Subgroups — C₃×#3481 #87365

Order 384.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 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	1	yes	maximal, characteristic, complemented
2	C₃×#554	C₂	1	2	yes	maximal, complemented
3	C₃×#553	C₂	1	4	yes	maximal, complemented
4	C₃×#551	C₂	1	4	yes	maximal, complemented
5	C₃×#546	C₂	1	4	yes	maximal, complemented
6	C₃×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#529	C₂	1	4	yes	maximal, complemented
9	C₃×#525	C₂	1	4	yes	maximal, complemented
10	C₃×C₄×#41	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×C₄×D₄	C₂	1	4	yes	maximal, complemented
12	#3481	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂×C₃×#41	C₂²	1	2	yes	complemented
15	C₂²×C₃×D₄	C₂²	1	4	yes	—
16	C₂³×C₃×C₄	C₂²	1	4	yes	complemented
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₃×#127	C₂²	1	1	yes	characteristic
19	C₃×#126	C₂²	1	8	yes	complemented
20	C₃×#125	C₂²	1	4	yes	complemented
21	C₃×#124	C₂²	1	4	yes	complemented
22	C₃×#124	C₂²	1	2	yes	complemented
23	C₃×#123	C₂²	1	4	yes	complemented
24	C₃×#123	C₂²	1	4	yes	complemented
25	C₃×#123	C₂²	1	8	yes	complemented
26	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
27	C₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
28	C₃×#121	C₂²	1	4	yes	—
29	C₃×#121	C₂²	1	8	yes	complemented
30	C₃×#121	C₂²	1	4	yes	complemented
31	C₃×#120	C₂²	1	4	yes	—
32	C₃×#120	C₂²	1	4	yes	complemented
33	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
34	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
35	C₃×C₄×D₄	C₂²	1	2	yes	—
36	C₃×C₄×D₄	C₂²	1	4	yes	—
37	C₃×C₄×D₄	C₂²	1	2	yes	—
38	C₃×C₄×D₄	C₂²	1	4	yes	complemented
39	C₃×C₄×D₄	C₂²	1	8	yes	complemented
40	C₃×C₄×D₄	C₂²	1	4	yes	complemented
41	C₃×C₄×D₄	C₂²	1	8	yes	complemented
42	C₃×C₄×D₄	C₂²	1	4	yes	complemented
43	C₃×C₄×D₄	C₂²	1	4	yes	complemented
44	C₃×#117	C₂²	1	4	yes	complemented
45	C₃×#117	C₂²	1	2	yes	complemented
46	C₃×#117	C₂²	1	4	yes	complemented
47	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
48	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
49	C₂×C₃×C₄²	C₂²	1	4	yes	complemented
50	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
51	#559	C₂×C₃	1	1	yes	characteristic, complemented
52	#554	C₂×C₃	1	2	yes	complemented
53	#553	C₂×C₃	1	4	yes	complemented
54	#551	C₂×C₃	1	4	yes	complemented
55	#546	C₂×C₃	1	4	yes	complemented
56	D₄²	C₂×C₃	1	1	yes	characteristic, complemented
57	#532	C₂×C₃	1	1	yes	characteristic, complemented
58	#529	C₂×C₃	1	4	yes	complemented
59	#525	C₂×C₃	1	4	yes	complemented
60	C₄×#41	C₂×C₃	1	2	yes	complemented
61	C₂×C₄×D₄	C₂×C₃	1	4	yes	complemented
62	C₂⁴×C₃	C₂³	1	4	yes	—
63	C₃×#41	—	2	8	no	—
64	C₃×#41	—	4	4	no	—
65	C₂×C₃×Q₈	C₂³	1	2	yes	—
66	C₂×C₃×Q₈	C₂³	1	2	yes	—
67	C₂×C₃×D₄	—	2	16	no	—
68	C₂×C₃×D₄	C₂³	1	8	yes	—
69	C₂×C₃×D₄	C₂³	1	4	yes	—
70	C₂×C₃×D₄	C₂³	1	4	yes	—
71	C₂×C₃×D₄	C₂³	1	2	yes	—
72	C₂×C₃×D₄	C₂³	1	4	yes	—
73	C₂×C₃×D₄	C₂³	1	2	yes	—
74	C₂²×C₃×C₄	—	4	8	no	—
75	C₂²×C₃×C₄	C₂³	1	4	yes	—
76	C₂²×C₃×C₄	C₂³	1	4	yes	—
