Subgroups — C₃×#3485 #87369

Order 384.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#547	C₂	1	8	yes	maximal, complemented
4	C₃×#546	C₂	1	8	yes	maximal, complemented
5	C₃×#536	C₂	1	2	yes	maximal, complemented
6	C₃×#534	C₂	1	4	yes	maximal, complemented
7	C₃×#525	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#123	C₂	1	4	yes	maximal, complemented
9	C₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
10	#3485	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×H(5,𝔽₂)	—	2	8	no	—
12	C₂×C₃×#41	—	2	8	no	—
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂×C₃×#41	C₂²	1	8	yes	complemented
15	C₂×C₃×#41	C₂²	1	4	yes	complemented
16	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₃×D₄	C₂²	1	4	yes	—
18	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
19	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
20	C₂²×C₃×D₄	C₂²	1	4	yes	complemented
21	C₂³×C₃×C₄	C₂²	1	2	yes	complemented
22	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
23	C₃×C₄⋊Q₈	C₂²	1	4	yes	complemented
24	C₃×#124	C₂²	1	4	yes	complemented
25	C₃×#123	C₂²	1	4	yes	complemented
26	C₃×#123	C₂²	1	8	yes	complemented
27	C₃×#123	C₂²	1	16	yes	complemented
28	C₃×#123	C₂²	1	8	yes	complemented
29	C₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
30	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
31	C₃×#121	C₂²	1	4	yes	complemented
32	C₃×#121	C₂²	1	16	yes	complemented
33	C₃×#120	C₂²	1	4	yes	complemented
34	C₃×#120	C₂²	1	8	yes	complemented
35	C₃×C₄×D₄	C₂²	1	8	yes	complemented
36	C₃×C₄×D₄	C₂²	1	8	yes	complemented
37	C₃×#117	C₂²	1	2	yes	complemented
38	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	complemented
39	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
40	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
41	C₂×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂²×#41	C₂×C₃	1	1	yes	characteristic, complemented
43	#547	C₂×C₃	1	8	yes	complemented
44	#546	C₂×C₃	1	8	yes	complemented
45	#536	C₂×C₃	1	2	yes	complemented
46	#534	C₂×C₃	1	4	yes	complemented
47	#525	C₂×C₃	1	2	yes	complemented
48	C₂×#123	C₂×C₃	1	4	yes	complemented
49	#518	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃	C₂³	1	4	yes	—
51	C₂⁴×C₃	C₂³	1	1	yes	characteristic
52	C₂⁴×C₃	C₂³	1	2	yes	—
53	C₃×#41	—	2	32	no	—
54	C₃×#41	—	2	16	no	—
55	C₃×#41	—	2	8	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	C₂³	1	4	yes	—
58	C₂×C₃×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×D₄	—	2	16	no	—
60	C₂×C₃×D₄	—	2	8	no	—
61	C₂×C₃×D₄	—	2	16	no	—
62	C₂×C₃×D₄	D₄	1	8	yes	complemented
63	C₂×C₃×D₄	C₂³	1	8	yes	—
64	C₂×C₃×D₄	C₂³	1	4	yes	—
65	C₂×C₃×D₄	C₂³	1	8	yes	—
66	C₂×C₃×D₄	C₂³	1	2	yes	—
67	C₂×C₃×D₄	C₂³	1	8	yes	—
68	C₂×C₃×D₄	C₂³	1	4	yes	—
69	C₂²×C₃×C₄	—	2	8	no	—
70	C₂²×C₃×C₄	—	2	4	no	—
71	C₂²×C₃×C₄	C₂³	1	8	yes	—
72	C₂²×C₃×C₄	C₂³	1	4	yes	—
73	C₂²×C₃×C₄	C₂³	1	4	yes	—
74	C₂²×C₃×C₄	C₂³	1	4	yes	—
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	2	yes	—
77	C₂²×C₃×C₄	C₂³	1	8	yes	—
78	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	8	yes	—
80	C₃×C₄⋊C₄	C₂³	1	16	yes	—
81	C₃×C₄⋊C₄	C₂³	1	4	yes	—
82	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
83	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
84	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
85	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
86	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
87	C₃×C₄²	C₂³	1	4	yes	—
88	H(5,𝔽₂)	—	2	8	no	—
89	C₂×#41	—	2	8	no	—
90	C₂×#41	C₂²×C₃	1	2	yes	—
91	C₂×#41	C₂²×C₃	1	8	yes	complemented
92	C₂×#41	C₂²×C₃	1	4	yes	complemented
93	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
