Subgroups — C₃×#3510 #87394

Order 384.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#559	C₂	1	4	yes	maximal, complemented
2	C₃×#556	C₂	1	4	yes	maximal, complemented
3	C₃×#551	C₂	1	8	yes	maximal, complemented
4	C₃×#542	C₂	1	4	yes	maximal, complemented
5	C₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#535	C₂	1	1	yes	maximal, characteristic
7	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#529	C₂	1	4	yes	maximal, complemented
9	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
13	#3510	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂×C₃×#41	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₃×#127	C₂²	1	2	yes	—
20	C₃×#126	C₂²	1	8	yes	complemented
21	C₃×#126	C₂²	1	8	yes	complemented
22	C₃×#125	C₂²	1	2	yes	complemented
23	C₃×#124	C₂²	1	4	yes	—
24	C₃×#124	C₂²	1	2	yes	—
25	C₃×#124	C₂²	1	2	yes	—
26	C₃×#124	C₂²	1	2	yes	—
27	C₃×#124	C₂²	1	2	yes	complemented
28	C₃×#124	C₂²	1	8	yes	complemented
29	C₃×#123	C₂²	1	4	yes	—
30	C₃×#123	C₂²	1	8	yes	complemented
31	C₃×#123	C₂²	1	8	yes	complemented
32	C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
33	C₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
34	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
35	C₃×#121	C₂²	1	4	yes	—
36	C₃×#120	C₂²	1	4	yes	—
37	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
38	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
39	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
40	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
41	C₃×C₄×D₄	C₂²	1	4	yes	—
42	C₃×C₄×D₄	C₂²	1	4	yes	—
43	C₃×C₄×D₄	C₂²	1	4	yes	—
44	C₃×C₄×D₄	C₂²	1	2	yes	complemented
45	C₃×C₄×D₄	C₂²	1	8	yes	complemented
46	C₃×#117	C₂²	1	1	yes	characteristic
47	C₃×#117	C₂²	1	1	yes	characteristic
48	C₃×#117	C₂²	1	1	yes	characteristic
49	C₃×#117	C₂²	1	4	yes	—
50	C₃×#117	C₂²	1	2	yes	complemented
51	C₃×#117	C₂²	1	8	yes	complemented
52	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
53	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
54	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
56	#559	C₂×C₃	1	4	yes	complemented
57	#556	C₂×C₃	1	4	yes	complemented
58	#551	C₂×C₃	1	8	yes	complemented
59	#542	C₂×C₃	1	4	yes	complemented
60	#536	C₂×C₃	1	1	yes	characteristic, complemented
61	#535	C₂×C₃	1	1	yes	characteristic
62	#533	C₂×C₃	1	1	yes	characteristic, complemented
63	#529	C₂×C₃	1	4	yes	complemented
64	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
65	#519	C₂×C₃	1	1	yes	characteristic, complemented
66	#518	C₂×C₃	1	1	yes	characteristic, complemented
67	C₄×#41	C₂×C₃	1	1	yes	characteristic, complemented
68	C₂⁴×C₃	C₂³	1	2	yes	—
69	C₃×#41	—	2	4	no	—
70	C₂×C₃×Q₈	—	2	4	no	—
71	C₂×C₃×Q₈	C₂³	1	2	yes	—
72	C₂×C₃×Q₈	C₂³	1	2	yes	—
73	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×Q₈	C₂³	1	4	yes	—
75	C₂×C₃×D₄	—	2	4	no	—
76	C₂×C₃×D₄	C₂³	1	2	yes	—
77	C₂×C₃×D₄	C₂³	1	2	yes	—
78	C₂×C₃×D₄	C₂³	1	2	yes	—
