Subgroups — C₂×C₃×#2234 #10652248

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2234	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#2234	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#121	—	2	1	no	—
13	C₂×C₃×#117	—	2	1	no	—
14	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×#466	—	2	2	no	—
17	C₃×#466	C₂²	1	4	yes	—
18	C₃×#427	—	2	2	no	—
19	C₃×#427	C₂²	1	4	yes	complemented
20	C₂×C₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃×#106	—	2	1	no	—
24	C₂×C₃×#102	—	2	1	no	—
25	C₃×#367	C₂²	1	4	yes	complemented
26	C₃×#366	C₂²	1	4	yes	complemented
27	C₃×#357	C₂²	1	4	yes	complemented
28	C₃×#357	C₂²	1	4	yes	complemented
29	C₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
30	C₂×#466	C₂×C₃	1	1	yes	characteristic
31	C₂×#427	C₂×C₃	1	1	yes	characteristic, complemented
32	#2234	C₂×C₃	1	8	yes	direct product factor
33	C₂×#367	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#366	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#357	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#357	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃×D₄	—	2	1	no	—
39	C₂²×C₃×D₄	—	4	1	no	—
40	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃×D₈	—	2	1	no	—
42	C₂×C₃×D₈	—	2	2	no	—
43	C₂×C₃×D₈	—	2	2	no	—
44	C₂×C₃×D₈	C₂³	1	2	yes	—
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	—	2	2	no	—
47	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₃×C₈	D₄	1	2	yes	—
49	C₂²×C₃×C₈	C₂³	1	2	yes	—
50	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₃×#121	—	2	4	no	—
52	C₃×#121	—	4	2	no	—
53	C₃×#117	—	2	4	no	—
54	C₃×#117	—	4	2	no	—
55	C₂×C₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₃×C₄⋊C₄	—	2	1	no	—
58	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
59	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
60	C₂×C₃×C₄²	—	4	1	no	—
61	C₂×C₃×C₁₆	—	2	2	no	—
62	C₂×C₃×C₁₆	—	2	2	no	—
63	C₂×C₃×C₁₆	C₂³	1	2	yes	—
64	C₂×C₃×C₁₆	C₂³	1	2	yes	—
65	C₃×#107	—	2	1	no	—
66	C₃×#107	—	2	1	no	—
67	C₃×#107	—	2	1	no	—
68	C₃×#107	C₂³	1	2	yes	—
69	C₃×#107	C₂³	1	2	yes	—
70	C₃×#107	C₂³	1	2	yes	—
71	C₃×#106	—	2	4	no	—
72	C₃×#102	—	2	4	no	—
73	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
74	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×#121	—	2	1	no	—
76	C₂×#117	—	2	1	no	—
77	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
79	#466	—	2	2	no	—
80	#466	C₂²×C₃	1	4	yes	—
81	#427	—	2	2	no	—
82	#427	C₂²×C₃	1	4	yes	complemented
83	C₂×#107	C₂²×C₃	1	1	yes	characteristic
84	C₂×#107	C₂²×C₃	1	1	yes	characteristic
85	C₂×#107	C₂²×C₃	1	1	yes	characteristic
86	C₂×#106	—	2	1	no	—
87	C₂×#102	—	2	1	no	—
88	#367	C₂²×C₃	1	4	yes	complemented
89	#366	C₂²×C₃	1	4	yes	complemented
90	#357	C₂²×C₃	1	4	yes	complemented
91	#357	C₂²×C₃	1	4	yes	complemented
92	C₂²⋊C₁₆	C₂²×C₃	1	4	yes	complemented
93	C₂⁴×C₃	—	4	1	no	—
94	C₂⁴×C₃	D₈	1	1	yes	characteristic
95	C₂×C₃×D₄	—	2	2	no	—
96	C₂×C₃×D₄	—	4	4	no	—
97	C₂×C₃×D₄	—	4	2	no	—
