Subgroups — C₂×C₃×#2237 #10652249

Order 768.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₃×#2237	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×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂×#2237	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₂²⋊Q₈	—	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×#467	—	2	2	no	—
17	C₃×#467	C₂²	1	4	yes	—
18	C₃×#427	—	2	2	no	—
19	C₃×#427	C₂²	1	4	yes	—
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×#103	—	2	1	no	—
25	C₃×#367	C₂²	1	4	yes	complemented
26	C₃×#366	C₂²	1	4	yes	complemented
27	C₃×#358	C₂²	1	4	yes	complemented
28	C₃×#358	C₂²	1	4	yes	complemented
29	C₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
30	C₂×#467	C₂×C₃	1	1	yes	characteristic
31	C₂×#427	C₂×C₃	1	1	yes	characteristic
32	#2237	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×#358	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#358	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic
38	C₂²×C₃×Q₈	—	2	1	no	—
39	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃×Q₁₆	—	2	1	no	—
41	C₂×C₃×Q₁₆	—	2	2	no	—
42	C₂×C₃×Q₁₆	—	2	2	no	—
43	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
44	C₂²×C₃×C₈	—	2	2	no	—
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₃×C₈	D₄	1	2	yes	—
48	C₂²×C₃×C₈	C₂³	1	2	yes	—
49	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₃×C₂²⋊Q₈	—	2	4	no	—
51	C₃×C₂²⋊Q₈	—	4	2	no	—
52	C₃×#117	—	2	4	no	—
53	C₃×#117	—	4	2	no	—
54	C₂×C₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₃×C₄⋊C₄	—	4	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×#103	—	2	4	no	—
73	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₂²⋊Q₈	—	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	#467	—	2	2	no	—
80	#467	C₂²×C₃	1	4	yes	—
81	#427	—	2	2	no	—
82	#427	C₂²×C₃	1	4	yes	—
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×#103	—	2	1	no	—
88	#367	C₂²×C₃	1	4	yes	complemented
89	#366	C₂²×C₃	1	4	yes	complemented
90	#358	C₂²×C₃	1	4	yes	complemented
91	#358	C₂²×C₃	1	4	yes	complemented
92	C₂²⋊C₁₆	C₂²×C₃	1	4	yes	—
93	C₂⁴×C₃	D₈	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	—	4	1	no	—
96	C₂×C₃×Q₈	—	4	2	no	—
97	C₂×C₃×Q₈	—	4	2	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	—	4	1	no	—
101	C₂²×C₃×C₄	—	4	1	no	—
102	C₂²×C₃×C₄	—	4	1	no	—
103	C₂²×C₃×C₄	—	4	1	no	—
104	C₂²×C₃×C₄	D₈	1	2	yes	—
105	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
107	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
108	C₃×Q₁₆	—	2	2	no	—
109	C₃×Q₁₆	—	2	2	no	—
110	C₃×Q₁₆	—	4	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₈	#41	1	2	yes	—
118	C₂×C₃×C₈	#41	1	2	yes	—
119	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	4	1	no	—
124	C₃×C₄⋊C₄	—	4	1	no	—
125	C₃×C₄⋊C₄	—	4	1	no	—
126	C₃×C₄⋊C₄	—	4	4	no	—
127	C₃×C₂²⋊C₄	—	4	4	no	—
128	C₃×C₂²⋊C₄	—	4	4	no	—
129	C₃×C₄²	—	4	4	no	—
130	C₃×C₁₆	—	2	2	no	—
131	C₃×C₁₆	—	2	2	no	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
141	C₂²×C₈	C₃×D₄	1	2	yes	—
142	C₂²×C₈	C₂³×C₃	1	2	yes	—
143	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
144	C₂²⋊Q₈	—	2	4	no	—
145	C₂²⋊Q₈	—	4	2	no	—
146	#117	—	2	4	no	—
147	#117	—	4	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	4	1	no	—
152	C₂×C₂²⋊C₄	—	4	1	no	—
153	C₂×C₂²⋊C₄	—	4	1	no	—
154	C₂×C₄²	—	4	1	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	C₂³×C₃	1	2	yes	—
158	C₂×C₁₆	C₂³×C₃	1	2	yes	—
159	#107	—	2	1	no	—
160	#107	—	2	1	no	—
161	#107	—	2	1	no	—
162	#107	C₂³×C₃	1	2	yes	—
163	#107	C₂³×C₃	1	2	yes	—
164	#107	C₂³×C₃	1	2	yes	—
165	#106	—	2	4	no	—
166	#103	—	2	4	no	—
167	C₂³×C₃	—	2	2	no	—
168	C₂³×C₃	—	2	2	no	—
169	C₂³×C₃	—	2	2	no	—
170	C₂³×C₃	C₂×D₈	1	2	yes	—
171	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