79	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
80	C₂×C₃×D₄	C₂³	1	8	yes	—
81	C₂²×C₃×C₄	C₂³	1	2	yes	—
82	C₂²×C₃×C₄	C₂³	1	2	yes	—
83	C₂²×C₃×C₄	C₂³	1	2	yes	—
84	C₂²×C₃×C₄	C₂³	1	4	yes	—
85	C₂²×C₃×C₄	C₂³	1	8	yes	—
86	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	2	yes	—
88	C₃×C₄⋊C₄	C₂³	1	2	yes	—
89	C₃×C₄⋊C₄	C₂³	1	2	yes	—
90	C₃×C₄⋊C₄	C₂³	1	8	yes	—
91	C₃×C₄⋊C₄	C₂³	1	4	yes	—
92	C₃×C₄⋊C₄	C₂³	1	4	yes	—
93	C₃×C₄⋊C₄	C₂³	1	8	yes	—
94	C₃×C₄⋊C₄	C₂³	1	4	yes	—
95	C₃×C₄⋊C₄	C₂³	1	4	yes	—
96	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
97	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
98	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
99	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
100	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
101	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
102	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
103	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
104	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
105	C₃×C₄²	C₂³	1	1	yes	characteristic
106	C₃×C₄²	C₂³	1	1	yes	characteristic
107	C₃×C₄²	C₂³	1	2	yes	—
108	C₃×C₄²	C₂³	1	2	yes	—
109	C₃×C₄²	C₂³	1	4	yes	—
110	C₃×C₄²	C₂³	1	8	yes	—
111	C₃×C₄²	C₂³	1	2	yes	—
112	C₃×C₄²	C₂³	1	2	yes	—
113	C₂×#41	C₂²×C₃	1	1	yes	characteristic
114	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
115	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
116	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
117	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
118	#127	C₂²×C₃	1	2	yes	—
119	#126	C₂²×C₃	1	8	yes	complemented
120	#126	C₂²×C₃	1	8	yes	complemented
121	#125	C₂²×C₃	1	2	yes	complemented
122	#124	C₂²×C₃	1	2	yes	—
123	#124	C₂²×C₃	1	2	yes	—
124	#124	C₂²×C₃	1	4	yes	—
125	#124	C₂²×C₃	1	2	yes	—
126	#124	C₂²×C₃	1	2	yes	complemented
127	#124	C₂²×C₃	1	8	yes	complemented
128	#123	C₂²×C₃	1	4	yes	—
129	#123	C₂²×C₃	1	8	yes	complemented
130	#123	C₂²×C₃	1	8	yes	complemented
131	C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
132	C₂²⋊Q₈	C₂²×C₃	1	4	yes	complemented
133	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
134	#121	C₂²×C₃	1	4	yes	—
135	#120	C₂²×C₃	1	4	yes	—
136	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
137	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
138	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
139	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
140	C₄×D₄	C₂²×C₃	1	4	yes	—
141	C₄×D₄	C₂²×C₃	1	4	yes	—
142	C₄×D₄	C₂²×C₃	1	4	yes	—
143	C₄×D₄	C₂²×C₃	1	2	yes	complemented
144	C₄×D₄	C₂²×C₃	1	8	yes	complemented
145	#117	C₂²×C₃	1	1	yes	characteristic
146	#117	C₂²×C₃	1	1	yes	characteristic
147	#117	C₂²×C₃	1	1	yes	characteristic
148	#117	C₂²×C₃	1	4	yes	—
149	#117	C₂²×C₃	1	2	yes	complemented
150	#117	C₂²×C₃	1	8	yes	complemented
151	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
153	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