98	C₂×C₃×D₄	—	4	1	no	—
99	C₂×C₃×D₄	—	4	2	no	—
100	C₂×C₃×D₄	—	8	4	no	—
101	C₂²×C₃×C₄	—	2	2	no	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	—	4	1	no	—
104	C₂²×C₃×C₄	—	4	1	no	—
105	C₂²×C₃×C₄	—	4	1	no	—
106	C₂²×C₃×C₄	D₈	1	2	yes	—
107	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
109	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₃×D₈	—	2	2	no	—
111	C₃×D₈	—	2	2	no	—
112	C₃×D₈	—	4	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₈	#41	1	2	yes	—
120	C₂×C₃×C₈	#41	1	2	yes	—
121	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	4	1	no	—
126	C₃×C₄⋊C₄	—	4	1	no	—
127	C₃×C₄⋊C₄	—	4	1	no	—
128	C₃×C₂²⋊C₄	—	4	4	no	—
129	C₃×C₂²⋊C₄	—	4	4	no	—
130	C₃×C₄²	—	4	4	no	—
131	C₃×C₁₆	—	2	2	no	—
132	C₃×C₁₆	—	2	2	no	—
133	C₂²×D₄	—	2	1	no	—
134	C₂²×D₄	—	4	1	no	—
135	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
136	C₂×D₈	—	2	1	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	C₂³×C₃	1	2	yes	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
143	C₂²×C₈	C₃×D₄	1	2	yes	—
144	C₂²×C₈	C₂³×C₃	1	2	yes	—
145	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
146	#121	—	2	4	no	—
147	#121	—	4	2	no	—
148	#117	—	2	4	no	—
149	#117	—	4	2	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₂²⋊C₄	—	4	1	no	—
154	C₂×C₂²⋊C₄	—	4	1	no	—
155	C₂×C₄²	—	4	1	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	C₂³×C₃	1	2	yes	—
159	C₂×C₁₆	C₂³×C₃	1	2	yes	—
160	#107	—	2	1	no	—
161	#107	—	2	1	no	—
162	#107	—	2	1	no	—
163	#107	C₂³×C₃	1	2	yes	—
164	#107	C₂³×C₃	1	2	yes	—
165	#107	C₂³×C₃	1	2	yes	—
166	#106	—	2	4	no	—
167	#102	—	2	4	no	—
168	C₂³×C₃	—	2	2	no	—
169	C₂³×C₃	—	2	2	no	—
170	C₂³×C₃	—	2	2	no	—
171	C₂³×C₃	—	4	2	no	—
172	C₂³×C₃	—	8	2	no	—
173	C₂³×C₃	—	8	2	no	—
174	C₂³×C₃	—	8	2	no	—
175	C₂³×C₃	C₂×D₈	1	2	yes	—
176	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
177	C₃×D₄	—	4	2	no	—
178	C₃×D₄	—	4	2	no	—
179	C₃×D₄	—	8	8	no	—
180	C₃×D₄	—	8	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	8	2	no	—
189	C₂×C₃×C₄	—	8	2	no	—
190	C₂×C₃×C₄	—	8	2	no	—
191	C₂×C₃×C₄	#123	1	2	yes	—
192	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
193	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	#137	1	2	yes	—
195	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	#123	1	2	yes	—
198	C₃×C₈	C₂×#41	1	1	yes	characteristic
199	C₃×C₈	C₂×#41	1	1	yes	characteristic
200	C₂⁴	—	4	1	no	—
201	C₂⁴	C₃×D₈	1	1	yes	characteristic
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	8	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
214	C₂²×C₄	C₃×D₈	1	2	yes	—
215	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
216	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	D₈	—	2	2	no	—
218	D₈	—	2	2	no	—
219	D₈	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
225	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
226	C₂×C₈	C₃×#41	1	2	yes	—
227	C₂×C₈	C₃×#41	1	2	yes	—
228	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	4	4	no	—
238	C₁₆	—	2	2	no	—