94	C₂²×D₄	C₂²×C₃	1	4	yes	—
95	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
96	C₂²×D₄	C₂²×C₃	1	2	yes	complemented
97	C₂²×D₄	C₂²×C₃	1	4	yes	complemented
98	C₂³×C₄	C₂²×C₃	1	2	yes	complemented
99	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
100	C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
101	#124	C₂²×C₃	1	4	yes	complemented
102	#123	C₂²×C₃	1	4	yes	complemented
103	#123	C₂²×C₃	1	8	yes	complemented
104	#123	C₂²×C₃	1	16	yes	complemented
105	#123	C₂²×C₃	1	8	yes	complemented
106	C₂²⋊Q₈	C₂²×C₃	1	4	yes	complemented
107	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
108	#121	C₂²×C₃	1	4	yes	complemented
109	#121	C₂²×C₃	1	16	yes	complemented
110	#120	C₂²×C₃	1	4	yes	complemented
111	#120	C₂²×C₃	1	8	yes	complemented
112	C₄×D₄	C₂²×C₃	1	8	yes	complemented
113	C₄×D₄	C₂²×C₃	1	8	yes	complemented
114	#117	C₂²×C₃	1	2	yes	complemented
115	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
116	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
117	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	—	2	8	no	—
120	C₂³×C₃	—	4	4	no	—
121	C₂³×C₃	—	4	2	no	—
122	C₂³×C₃	—	4	8	no	—
123	C₂³×C₃	C₂×D₄	1	8	yes	—
124	C₂³×C₃	C₂⁴	1	4	yes	—
125	C₂³×C₃	C₂⁴	1	2	yes	—
126	C₂³×C₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₃	C₂⁴	1	4	yes	—
128	C₃×Q₈	—	2	8	no	—
129	C₃×Q₈	—	2	4	no	—
130	C₃×D₄	—	2	8	no	—
131	C₃×D₄	—	2	16	no	—
132	C₃×D₄	—	2	16	no	—
133	C₃×D₄	—	2	4	no	—
134	C₃×D₄	—	4	8	no	—
135	C₃×D₄	C₂×D₄	1	16	yes	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	2	16	no	—
139	C₂×C₃×C₄	—	2	8	no	—
140	C₂×C₃×C₄	—	2	8	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	—	4	8	no	—
143	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
144	C₂×C₃×C₄	C₂⁴	1	8	yes	—
145	C₂×C₃×C₄	C₂⁴	1	2	yes	—
146	C₂×C₃×C₄	C₂⁴	1	4	yes	—
147	C₂×C₃×C₄	C₂⁴	1	4	yes	—
148	C₂×C₃×C₄	C₂⁴	1	2	yes	—
149	C₂⁴	C₂³×C₃	1	4	yes	—
150	C₂⁴	C₂³×C₃	1	1	yes	characteristic
151	C₂⁴	C₂³×C₃	1	2	yes	—
152	#41	—	2	32	no	—
153	#41	—	2	16	no	—
154	#41	—	2	8	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	C₂³×C₃	1	4	yes	—
157	C₂×Q₈	C₂³×C₃	1	2	yes	—
158	C₂×D₄	—	2	16	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	16	no	—
161	C₂×D₄	C₃×D₄	1	8	yes	complemented
162	C₂×D₄	C₂³×C₃	1	8	yes	—
163	C₂×D₄	C₂³×C₃	1	4	yes	—
164	C₂×D₄	C₂³×C₃	1	8	yes	—
165	C₂×D₄	C₂³×C₃	1	2	yes	—
166	C₂×D₄	C₂³×C₃	1	8	yes	—
167	C₂×D₄	C₂³×C₃	1	4	yes	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	C₂³×C₃	1	8	yes	—
171	C₂²×C₄	C₂³×C₃	1	4	yes	—
172	C₂²×C₄	C₂³×C₃	1	4	yes	—
173	C₂²×C₄	C₂³×C₃	1	4	yes	—
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₃	1	8	yes	—
177	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₃	1	8	yes	—
179	C₄⋊C₄	C₂³×C₃	1	16	yes	—
180	C₄⋊C₄	C₂³×C₃	1	4	yes	—
181	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
182	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
183	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
184	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
185	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
186	C₄²	C₂³×C₃	1	4	yes	—
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	8	no	—
193	C₂²×C₃	—	4	8	no	—
194	C₂²×C₃	—	4	4	no	—
195	C₂²×C₃	—	8	8	no	—
196	C₂²×C₃	C₂²×D₄	1	2	yes	—
197	C₂²×C₃	C₂²×D₄	1	8	yes	—
198	C₂²×C₃	C₂⁵	1	1	yes	characteristic
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	8	no	—
203	C₃×C₄	C₂²×D₄	1	4	yes	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	8	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	4	2	no	—
208	C₂³	—	4	8	no	—
209	C₂³	C₂×C₃×D₄	1	8	yes	—
210	C₂³	C₂⁴×C₃	1	4	yes	—
211	C₂³	C₂⁴×C₃	1	2	yes	—
212	C₂³	C₂⁴×C₃	1	1	yes	characteristic
