Subgroups — C₂×C₃²×#485 #11770137

Order 1152.

1248 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#117	C₂	1	1	yes	maximal, characteristic
3	C₃²×#485	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃²×#103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃×#485	C₃	1	4	yes	maximal, direct product factor
10	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₂²⋊Q₈	—	2	2	no	—
15	C₃²×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₃²×#117	—	2	2	no	—
17	C₃²×#117	C₂²	1	4	yes	—
18	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
19	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃²×C₂²⋊C₄	—	2	1	no	—
23	C₂×C₃²×C₂²⋊C₄	—	2	1	no	—
24	C₂×C₃²×C₄²	—	2	1	no	—
25	C₃²×#107	C₂²	1	4	yes	complemented
26	C₃²×#106	C₂²	1	4	yes	complemented
27	C₃²×#103	C₂²	1	4	yes	complemented
28	C₃²×#103	C₂²	1	4	yes	complemented
29	C₃²×C₂²⋊C₈	C₂²	1	4	yes	—
30	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	4	yes	—
31	C₂×C₃×#117	C₂×C₃	1	4	yes	—
32	C₃×#485	C₂×C₃	1	32	yes	direct product factor
33	C₂×C₃×#107	C₂×C₃	1	4	yes	complemented
34	C₂×C₃×#106	C₂×C₃	1	4	yes	complemented
35	C₂×C₃×#103	C₂×C₃	1	4	yes	complemented
36	C₂×C₃×#103	C₂×C₃	1	4	yes	complemented
37	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	4	yes	—
38	C₂⁴×C₃²	D₄	1	1	yes	characteristic
39	C₂×C₃²×Q₈	—	2	1	no	—
40	C₂×C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₂×C₃²×Q₈	C₂³	1	2	yes	—
43	C₂²×C₃²×C₄	—	2	2	no	—
44	C₂²×C₃²×C₄	—	2	2	no	—
45	C₂²×C₃²×C₄	—	2	1	no	—
46	C₂²×C₃²×C₄	—	2	1	no	—
47	C₂²×C₃²×C₄	—	2	1	no	—
48	C₂²×C₃²×C₄	—	2	1	no	—
49	C₂²×C₃²×C₄	D₄	1	2	yes	—
50	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
51	C₂²×C₃²×C₄	C₂³	1	2	yes	—
52	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
53	C₂×C₃²×C₈	—	2	2	no	—
54	C₂×C₃²×C₈	—	2	2	no	—
55	C₂×C₃²×C₈	C₂³	1	2	yes	—
56	C₂×C₃²×C₈	C₂³	1	2	yes	—
57	C₃²×C₄⋊C₄	—	2	4	no	—
58	C₃²×C₄⋊C₄	—	2	1	no	—
59	C₃²×C₄⋊C₄	—	2	1	no	—
60	C₃²×C₄⋊C₄	—	2	1	no	—
61	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
62	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
63	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃²×C₂²⋊C₄	—	2	4	no	—
65	C₃²×C₂²⋊C₄	—	2	4	no	—
66	C₃²×C₄²	—	2	4	no	—
67	C₂×#485	C₃²	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
69	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
70	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
71	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
72	C₃×C₂²⋊Q₈	—	2	8	no	—
73	C₃×C₂²⋊Q₈	C₂²×C₃	1	16	yes	—
74	C₃×#117	—	2	8	no	—
75	C₃×#117	C₂²×C₃	1	16	yes	—
76	C₂×C₃×C₄⋊C₄	—	2	4	no	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
80	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
81	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
82	C₂×C₃×C₄²	—	2	4	no	—
83	C₃×#107	C₂²×C₃	1	16	yes	complemented
84	C₃×#106	C₂²×C₃	1	16	yes	complemented
85	C₃×#103	C₂²×C₃	1	16	yes	complemented
86	C₃×#103	C₂²×C₃	1	16	yes	complemented
87	C₃×C₂²⋊C₈	C₂²×C₃	1	16	yes	—
88	C₂³×C₃²	—	2	2	no	—
89	C₂³×C₃²	—	2	2	no	—
90	C₂³×C₃²	—	2	2	no	—
91	C₂³×C₃²	C₂×D₄	1	2	yes	—
92	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
93	C₃²×Q₈	—	2	2	no	—
94	C₃²×Q₈	—	2	2	no	—
