Subgroups — C₂×C₃×#4693 #11874147

Order 1152.

1393 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#204	C₂	1	4	yes	maximal, complemented
2	C₂×C₃×#874	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4693	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#492	—	3	1	no	maximal
7	C₂×#4693	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×D₁₂	—	2	2	no	—
9	C₂²×C₃×D₁₂	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
11	C₂²×C₃²×C₈	C₂²	1	2	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
14	C₂×C₃×#204	—	2	16	no	—
15	C₂×C₃×#204	C₂²	1	16	yes	complemented
16	C₃×#874	—	2	2	no	—
17	C₃×#874	C₂²	1	4	yes	—
18	C₃×#869	—	2	2	no	—
19	C₃×#869	C₂²	1	4	yes	complemented
20	C₃²×C₄×C₈	C₂²	1	4	yes	complemented
21	C₂²×C₃×SD₁₆	—	3	4	no	—
22	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
23	C₂×C₃×#127	—	3	1	no	—
24	C₂²×#204	C₂×C₃	1	4	yes	complemented
25	C₂×#874	C₂×C₃	1	1	yes	characteristic
26	C₂×#869	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#492	—	3	8	no	—
28	C₂×C₃×C₄×C₈	—	2	1	no	—
29	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
31	#4693	C₂×C₃	1	8	yes	direct product factor
32	C₂³×C₃×D₃	—	2	2	no	—
33	C₂²×C₃²×C₄	D₄	1	2	yes	—
34	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₃×Dic₃	—	2	2	no	—
36	C₂×C₃×D₁₂	—	2	8	no	—
37	C₂×C₃×D₁₂	—	2	8	no	—
38	C₂×C₃×D₁₂	—	2	2	no	—
39	C₂×C₃×D₁₂	—	4	8	no	—
40	C₂×C₃×D₁₂	C₂³	1	4	yes	—
41	C₂×C₃×Dic₆	—	2	8	no	—
42	C₂×C₃×Dic₆	—	2	2	no	—
43	C₂×C₃×Dic₆	C₂³	1	4	yes	—
44	C₂×C₃²×C₈	D₄	1	8	yes	complemented
45	C₂×C₃²×C₈	C₂³	1	4	yes	—
46	C₃²×C₄²	D₄	1	2	yes	—
47	C₃²×C₄²	C₂³	1	2	yes	—
48	C₃×C₄⋊Dic₃	—	2	8	no	—
49	C₃×#204	—	2	32	no	—
50	C₂×#492	—	3	1	no	Sylow
51	C₂²×C₃×Q₈	—	3	2	no	—
52	C₂²×C₃×D₄	—	3	2	no	—
53	C₂²×C₃×D₄	—	6	2	no	—
54	C₂²×D₁₂	—	2	2	no	—
55	C₂²×D₁₂	C₂²×C₃	1	2	yes	—
56	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
57	C₂×C₃×SD₁₆	—	3	16	no	—
58	C₂×C₃×SD₁₆	—	6	16	no	—
59	C₂²×C₃×C₈	—	2	2	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
61	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
62	C₃×C₄⋊Q₈	—	3	4	no	—
63	C₃×C₄⋊Q₈	—	6	2	no	—
64	C₃×#127	—	3	4	no	—
65	C₃×#127	—	6	2	no	—
66	C₂×C₃×C₄⋊C₄	—	6	2	no	—
67	C₂×C₃×C₄²	—	2	1	no	—
68	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄⋊Dic₃	—	2	2	no	—
71	C₂×#204	—	2	16	no	—
72	C₂×#204	C₂²×C₃	1	16	yes	complemented
73	#874	—	2	2	no	—
74	#874	C₂²×C₃	1	4	yes	—
75	#869	—	2	2	no	—
76	#869	C₂²×C₃	1	4	yes	complemented
77	C₃×C₄×C₈	—	2	4	no	—
78	C₃×C₄×C₈	C₂×D₃	1	4	yes	complemented
79	C₃×C₄×C₈	C₂²×C₃	1	4	yes	complemented
80	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×D₃	—	2	4	no	—
82	C₂²×C₃×D₃	—	4	4	no	—
83	C₂²×C₃×D₃	—	4	8	no	—
84	C₂×C₃²×C₄	SD₁₆	1	8	yes	complemented
