Subgroups — C₂×C₃×#4709 #11858780

Order 1152.

897 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₃×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4709	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃×#433	—	3	1	no	maximal
8	C₂×#4709	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	2	yes	—
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×#872	C₄	1	4	yes	—
17	C₃×#872	C₂²	1	4	yes	—
18	C₃²×#97	C₂²	1	4	yes	complemented
19	C₃×(C₂×D₃)⋊C₈	C₂²	1	8	yes	—
20	C₃×Dic₃⋊C₈	C₂²	1	8	yes	complemented
21	C₃×C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
22	C₂×C₃×#117	—	3	1	no	—
23	C₂×#872	C₂×C₃	1	1	yes	characteristic
24	C₃×#433	—	3	8	no	—
25	C₂×C₃×C₄⋊C₈	—	3	2	no	—
26	C₂×C₃×C₂²⋊C₈	—	3	2	no	—
27	C₂×C₃×#97	—	2	1	no	—
28	C₂×C₃×#97	D₃	1	1	yes	characteristic, complemented
29	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×C₈	—	3	1	no	—
31	C₂×(C₂×D₃)⋊C₈	C₂×C₃	1	2	yes	—
32	C₂×Dic₃⋊C₈	C₂×C₃	1	2	yes	complemented
33	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
34	#4709	C₂×C₃	1	8	yes	direct product factor
35	C₂³×C₃×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
37	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
39	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₃	—	2	4	no	—
41	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₄×D₃	C₂³	1	2	yes	—
43	C₂×C₃²×C₈	—	2	4	no	—
44	C₂×C₃²×C₈	C₂³	1	4	yes	—
45	C₃²×C₄²	C₂×C₄	1	2	yes	—
46	C₃²×C₄²	C₂³	1	2	yes	—
47	C₃×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
48	C₃×Dic₃⋊C₄	C₂×C₄	1	8	yes	—
49	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
50	C₃×C₄×Dic₃	C₂³	1	2	yes	—
51	C₂×C₃×C₃⋊C₈	—	2	4	no	—
52	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
53	C₂×#433	—	3	1	no	Sylow
54	C₂³×C₃×C₄	—	3	1	no	—
55	C₂²×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₃×C₈	—	2	2	no	—
57	C₂²×C₃×C₈	—	3	2	no	—
58	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
59	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
60	C₃×#117	—	3	4	no	—
61	C₃×#117	—	3	4	no	—
62	C₂×C₃×C₄⋊C₄	—	3	2	no	—
63	C₂×C₃×C₂²⋊C₄	—	3	2	no	—
64	C₂×C₃×C₄²	—	2	1	no	—
65	C₂×C₃×C₄²	—	3	1	no	—
66	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₂×(C₂×D₃)⋊C₄	C₃×C₄	1	2	yes	—
69	C₂×Dic₃⋊C₄	C₃×C₄	1	2	yes	—
70	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
72	#872	C₃×C₄	1	4	yes	—
73	#872	C₂²×C₃	1	4	yes	—
74	C₃×C₄⋊C₈	—	3	8	no	—
75	C₃×C₂²⋊C₈	—	3	8	no	—
76	C₃×#97	—	2	4	no	—
77	C₃×#97	C₂×D₃	1	4	yes	complemented
78	C₃×#97	C₂²×C₃	1	4	yes	complemented
79	C₃×C₄×C₈	—	3	4	no	—
80	(C₂×D₃)⋊C₈	C₂²×C₃	1	8	yes	—
81	Dic₃⋊C₈	C₂²×C₃	1	8	yes	complemented
82	C₄×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
83	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₃×D₃	—	2	2	no	—
85	C₂²×C₃×D₃	—	2	4	no	—
86	C₂²×C₃×D₃	C₂²×C₄	1	2	yes	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
89	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	#41	1	4	yes	—
91	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
92	C₂×C₃×Dic₃	—	2	2	no	—
93	C₂×C₃×Dic₃	—	2	4	no	—
94	C₂×C₃×Dic₃	C₂²×C₄	1	4	yes	—
95	C₂×C₃×Dic₃	C₂²×C₄	1	2	yes	—
96	C₃×C₄×D₃	—	2	4	no	—
97	C₃×C₄×D₃	—	2	4	no	—
98	C₃²×C₈	—	2	4	no	—
99	C₃×C₃⋊C₈	—	2	4	no	—
100	C₂×#117	—	3	1	no	—
101	#433	—	3	8	no	—
102	C₂×C₄⋊C₈	—	3	2	no	—
103	C₂×C₂²⋊C₈	—	3	2	no	—
104	C₂×#97	C₃×D₃	1	1	yes	characteristic, complemented
105	C₂×C₄×C₈	—	3	1	no	—
106	C₂⁴×C₃	—	3	1	no	—
107	C₂³×D₃	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	3	2	no	—
