Subgroups — C₂×C₃×#918 #10595861

Order 576.

546 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#239	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#216	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#215	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#918	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃×#204	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#137	—	3	1	no	maximal
10	C₂×#918	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₃×#239	—	2	2	no	—
14	C₃×#239	C₂²	1	4	yes	—
15	C₃×D₃×Q₈	—	2	2	no	—
16	C₃×D₃×Q₈	C₂²	1	4	yes	complemented
17	C₂×C₃×D₁₂	—	2	1	no	—
18	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₃	—	2	1	no	—
20	C₂×C₃×C₄×D₃	—	2	1	no	—
21	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₃×Dic₆	—	2	1	no	—
23	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
24	C₃²×Q₁₆	C₂²	1	4	yes	complemented
25	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
26	C₃×#216	C₂²	1	4	yes	complemented
27	C₃×#215	C₂²	1	4	yes	complemented
28	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×#204	C₂²	1	4	yes	complemented
30	C₃×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	complemented
31	C₂×C₃×#41	—	3	1	no	—
32	C₂²×C₃×Q₈	—	3	1	no	—
33	C₂×#239	C₂×C₃	1	1	yes	characteristic
34	C₂×D₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×#137	—	3	8	no	—
36	C₂×C₃×Q₁₆	—	2	1	no	—
37	C₂×C₃×Q₁₆	—	3	1	no	—
38	C₂×C₃×Q₁₆	D₃	1	1	yes	characteristic, complemented
39	C₂×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×SD₁₆	—	3	1	no	—
41	C₂×C₃×SD₁₆	—	3	1	no	—
42	C₂×C₃×M₁₆	—	3	1	no	—
43	C₂×#216	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×#215	C₂×C₃	1	1	yes	characteristic, complemented
45	#918	C₂×C₃	1	8	yes	direct product factor
46	C₂×#204	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₃	—	2	1	no	—
49	C₂²×C₃×D₃	D₄	1	1	yes	characteristic
50	C₃²×Q₈	—	2	1	no	—
51	C₃²×Q₈	—	2	1	no	—
52	C₃²×Q₈	C₂³	1	2	yes	—
53	C₃²×Q₈	C₂³	1	2	yes	—
54	C₂×C₃²×C₄	—	2	1	no	—
55	C₂×C₃²×C₄	—	2	1	no	—
56	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃×Dic₃	—	2	1	no	—
58	C₂×C₃×Dic₃	D₄	1	1	yes	characteristic
59	C₃×D₁₂	—	2	4	no	—
60	C₃×D₁₂	—	2	1	no	—
61	C₃×D₁₂	C₂³	1	2	yes	—
62	C₃×C₄×D₃	—	2	4	no	—
63	C₃×C₄×D₃	—	2	4	no	—
64	C₃×C₄×D₃	D₄	1	2	yes	—
65	C₃×C₄×D₃	C₂³	1	2	yes	—
66	C₃×Dic₆	—	2	1	no	—
67	C₃×Dic₆	—	2	4	no	—
68	C₃×Dic₆	C₂³	1	2	yes	—
69	C₃²×C₈	C₂³	1	2	yes	—
70	C₃×C₃⋊C₈	C₂³	1	2	yes	—
71	C₂×#137	—	3	1	no	Sylow
72	C₃×#41	—	3	4	no	—
73	C₃×#41	—	6	2	no	—
74	C₂×C₃×Q₈	—	2	1	no	—
75	C₂×C₃×Q₈	—	2	1	no	—
76	C₂×C₃×Q₈	—	3	4	no	—
77	C₂×C₃×Q₈	—	3	1	no	—
78	C₂×C₃×Q₈	—	6	1	no	—
79	C₂×C₃×Q₈	—	6	2	no	—
80	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
82	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×D₄	—	3	1	no	—
85	C₂×C₃×D₄	—	6	1	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	—	6	1	no	—
88	C₂²×C₃×C₄	—	6	1	no	—
89	#239	—	2	2	no	—
90	#239	C₂²×C₃	1	4	yes	—
91	D₃×Q₈	—	2	2	no	—
92	D₃×Q₈	C₂²×C₃	1	4	yes	complemented
93	C₂×D₁₂	—	2	1	no	—
94	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₄×D₃	—	2	1	no	—
96	C₂×C₄×D₃	—	2	1	no	—
97	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
98	C₂×Dic₆	—	2	1	no	—
99	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
100	C₃×Q₁₆	—	2	4	no	—
101	C₃×Q₁₆	—	3	4	no	—
