Subgroups — C₂×C₃×#4901 #11899379

Order 1152.

785 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#901	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4901	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#798	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×D₃×C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#504	—	3	1	no	maximal
8	C₂×#4901	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃×#235	C₄	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₄	1	1	yes	characteristic
12	C₃×#901	C₄	1	4	yes	—
13	C₃×#901	C₂²	1	4	yes	—
14	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
15	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
16	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃²×C₁₆	C₂²	1	4	yes	complemented
19	C₃×#812	C₂²	1	4	yes	complemented
20	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
21	C₃×#798	C₂²	1	8	yes	complemented
22	C₃×D₃×C₁₆	C₂²	1	8	yes	complemented
23	C₂×C₃×#131	—	3	1	no	—
24	C₂×#901	C₂×C₃	1	1	yes	characteristic
25	C₃×#504	—	3	8	no	—
26	C₂×C₃×M₃₂	—	3	1	no	—
27	C₂×C₃×M₃₂	—	3	2	no	—
28	C₂²×C₃×C₁₆	—	2	1	no	—
29	C₂²×C₃×C₁₆	—	3	2	no	—
30	C₂²×C₃×C₁₆	D₃	1	1	yes	characteristic, complemented
31	C₂²×C₃×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#812	C₂×C₃	1	1	yes	characteristic, complemented
33	#4901	C₂×C₃	1	8	yes	direct product factor
34	C₂×#798	C₂×C₃	1	2	yes	complemented
35	C₂×D₃×C₁₆	C₂×C₃	1	2	yes	complemented
36	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#67	C₈	1	2	yes	—
38	C₃×#235	C₈	1	4	yes	—
39	C₃×#235	C₂×C₄	1	4	yes	—
40	C₂×C₃×D₁₂	C₈	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
42	C₂×C₃×Dic₆	C₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₃²×C₈	C₂³	1	2	yes	—
47	C₃×#208	C₂×C₄	1	4	yes	—
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₃×ΓL(1,𝔽₂₅)	C₂×C₄	1	8	yes	—
50	C₃×D₃×C₈	C₂×C₄	1	4	yes	—
51	C₃×D₃×C₈	C₂³	1	4	yes	—
52	C₃²×C₁₆	C₂³	1	2	yes	—
53	C₃²×C₁₆	C₂³	1	2	yes	—
54	C₃×C₃⋊C₁₆	C₂³	1	4	yes	—
55	C₂×#504	—	3	1	no	Sylow
56	C₂×C₃×#41	—	3	1	no	—
57	C₂×#235	C₃×C₄	1	1	yes	characteristic
58	C₃×#131	—	3	4	no	—
59	C₃×#131	—	3	4	no	—
60	C₂×C₃×M₁₆	—	3	1	no	—
61	C₂×C₃×M₁₆	—	3	2	no	—
62	C₂²×C₃×C₈	—	2	1	no	—
63	C₂²×C₃×C₈	—	3	2	no	—
64	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₂×#208	C₃×C₄	1	1	yes	characteristic
67	#901	C₃×C₄	1	4	yes	—
68	#901	C₂²×C₃	1	4	yes	—
69	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
70	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
71	C₃×M₃₂	—	3	4	no	—
72	C₃×M₃₂	—	3	8	no	—
73	C₂×C₃×C₁₆	—	2	1	no	—
74	C₂×C₃×C₁₆	—	2	1	no	—
75	C₂×C₃×C₁₆	—	2	4	no	—
76	C₂×C₃×C₁₆	—	3	2	no	—
77	C₂×C₃×C₁₆	—	3	8	no	—
78	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
80	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	complemented
81	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	complemented
84	#812	C₂²×C₃	1	4	yes	complemented
85	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
86	#798	C₂²×C₃	1	8	yes	complemented
87	D₃×C₁₆	C₂²×C₃	1	8	yes	complemented
88	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₃×D₃	C₂×C₈	1	2	yes	—
90	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
92	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
94	C₃×#67	C₂×C₈	1	8	yes	—
95	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
96	C₃×D₁₂	C₂×C₈	1	4	yes	—
97	C₃×C₄×D₃	C₂×C₈	1	4	yes	—
98	C₃×C₄×D₃	C₂²×C₄	1	4	yes	—
99	C₃×Dic₆	C₂×C₈	1	4	yes	—
100	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₃²×C₈	C₂²×C₄	1	2	yes	—
102	C₃²×C₈	C₂⁴	1	1	yes	characteristic
103	C₃×C₃⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×#131	—	3	1	no	—
105	#504	—	3	8	no	—
106	C₂×M₃₂	—	3	1	no	—
107	C₂×M₃₂	—	3	2	no	—
108	C₂²×C₁₆	—	3	2	no	—
109	C₂²×C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
110	C₃×#41	—	3	4	no	—
111	C₃×#41	—	3	4	no	—
