Subgroups — C₂×C₃²×#473 #11770115

Order 1152.

1344 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#137	C₂	1	2	yes	maximal
2	C₂²×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
4	C₃²×#473	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#101	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#473	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
10	C₃²×#137	—	2	4	no	—
11	C₃²×#137	C₂²	1	8	yes	—
12	C₂×C₃²×Q₁₆	—	2	2	no	—
13	C₂×C₃²×Q₁₆	—	2	1	no	—
14	C₂×C₃²×Q₁₆	—	2	1	no	—
15	C₂×C₃²×Q₁₆	—	2	2	no	—
16	C₂×C₃²×Q₁₆	C₂²	1	4	yes	complemented
17	C₂×C₃²×SD₁₆	—	2	2	no	—
18	C₃²×#131	—	2	2	no	—
19	C₃²×#131	C₂²	1	4	yes	—
20	C₂×C₃²×M₁₆	—	2	1	no	—
21	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
23	C₂²×C₃²×C₈	—	2	1	no	—
24	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
25	C₃²×#108	C₂²	1	4	yes	complemented
26	C₃²×#101	C₂²	1	8	yes	complemented
27	C₂×C₃×#137	C₂×C₃	1	8	yes	—
28	C₂²×C₃×Q₁₆	C₂×C₃	1	4	yes	complemented
29	C₂×C₃×#131	C₂×C₃	1	4	yes	—
30	C₃×#473	C₂×C₃	1	32	yes	direct product factor
31	C₂×C₃×#108	C₂×C₃	1	4	yes	complemented
32	C₂×C₃×#101	C₂×C₃	1	8	yes	complemented
33	C₃²×#41	—	2	1	no	—
34	C₃²×#41	—	2	1	no	—
35	C₃²×#41	D₄	1	2	yes	—
36	C₃²×#41	C₂³	1	2	yes	—
37	C₂×C₃²×Q₈	—	2	2	no	—
38	C₂×C₃²×Q₈	—	2	2	no	—
39	C₂×C₃²×Q₈	—	2	2	no	—
40	C₂×C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
43	C₂×C₃²×Q₈	C₂³	1	4	yes	—
44	C₂×C₃²×D₄	—	2	1	no	—
45	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
46	C₂²×C₃²×C₄	—	2	1	no	—
47	C₂²×C₃²×C₄	—	2	2	no	—
48	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₃²×Q₁₆	—	2	8	no	—
50	C₃²×Q₁₆	—	2	4	no	—
51	C₃²×Q₁₆	—	2	4	no	—
52	C₃²×SD₁₆	—	2	8	no	—
53	C₃²×M₁₆	—	2	1	no	—
54	C₃²×M₁₆	—	2	2	no	—
55	C₃²×M₁₆	—	2	4	no	—
56	C₃²×M₁₆	C₂³	1	2	yes	—
57	C₃²×M₁₆	C₂³	1	4	yes	—
58	C₂×C₃²×C₈	—	2	1	no	—
59	C₂×C₃²×C₈	—	2	2	no	—
60	C₂×C₃²×C₈	—	2	4	no	—
61	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
62	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
63	C₂×C₃²×C₈	D₄	1	2	yes	—
64	C₂×C₃²×C₈	C₂³	1	2	yes	—
65	C₂×#473	C₃²	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
67	C₂²×C₃×Q₈	C₂²×C₃	1	8	yes	—
68	C₃×#137	—	2	16	no	—
69	C₃×#137	C₂²×C₃	1	32	yes	—
70	C₂×C₃×Q₁₆	—	2	8	no	—
71	C₂×C₃×Q₁₆	—	2	4	no	—
72	C₂×C₃×Q₁₆	—	2	4	no	—
73	C₂×C₃×Q₁₆	—	2	8	no	—
74	C₂×C₃×Q₁₆	C₂²×C₃	1	16	yes	complemented
75	C₂×C₃×SD₁₆	—	2	8	no	—
76	C₃×#131	—	2	8	no	—
77	C₃×#131	C₂²×C₃	1	16	yes	—
78	C₂×C₃×M₁₆	—	2	4	no	—
79	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
80	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
81	C₂²×C₃×C₈	—	2	4	no	—
82	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
83	C₃×#108	C₂²×C₃	1	16	yes	complemented
84	C₃×#101	C₂²×C₃	1	32	yes	complemented
85	C₂³×C₃²	—	2	1	no	—
86	C₂³×C₃²	#41	1	1	yes	characteristic
87	C₃²×Q₈	—	2	1	no	—
88	C₃²×Q₈	—	2	4	no	—
89	C₃²×Q₈	—	2	4	no	—
90	C₃²×Q₈	—	4	2	no	—
91	C₃²×Q₈	—	4	2	no	—
92	C₃²×Q₈	C₂×D₄	1	2	yes	—
93	C₃²×D₄	—	2	2	no	—
94	C₃²×D₄	—	2	2	no	—
95	C₃²×D₄	—	2	1	no	—
96	C₃²×D₄	C₂×D₄	1	2	yes	—
97	C₂×C₃²×C₄	—	2	1	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	4	2	no	—
102	C₂×C₃²×C₄	—	4	2	no	—
103	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃²×C₄	#41	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
109	C₃²×C₈	—	2	2	no	—
110	C₃²×C₈	—	2	4	no	—
111	C₃²×C₈	C₂×D₄	1	2	yes	—
112	C₃²×C₈	C₂×D₄	1	2	yes	—
113	C₂×#137	C₂×C₃²	1	2	yes	—
