Subgroups — C₃×#3280 #85144

Order 384.

526 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	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#546	C₂	1	2	yes	maximal, complemented
4	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#463	C₂	1	2	yes	maximal, complemented
8	C₃×#450	C₂	1	2	yes	maximal, complemented
9	C₃×#449	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#3280	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	complemented
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₃×#135	—	2	4	no	—
16	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×D₈	—	2	4	no	—
20	C₂×C₃×D₈	C₂²	1	2	yes	—
21	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
25	C₃×#124	C₂²	1	2	yes	complemented
26	C₃×#123	—	2	2	no	—
27	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₃×#121	—	2	2	no	—
29	C₃×#121	C₂²	1	2	yes	complemented
30	C₃×#120	—	2	2	no	—
31	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
32	C₃×C₄×D₄	C₂²	1	2	yes	complemented
33	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
34	C₃×#107	C₂²	1	1	yes	characteristic, complemented
35	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₃×#103	C₂²	1	2	yes	complemented
37	C₃×#102	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	2	yes	complemented
39	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂×#135	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂²×D₈	C₂×C₃	1	1	yes	characteristic, complemented
44	#546	C₂×C₃	1	2	yes	complemented
45	#495	C₂×C₃	1	1	yes	characteristic, complemented
46	#467	C₂×C₃	1	1	yes	characteristic, complemented
47	#466	C₂×C₃	1	1	yes	characteristic, complemented
48	#463	C₂×C₃	1	2	yes	complemented
49	#450	C₂×C₃	1	2	yes	complemented
50	#449	C₂×C₃	1	2	yes	complemented
51	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
52	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂⁴×C₃	—	2	2	no	—
54	C₃×#41	—	2	4	no	—
55	C₃×#41	—	4	2	no	—
56	C₂×C₃×Q₈	C₂³	1	2	yes	—
57	C₂×C₃×D₄	—	2	4	no	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂×C₃×D₄	D₄	1	1	yes	characteristic
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂×C₃×D₄	C₂³	1	2	yes	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	—	2	2	no	—
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₃×Q₁₆	—	2	2	no	—
70	C₃×SD₁₆	—	2	4	no	—
71	C₃×SD₁₆	—	4	2	no	—
72	C₃×D₈	—	2	4	no	—
73	C₃×D₈	—	2	2	no	—
74	C₃×D₈	D₄	1	4	yes	complemented
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂³	1	2	yes	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₂²⋊C₄	—	2	2	no	—
87	C₃×C₂²⋊C₄	—	2	2	no	—
88	C₃×C₂²⋊C₄	—	2	2	no	—
89	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
90	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	C₂×#41	C₂²×C₃	1	2	yes	complemented
93	C₂²×D₄	C₂²×C₃	1	2	yes	—
94	#135	—	2	4	no	—
95	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
96	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
97	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
98	C₂×D₈	—	2	4	no	—
99	C₂×D₈	C₂²×C₃	1	2	yes	—
100	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
101	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
104	#124	C₂²×C₃	1	2	yes	complemented
105	#123	—	2	2	no	—
106	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
107	#121	—	2	2	no	—
108	#121	C₂²×C₃	1	2	yes	complemented
109	#120	—	2	2	no	—
110	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
111	C₄×D₄	C₂²×C₃	1	2	yes	complemented
112	C₂×C₂²⋊C₄	—	2	2	no	—
113	#107	C₂²×C₃	1	1	yes	characteristic, complemented
114	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
115	#103	C₂²×C₃	1	2	yes	complemented
116	#102	C₂²×C₃	1	2	yes	complemented
117	#102	C₂²×C₃	1	2	yes	complemented
118	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
119	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	2	2	no	—
123	C₂³×C₃	—	4	4	no	—
124	C₂³×C₃	—	4	4	no	—
125	C₂³×C₃	—	8	2	no	—
126	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	2	2	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	2	2	no	—
