Subgroups — C₃×#20514 #10639121

Order 768.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#443	C₂	1	2	yes	maximal
2	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×#2875	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1772	C₂	1	2	yes	maximal, complemented
6	C₃×#1765	C₂	1	4	yes	maximal, complemented
7	C₃×#1759	C₂	1	2	yes	maximal, complemented
8	C₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	#20514	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#135	C₂²	1	1	yes	characteristic
11	C₂×C₃×#131	C₂²	1	4	yes	—
12	C₃×C₄×#41	C₂²	1	2	yes	—
13	C₃×#443	C₄	1	16	yes	complemented
14	C₃×C₄×Q₁₆	C₄	1	2	yes	complemented
15	C₃×C₄×SD₁₆	C₄	1	4	yes	complemented
16	C₃×C₄×D₈	C₄	1	2	yes	complemented
17	C₂×C₃×#108	C₂²	1	2	yes	—
18	C₃×#427	C₂²	1	1	yes	characteristic
19	C₂×C₃×#104	C₂²	1	4	yes	—
20	C₃×#416	C₂²	1	2	yes	—
21	C₃×#405	—	2	4	no	—
22	C₃×#405	C₂²	1	4	yes	—
23	C₃×C₄×M₁₆	—	2	2	no	—
24	C₃×C₄×M₁₆	C₂²	1	2	yes	—
25	C₂×C₃×#97	—	2	2	no	—
26	C₂×C₃×#97	C₂²	1	2	yes	—
27	C₂×C₃×#97	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
30	C₃×#376	C₂²	1	1	yes	characteristic
31	C₃×#341	C₂²	1	2	yes	—
32	C₃×#339	C₂²	1	4	yes	—
33	C₂×#443	C₂×C₃	1	2	yes	—
34	C₄×#135	C₂×C₃	1	1	yes	characteristic
35	#2875	C₂×C₃	1	2	yes	complemented
36	C₄×#108	C₂×C₃	1	1	yes	characteristic, complemented
37	#1772	C₂×C₃	1	2	yes	complemented
38	#1765	C₂×C₃	1	4	yes	complemented
39	#1759	C₂×C₃	1	2	yes	complemented
40	#1726	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃×#41	C₂³	1	2	yes	—
42	C₃×#135	C₂×C₄	1	8	yes	—
43	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic
46	C₃×#131	—	2	8	no	—
47	C₃×#131	C₂×C₄	1	16	yes	complemented
48	C₂×C₃×M₁₆	—	2	4	no	—
49	C₂×C₃×M₁₆	C₂³	1	4	yes	—
50	C₂²×C₃×C₈	—	2	4	no	—
51	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂³	1	2	yes	—
54	C₃×C₄×Q₈	C₂×C₄	1	4	yes	—
55	C₃×C₄×D₄	—	2	4	no	—
56	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
57	C₃×#117	—	2	2	no	—
58	C₃×#117	C₂³	1	2	yes	—
59	C₂×C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₄²	C₂³	1	2	yes	—
61	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₃×#108	C₂×C₄	1	8	yes	complemented
63	C₃×#107	C₂×C₄	1	2	yes	—
64	C₃×#106	C₂×C₄	1	2	yes	—
65	C₃×#104	C₂×C₄	1	16	yes	complemented
66	C₃×#103	C₂×C₄	1	4	yes	—
67	C₃×#102	C₂×C₄	1	4	yes	—
68	C₃×#97	—	2	4	no	—
69	C₃×#97	—	2	4	no	—
70	C₃×#97	—	2	2	no	—
71	C₃×#97	D₄	1	1	yes	characteristic
72	C₃×#97	D₄	1	1	yes	characteristic
73	C₃×#97	D₄	1	2	yes	—
74	C₃×C₄×C₈	—	2	4	no	—
75	C₃×C₄×C₈	—	2	2	no	—
76	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
77	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
78	C₃×C₄×C₈	C₂×C₄	1	8	yes	complemented
79	C₃×#95	—	2	2	no	—
80	C₂×#135	C₂²×C₃	1	1	yes	characteristic
81	C₂×#131	C₂²×C₃	1	4	yes	—
82	C₄×#41	C₂²×C₃	1	2	yes	—
83	#443	C₃×C₄	1	16	yes	complemented
84	C₄×Q₁₆	C₃×C₄	1	2	yes	complemented
85	C₄×SD₁₆	C₃×C₄	1	4	yes	complemented
86	C₄×D₈	C₃×C₄	1	2	yes	complemented
87	C₂×#108	C₂²×C₃	1	2	yes	—
88	#427	C₂²×C₃	1	1	yes	characteristic
89	C₂×#104	C₂²×C₃	1	4	yes	—
90	#416	C₂²×C₃	1	2	yes	—
91	#405	—	2	4	no	—
92	#405	C₂²×C₃	1	4	yes	—
93	C₄×M₁₆	—	2	2	no	—
94	C₄×M₁₆	C₂²×C₃	1	2	yes	—
95	C₂×#97	—	2	2	no	—
96	C₂×#97	C₂²×C₃	1	2	yes	—
97	C₂×#97	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
100	#376	C₂²×C₃	1	1	yes	characteristic
101	#341	C₂²×C₃	1	2	yes	—