213	C₂³	C₂⁴×C₃	1	4	yes	—
214	Q₈	—	2	8	no	—
215	Q₈	—	2	4	no	—
216	D₄	—	2	8	no	—
217	D₄	—	2	16	no	—
218	D₄	—	2	16	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	8	no	—
221	D₄	C₂×C₃×D₄	1	16	yes	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
230	C₂×C₄	C₂⁴×C₃	1	8	yes	—
231	C₂×C₄	C₂⁴×C₃	1	2	yes	—
232	C₂×C₄	C₂⁴×C₃	1	4	yes	—
233	C₂×C₄	C₂⁴×C₃	1	4	yes	—
234	C₂×C₄	C₂⁴×C₃	1	2	yes	—
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	—	2	8	no	—
237	C₂×C₃	—	4	2	no	—
238	C₂×C₃	—	4	4	no	—
239	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
240	C₂×C₃	#585	1	2	yes	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	8	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	8	8	no	—
250	C₂²	C₂²×C₃×D₄	1	2	yes	—
251	C₂²	C₂²×C₃×D₄	1	8	yes	—
252	C₂²	C₂⁵×C₃	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	8	no	—
257	C₄	C₂²×C₃×D₄	1	4	yes	—
258	C₃	#3485	1	1	yes	characteristic, direct product factor, Sylow
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	2	no	—
262	C₂	—	4	4	no	—
263	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
264	C₂	C₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
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. C₂×C₃×H(5,𝔽₂)	C₂	semidirect
2. C₂²×C₃×#41	C₂	semidirect
3. C₃×#547	C₂	semidirect
4. C₃×#546	C₂	semidirect
5. C₃×#536	C₂	semidirect
6. C₃×#534	C₂	semidirect
7. C₃×#525	C₂	semidirect
8. C₂×C₃×#123	C₂	semidirect
9. C₃×#518	C₂	semidirect
10. #3485	C₃	direct (N × H)
14. C₂×C₃×#41	C₂²	semidirect
15. C₂×C₃×#41	C₂²	semidirect
16. C₂²×C₃×Q₈	C₂²	semidirect
19. C₂²×C₃×D₄	C₂²	semidirect
20. C₂²×C₃×D₄	C₂²	semidirect
21. C₂³×C₃×C₄	C₂²	semidirect
22. C₂³×C₃×C₄	C₂²	semidirect
23. C₃×C₄⋊Q₈	C₂²	semidirect
24. C₃×#124	C₂²	semidirect
25. C₃×#123	C₂²	semidirect
26. C₃×#123	C₂²	semidirect
27. C₃×#123	C₂²	semidirect
28. C₃×#123	C₂²	semidirect
29. C₃×C₂²⋊Q₈	C₂²	semidirect
30. C₃×C₂²⋊Q₈	C₂²	semidirect
31. C₃×#121	C₂²	semidirect
32. C₃×#121	C₂²	semidirect
33. C₃×#120	C₂²	semidirect
34. C₃×#120	C₂²	semidirect
35. C₃×C₄×D₄	C₂²	semidirect
36. C₃×C₄×D₄	C₂²	semidirect
37. C₃×#117	C₂²	semidirect
38. C₂×C₃×C₄⋊C₄	C₂²	semidirect
39. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
40. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
41. C₂×H(5,𝔽₂)	C₂×C₃	semidirect
42. C₂²×#41	C₂×C₃	semidirect
43. #547	C₂×C₃	semidirect
44. #546	C₂×C₃	semidirect
45. #536	C₂×C₃	semidirect
46. #534	C₂×C₃	semidirect
47. #525	C₂×C₃	semidirect
48. C₂×#123	C₂×C₃	semidirect
49. #518	C₂×C₃	semidirect
62. C₂×C₃×D₄	D₄	semidirect
91. C₂×#41	C₂²×C₃	semidirect
92. C₂×#41	C₂²×C₃	semidirect
93. C₂²×Q₈	C₂²×C₃	semidirect
96. C₂²×D₄	C₂²×C₃	semidirect
97. C₂²×D₄	C₂²×C₃	semidirect
98. C₂³×C₄	C₂²×C₃	semidirect
99. C₂³×C₄	C₂²×C₃	semidirect
100. C₄⋊Q₈	C₂²×C₃	semidirect
101. #124	C₂²×C₃	semidirect
102. #123	C₂²×C₃	semidirect
103. #123	C₂²×C₃	semidirect
104. #123	C₂²×C₃	semidirect
105. #123	C₂²×C₃	semidirect
106. C₂²⋊Q₈	C₂²×C₃	semidirect
107. C₂²⋊Q₈	C₂²×C₃	semidirect
108. #121	C₂²×C₃	semidirect
109. #121	C₂²×C₃	semidirect
110. #120	C₂²×C₃	semidirect
111. #120	C₂²×C₃	semidirect
112. C₄×D₄	C₂²×C₃	semidirect
113. C₄×D₄	C₂²×C₃	semidirect
114. #117	C₂²×C₃	semidirect
115. C₂×C₄⋊C₄	C₂²×C₃	semidirect
116. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
117. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
161. C₂×D₄	C₃×D₄	semidirect
258. C₃	#3485	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×H(5,𝔽₂)	2
C₂²×C₃×#41	2
C₃×#547	2
C₃×#546	2
C₃×#536	2
C₃×#534	2
C₃×#525	2
C₂×C₃×#123	2
C₃×#518	2
#3485	3

Subgroups — C3×#3485 #87369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3485 #87369