133	C₃×D₄	—	2	2	no	—
134	C₃×D₄	—	4	4	no	—
135	C₃×D₄	—	4	4	no	—
136	C₃×D₄	—	8	2	no	—
137	C₃×D₄	C₂×D₄	1	4	yes	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	4	no	—
145	C₂×C₃×C₄	—	8	2	no	—
146	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	C₂×D₄	1	2	yes	—
155	C₂⁴	—	2	2	no	—
156	#41	—	2	4	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	C₂³×C₃	1	2	yes	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	C₃×D₄	1	1	yes	characteristic
165	C₂×D₄	C₂³×C₃	1	2	yes	—
166	C₂×D₄	C₂³×C₃	1	2	yes	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	Q₁₆	—	2	2	no	—
172	SD₁₆	—	2	4	no	—
173	SD₁₆	—	4	2	no	—
174	D₈	—	2	4	no	—
175	D₈	—	2	2	no	—
176	D₈	C₃×D₄	1	4	yes	complemented
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
181	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₃	1	2	yes	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
192	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
193	C₄²	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	4	2	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	8	8	no	—
206	C₂²×C₃	#135	1	2	yes	complemented
207	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	4	2	no	—
213	C₃×C₄	—	4	2	no	—
214	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	8	2	no	—
221	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	2	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	8	2	no	—
232	D₄	C₂×C₃×D₄	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	—	8	2	no	—
254	C₂×C₃	D₄²	1	1	yes	characteristic
255	C₂×C₃	C₂×#135	1	1	yes	characteristic
256	C₂×C₃	#576	1	1	yes	characteristic
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	8	no	—
269	C₂²	C₃×#135	1	2	yes	complemented
270	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
278	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
279	C₃	#3280	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	2	2	no	—
281	C₂	—	4	1	no	—
282	C₂	—	4	4	no	—
283	C₂	—	8	2	no	—
284	C₂	C₃×D₄²	1	1	yes	characteristic
285	C₂	C₂×C₃×#135	1	1	yes	characteristic
286	C₂	C₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#135	C₂	semidirect
2. C₂²×C₃×D₈	C₂	semidirect
3. C₃×#546	C₂	semidirect
4. C₃×#495	C₂	semidirect
5. C₃×#467	C₂	semidirect
6. C₃×#466	C₂	semidirect
7. C₃×#463	C₂	semidirect
8. C₃×#450	C₂	semidirect
9. C₃×#449	C₂	semidirect
10. C₃×C₄×D₈	C₂	semidirect
11. C₃×C₈×D₄	C₂	semidirect
12. #3280	C₃	direct (N × H)
13. C₂×C₃×#41	C₂²	semidirect
16. C₂×C₃×Q₁₆	C₂²	semidirect
17. C₂×C₃×SD₁₆	C₂²	semidirect
18. C₂×C₃×SD₁₆	C₂²	semidirect
24. C₂²×C₃×C₈	C₂²	semidirect
25. C₃×#124	C₂²	semidirect
29. C₃×#121	C₂²	semidirect
32. C₃×C₄×D₄	C₂²	semidirect
34. C₃×#107	C₂²	semidirect
35. C₃×C₄⋊C₈	C₂²	semidirect
36. C₃×#103	C₂²	semidirect
37. C₃×#102	C₂²	semidirect
38. C₃×#102	C₂²	semidirect
40. C₃×C₂²⋊C₈	C₂²	semidirect
41. C₃×C₄×C₈	C₂²	semidirect
42. C₂×#135	C₂×C₃	semidirect
43. C₂²×D₈	C₂×C₃	semidirect
44. #546	C₂×C₃	semidirect
45. #495	C₂×C₃	semidirect
46. #467	C₂×C₃	semidirect
47. #466	C₂×C₃	semidirect
48. #463	C₂×C₃	semidirect
49. #450	C₂×C₃	semidirect
50. #449	C₂×C₃	semidirect
51. C₄×D₈	C₂×C₃	semidirect
52. C₈×D₄	C₂×C₃	semidirect
74. C₃×D₈	D₄	semidirect
92. C₂×#41	C₂²×C₃	semidirect
95. C₂×Q₁₆	C₂²×C₃	semidirect
96. C₂×SD₁₆	C₂²×C₃	semidirect
97. C₂×SD₁₆	C₂²×C₃	semidirect
103. C₂²×C₈	C₂²×C₃	semidirect
104. #124	C₂²×C₃	semidirect
108. #121	C₂²×C₃	semidirect
111. C₄×D₄	C₂²×C₃	semidirect
113. #107	C₂²×C₃	semidirect
114. C₄⋊C₈	C₂²×C₃	semidirect
115. #103	C₂²×C₃	semidirect
116. #102	C₂²×C₃	semidirect
117. #102	C₂²×C₃	semidirect
119. C₂²⋊C₈	C₂²×C₃	semidirect
120. C₄×C₈	C₂²×C₃	semidirect
176. D₈	C₃×D₄	semidirect
206. C₂²×C₃	#135	semidirect
269. C₂²	C₃×#135	semidirect
279. C₃	#3280	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×#135	2
C₂²×C₃×D₈	2
C₃×#546	2
C₃×#495	2
C₃×#467	2
C₃×#466	2
C₃×#463	2
C₃×#450	2
C₃×#449	2
C₃×C₄×D₈	2
C₃×C₈×D₄	2
#3280	3

Subgroups — C3×#3280 #85144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3280 #85144