239	C₁₆	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	4	4	no	—
244	C₂²×C₃	—	8	2	no	—
245	C₂²×C₃	—	8	2	no	—
246	C₂²×C₃	—	8	2	no	—
247	C₂²×C₃	—	16	4	no	—
248	C₂²×C₃	#480	1	2	yes	—
249	C₂²×C₃	#508	1	2	yes	—
250	C₂²×C₃	#509	1	2	yes	—
251	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	8	2	no	—
254	C₃×C₄	—	8	2	no	—
255	C₃×C₄	—	8	2	no	—
256	C₃×C₄	#480	1	2	yes	—
257	C₃×C₄	C₂×#123	1	1	yes	characteristic
258	C₃×C₄	C₂×#137	1	1	yes	characteristic
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	8	2	no	—
264	C₂³	—	8	2	no	—
265	C₂³	—	8	2	no	—
266	C₂³	C₂×C₃×D₈	1	2	yes	—
267	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	C₃×#123	1	2	yes	—
283	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
284	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
285	C₂×C₄	C₃×#137	1	2	yes	—
286	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₃×#123	1	2	yes	—
289	C₈	C₂×C₃×#41	1	1	yes	characteristic
290	C₈	C₂×C₃×#41	1	1	yes	characteristic
291	C₂×C₃	—	4	2	no	—
292	C₂×C₃	—	16	2	no	—
293	C₂×C₃	#2234	1	4	yes	direct product factor
294	C₂×C₃	C₂×#480	1	1	yes	characteristic
295	C₂×C₃	C₂×#508	1	1	yes	characteristic
296	C₂×C₃	C₂×#509	1	1	yes	characteristic
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	16	4	no	—
305	C₂²	C₃×#480	1	2	yes	—
306	C₂²	C₃×#508	1	2	yes	—
307	C₂²	C₃×#509	1	2	yes	—
308	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₃×#480	1	2	yes	—
314	C₄	C₂×C₃×#123	1	1	yes	characteristic
315	C₄	C₂×C₃×#137	1	1	yes	characteristic
316	C₃	C₂×#2234	1	1	yes	characteristic, direct product factor, Sylow
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	C₃×#2234	1	4	yes	direct product factor
320	C₂	C₂×C₃×#480	1	1	yes	characteristic
321	C₂	C₂×C₃×#508	1	1	yes	characteristic
322	C₂	C₂×C₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#427	C₂	semidirect
3. C₃×#2234	C₂	direct (N × H)
4. C₂×C₃×#367	C₂	semidirect
5. C₂×C₃×#366	C₂	semidirect
6. C₂×C₃×#357	C₂	semidirect
7. C₂×C₃×#357	C₂	semidirect
8. C₂×C₃×C₂²⋊C₁₆	C₂	semidirect
9. C₂×#2234	C₃	direct (N × H)
19. C₃×#427	C₂²	semidirect
25. C₃×#367	C₂²	semidirect
26. C₃×#366	C₂²	semidirect
27. C₃×#357	C₂²	semidirect
28. C₃×#357	C₂²	semidirect
29. C₃×C₂²⋊C₁₆	C₂²	semidirect
31. C₂×#427	C₂×C₃	semidirect
32. #2234	C₂×C₃	direct (N × H)
33. C₂×#367	C₂×C₃	semidirect
34. C₂×#366	C₂×C₃	semidirect
35. C₂×#357	C₂×C₃	semidirect
36. C₂×#357	C₂×C₃	semidirect
37. C₂×C₂²⋊C₁₆	C₂×C₃	semidirect
82. #427	C₂²×C₃	semidirect
88. #367	C₂²×C₃	semidirect
89. #366	C₂²×C₃	semidirect
90. #357	C₂²×C₃	semidirect
91. #357	C₂²×C₃	semidirect
92. C₂²⋊C₁₆	C₂²×C₃	semidirect
293. C₂×C₃	#2234	direct (N × H)
316. C₃	C₂×#2234	direct (N × H)
319. C₂	C₃×#2234	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#466	2
C₂×C₃×#427	2
C₃×#2234	2
C₂×C₃×#367	2
C₂×C₃×#366	2
C₂×C₃×#357	2
C₂×C₃×C₂²⋊C₁₆	2
C₂×#2234	3

Subgroups — C2×C3×#2234 #10652248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2234 #10652248