172	C₃×Q₈	—	4	2	no	—
173	C₃×Q₈	—	4	2	no	—
174	C₃×Q₈	—	8	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	8	2	no	—
184	C₂×C₃×C₄	—	8	2	no	—
185	C₂×C₃×C₄	—	8	2	no	—
186	C₂×C₃×C₄	—	8	2	no	—
187	C₂×C₃×C₄	#123	1	2	yes	—
188	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
189	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
190	C₂×C₃×C₄	#137	1	2	yes	—
191	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	#123	1	2	yes	—
194	C₃×C₈	C₂×#41	1	1	yes	characteristic
195	C₃×C₈	C₂×#41	1	1	yes	characteristic
196	C₂⁴	C₃×D₈	1	1	yes	characteristic
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
208	C₂²×C₄	C₃×D₈	1	2	yes	—
209	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
210	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	Q₁₆	—	2	2	no	—
212	Q₁₆	—	2	2	no	—
213	Q₁₆	—	4	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
219	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂×C₈	C₃×#41	1	2	yes	—
221	C₂×C₈	C₃×#41	1	2	yes	—
222	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	4	4	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	4	4	no	—
239	C₂²×C₃	#480	1	2	yes	—
240	C₂²×C₃	#508	1	2	yes	—
241	C₂²×C₃	#510	1	2	yes	—
242	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	8	2	no	—
245	C₃×C₄	—	8	2	no	—
246	C₃×C₄	—	8	2	no	—
247	C₃×C₄	—	8	2	no	—
248	C₃×C₄	#480	1	2	yes	—
249	C₃×C₄	C₂×#123	1	1	yes	characteristic
250	C₃×C₄	C₂×#137	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₃×D₈	1	2	yes	—
255	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
256	Q₈	—	4	2	no	—
257	Q₈	—	4	2	no	—
258	Q₈	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	C₃×#123	1	2	yes	—
272	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
273	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
274	C₂×C₄	C₃×#137	1	2	yes	—
275	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	C₃×#123	1	2	yes	—
278	C₈	C₂×C₃×#41	1	1	yes	characteristic
279	C₈	C₂×C₃×#41	1	1	yes	characteristic
280	C₂×C₃	—	4	2	no	—
281	C₂×C₃	#2237	1	4	yes	direct product factor
282	C₂×C₃	C₂×#480	1	1	yes	characteristic
283	C₂×C₃	C₂×#508	1	1	yes	characteristic
284	C₂×C₃	C₂×#510	1	1	yes	characteristic
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₃×#480	1	2	yes	—
290	C₂²	C₃×#508	1	2	yes	—
291	C₂²	C₃×#510	1	2	yes	—
292	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₃×#480	1	2	yes	—
299	C₄	C₂×C₃×#123	1	1	yes	characteristic
300	C₄	C₂×C₃×#137	1	1	yes	characteristic
301	C₃	C₂×#2237	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	4	2	no	—
303	C₂	C₃×#2237	1	4	yes	direct product factor
304	C₂	C₂×C₃×#480	1	1	yes	characteristic
305	C₂	C₂×C₃×#508	1	1	yes	characteristic
306	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	C₂⁴×C₃	D₈
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
3. C₃×#2237	C₂	direct (N × H)
4. C₂×C₃×#367	C₂	semidirect
5. C₂×C₃×#366	C₂	semidirect
6. C₂×C₃×#358	C₂	semidirect
7. C₂×C₃×#358	C₂	semidirect
9. C₂×#2237	C₃	direct (N × H)
25. C₃×#367	C₂²	semidirect
26. C₃×#366	C₂²	semidirect
27. C₃×#358	C₂²	semidirect
28. C₃×#358	C₂²	semidirect
32. #2237	C₂×C₃	direct (N × H)
33. C₂×#367	C₂×C₃	semidirect
34. C₂×#366	C₂×C₃	semidirect
35. C₂×#358	C₂×C₃	semidirect
36. C₂×#358	C₂×C₃	semidirect
88. #367	C₂²×C₃	semidirect
89. #366	C₂²×C₃	semidirect
90. #358	C₂²×C₃	semidirect
91. #358	C₂²×C₃	semidirect
281. C₂×C₃	#2237	direct (N × H)
301. C₃	C₂×#2237	direct (N × H)
303. C₂	C₃×#2237	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×#467	2
C₂×C₃×#427	2
C₃×#2237	2
C₂×C₃×#367	2
C₂×C₃×#366	2
C₂×C₃×#358	2
C₂×C₃×C₂²⋊C₁₆	2
C₂×#2237	3

Subgroups — C2×C3×#2237 #10652249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2237 #10652249