77	C₂²×C₃×C₄	C₂³	1	2	yes	—
78	C₂²×C₃×C₄	C₂³	1	4	yes	—
79	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂³	1	4	yes	—
81	C₂²×C₃×C₄	C₂³	1	8	yes	—
82	C₂²×C₃×C₄	C₂³	1	4	yes	—
83	C₃×C₄⋊C₄	C₂³	1	2	yes	—
84	C₃×C₄⋊C₄	C₂³	1	4	yes	—
85	C₃×C₄⋊C₄	C₂³	1	8	yes	—
86	C₃×C₄⋊C₄	C₂³	1	2	yes	—
87	C₃×C₄⋊C₄	C₂³	1	4	yes	—
88	C₃×C₄⋊C₄	C₂³	1	4	yes	—
89	C₃×C₄⋊C₄	C₂³	1	8	yes	—
90	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
91	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
94	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
95	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
96	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
97	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
98	C₃×C₄²	C₂³	1	1	yes	characteristic
99	C₃×C₄²	C₂³	1	4	yes	—
100	C₃×C₄²	C₂³	1	1	yes	characteristic
101	C₃×C₄²	C₂³	1	4	yes	—
102	C₃×C₄²	C₂³	1	2	yes	—
103	C₃×C₄²	C₂³	1	4	yes	—
104	C₂×#41	C₂²×C₃	1	2	yes	—
105	C₂×#41	C₂²×C₃	1	2	yes	complemented
106	C₂²×D₄	C₂²×C₃	1	4	yes	—
107	C₂³×C₄	C₂²×C₃	1	4	yes	complemented
108	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
109	#127	C₂²×C₃	1	1	yes	characteristic
110	#126	C₂²×C₃	1	8	yes	complemented
111	#125	C₂²×C₃	1	4	yes	complemented
112	#124	C₂²×C₃	1	4	yes	complemented
113	#124	C₂²×C₃	1	2	yes	complemented
114	#123	C₂²×C₃	1	4	yes	complemented
115	#123	C₂²×C₃	1	4	yes	complemented
116	#123	C₂²×C₃	1	8	yes	complemented
117	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
118	C₂²⋊Q₈	C₂²×C₃	1	4	yes	complemented
119	#121	C₂²×C₃	1	4	yes	—
120	#121	C₂²×C₃	1	8	yes	complemented
121	#121	C₂²×C₃	1	4	yes	complemented
122	#120	C₂²×C₃	1	4	yes	—
123	#120	C₂²×C₃	1	4	yes	complemented
124	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
125	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
126	C₄×D₄	C₂²×C₃	1	2	yes	—
127	C₄×D₄	C₂²×C₃	1	4	yes	—
128	C₄×D₄	C₂²×C₃	1	2	yes	—
129	C₄×D₄	C₂²×C₃	1	8	yes	complemented
130	C₄×D₄	C₂²×C₃	1	4	yes	complemented
131	C₄×D₄	C₂²×C₃	1	8	yes	complemented
132	C₄×D₄	C₂²×C₃	1	4	yes	complemented
133	C₄×D₄	C₂²×C₃	1	4	yes	complemented
134	C₄×D₄	C₂²×C₃	1	4	yes	complemented
135	#117	C₂²×C₃	1	4	yes	complemented
136	#117	C₂²×C₃	1	2	yes	complemented
137	#117	C₂²×C₃	1	4	yes	complemented
138	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
139	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
140	C₂×C₄²	C₂²×C₃	1	4	yes	complemented
141	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
142	C₂³×C₃	—	2	16	no	—
143	C₂³×C₃	—	4	4	no	—
144	C₂³×C₃	C₂⁴	1	4	yes	—
145	C₂³×C₃	C₂⁴	1	1	yes	characteristic
146	C₂³×C₃	C₂⁴	1	4	yes	—
147	C₃×Q₈	—	2	4	no	—
148	C₃×Q₈	—	4	2	no	—
149	C₃×D₄	—	2	16	no	—
150	C₃×D₄	—	2	8	no	—
151	C₃×D₄	—	2	8	no	—
152	C₃×D₄	—	4	4	no	—
153	C₃×D₄	—	4	2	no	—
154	C₃×D₄	#41	1	8	yes	complemented
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	4	4	no	—
162	C₂×C₃×C₄	—	4	8	no	—
163	C₂×C₃×C₄	C₂⁴	1	2	yes	—
164	C₂×C₃×C₄	C₂⁴	1	4	yes	—
165	C₂×C₃×C₄	C₂⁴	1	4	yes	—
166	C₂×C₃×C₄	C₂⁴	1	4	yes	—