155	C₂³×C₃	—	2	4	no	—
156	C₂³×C₃	—	4	4	no	—
157	C₂³×C₃	C₂⁴	1	4	yes	—
158	C₂³×C₃	C₂⁴	1	2	yes	—
159	C₂³×C₃	C₂⁴	1	1	yes	characteristic
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	—	2	4	no	—
162	C₃×Q₈	—	2	2	no	—
163	C₃×Q₈	—	4	4	no	—
164	C₃×D₄	—	2	4	no	—
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	4	8	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	4	4	no	—
172	C₂×C₃×C₄	—	4	8	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	#41	1	4	yes	—
175	C₂×C₃×C₄	C₂⁴	1	2	yes	—
176	C₂×C₃×C₄	C₂⁴	1	4	yes	—
177	C₂×C₃×C₄	C₂⁴	1	4	yes	—
178	C₂×C₃×C₄	C₂⁴	1	4	yes	—
179	C₂×C₃×C₄	C₂⁴	1	2	yes	—
180	C₂×C₃×C₄	C₂⁴	1	2	yes	—
181	C₂×C₃×C₄	C₂⁴	1	4	yes	—
182	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
184	C₂⁴	C₂³×C₃	1	2	yes	—
185	#41	—	2	4	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₂³×C₃	1	2	yes	—
188	C₂×Q₈	C₂³×C₃	1	2	yes	—
189	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
190	C₂×Q₈	C₂³×C₃	1	4	yes	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₂³×C₃	1	2	yes	—
193	C₂×D₄	C₂³×C₃	1	2	yes	—
194	C₂×D₄	C₂³×C₃	1	2	yes	—
195	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
196	C₂×D₄	C₂³×C₃	1	8	yes	—
197	C₂²×C₄	C₂³×C₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₃	1	2	yes	—
200	C₂²×C₄	C₂³×C₃	1	4	yes	—
201	C₂²×C₄	C₂³×C₃	1	8	yes	—
202	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₃	1	2	yes	—
204	C₄⋊C₄	C₂³×C₃	1	2	yes	—
205	C₄⋊C₄	C₂³×C₃	1	2	yes	—
206	C₄⋊C₄	C₂³×C₃	1	8	yes	—
207	C₄⋊C₄	C₂³×C₃	1	4	yes	—
208	C₄⋊C₄	C₂³×C₃	1	4	yes	—
209	C₄⋊C₄	C₂³×C₃	1	8	yes	—
210	C₄⋊C₄	C₂³×C₃	1	4	yes	—
211	C₄⋊C₄	C₂³×C₃	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₃	1	2	yes	—
213	C₂²⋊C₄	C₂³×C₃	1	2	yes	—
214	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
217	C₂²⋊C₄	C₂³×C₃	1	2	yes	—
218	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
219	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
220	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
221	C₄²	C₂³×C₃	1	1	yes	characteristic
222	C₄²	C₂³×C₃	1	1	yes	characteristic
223	C₄²	C₂³×C₃	1	2	yes	—
224	C₄²	C₂³×C₃	1	2	yes	—
225	C₄²	C₂³×C₃	1	4	yes	—
226	C₄²	C₂³×C₃	1	8	yes	—
227	C₄²	C₂³×C₃	1	2	yes	—
228	C₄²	C₂³×C₃	1	2	yes	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	2	4	no	—
233	C₂²×C₃	—	2	8	no	—
234	C₂²×C₃	—	8	4	no	—
235	C₂²×C₃	C₂×#41	1	2	yes	—
236	C₂²×C₃	C₂⁵	1	1	yes	characteristic
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	4	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	C₂×#41	1	2	yes	—
245	C₃×C₄	C₂×#41	1	2	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	C₂⁴×C₃	1	4	yes	—
249	C₂³	C₂⁴×C₃	1	2	yes	—
250	C₂³	C₂⁴×C₃	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	4	no	—
253	Q₈	—	2	2	no	—
254	Q₈	—	4	4	no	—
255	D₄	—	2	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	2	no	—
259	D₄	—	4	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₃×#41	1	4	yes	—
266	C₂×C₄	C₂⁴×C₃	1	2	yes	—