102	C₃×Q₁₆	C₂×D₃	1	4	yes	complemented
103	C₃×Q₁₆	C₂²×C₃	1	4	yes	complemented
104	C₃×SD₁₆	—	3	4	no	—
105	C₃×SD₁₆	—	3	4	no	—
106	C₃×M₁₆	—	3	4	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	3	1	no	—
109	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
111	#216	C₂²×C₃	1	4	yes	complemented
112	#215	C₂²×C₃	1	4	yes	complemented
113	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
114	#204	C₂²×C₃	1	4	yes	complemented
115	ΓL(1,𝔽₂₅)	C₂²×C₃	1	4	yes	complemented
116	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×D₃	—	2	2	no	—
118	C₂×C₃×D₃	—	2	2	no	—
119	C₂×C₃×D₃	—	4	2	no	—
120	C₂×C₃×D₃	C₂×D₄	1	2	yes	—
121	C₃²×C₄	—	2	2	no	—
122	C₃²×C₄	—	2	2	no	—
123	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
124	C₃²×C₄	C₂⁴	1	1	yes	characteristic
125	C₃×Dic₃	—	2	2	no	—
126	C₃×Dic₃	C₂×D₄	1	2	yes	—
127	C₂×#41	—	3	1	no	—
128	C₂²×Q₈	—	3	1	no	—
129	#137	—	3	8	no	—
130	C₂×Q₁₆	—	3	1	no	—
131	C₂×Q₁₆	C₃×D₃	1	1	yes	characteristic, complemented
132	C₂×SD₁₆	—	3	1	no	—
133	C₂×SD₁₆	—	3	1	no	—
134	C₂×M₁₆	—	3	1	no	—
135	C₂³×C₃	—	3	1	no	—
136	C₂³×C₃	—	6	1	no	—
137	C₂²×D₃	—	2	1	no	—
138	C₂²×D₃	C₃×D₄	1	1	yes	characteristic
139	C₃×Q₈	—	2	1	no	—
140	C₃×Q₈	—	2	1	no	—
141	C₃×Q₈	—	2	1	no	—
142	C₃×Q₈	—	2	1	no	—
143	C₃×Q₈	—	2	2	no	—
144	C₃×Q₈	—	2	2	no	—
145	C₃×Q₈	—	3	2	no	—
146	C₃×Q₈	—	4	1	no	—
147	C₃×Q₈	—	4	1	no	—
148	C₃×Q₈	—	6	4	no	—
149	C₃×Q₈	—	6	1	no	—
150	C₃×Q₈	C₂²×D₃	1	2	yes	—
151	C₃×Q₈	C₂²×D₃	1	2	yes	—
152	C₃×Q₈	C₂³×C₃	1	2	yes	—
153	C₃×Q₈	C₂³×C₃	1	2	yes	—
154	C₃×D₄	—	3	2	no	—
155	C₃×D₄	—	6	4	no	—
156	C₃×D₄	—	6	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	3	1	no	—
163	C₂×C₃×C₄	—	3	2	no	—
164	C₂×C₃×C₄	—	3	2	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	6	4	no	—
168	C₂×C₃×C₄	—	6	4	no	—
169	C₂×C₃×C₄	—	6	1	no	—
170	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₂×Dic₃	—	2	1	no	—
173	C₂×Dic₃	C₃×D₄	1	1	yes	characteristic
174	D₁₂	—	2	4	no	—
175	D₁₂	—	2	1	no	—
176	D₁₂	C₂³×C₃	1	2	yes	—
177	C₄×D₃	—	2	4	no	—
178	C₄×D₃	—	2	4	no	—
179	C₄×D₃	C₃×D₄	1	2	yes	—
180	C₄×D₃	C₂³×C₃	1	2	yes	—
181	Dic₆	—	2	1	no	—
182	Dic₆	—	2	4	no	—
183	Dic₆	C₂³×C₃	1	2	yes	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	3	2	no	—
186	C₃×C₈	C₂²×D₃	1	2	yes	—
187	C₃×C₈	C₂³×C₃	1	2	yes	—
188	C₃⋊C₈	C₂³×C₃	1	2	yes	—
189	C₂×C₃²	#137	1	2	yes	complemented
190	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
191	C₃×D₃	—	2	2	no	—
192	C₃×D₃	—	4	2	no	—
193	#41	—	3	4	no	—
194	#41	—	6	2	no	—
195	C₂×Q₈	—	3	4	no	—
196	C₂×Q₈	—	3	1	no	—
197	C₂×Q₈	—	6	1	no	—
198	C₂×Q₈	—	6	2	no	—
199	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
200	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
201	C₂×D₄	—	3	1	no	—
202	C₂×D₄	—	6	1	no	—
203	C₂²×C₄	—	3	1	no	—
204	C₂²×C₄	—	6	1	no	—
205	C₂²×C₄	—	6	1	no	—
206	Q₁₆	—	3	4	no	—
207	Q₁₆	C₂×C₃×D₃	1	4	yes	complemented
208	SD₁₆	—	3	4	no	—
209	SD₁₆	—	3	4	no	—
210	M₁₆	—	3	4	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	3	2	no	—
215	C₂²×C₃	—	6	2	no	—
216	C₂²×C₃	—	6	2	no	—
217	C₂²×C₃	—	12	2	no	—
218	C₂²×C₃	D₃×D₄	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂×D₃	—	2	2	no	—
221	C₂×D₃	—	2	2	no	—
222	C₂×D₃	—	4	2	no	—
223	C₂×D₃	C₂×C₃×D₄	1	2	yes	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	3	2	no	—
231	C₃×C₄	—	4	2	no	—
232	C₃×C₄	—	4	2	no	—