114	C₂²×Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
115	C₂×#131	C₂×C₃²	1	1	yes	characteristic
116	#473	C₂×C₃²	1	8	yes	direct product factor
117	C₂×#108	C₂×C₃²	1	1	yes	characteristic, complemented
118	C₂×#101	C₂×C₃²	1	2	yes	complemented
119	C₃×#41	—	2	4	no	—
120	C₃×#41	—	2	4	no	—
121	C₃×#41	C₃×D₄	1	8	yes	—
122	C₃×#41	C₂³×C₃	1	8	yes	—
123	C₂×C₃×Q₈	—	2	8	no	—
124	C₂×C₃×Q₈	—	2	8	no	—
125	C₂×C₃×Q₈	—	2	8	no	—
126	C₂×C₃×Q₈	—	2	8	no	—
127	C₂×C₃×Q₈	—	2	8	no	—
128	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
129	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
130	C₂×C₃×D₄	—	2	4	no	—
131	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
132	C₂²×C₃×C₄	—	2	4	no	—
133	C₂²×C₃×C₄	—	2	8	no	—
134	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
135	C₃×Q₁₆	—	2	32	no	—
136	C₃×Q₁₆	—	2	16	no	—
137	C₃×Q₁₆	—	2	16	no	—
138	C₃×SD₁₆	—	2	32	no	—
139	C₃×M₁₆	—	2	4	no	—
140	C₃×M₁₆	—	2	8	no	—
141	C₃×M₁₆	—	2	16	no	—
142	C₃×M₁₆	C₂³×C₃	1	8	yes	—
143	C₃×M₁₆	C₂³×C₃	1	16	yes	—
144	C₂×C₃×C₈	—	2	4	no	—
145	C₂×C₃×C₈	—	2	8	no	—
146	C₂×C₃×C₈	—	2	16	no	—
147	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
148	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
149	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
150	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
151	C₂²×C₃²	—	2	2	no	—
152	C₂²×C₃²	—	2	2	no	—
153	C₂²×C₃²	—	4	2	no	—
154	C₂²×C₃²	#121	1	1	yes	characteristic
155	C₂²×C₃²	#121	1	1	yes	characteristic
156	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
157	C₃²×C₄	—	2	2	no	—
158	C₃²×C₄	—	4	4	no	—
159	C₃²×C₄	#121	1	1	yes	characteristic
160	C₃²×C₄	#121	1	1	yes	characteristic
161	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
164	C₂²×Q₈	C₂²×C₃²	1	2	yes	—
165	#137	—	2	4	no	—
166	#137	C₂²×C₃²	1	8	yes	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	2	1	no	—
169	C₂×Q₁₆	—	2	1	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	C₂²×C₃²	1	4	yes	complemented
172	C₂×SD₁₆	—	2	2	no	—
173	#131	—	2	2	no	—
174	#131	C₂²×C₃²	1	4	yes	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
177	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
180	#108	C₂²×C₃²	1	4	yes	complemented
181	#101	C₂²×C₃²	1	8	yes	complemented
182	C₂³×C₃	—	2	4	no	—
183	C₂³×C₃	C₃×#41	1	4	yes	—
184	C₃×Q₈	—	2	4	no	—
185	C₃×Q₈	—	2	16	no	—
186	C₃×Q₈	—	2	16	no	—
187	C₃×Q₈	—	4	8	no	—
188	C₃×Q₈	—	4	8	no	—
189	C₃×Q₈	C₂×C₃×D₄	1	8	yes	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	—	2	8	no	—
192	C₃×D₄	—	2	4	no	—
193	C₃×D₄	C₂×C₃×D₄	1	8	yes	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	8	no	—
196	C₂×C₃×C₄	—	2	8	no	—
197	C₂×C₃×C₄	—	2	16	no	—
198	C₂×C₃×C₄	—	4	8	no	—
199	C₂×C₃×C₄	—	4	8	no	—
200	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
201	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
202	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
203	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
204	C₂×C₃×C₄	C₃×#41	1	4	yes	—
205	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
206	C₃×C₈	—	2	8	no	—
207	C₃×C₈	—	2	16	no	—
208	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
209	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
210	C₂×C₃²	—	2	1	no	—
211	C₂×C₃²	—	2	1	no	—
212	C₂×C₃²	—	4	2	no	—
213	C₂×C₃²	#473	1	2	yes	direct product factor
214	C₂×C₃²	C₂×#121	1	1	yes	characteristic
215	#41	—	2	1	no	—
216	#41	—	2	1	no	—
217	#41	C₃²×D₄	1	2	yes	—