102	#339	C₂²×C₃	1	4	yes	—
103	C₃×#41	—	2	4	no	—
104	C₃×#41	C₂²×C₄	1	8	yes	—
105	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
106	C₂×C₃×D₄	—	2	2	no	—
107	C₂×C₃×D₄	C₂²×C₄	1	2	yes	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	#41	1	2	yes	—
112	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
113	C₃×Q₁₆	C₄²	1	4	yes	—
114	C₃×SD₁₆	C₄²	1	8	yes	—
115	C₃×D₈	C₄²	1	4	yes	—
116	C₃×M₁₆	—	2	4	no	—
117	C₃×M₁₆	—	2	16	no	—
118	C₃×M₁₆	C₂²×C₄	1	8	yes	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	16	no	—
122	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
126	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
127	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
128	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
129	C₂×C₃×C₈	#41	1	4	yes	—
130	C₃×C₄⋊C₄	—	2	4	no	—
131	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
132	C₃×C₂²⋊C₄	—	2	2	no	—
133	C₃×C₂²⋊C₄	—	2	4	no	—
134	C₃×C₄²	—	2	4	no	—
135	C₃×C₄²	—	2	2	no	—
136	C₃×C₄²	—	2	2	no	—
137	C₃×C₄²	C₂²×C₄	1	4	yes	—
138	C₃×C₄²	C₂²×C₄	1	2	yes	—
139	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
140	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
141	C₂×#41	C₂³×C₃	1	2	yes	—
142	#135	C₂×C₃×C₄	1	8	yes	—
143	C₂×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic
144	C₂×SD₁₆	C₂×C₃×C₄	1	2	yes	—
145	C₂×D₈	C₂×C₃×C₄	1	1	yes	characteristic
146	#131	—	2	8	no	—
147	#131	C₂×C₃×C₄	1	16	yes	complemented
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	C₂³×C₃	1	4	yes	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₃	1	2	yes	—
154	C₄×Q₈	C₂×C₃×C₄	1	4	yes	—
155	C₄×D₄	—	2	4	no	—
156	C₄×D₄	C₂×C₃×C₄	1	4	yes	—
157	#117	—	2	2	no	—
158	#117	C₂³×C₃	1	2	yes	—
159	C₂×C₄²	—	2	2	no	—
160	C₂×C₄²	C₂³×C₃	1	2	yes	—
161	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
162	#108	C₂×C₃×C₄	1	8	yes	complemented
163	#107	C₂×C₃×C₄	1	2	yes	—
164	#106	C₂×C₃×C₄	1	2	yes	—
165	#104	C₂×C₃×C₄	1	16	yes	complemented
166	#103	C₂×C₃×C₄	1	4	yes	—
167	#102	C₂×C₃×C₄	1	4	yes	—
168	#97	—	2	4	no	—
169	#97	—	2	4	no	—
170	#97	—	2	2	no	—
171	#97	C₃×D₄	1	1	yes	characteristic
172	#97	C₃×D₄	1	1	yes	characteristic
173	#97	C₃×D₄	1	2	yes	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
177	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
178	C₄×C₈	C₂×C₃×C₄	1	8	yes	complemented
179	#95	—	2	2	no	—
180	C₂³×C₃	—	2	2	no	—
181	C₂³×C₃	C₂×#41	1	1	yes	characteristic
182	C₃×Q₈	—	2	2	no	—
183	C₃×Q₈	C₂×C₄²	1	4	yes	—
184	C₃×D₄	—	2	8	no	—
185	C₃×D₄	—	2	2	no	—
186	C₃×D₄	C₂×C₄²	1	4	yes	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	—	8	4	no	—
194	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
195	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
196	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
197	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×#41	1	2	yes	—
200	C₃×C₈	—	2	8	no	—
201	C₃×C₈	—	2	2	no	—
202	C₃×C₈	C₂×C₄²	1	2	yes	—
203	C₃×C₈	C₂×C₄²	1	2	yes	—
204	C₃×C₈	C₄×D₄	1	4	yes	—
205	C₃×C₈	C₄×D₄	1	4	yes	—
206	#41	—	2	4	no	—
207	#41	C₂²×C₃×C₄	1	8	yes	—
208	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×C₃×C₄	1	2	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₃×#41	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	Q₁₆	C₃×C₄²	1	4	yes	—
217	SD₁₆	C₃×C₄²	1	8	yes	—
218	D₈	C₃×C₄²	1	4	yes	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	16	no	—