112	C₂×C₃×Q₈	—	3	1	no	—
113	C₂×C₃×D₄	—	3	2	no	—
114	C₂×C₃×D₄	—	3	1	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	—	3	2	no	—
117	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂×#67	C₃×C₈	1	2	yes	—
120	#235	C₃×C₈	1	4	yes	—
121	#235	C₂×C₃×C₄	1	4	yes	—
122	C₂×D₁₂	C₃×C₈	1	1	yes	characteristic
123	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
124	C₂×Dic₆	C₃×C₈	1	1	yes	characteristic
125	C₃×M₁₆	—	3	4	no	—
126	C₃×M₁₆	—	3	8	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	3	2	no	—
132	C₂×C₃×C₈	—	3	4	no	—
133	C₂×C₃×C₈	—	3	4	no	—
134	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
135	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
136	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
138	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
140	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
142	#208	C₂×C₃×C₄	1	4	yes	—
143	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
144	ΓL(1,𝔽₂₅)	C₂×C₃×C₄	1	8	yes	—
145	D₃×C₈	C₂×C₃×C₄	1	4	yes	—
146	D₃×C₈	C₂³×C₃	1	4	yes	—
147	C₃×C₁₆	—	2	2	no	—
148	C₃×C₁₆	—	2	2	no	—
149	C₃×C₁₆	—	3	4	no	—
150	C₃×C₁₆	C₂²×D₃	1	2	yes	—
151	C₃×C₁₆	C₂²×D₃	1	2	yes	—
152	C₃×C₁₆	C₂³×C₃	1	2	yes	—
153	C₃×C₁₆	C₂³×C₃	1	2	yes	—
154	C₃⋊C₁₆	C₂³×C₃	1	4	yes	—
155	C₂²×C₃²	—	2	2	no	—
156	C₂²×C₃²	C₂²×C₈	1	2	yes	—
157	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₃×D₃	—	2	4	no	—
159	C₂×C₃×D₃	C₂²×C₈	1	4	yes	—
160	C₃²×C₄	C₂²×C₈	1	2	yes	—
161	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
162	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
163	C₃×Dic₃	C₂²×C₈	1	4	yes	—
164	C₂×#41	—	3	1	no	—
165	#131	—	3	4	no	—
166	#131	—	3	4	no	—
167	C₂×M₁₆	—	3	1	no	—
168	C₂×M₁₆	—	3	2	no	—
169	C₂²×C₈	—	3	2	no	—
170	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
171	M₃₂	—	3	4	no	—
172	M₃₂	—	3	8	no	—
173	C₂×C₁₆	—	3	2	no	—
174	C₂×C₁₆	—	3	8	no	—
175	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
176	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂×C₃×D₃	1	4	yes	complemented
178	C₂³×C₃	—	2	1	no	—
179	C₂³×C₃	—	3	2	no	—
180	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
181	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
182	C₂²×D₃	C₂×C₃×C₈	1	2	yes	—
183	C₃×Q₈	—	3	4	no	—
184	C₃×D₄	—	3	8	no	—
185	C₃×D₄	—	3	4	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	3	2	no	—
191	C₂×C₃×C₄	—	3	4	no	—
192	C₂×C₃×C₄	—	3	4	no	—
193	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
194	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
195	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
199	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
201	#67	C₂×C₃×C₈	1	8	yes	—
202	C₂×Dic₃	C₂×C₃×C₈	1	2	yes	—
203	D₁₂	C₂×C₃×C₈	1	4	yes	—
204	C₄×D₃	C₂×C₃×C₈	1	4	yes	—
205	C₄×D₃	C₂²×C₃×C₄	1	4	yes	—
206	Dic₆	C₂×C₃×C₈	1	4	yes	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	—	2	2	no	—
210	C₃×C₈	—	3	4	no	—
211	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
212	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
213	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
214	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
215	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
216	C₃×C₈	C₂⁴×C₃	1	1	yes	characteristic
217	C₃⋊C₈	C₂²×C₃×C₄	1	4	yes	—
218	C₂×C₃²	—	2	2	no	—
219	C₂×C₃²	#504	1	2	yes	complemented
220	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
221	C₃×D₃	—	2	4	no	—
222	#41	—	3	4	no	—
223	#41	—	3	4	no	—
224	C₂×Q₈	—	3	1	no	—
225	C₂×D₄	—	3	2	no	—
226	C₂×D₄	—	3	1	no	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
229	M₁₆	—	3	4	no	—
230	M₁₆	—	3	8	no	—
231	C₂×C₈	—	3	2	no	—
232	C₂×C₈	—	3	4	no	—
233	C₂×C₈	—	3	4	no	—
234	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
235	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
236	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