233	C₃×C₄	—	6	2	no	—
234	C₃×C₄	D₃×D₄	1	1	yes	characteristic
235	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
236	C₃×C₄	C₂³×D₃	1	1	yes	characteristic
237	C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
238	Dic₃	—	2	2	no	—
239	Dic₃	C₂×C₃×D₄	1	2	yes	—
240	C₃²	C₂×#137	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	3	1	no	—
242	C₂³	—	6	1	no	—
243	Q₈	—	2	1	no	—
244	Q₈	—	2	1	no	—
245	Q₈	—	3	2	no	—
246	Q₈	—	6	4	no	—
247	Q₈	—	6	1	no	—
248	Q₈	C₂²×C₃×D₃	1	2	yes	—
249	Q₈	C₂²×C₃×D₃	1	2	yes	—
250	D₄	—	3	2	no	—
251	D₄	—	6	4	no	—
252	D₄	—	6	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	6	4	no	—
259	C₂×C₄	—	6	4	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
262	C₈	—	3	2	no	—
263	C₈	C₂²×C₃×D₃	1	2	yes	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	6	2	no	—
267	C₂×C₃	—	12	2	no	—
268	C₂×C₃	#918	1	2	yes	direct product factor
269	C₂×C₃	C₃×#137	1	2	yes	complemented
270	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
271	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
272	D₃	—	2	2	no	—
273	D₃	—	4	2	no	—
274	C₂²	—	3	2	no	—
275	C₂²	—	6	2	no	—
276	C₂²	—	6	2	no	—
277	C₂²	—	12	2	no	—
278	C₂²	C₃×D₃×D₄	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	3	2	no	—
282	C₄	—	6	2	no	—
283	C₄	C₃×D₃×D₄	1	1	yes	characteristic
284	C₄	C₂³×C₃×D₃	1	1	yes	characteristic
285	C₃	—	2	1	no	—
286	C₃	C₂×#918	1	1	yes	characteristic, direct product factor
287	C₃	C₂×C₃×#137	1	1	yes	characteristic, complemented
288	C₂	—	6	2	no	—
289	C₂	—	12	2	no	—
290	C₂	C₃×#918	1	2	yes	direct product factor
291	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×Q₁₆	C₂
socle	C₂²×C₃²	C₂×D₄
prime radical	C₂×C₃×#239	C₂
hypercenter	C₂×C₃×Q₁₆	D₃
hypocenter	C₃	C₂×C₃×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×D₃×Q₈	C₂	semidirect
3. C₂×C₃²×Q₁₆	C₂	semidirect
4. C₂×C₃×#216	C₂	semidirect
5. C₂×C₃×#215	C₂	semidirect
6. C₃×#918	C₂	direct (N × H)
7. C₂×C₃×#204	C₂	semidirect
8. C₂×C₃×ΓL(1,𝔽₂₅)	C₂	semidirect
10. C₂×#918	C₃	direct (N × H)
16. C₃×D₃×Q₈	C₂²	semidirect
24. C₃²×Q₁₆	C₂²	semidirect
26. C₃×#216	C₂²	semidirect
27. C₃×#215	C₂²	semidirect
29. C₃×#204	C₂²	semidirect
30. C₃×ΓL(1,𝔽₂₅)	C₂²	semidirect
34. C₂×D₃×Q₈	C₂×C₃	semidirect
38. C₂×C₃×Q₁₆	D₃	semidirect
39. C₂×C₃×Q₁₆	C₂×C₃	semidirect
43. C₂×#216	C₂×C₃	semidirect
44. C₂×#215	C₂×C₃	semidirect
45. #918	C₂×C₃	direct (N × H)
46. C₂×#204	C₂×C₃	semidirect
47. C₂×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
92. D₃×Q₈	C₂²×C₃	semidirect
102. C₃×Q₁₆	C₂×D₃	semidirect
103. C₃×Q₁₆	C₂²×C₃	semidirect
111. #216	C₂²×C₃	semidirect
112. #215	C₂²×C₃	semidirect
114. #204	C₂²×C₃	semidirect
115. ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
131. C₂×Q₁₆	C₃×D₃	semidirect
189. C₂×C₃²	#137	semidirect
207. Q₁₆	C₂×C₃×D₃	semidirect
240. C₃²	C₂×#137	semidirect
268. C₂×C₃	#918	direct (N × H)
269. C₂×C₃	C₃×#137	semidirect
286. C₃	C₂×#918	direct (N × H)
287. C₃	C₂×C₃×#137	semidirect
290. C₂	C₃×#918	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×#239	2
C₂×C₃×D₃×Q₈	2
C₂×C₃²×Q₁₆	2
C₂×C₃×#216	2
C₂×C₃×#215	2
C₃×#918	2
C₂×C₃×#204	2
C₂×C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃×#137	3
C₂×#918	3

Subgroups — C2×C3×#918 #10595861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#918 #10595861