218	#41	C₂³×C₃²	1	2	yes	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
225	C₂×Q₈	C₂³×C₃²	1	4	yes	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
231	Q₁₆	—	2	8	no	—
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	2	4	no	—
234	SD₁₆	—	2	8	no	—
235	M₁₆	—	2	1	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	C₂³×C₃²	1	2	yes	—
239	M₁₆	C₂³×C₃²	1	4	yes	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
244	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
245	C₂×C₈	C₃²×D₄	1	2	yes	—
246	C₂×C₈	C₂³×C₃²	1	2	yes	—
247	C₂²×C₃	—	2	8	no	—
248	C₂²×C₃	—	2	8	no	—
249	C₂²×C₃	—	4	8	no	—
250	C₂²×C₃	C₃×#121	1	4	yes	—
251	C₂²×C₃	C₃×#121	1	4	yes	—
252	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
253	C₃×C₄	—	2	8	no	—
254	C₃×C₄	—	4	16	no	—
255	C₃×C₄	C₃×#121	1	4	yes	—
256	C₃×C₄	C₃×#121	1	4	yes	—
257	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
258	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
259	C₃²	C₂×#473	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	—	2	1	no	—
261	C₂³	C₃²×#41	1	1	yes	characteristic
262	Q₈	—	2	1	no	—
263	Q₈	—	2	4	no	—
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	Q₈	C₂×C₃²×D₄	1	2	yes	—
268	D₄	—	2	2	no	—
269	D₄	—	2	2	no	—
270	D₄	—	2	1	no	—
271	D₄	C₂×C₃²×D₄	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
282	C₂×C₄	C₃²×#41	1	1	yes	characteristic
283	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₈	C₂×C₃²×D₄	1	2	yes	—
287	C₈	C₂×C₃²×D₄	1	2	yes	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	2	4	no	—
290	C₂×C₃	—	4	8	no	—
291	C₂×C₃	C₃×#473	1	8	yes	direct product factor
292	C₂×C₃	C₂×C₃×#121	1	4	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	C₃²×#121	1	1	yes	characteristic
297	C₂²	C₃²×#121	1	1	yes	characteristic
298	C₂²	C₂×C₃²×#41	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	4	no	—
301	C₄	C₃²×#121	1	1	yes	characteristic
302	C₄	C₃²×#121	1	1	yes	characteristic
303	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
304	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
305	C₃	C₂×C₃×#473	1	4	yes	direct product factor
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	4	2	no	—
309	C₂	C₃²×#473	1	2	yes	direct product factor
310	C₂	C₂×C₃²×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×Q₁₆	C₂	semidirect
4. C₃²×#473	C₂	direct (N × H)
5. C₂×C₃²×#108	C₂	semidirect
6. C₂×C₃²×#101	C₂	semidirect
7. C₂×C₃×#473	C₃	direct (N × H)
16. C₂×C₃²×Q₁₆	C₂²	semidirect
25. C₃²×#108	C₂²	semidirect
26. C₃²×#101	C₂²	semidirect
28. C₂²×C₃×Q₁₆	C₂×C₃	semidirect
30. C₃×#473	C₂×C₃	direct (N × H)
31. C₂×C₃×#108	C₂×C₃	semidirect
32. C₂×C₃×#101	C₂×C₃	semidirect
65. C₂×#473	C₃²	direct (N × H)
74. C₂×C₃×Q₁₆	C₂²×C₃	semidirect
83. C₃×#108	C₂²×C₃	semidirect
84. C₃×#101	C₂²×C₃	semidirect
114. C₂²×Q₁₆	C₂×C₃²	semidirect
116. #473	C₂×C₃²	direct (N × H)
117. C₂×#108	C₂×C₃²	semidirect
118. C₂×#101	C₂×C₃²	semidirect
171. C₂×Q₁₆	C₂²×C₃²	semidirect
180. #108	C₂²×C₃²	semidirect
181. #101	C₂²×C₃²	semidirect
213. C₂×C₃²	#473	direct (N × H)
259. C₃²	C₂×#473	direct (N × H)
291. C₂×C₃	C₃×#473	direct (N × H)
305. C₃	C₂×C₃×#473	direct (N × H)
309. C₂	C₃²×#473	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×C₃²×#137	2
C₂²×C₃²×Q₁₆	2
C₂×C₃²×#131	2
C₃²×#473	2
C₂×C₃²×#108	2
C₂×C₃²×#101	2
C₂×C₃×#473	3

Subgroups — C2×C32×#473 #11770115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#473 #11770115