111	C₂²×C₃×C₄	—	3	1	no	—
112	C₂²×C₃×C₄	—	3	2	no	—
113	C₂²×C₃×C₄	—	3	2	no	—
114	C₂²×C₃×C₄	—	6	4	no	—
115	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
116	C₂²×C₃×C₄	C₂×C₃×C₄	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	C₂×C₃×C₄	1	2	yes	—
120	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×C₄×D₃	—	2	4	no	—
122	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
123	C₂×C₄×D₃	C₂³×C₃	1	2	yes	—
124	C₂×C₃×C₈	—	2	8	no	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	—	2	4	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	—	3	4	no	—
129	C₂×C₃×C₈	—	6	4	no	—
130	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
131	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
132	C₃×C₄⋊C₄	—	3	8	no	—
133	C₃×C₂²⋊C₄	—	3	8	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	2	2	no	—
136	C₃×C₄²	—	3	2	no	—
137	C₃×C₄²	—	3	2	no	—
138	C₃×C₄²	C₄×D₃	1	2	yes	—
139	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
140	C₃×C₄²	C₂²×D₃	1	2	yes	—
141	C₃×C₄²	C₂³×C₃	1	2	yes	—
142	(C₂×D₃)⋊C₄	C₂×C₃×C₄	1	8	yes	—
143	Dic₃⋊C₄	C₂×C₃×C₄	1	8	yes	—
144	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
145	C₄×Dic₃	C₂³×C₃	1	2	yes	—
146	C₂×C₃⋊C₈	—	2	4	no	—
147	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
148	C₂²×C₃²	#117	1	2	yes	—
149	C₂²×C₃²	#131	1	4	yes	—
150	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₃×D₃	—	2	2	no	—
152	C₂×C₃×D₃	—	2	4	no	—
153	C₂×C₃×D₃	—	4	4	no	—
154	C₃²×C₄	—	2	4	no	—
155	C₃²×C₄	#117	1	2	yes	—
156	C₃²×C₄	C₂×#41	1	2	yes	—
157	C₃×Dic₃	—	2	2	no	—
158	C₃×Dic₃	—	2	4	no	—
159	C₂³×C₄	—	3	1	no	—
160	C₂²×C₈	—	3	2	no	—
161	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
162	#117	—	3	4	no	—
163	#117	—	3	4	no	—
164	C₂×C₄⋊C₄	—	3	2	no	—
165	C₂×C₂²⋊C₄	—	3	2	no	—
166	C₂×C₄²	—	3	1	no	—
167	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
168	C₄⋊C₈	—	3	8	no	—
169	C₂²⋊C₈	—	3	8	no	—
170	#97	C₂×C₃×D₃	1	4	yes	complemented
171	C₄×C₈	—	3	4	no	—
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	3	2	no	—
174	C₂³×C₃	—	6	2	no	—
175	C₂³×C₃	—	6	4	no	—
176	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂²×D₃	—	2	2	no	—
179	C₂²×D₃	—	2	4	no	—
180	C₂²×D₃	C₂²×C₃×C₄	1	2	yes	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	3	4	no	—
188	C₂×C₃×C₄	—	3	2	no	—
189	C₂×C₃×C₄	—	4	4	no	—
190	C₂×C₃×C₄	—	6	2	no	—
191	C₂×C₃×C₄	—	6	4	no	—
192	C₂×C₃×C₄	—	6	4	no	—
193	C₂×C₃×C₄	—	6	4	no	—
194	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	#235	1	4	yes	—
197	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
198	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₃×#41	1	4	yes	—
200	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
202	C₂×Dic₃	—	2	2	no	—
203	C₂×Dic₃	—	2	4	no	—
204	C₂×Dic₃	C₂²×C₃×C₄	1	4	yes	—
205	C₂×Dic₃	C₂²×C₃×C₄	1	2	yes	—
206	C₄×D₃	—	2	4	no	—
207	C₄×D₃	—	2	4	no	—
208	C₃×C₈	—	2	8	no	—
209	C₃×C₈	—	2	4	no	—
210	C₃×C₈	—	2	4	no	—
211	C₃×C₈	—	6	4	no	—
212	C₃⋊C₈	—	2	4	no	—
213	C₂×C₃²	#433	1	4	yes	complemented
214	C₂×C₃²	C₂×#117	1	1	yes	characteristic
215	C₂×C₃²	C₂×#131	1	2	yes	—
216	C₃×D₃	—	4	2	no	—
217	C₂⁴	—	3	1	no	—
218	C₂²×C₄	—	3	2	no	—
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	3	2	no	—
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	—	6	4	no	—
223	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	3	4	no	—
227	C₂×C₈	—	6	4	no	—
228	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
229	C₄⋊C₄	—	3	8	no	—
230	C₂²⋊C₄	—	3	8	no	—
231	C₄²	—	3	2	no	—