267	C₂×C₄	C₂⁴×C₃	1	4	yes	—
268	C₂×C₄	C₂⁴×C₃	1	4	yes	—
269	C₂×C₄	C₂⁴×C₃	1	4	yes	—
270	C₂×C₄	C₂⁴×C₃	1	2	yes	—
271	C₂×C₄	C₂⁴×C₃	1	2	yes	—
272	C₂×C₄	C₂⁴×C₃	1	4	yes	—
273	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
274	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	4	2	no	—
277	C₂×C₃	—	4	4	no	—
278	C₂×C₃	C₂²×#41	1	1	yes	characteristic
279	C₂×C₃	#585	1	2	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	8	no	—
285	C₂²	—	8	4	no	—
286	C₂²	C₂×C₃×#41	1	2	yes	—
287	C₂²	C₂⁵×C₃	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	C₂×C₃×#41	1	2	yes	—
296	C₄	C₂×C₃×#41	1	2	yes	—
297	C₃	#3510	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	4	2	no	—
301	C₂	C₂²×C₃×#41	1	1	yes	characteristic
302	C₂	C₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₃×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#559	C₂	semidirect
2. C₃×#556	C₂	semidirect
3. C₃×#551	C₂	semidirect
4. C₃×#542	C₂	semidirect
5. C₃×#536	C₂	semidirect
7. C₃×#533	C₂	semidirect
8. C₃×#529	C₂	semidirect
9. C₂×C₃×#124	C₂	semidirect
10. C₃×#519	C₂	semidirect
11. C₃×#518	C₂	semidirect
12. C₃×C₄×#41	C₂	semidirect
13. #3510	C₃	direct (N × H)
17. C₃×C₄⋊Q₈	C₂²	semidirect
18. C₃×C₄⋊Q₈	C₂²	semidirect
20. C₃×#126	C₂²	semidirect
21. C₃×#126	C₂²	semidirect
22. C₃×#125	C₂²	semidirect
27. C₃×#124	C₂²	semidirect
28. C₃×#124	C₂²	semidirect
30. C₃×#123	C₂²	semidirect
31. C₃×#123	C₂²	semidirect
33. C₃×C₂²⋊Q₈	C₂²	semidirect
34. C₃×C₂²⋊Q₈	C₂²	semidirect
37. C₃×C₄×Q₈	C₂²	semidirect
38. C₃×C₄×Q₈	C₂²	semidirect
39. C₃×C₄×Q₈	C₂²	semidirect
40. C₃×C₄×Q₈	C₂²	semidirect
44. C₃×C₄×D₄	C₂²	semidirect
45. C₃×C₄×D₄	C₂²	semidirect
50. C₃×#117	C₂²	semidirect
51. C₃×#117	C₂²	semidirect
55. C₂×C₃×C₄²	C₂²	semidirect
56. #559	C₂×C₃	semidirect
57. #556	C₂×C₃	semidirect
58. #551	C₂×C₃	semidirect
59. #542	C₂×C₃	semidirect
60. #536	C₂×C₃	semidirect
62. #533	C₂×C₃	semidirect
63. #529	C₂×C₃	semidirect
64. C₂×#124	C₂×C₃	semidirect
65. #519	C₂×C₃	semidirect
66. #518	C₂×C₃	semidirect
67. C₄×#41	C₂×C₃	semidirect
116. C₄⋊Q₈	C₂²×C₃	semidirect
117. C₄⋊Q₈	C₂²×C₃	semidirect
119. #126	C₂²×C₃	semidirect
120. #126	C₂²×C₃	semidirect
121. #125	C₂²×C₃	semidirect
126. #124	C₂²×C₃	semidirect
127. #124	C₂²×C₃	semidirect
129. #123	C₂²×C₃	semidirect
130. #123	C₂²×C₃	semidirect
132. C₂²⋊Q₈	C₂²×C₃	semidirect
133. C₂²⋊Q₈	C₂²×C₃	semidirect
136. C₄×Q₈	C₂²×C₃	semidirect
137. C₄×Q₈	C₂²×C₃	semidirect
138. C₄×Q₈	C₂²×C₃	semidirect
139. C₄×Q₈	C₂²×C₃	semidirect
143. C₄×D₄	C₂²×C₃	semidirect
144. C₄×D₄	C₂²×C₃	semidirect
149. #117	C₂²×C₃	semidirect
150. #117	C₂²×C₃	semidirect
154. C₂×C₄²	C₂²×C₃	semidirect
297. C₃	#3510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#559	2
C₃×#556	2
C₃×#551	2
C₃×#542	2
C₃×#536	2
C₃×#535	2
C₃×#533	2
C₃×#529	2
C₂×C₃×#124	2
C₃×#519	2
C₃×#518	2
C₃×C₄×#41	2
#3510	3

Subgroups — C3×#3510 #87394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3510 #87394