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
87	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
88	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
89	C₂×C₃×Dic₃	—	2	4	no	—
90	C₂×C₃×Dic₃	—	4	4	no	—
91	C₃×D₁₂	—	2	8	no	—
92	C₃×D₁₂	—	4	4	no	—
93	C₃×D₁₂	—	4	16	no	—
94	C₃×Dic₆	—	2	8	no	—
95	C₃×Dic₆	—	4	4	no	—
96	C₃²×C₈	C₂×D₄	1	8	yes	complemented
97	C₂²×SD₁₆	—	3	4	no	—
98	C₂×C₄⋊Q₈	—	3	1	no	—
99	C₂×#127	—	3	1	no	—
100	#492	—	3	8	no	—
101	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic, complemented
102	C₂⁴×C₃	—	6	2	no	—
103	C₂³×D₃	—	2	2	no	—
104	C₂×C₃×Q₈	—	3	4	no	—
105	C₂×C₃×Q₈	—	6	2	no	—
106	C₂×C₃×Q₈	—	6	8	no	—
107	C₂×C₃×D₄	—	3	4	no	—
108	C₂×C₃×D₄	—	6	8	no	—
109	C₂×C₃×D₄	—	6	8	no	—
110	C₂×C₃×D₄	—	6	2	no	—
111	C₂×C₃×D₄	—	12	8	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	6	2	no	—
115	C₂²×C₃×C₄	D₁₂	1	2	yes	—
116	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
117	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₂²×Dic₃	—	2	2	no	—
120	C₂×D₁₂	—	2	8	no	—
121	C₂×D₁₂	—	2	2	no	—
122	C₂×D₁₂	—	2	8	no	—
123	C₂×D₁₂	—	4	8	no	—
124	C₂×D₁₂	C₂³×C₃	1	4	yes	—
125	C₂×Dic₆	—	2	8	no	—
126	C₂×Dic₆	—	2	2	no	—
127	C₂×Dic₆	C₂³×C₃	1	4	yes	—
128	C₃×SD₁₆	—	6	32	no	—
129	C₂×C₃×C₈	—	2	8	no	—
130	C₂×C₃×C₈	—	2	4	no	—
131	C₂×C₃×C₈	D₁₂	1	8	yes	complemented
132	C₂×C₃×C₈	C₃×D₄	1	8	yes	complemented
133	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
134	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
135	C₃×C₄⋊C₄	—	6	8	no	—
136	C₃×C₄²	—	2	2	no	—
137	C₃×C₄²	—	2	2	no	—
138	C₃×C₄²	D₁₂	1	2	yes	—
139	C₃×C₄²	C₃×D₄	1	2	yes	—
140	C₃×C₄²	C₂²×D₃	1	2	yes	—
141	C₃×C₄²	C₂³×C₃	1	2	yes	—
142	C₄⋊Dic₃	—	2	8	no	—
143	#204	—	2	32	no	—
144	C₂²×C₃²	#127	1	2	yes	—
145	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
146	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃×D₃	—	4	4	no	—
148	C₂×C₃×D₃	—	4	8	no	—
149	C₂×C₃×D₃	—	8	8	no	—
150	C₃²×C₄	#127	1	2	yes	—
151	C₃²×C₄	C₂×SD₁₆	1	8	yes	complemented
152	C₃²×C₄	C₂²×D₄	1	2	yes	—
153	C₃×Dic₃	—	4	4	no	—
154	C₂²×Q₈	—	3	2	no	—
155	C₂²×D₄	—	3	2	no	—
156	C₂²×D₄	—	6	2	no	—
157	C₂×SD₁₆	—	3	16	no	—
158	C₂×SD₁₆	—	6	16	no	—
159	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
160	C₄⋊Q₈	—	3	4	no	—
161	C₄⋊Q₈	—	6	2	no	—
162	#127	—	3	4	no	—
163	#127	—	6	2	no	—
164	C₂×C₄⋊C₄	—	6	2	no	—
165	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
166	C₄×C₈	C₂×C₃×D₃	1	4	yes	complemented
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	—	6	4	no	—
169	C₂³×C₃	—	12	4	no	—
170	C₂³×C₃	—	12	8	no	—
171	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
172	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂²×D₃	—	2	4	no	—
174	C₂²×D₃	—	4	4	no	—
175	C₂²×D₃	—	4	8	no	—
176	C₃×Q₈	—	6	8	no	—
