1	no	—
83	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
84	C₂×#124	—	2	1	no	—
85	C₂×C₄×Q₈	—	2	1	no	—
86	C₂×Q₃₂	—	2	4	no	—
87	C₂×Q₃₂	C₂²×C₃	1	4	yes	complemented
88	C₂×SD₃₂	—	2	4	no	—
89	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
90	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
91	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
92	#495	—	2	2	no	—
93	#495	C₂²×C₃	1	4	yes	—
94	C₄×Q₁₆	—	2	2	no	—
95	C₄×Q₁₆	C₂²×C₃	1	4	yes	complemented
96	C₂×#107	C₂²×C₃	1	1	yes	characteristic
97	C₂×#103	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₄⋊C₁₆	C₂²×C₃	1	4	yes	complemented
100	#358	C₂²×C₃	1	4	yes	complemented
101	#357	C₂²×C₃	1	4	yes	—
102	C₂⁴×C₃	—	4	1	no	—
103	C₂×C₃×Q₈	—	2	4	no	—
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	—	2	2	no	—
106	C₂×C₃×Q₈	—	4	1	no	—
107	C₂×C₃×Q₈	—	4	1	no	—
108	C₂×C₃×Q₈	—	4	2	no	—
109	C₂×C₃×Q₈	—	4	2	no	—
110	C₂×C₃×Q₈	—	4	2	no	—
111	C₂×C₃×D₄	—	2	2	no	—
112	C₂×C₃×D₄	—	4	2	no	—
113	C₂×C₃×D₄	—	4	1	no	—
114	C₂×C₃×D₄	—	4	2	no	—
115	C₂²×C₃×C₄	—	4	1	no	—
116	C₂²×C₃×C₄	—	4	1	no	—
117	C₂²×C₃×C₄	—	4	1	no	—
118	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₃×Q₁₆	—	2	4	no	—
122	C₃×Q₁₆	—	2	2	no	—
123	C₃×Q₁₆	—	2	2	no	—
124	C₃×Q₁₆	—	2	2	no	—
125	C₃×Q₁₆	—	4	2	no	—
126	C₃×Q₁₆	C₂×D₄	1	4	yes	—
127	C₃×SD₁₆	—	4	8	no	—
128	C₃×D₈	—	2	2	no	—
129	C₃×D₈	—	2	2	no	—
130	C₃×D₈	—	4	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₈	#41	1	2	yes	—
136	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	4	1	no	—
139	C₃×C₄⋊C₄	—	4	4	no	—
140	C₃×C₂²⋊C₄	—	4	4	no	—
141	C₃×C₂²⋊C₄	—	4	4	no	—
142	C₃×C₄²	—	4	4	no	—
143	C₃×C₄²	D₈	1	2	yes	—
144	C₃×C₄²	C₂×D₄	1	2	yes	—
145	C₃×C₁₆	—	2	2	no	—
146	C₃×C₁₆	—	2	2	no	—
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×Q₈	—	2	1	no	—
149	C₂²×D₄	—	2	1	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	C₃×D₄	1	4	yes	—
156	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
157	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
158	C₂×SD₁₆	—	2	4	no	—
159	C₂×SD₁₆	—	4	4	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	C₂³×C₃	1	2	yes	—
164	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
166	#124	—	2	4	no	—
167	#124	—	4	2	no	—
168	C₄×Q₈	—	2	4	no	—
169	C₄×Q₈	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₂²⋊C₄	—	4	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₄²	—	4	1	no	—
175	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
176	Q₃₂	—	2	8	no	—
177	SD₃₂	—	2	8	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	C₂³×C₃	1	2	yes	—
181	C₂×C₁₆	C₂³×C₃	1	2	yes	—
182	#107	—	2	1	no	—
183	#107	C₂³×C₃	1	2	yes	—
184	#103	—	2	4	no	—
185	C₄×C₈	C₃×D₄	1	2	yes	—
186	C₄×C₈	C₂³×C₃	1	2	yes	—
187	C₂³×C₃	—	4	2	no	—
188	C₂³×C₃	—	8	2	no	—
189	C₂³×C₃	—	8	2	no	—
190	C₂³×C₃	—	8	2	no	—
191	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
192	C₃×Q₈	—	2	4	no	—
193	C₃×Q₈	—	4	2	no	—
194	C₃×Q₈	—	4	4	no	—
195	C₃×Q₈	—	4	2	no	—
196	C₃×Q₈	—	4	2	no	—
197	C₃×Q₈	—	8	2	no	—
198	C₃×D₄	—	4	2	no	—
199	C₃×D₄	—	4	2	no	—
200	C₃×D₄	—	8	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	4	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	4	2	no	—
207	C₂×C₃×C₄	—	8	2	no	—
208	C₂×C₃×C₄	—	8	2	no	—
209	C₂×C₃×C₄	#121	1	2	yes	—
210	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
211	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
212	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