167	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂⁴	1	4	yes	—
169	C₂×C₃×C₄	C₂⁴	1	2	yes	—
170	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
171	C₂⁴	C₂³×C₃	1	4	yes	—
172	#41	—	2	8	no	—
173	#41	—	4	4	no	—
174	C₂×Q₈	C₂³×C₃	1	2	yes	—
175	C₂×Q₈	C₂³×C₃	1	2	yes	—
176	C₂×D₄	—	2	16	no	—
177	C₂×D₄	C₂³×C₃	1	8	yes	—
178	C₂×D₄	C₂³×C₃	1	4	yes	—
179	C₂×D₄	C₂³×C₃	1	4	yes	—
180	C₂×D₄	C₂³×C₃	1	2	yes	—
181	C₂×D₄	C₂³×C₃	1	4	yes	—
182	C₂×D₄	C₂³×C₃	1	2	yes	—
183	C₂²×C₄	—	4	8	no	—
184	C₂²×C₄	C₂³×C₃	1	4	yes	—
185	C₂²×C₄	C₂³×C₃	1	4	yes	—
186	C₂²×C₄	C₂³×C₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₃	1	4	yes	—
188	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₃	1	4	yes	—
190	C₂²×C₄	C₂³×C₃	1	8	yes	—
191	C₂²×C₄	C₂³×C₃	1	4	yes	—
192	C₄⋊C₄	C₂³×C₃	1	2	yes	—
193	C₄⋊C₄	C₂³×C₃	1	4	yes	—
194	C₄⋊C₄	C₂³×C₃	1	8	yes	—
195	C₄⋊C₄	C₂³×C₃	1	2	yes	—
196	C₄⋊C₄	C₂³×C₃	1	4	yes	—
197	C₄⋊C₄	C₂³×C₃	1	4	yes	—
198	C₄⋊C₄	C₂³×C₃	1	8	yes	—
199	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
200	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
201	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
202	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
203	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
204	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
205	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
206	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
207	C₄²	C₂³×C₃	1	2	yes	—
208	C₄²	C₂³×C₃	1	1	yes	characteristic
209	C₄²	C₂³×C₃	1	4	yes	—
210	C₄²	C₂³×C₃	1	4	yes	—
211	C₄²	C₂³×C₃	1	1	yes	characteristic
212	C₄²	C₂³×C₃	1	4	yes	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	8	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	4	16	no	—
220	C₂²×C₃	C₂×#41	1	8	yes	—
221	C₂²×C₃	C₂⁵	1	1	yes	characteristic
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	4	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	C₂×#41	1	4	yes	—
229	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
230	C₂³	—	2	16	no	—
231	C₂³	—	4	4	no	—
232	C₂³	C₂⁴×C₃	1	4	yes	—
233	C₂³	C₂⁴×C₃	1	1	yes	characteristic
234	C₂³	C₂⁴×C₃	1	4	yes	—
235	Q₈	—	2	4	no	—
236	Q₈	—	4	2	no	—
237	D₄	—	2	16	no	—
238	D₄	—	2	8	no	—
239	D₄	—	2	8	no	—
240	D₄	—	4	4	no	—
241	D₄	—	4	2	no	—
242	D₄	C₃×#41	1	8	yes	complemented
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	C₂⁴×C₃	1	2	yes	—
252	C₂×C₄	C₂⁴×C₃	1	4	yes	—
253	C₂×C₄	C₂⁴×C₃	1	4	yes	—
254	C₂×C₄	C₂⁴×C₃	1	4	yes	—
255	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
256	C₂×C₄	C₂⁴×C₃	1	4	yes	—
257	C₂×C₄	C₂⁴×C₃	1	2	yes	—
258	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
259	C₂×C₃	—	2	8	no	—
260	C₂×C₃	—	4	1	no	—
261	C₂×C₃	—	4	4	no	—
262	C₂×C₃	C₂²×#41	1	2	yes	—
263	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	16	no	—
271	C₂²	C₂×C₃×#41	1	8	yes	—
272	C₂²	C₂⁵×C₃	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	C₂×C₃×#41	1	4	yes	—
280	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
281	C₃	#3481	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	2	8	no	—