177	C₃×Q₈	—	12	4	no	—
178	C₃×D₄	—	6	8	no	—
179	C₃×D₄	—	12	4	no	—
180	C₃×D₄	—	12	16	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	6	4	no	—
187	C₂×C₃×C₄	—	12	4	no	—
188	C₂×C₃×C₄	#204	1	8	yes	complemented
189	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	complemented
190	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
192	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
193	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
196	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
198	C₂×Dic₃	—	2	4	no	—
199	C₂×Dic₃	—	4	4	no	—
200	D₁₂	—	2	8	no	—
201	D₁₂	—	4	4	no	—
202	D₁₂	—	4	16	no	—
203	Dic₆	—	2	8	no	—
204	Dic₆	—	4	4	no	—
205	C₃×C₈	—	2	8	no	—
206	C₃×C₈	C₂×D₁₂	1	8	yes	complemented
207	C₃×C₈	C₂×C₃×D₄	1	8	yes	complemented
208	C₂×C₃²	#492	1	4	yes	complemented
209	C₂×C₃²	C₂×#127	1	1	yes	characteristic
210	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
211	C₃×D₃	—	8	4	no	—
212	C₂⁴	—	6	2	no	—
213	C₂×Q₈	—	3	4	no	—
214	C₂×Q₈	—	6	2	no	—
215	C₂×Q₈	—	6	8	no	—
216	C₂×D₄	—	3	4	no	—
217	C₂×D₄	—	6	8	no	—
218	C₂×D₄	—	6	8	no	—
219	C₂×D₄	—	6	2	no	—
220	C₂×D₄	—	12	8	no	—
221	C₂²×C₄	—	6	2	no	—
222	C₂²×C₄	C₃×D₁₂	1	2	yes	—
223	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
224	SD₁₆	—	6	32	no	—
225	C₂×C₈	C₃×D₁₂	1	8	yes	complemented
226	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
227	C₄⋊C₄	—	6	8	no	—
228	C₄²	C₃×D₁₂	1	2	yes	—
229	C₄²	C₂²×C₃×D₃	1	2	yes	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	12	4	no	—
234	C₂²×C₃	—	12	8	no	—
235	C₂²×C₃	—	24	8	no	—
236	C₂²×C₃	#874	1	2	yes	—
237	C₂²×C₃	C₂×#204	1	4	yes	—
238	C₂²×C₃	C₃×#127	1	2	yes	—
239	C₂²×C₃	C₂×C₃×SD₁₆	1	4	yes	—
240	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
241	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
242	C₂×D₃	—	4	8	no	—
243	C₂×D₃	—	4	4	no	—
244	C₂×D₃	—	8	8	no	—
245	C₃×C₄	—	2	8	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	12	4	no	—
249	C₃×C₄	#874	1	2	yes	—
250	C₃×C₄	C₂×#204	1	8	yes	complemented
251	C₃×C₄	C₃×#127	1	2	yes	—
252	C₃×C₄	C₂×C₃×SD₁₆	1	8	yes	complemented
253	C₃×C₄	C₂²×D₁₂	1	2	yes	—
254	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
255	Dic₃	—	4	4	no	—
256	C₃²	C₂×#492	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	6	4	no	—
258	C₂³	—	12	8	no	—
259	C₂³	—	12	4	no	—
260	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
261	Q₈	—	6	8	no	—
262	Q₈	—	12	4	no	—
263	D₄	—	6	8	no	—
264	D₄	—	12	16	no	—
265	D₄	—	12	4	no	—
266	C₂×C₄	—	6	4	no	—
267	C₂×C₄	—	12	4	no	—
268	C₂×C₄	C₃×#204	1	8	yes	complemented
269	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
270	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
271	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