238	C₁₆	—	3	4	no	—
239	C₁₆	C₂²×C₃×D₃	1	2	yes	—
240	C₁₆	C₂²×C₃×D₃	1	2	yes	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	3	4	no	—
247	C₂²×C₃	—	6	4	no	—
248	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
249	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
250	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
251	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
252	C₂×D₃	—	2	4	no	—
253	C₂×D₃	C₂²×C₃×C₈	1	4	yes	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	3	4	no	—
258	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
259	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
260	C₃×C₄	C₂²×C₃×C₈	1	2	yes	—
261	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
262	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
263	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
264	Dic₃	C₂²×C₃×C₈	1	4	yes	—
265	C₃²	C₂×#504	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	3	2	no	—
267	C₂³	C₃×D₃×C₈	1	1	yes	characteristic
268	Q₈	—	3	4	no	—
269	D₄	—	3	8	no	—
270	D₄	—	3	4	no	—
271	C₂×C₄	—	3	2	no	—
272	C₂×C₄	—	3	4	no	—
273	C₂×C₄	—	3	4	no	—
274	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
275	C₂×C₄	C₃×D₃×C₈	1	2	yes	—
276	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
278	C₈	—	3	4	no	—
279	C₈	C₂×C₃×C₄×D₃	1	1	yes	characteristic
280	C₈	C₂×C₃×C₄×D₃	1	2	yes	—
281	C₈	C₂³×C₃×D₃	1	1	yes	characteristic
282	C₂×C₃	—	2	4	no	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	2	no	—
287	C₂×C₃	—	6	4	no	—
288	C₂×C₃	#4901	1	2	yes	direct product factor
289	C₂×C₃	C₃×#504	1	2	yes	complemented
290	C₂×C₃	C₂²×D₃×C₈	1	1	yes	characteristic
291	C₂×C₃	C₂³×C₃×C₈	1	1	yes	characteristic
292	D₃	—	2	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	3	4	no	—
295	C₂²	—	6	4	no	—
296	C₂²	C₂×C₃×D₃×C₈	1	2	yes	—
297	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
298	C₄	—	3	4	no	—
299	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
300	C₄	C₂×C₃×D₃×C₈	1	2	yes	—
301	C₄	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
302	C₃	—	2	1	no	—
303	C₃	C₂×#4901	1	1	yes	characteristic, direct product factor
304	C₃	C₂×C₃×#504	1	1	yes	characteristic, complemented
305	C₂	—	2	2	no	—
306	C₂	—	6	4	no	—
307	C₂	C₃×#4901	1	2	yes	direct product factor
308	C₂	C₂²×C₃×D₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₁₆	C₂×D₃
commutator	C₂×C₃	C₂³×C₃×C₈
Frattini	C₈	C₂³×C₃×D₃
Fitting	C₂²×C₃²×C₁₆	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃×#235	C₄
cosocle	C₃×C₈	C₂⁴×C₃
hypercenter	C₂²×C₃×C₁₆	D₃
hypocenter	C₃	C₂×C₃×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₁₆ ◁ C₂²×C₃×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₁₆	C₂	semidirect
3. C₂×C₃×#812	C₂	semidirect
4. C₃×#4901	C₂	direct (N × H)
5. C₂×C₃×#798	C₂	semidirect
6. C₂×C₃×D₃×C₁₆	C₂	semidirect
8. C₂×#4901	C₃	direct (N × H)
18. C₂×C₃²×C₁₆	C₂²	semidirect
19. C₃×#812	C₂²	semidirect
21. C₃×#798	C₂²	semidirect
22. C₃×D₃×C₁₆	C₂²	semidirect
30. C₂²×C₃×C₁₆	D₃	semidirect
31. C₂²×C₃×C₁₆	C₂×C₃	semidirect
32. C₂×#812	C₂×C₃	semidirect
33. #4901	C₂×C₃	direct (N × H)
34. C₂×#798	C₂×C₃	semidirect
35. C₂×D₃×C₁₆	C₂×C₃	semidirect
80. C₂×C₃×C₁₆	C₂×D₃	semidirect
83. C₂×C₃×C₁₆	C₂²×C₃	semidirect
84. #812	C₂²×C₃	semidirect
86. #798	C₂²×C₃	semidirect
87. D₃×C₁₆	C₂²×C₃	semidirect
109. C₂²×C₁₆	C₃×D₃	semidirect
177. C₂×C₁₆	C₂×C₃×D₃	semidirect
219. C₂×C₃²	#504	semidirect
265. C₃²	C₂×#504	semidirect
288. C₂×C₃	#4901	direct (N × H)
289. C₂×C₃	C₃×#504	semidirect
303. C₃	C₂×#4901	direct (N × H)
304. C₃	C₂×C₃×#504	semidirect
307. C₂	C₃×#4901	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×#901	2
C₂²×C₃²×C₁₆	2
C₂×C₃×#812	2
C₃×#4901	2
C₂×C₃×#798	2
C₂×C₃×D₃×C₁₆	2
C₂×C₃×#504	3
C₂×#4901	3

Subgroups — C2×C3×#4901 #11899379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4901 #11899379