221	M₁₆	C₂²×C₃×C₄	1	8	yes	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	16	no	—
225	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
226	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
229	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
230	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
231	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
232	C₂×C₈	C₃×#41	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	C₂²×C₃×C₄	1	4	yes	—
241	C₄²	C₂²×C₃×C₄	1	2	yes	—
242	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
243	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	4	no	—
246	C₂²×C₃	—	4	4	no	—
247	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
248	C₂²×C₃	C₄×#41	1	2	yes	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	4	4	no	—
253	C₃×C₄	—	4	4	no	—
254	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
255	C₃×C₄	C₂×C₄×D₄	1	2	yes	—
256	C₃×C₄	C₄×#41	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₃×#41	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	Q₈	C₂×C₃×C₄²	1	4	yes	—
261	D₄	—	2	8	no	—
262	D₄	—	2	2	no	—
263	D₄	C₂×C₃×C₄²	1	4	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
272	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
273	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
274	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₃×#41	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	2	2	no	—
279	C₈	C₂×C₃×C₄²	1	2	yes	—
280	C₈	C₂×C₃×C₄²	1	2	yes	—
281	C₈	C₃×C₄×D₄	1	4	yes	—
282	C₈	C₃×C₄×D₄	1	4	yes	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	4	4	no	—
285	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
286	C₂×C₃	#2957	1	2	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
291	C₂²	C₃×C₄×#41	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	4	no	—
296	C₄	—	4	4	no	—
297	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
298	C₄	C₂×C₃×C₄×D₄	1	2	yes	—
299	C₄	C₃×C₄×#41	1	1	yes	characteristic
300	C₃	#20514	1	1	yes	characteristic, direct product factor, Sylow
301	C₂	—	2	2	no	—
302	C₂	—	4	4	no	—
303	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
304	C₂	C₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#2875	C₂	semidirect
4. C₃×C₄×#108	C₂	semidirect
5. C₃×#1772	C₂	semidirect
6. C₃×#1765	C₂	semidirect
7. C₃×#1759	C₂	semidirect
8. C₃×#1726	C₂	semidirect
9. #20514	C₃	direct (N × H)
13. C₃×#443	C₄	semidirect
14. C₃×C₄×Q₁₆	C₄	semidirect
15. C₃×C₄×SD₁₆	C₄	semidirect
16. C₃×C₄×D₈	C₄	semidirect
35. #2875	C₂×C₃	semidirect
36. C₄×#108	C₂×C₃	semidirect
37. #1772	C₂×C₃	semidirect
38. #1765	C₂×C₃	semidirect
39. #1759	C₂×C₃	semidirect
40. #1726	C₂×C₃	semidirect
47. C₃×#131	C₂×C₄	semidirect
62. C₃×#108	C₂×C₄	semidirect
65. C₃×#104	C₂×C₄	semidirect
78. C₃×C₄×C₈	C₂×C₄	semidirect
83. #443	C₃×C₄	semidirect
84. C₄×Q₁₆	C₃×C₄	semidirect
85. C₄×SD₁₆	C₃×C₄	semidirect
86. C₄×D₈	C₃×C₄	semidirect
147. #131	C₂×C₃×C₄	semidirect
162. #108	C₂×C₃×C₄	semidirect
165. #104	C₂×C₃×C₄	semidirect
178. C₄×C₈	C₂×C₃×C₄	semidirect
300. C₃	#20514	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×#443	2
C₃×C₄×#135	2
C₃×#2875	2
C₃×C₄×#108	2
C₃×#1772	2
C₃×#1765	2
C₃×#1759	2
C₃×#1726	2
#20514	3

Subgroups — C3×#20514 #10639121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20514 #10639121