283	C₂	—	4	4	no	—
284	C₂	—	4	1	no	—
285	C₂	C₂²×C₃×#41	1	2	yes	—
286	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂²×C₃	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#559	C₂	semidirect
2. C₃×#554	C₂	semidirect
3. C₃×#553	C₂	semidirect
4. C₃×#551	C₂	semidirect
5. C₃×#546	C₂	semidirect
6. C₃×D₄²	C₂	semidirect
7. C₃×#532	C₂	semidirect
8. C₃×#529	C₂	semidirect
9. C₃×#525	C₂	semidirect
10. C₃×C₄×#41	C₂	semidirect
11. C₂×C₃×C₄×D₄	C₂	semidirect
12. #3481	C₃	direct (N × H)
14. C₂×C₃×#41	C₂²	semidirect
16. C₂³×C₃×C₄	C₂²	semidirect
17. C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×#126	C₂²	semidirect
20. C₃×#125	C₂²	semidirect
21. C₃×#124	C₂²	semidirect
22. C₃×#124	C₂²	semidirect
23. C₃×#123	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
25. C₃×#123	C₂²	semidirect
26. C₃×C₂²⋊Q₈	C₂²	semidirect
27. C₃×C₂²⋊Q₈	C₂²	semidirect
29. C₃×#121	C₂²	semidirect
30. C₃×#121	C₂²	semidirect
32. C₃×#120	C₂²	semidirect
33. C₃×C₄×Q₈	C₂²	semidirect
34. C₃×C₄×Q₈	C₂²	semidirect
38. C₃×C₄×D₄	C₂²	semidirect
39. C₃×C₄×D₄	C₂²	semidirect
40. C₃×C₄×D₄	C₂²	semidirect
41. C₃×C₄×D₄	C₂²	semidirect
42. C₃×C₄×D₄	C₂²	semidirect
43. C₃×C₄×D₄	C₂²	semidirect
44. C₃×#117	C₂²	semidirect
45. C₃×#117	C₂²	semidirect
46. C₃×#117	C₂²	semidirect
47. C₂×C₃×C₄⋊C₄	C₂²	semidirect
48. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
49. C₂×C₃×C₄²	C₂²	semidirect
50. C₂×C₃×C₄²	C₂²	semidirect
51. #559	C₂×C₃	semidirect
52. #554	C₂×C₃	semidirect
53. #553	C₂×C₃	semidirect
54. #551	C₂×C₃	semidirect
55. #546	C₂×C₃	semidirect
56. D₄²	C₂×C₃	semidirect
57. #532	C₂×C₃	semidirect
58. #529	C₂×C₃	semidirect
59. #525	C₂×C₃	semidirect
60. C₄×#41	C₂×C₃	semidirect
61. C₂×C₄×D₄	C₂×C₃	semidirect
105. C₂×#41	C₂²×C₃	semidirect
107. C₂³×C₄	C₂²×C₃	semidirect
108. C₄⋊Q₈	C₂²×C₃	semidirect
110. #126	C₂²×C₃	semidirect
111. #125	C₂²×C₃	semidirect
112. #124	C₂²×C₃	semidirect
113. #124	C₂²×C₃	semidirect
114. #123	C₂²×C₃	semidirect
115. #123	C₂²×C₃	semidirect
116. #123	C₂²×C₃	semidirect
117. C₂²⋊Q₈	C₂²×C₃	semidirect
118. C₂²⋊Q₈	C₂²×C₃	semidirect
120. #121	C₂²×C₃	semidirect
121. #121	C₂²×C₃	semidirect
123. #120	C₂²×C₃	semidirect
124. C₄×Q₈	C₂²×C₃	semidirect
125. C₄×Q₈	C₂²×C₃	semidirect
129. C₄×D₄	C₂²×C₃	semidirect
130. C₄×D₄	C₂²×C₃	semidirect
131. C₄×D₄	C₂²×C₃	semidirect
132. C₄×D₄	C₂²×C₃	semidirect
133. C₄×D₄	C₂²×C₃	semidirect
134. C₄×D₄	C₂²×C₃	semidirect
135. #117	C₂²×C₃	semidirect
136. #117	C₂²×C₃	semidirect
137. #117	C₂²×C₃	semidirect
138. C₂×C₄⋊C₄	C₂²×C₃	semidirect
139. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
140. C₂×C₄²	C₂²×C₃	semidirect
141. C₂×C₄²	C₂²×C₃	semidirect
154. C₃×D₄	#41	semidirect
242. D₄	C₃×#41	semidirect
281. C₃	#3481	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×#554	2
C₃×#553	2
C₃×#551	2
C₃×#546	2
C₃×D₄²	2
C₃×#532	2
C₃×#529	2
C₃×#525	2
C₃×C₄×#41	2
C₂×C₃×C₄×D₄	2
#3481	3

Subgroups — C3×#3481 #87365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3481 #87365