213	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
214	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	#121	1	2	yes	—
217	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
218	C₃×C₈	C₂×#41	1	1	yes	characteristic
219	C₂⁴	—	4	1	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
236	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
238	Q₁₆	—	2	4	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	4	2	no	—
243	Q₁₆	C₂×C₃×D₄	1	4	yes	—
244	SD₁₆	—	4	8	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	4	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
252	C₂×C₈	C₃×#41	1	2	yes	—
253	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	C₃×D₈	1	2	yes	—
261	C₄²	C₂×C₃×D₄	1	2	yes	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	2	2	no	—
264	C₂²×C₃	—	8	2	no	—
265	C₂²×C₃	—	8	2	no	—
266	C₂²×C₃	—	8	2	no	—
267	C₂²×C₃	—	16	4	no	—
268	C₂²×C₃	#459	1	2	yes	—
269	C₂²×C₃	#508	1	2	yes	—
270	C₂²×C₃	#510	1	2	yes	—
271	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	4	4	no	—
275	C₃×C₄	—	8	2	no	—
276	C₃×C₄	—	8	2	no	—
277	C₃×C₄	#459	1	2	yes	—
278	C₃×C₄	C₂×#121	1	1	yes	characteristic
279	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂³	—	4	2	no	—
281	C₂³	—	8	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	C₃×#121	1	2	yes	—
303	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
304	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
305	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
307	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	C₃×#121	1	2	yes	—
310	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
311	C₈	C₂×C₃×#41	1	1	yes	characteristic
312	C₂×C₃	—	16	2	no	—
313	C₂×C₃	#2211	1	4	yes	direct product factor
314	C₂×C₃	C₂×#459	1	1	yes	characteristic
315	C₂×C₃	C₂×#508	1	1	yes	characteristic
316	C₂×C₃	C₂×#510	1	1	yes	characteristic
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	16	4	no	—
321	C₂²	C₃×#459	1	2	yes	—
322	C₂²	C₃×#508	1	2	yes	—
323	C₂²	C₃×#510	1	2	yes	—
324	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₃×#459	1	2	yes	—
331	C₄	C₂×C₃×#121	1	1	yes	characteristic
332	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₃	C₂×#2211	1	1	yes	characteristic, direct product factor, Sylow
334	C₂	—	16	2	no	—
335	C₂	C₃×#2211	1	4	yes	direct product factor
336	C₂	C₂×C₃×#459	1	1	yes	characteristic
337	C₂	C₂×C₃×#508	1	1	yes	characteristic
338	C₂	C₂×C₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂²×C₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₃₂	C₂	semidirect
4. C₂×C₃×C₄×Q₁₆	C₂	semidirect
5. C₃×#2211	C₂	direct (N × H)
6. C₂×C₃×C₄⋊C₁₆	C₂	semidirect
7. C₂×C₃×#358	C₂	semidirect
9. C₂×#2211	C₃	direct (N × H)
17. C₂×C₃×Q₃₂	C₂²	semidirect
25. C₃×C₄×Q₁₆	C₂²	semidirect
29. C₃×C₄⋊C₁₆	C₂²	semidirect
30. C₃×#358	C₂²	semidirect
32. C₂²×Q₃₂	C₂×C₃	semidirect
35. C₂×C₄×Q₁₆	C₂×C₃	semidirect
36. #2211	C₂×C₃	direct (N × H)
37. C₂×C₄⋊C₁₆	C₂×C₃	semidirect
38. C₂×#358	C₂×C₃	semidirect
87. C₂×Q₃₂	C₂²×C₃	semidirect
95. C₄×Q₁₆	C₂²×C₃	semidirect
99. C₄⋊C₁₆	C₂²×C₃	semidirect
100. #358	C₂²×C₃	semidirect
313. C₂×C₃	#2211	direct (N × H)
333. C₃	C₂×#2211	direct (N × H)
335. C₂	C₃×#2211	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×Q₃₂	2
C₂²×C₃×SD₃₂	2
C₂×C₃×#495	2
C₂×C₃×C₄×Q₁₆	2
C₃×#2211	2
C₂×C₃×C₄⋊C₁₆	2
C₂×C₃×#358	2
C₂×C₃×#357	2
C₂×#2211	3

Subgroups — C2×C3×#2211 #10652185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2211 #10652185