232	C₄²	—	3	2	no	—
233	C₄²	C₃×C₄×D₃	1	2	yes	—
234	C₄²	C₂²×C₃×D₃	1	2	yes	—
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	6	2	no	—
239	C₂²×C₃	—	6	4	no	—
240	C₂²×C₃	—	12	4	no	—
241	C₂²×C₃	#872	1	2	yes	—
242	C₂²×C₃	#907	1	4	yes	—
243	C₂²×C₃	C₃×#117	1	2	yes	—
244	C₂²×C₃	C₃×#131	1	4	yes	—
245	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
246	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
247	C₂×D₃	—	2	4	no	—
248	C₂×D₃	—	2	2	no	—
249	C₂×D₃	—	4	4	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	4	4	no	—
255	C₃×C₄	—	6	2	no	—
256	C₃×C₄	—	6	4	no	—
257	C₃×C₄	#872	1	2	yes	—
258	C₃×C₄	C₃×#117	1	2	yes	—
259	C₃×C₄	C₂×#235	1	2	yes	—
260	C₃×C₄	C₂×C₃×#41	1	2	yes	—
261	Dic₃	—	2	4	no	—
262	Dic₃	—	2	2	no	—
263	C₃²	C₂×#433	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	3	2	no	—
265	C₂³	—	6	4	no	—
266	C₂³	—	6	2	no	—
267	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	3	4	no	—
270	C₂×C₄	—	3	2	no	—
271	C₂×C₄	—	6	4	no	—
272	C₂×C₄	—	6	2	no	—
273	C₂×C₄	—	6	4	no	—
274	C₂×C₄	—	6	4	no	—
275	C₂×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
276	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
277	C₂×C₄	C₃×#235	1	4	yes	—
278	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
279	C₈	—	2	4	no	—
280	C₈	—	6	4	no	—
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	4	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	12	2	no	—
285	C₂×C₃	#4709	1	4	yes	direct product factor
286	C₂×C₃	C₃×#433	1	4	yes	complemented
287	C₂×C₃	C₂×#872	1	1	yes	characteristic
288	C₂×C₃	C₂×#907	1	2	yes	—
289	C₂×C₃	C₂×C₃×#117	1	1	yes	characteristic
290	C₂×C₃	C₂×C₃×#131	1	2	yes	—
291	D₃	—	4	2	no	—
292	C₂²	—	6	4	no	—
293	C₂²	—	6	2	no	—
294	C₂²	—	12	4	no	—
295	C₂²	C₃×#872	1	2	yes	—
296	C₂²	C₃×#907	1	4	yes	—
297	C₂²	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	6	4	no	—
300	C₄	—	6	2	no	—
301	C₄	C₃×#872	1	2	yes	—
302	C₄	C₂×C₃×#235	1	2	yes	—
303	C₃	—	2	1	no	—
304	C₃	C₂×#4709	1	1	yes	characteristic, direct product factor
305	C₃	C₂×C₃×#433	1	1	yes	characteristic, complemented
306	C₂	—	12	2	no	—
307	C₂	C₃×#4709	1	4	yes	direct product factor
308	C₂	C₂×C₃×#872	1	1	yes	characteristic
309	C₂	C₂×C₃×#907	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂³×C₃²	C₂²×C₄
prime radical	C₂³×C₃×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₂×C₃×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#97	C₂	semidirect
4. C₂×C₃×Dic₃⋊C₈	C₂	semidirect
5. C₂×C₃×C₄×C₃⋊C₈	C₂	semidirect
6. C₃×#4709	C₂	direct (N × H)
8. C₂×#4709	C₃	direct (N × H)
18. C₃²×#97	C₂²	semidirect
20. C₃×Dic₃⋊C₈	C₂²	semidirect
21. C₃×C₄×C₃⋊C₈	C₂²	semidirect
28. C₂×C₃×#97	D₃	semidirect
29. C₂×C₃×#97	C₂×C₃	semidirect
32. C₂×Dic₃⋊C₈	C₂×C₃	semidirect
33. C₂×C₄×C₃⋊C₈	C₂×C₃	semidirect
34. #4709	C₂×C₃	direct (N × H)
77. C₃×#97	C₂×D₃	semidirect
78. C₃×#97	C₂²×C₃	semidirect
81. Dic₃⋊C₈	C₂²×C₃	semidirect
82. C₄×C₃⋊C₈	C₂²×C₃	semidirect
104. C₂×#97	C₃×D₃	semidirect
170. #97	C₂×C₃×D₃	semidirect
213. C₂×C₃²	#433	semidirect
263. C₃²	C₂×#433	semidirect
285. C₂×C₃	#4709	direct (N × H)
286. C₂×C₃	C₃×#433	semidirect
304. C₃	C₂×#4709	direct (N × H)
305. C₃	C₂×C₃×#433	semidirect
307. C₂	C₃×#4709	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×#872	2
C₂×C₃²×#97	2
C₂×C₃×(C₂×D₃)⋊C₈	2
C₂×C₃×Dic₃⋊C₈	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×#4709	2
C₂×C₃×#433	3
C₂×#4709	3

Subgroups — C2×C3×#4709 #11858780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4709 #11858780