272	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
273	C₈	C₂×C₃×D₁₂	1	8	yes	complemented
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	—	24	4	no	—
278	C₂×C₃	#4693	1	4	yes	direct product factor
279	C₂×C₃	C₃×#492	1	4	yes	complemented
280	C₂×C₃	C₂×#874	1	1	yes	characteristic
281	C₂×C₃	C₂²×#204	1	2	yes	—
282	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
283	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
284	D₃	—	8	4	no	—
285	C₂²	—	12	4	no	—
286	C₂²	—	12	8	no	—
287	C₂²	—	24	8	no	—
288	C₂²	C₃×#874	1	2	yes	—
289	C₂²	C₂×C₃×#204	1	4	yes	—
290	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
291	C₄	—	12	4	no	—
292	C₄	C₃×#874	1	2	yes	—
293	C₄	C₂×C₃×#204	1	8	yes	complemented
294	C₄	C₂²×C₃×D₁₂	1	2	yes	—
295	C₃	—	2	1	no	—
296	C₃	C₂×#4693	1	1	yes	characteristic, direct product factor
297	C₃	C₂×C₃×#492	1	1	yes	characteristic, complemented
298	C₂	—	24	4	no	—
299	C₂	C₃×#4693	1	4	yes	direct product factor
300	C₂	C₂×C₃×#874	1	1	yes	characteristic
301	C₂	C₂²×C₃×#204	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂×C₃×#874	C₂
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₂×C₃×#492
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#204	C₂	semidirect
3. C₂×C₃×#869	C₂	semidirect
4. C₂×C₃²×C₄×C₈	C₂	semidirect
5. C₃×#4693	C₂	direct (N × H)
7. C₂×#4693	C₃	direct (N × H)
15. C₂×C₃×#204	C₂²	semidirect
19. C₃×#869	C₂²	semidirect
20. C₃²×C₄×C₈	C₂²	semidirect
24. C₂²×#204	C₂×C₃	semidirect
26. C₂×#869	C₂×C₃	semidirect
29. C₂×C₃×C₄×C₈	D₃	semidirect
30. C₂×C₃×C₄×C₈	C₂×C₃	semidirect
31. #4693	C₂×C₃	direct (N × H)
44. C₂×C₃²×C₈	D₄	semidirect
72. C₂×#204	C₂²×C₃	semidirect
76. #869	C₂²×C₃	semidirect
78. C₃×C₄×C₈	C₂×D₃	semidirect
79. C₃×C₄×C₈	C₂²×C₃	semidirect
84. C₂×C₃²×C₄	SD₁₆	semidirect
96. C₃²×C₈	C₂×D₄	semidirect
101. C₂×C₄×C₈	C₃×D₃	semidirect
131. C₂×C₃×C₈	D₁₂	semidirect
132. C₂×C₃×C₈	C₃×D₄	semidirect
151. C₃²×C₄	C₂×SD₁₆	semidirect
166. C₄×C₈	C₂×C₃×D₃	semidirect
188. C₂×C₃×C₄	#204	semidirect
189. C₂×C₃×C₄	C₃×SD₁₆	semidirect
206. C₃×C₈	C₂×D₁₂	semidirect
207. C₃×C₈	C₂×C₃×D₄	semidirect
208. C₂×C₃²	#492	semidirect
225. C₂×C₈	C₃×D₁₂	semidirect
250. C₃×C₄	C₂×#204	semidirect
252. C₃×C₄	C₂×C₃×SD₁₆	semidirect
256. C₃²	C₂×#492	semidirect
268. C₂×C₄	C₃×#204	semidirect
273. C₈	C₂×C₃×D₁₂	semidirect
278. C₂×C₃	#4693	direct (N × H)
279. C₂×C₃	C₃×#492	semidirect
293. C₄	C₂×C₃×#204	semidirect
296. C₃	C₂×#4693	direct (N × H)
297. C₃	C₂×C₃×#492	semidirect
299. C₂	C₃×#4693	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#204	2
C₂×C₃×#874	2
C₂×C₃×#869	2
C₂×C₃²×C₄×C₈	2
C₃×#4693	2
C₂×C₃×#492	3
C₂×#4693	3

Subgroups — C2×C3×#4693 #11874147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4693 #11874147