95	C₃²×Q₈	—	4	2	no	—
96	C₂×C₃²×C₄	—	2	2	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	—	2	2	no	—
101	C₂×C₃²×C₄	—	2	2	no	—
102	C₂×C₃²×C₄	—	2	2	no	—
103	C₂×C₃²×C₄	—	2	2	no	—
104	C₂×C₃²×C₄	—	4	2	no	—
105	C₂×C₃²×C₄	—	4	2	no	—
106	C₂×C₃²×C₄	—	4	2	no	—
107	C₂×C₃²×C₄	—	4	2	no	—
108	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
109	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂×C₃²×C₄	#41	1	2	yes	—
111	C₂×C₃²×C₄	#41	1	2	yes	—
112	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
113	C₃²×C₈	—	2	2	no	—
114	C₃²×C₈	—	2	2	no	—
115	C₂×C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic
116	C₂×#117	C₂×C₃²	1	1	yes	characteristic
117	#485	C₂×C₃²	1	8	yes	direct product factor
118	C₂×#107	C₂×C₃²	1	1	yes	characteristic, complemented
119	C₂×#106	C₂×C₃²	1	1	yes	characteristic, complemented
120	C₂×#103	C₂×C₃²	1	1	yes	characteristic, complemented
121	C₂×#103	C₂×C₃²	1	1	yes	characteristic, complemented
122	C₂×C₂²⋊C₈	C₂×C₃²	1	1	yes	characteristic
123	C₂⁴×C₃	C₃×D₄	1	4	yes	—
124	C₂×C₃×Q₈	—	2	4	no	—
125	C₂×C₃×Q₈	—	2	8	no	—
126	C₂×C₃×Q₈	—	2	8	no	—
127	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
128	C₂²×C₃×C₄	—	2	8	no	—
129	C₂²×C₃×C₄	—	2	8	no	—
130	C₂²×C₃×C₄	—	2	4	no	—
131	C₂²×C₃×C₄	—	2	4	no	—
132	C₂²×C₃×C₄	—	2	4	no	—
133	C₂²×C₃×C₄	—	2	4	no	—
134	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
135	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
136	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
137	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
138	C₂×C₃×C₈	—	2	8	no	—
139	C₂×C₃×C₈	—	2	8	no	—
140	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
141	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
142	C₃×C₄⋊C₄	—	2	16	no	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₄⋊C₄	—	2	4	no	—
146	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
147	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
148	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
149	C₃×C₂²⋊C₄	—	2	16	no	—
150	C₃×C₂²⋊C₄	—	2	16	no	—
151	C₃×C₄²	—	2	16	no	—
152	C₂²×C₃²	—	2	2	no	—
153	C₂²×C₃²	—	2	2	no	—
154	C₂²×C₃²	—	2	2	no	—
155	C₂²×C₃²	—	4	4	no	—
156	C₂²×C₃²	#123	1	2	yes	—
157	C₂²×C₃²	#135	1	2	yes	—
158	C₂²×C₃²	#137	1	2	yes	—
159	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
160	C₃²×C₄	—	2	2	no	—
161	C₃²×C₄	—	4	2	no	—
162	C₃²×C₄	—	4	2	no	—
163	C₃²×C₄	—	4	2	no	—
164	C₃²×C₄	—	4	2	no	—
165	C₃²×C₄	#123	1	2	yes	—
166	C₃²×C₄	C₂×#41	1	1	yes	characteristic
167	C₃²×C₄	C₂×#41	1	1	yes	characteristic
168	C₂²×Q₈	C₂²×C₃²	1	1	yes	characteristic
169	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
170	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
171	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
172	C₂²⋊Q₈	—	2	2	no	—
173	C₂²⋊Q₈	C₂²×C₃²	1	4	yes	—
174	#117	—	2	2	no	—
175	#117	C₂²×C₃²	1	4	yes	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	2	1	no	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	—	2	1	no	—
183	#107	C₂²×C₃²	1	4	yes	complemented
184	#106	C₂²×C₃²	1	4	yes	complemented
185	#103	C₂²×C₃²	1	4	yes	complemented
186	#103	C₂²×C₃²	1	4	yes	complemented
187	C₂²⋊C₈	C₂²×C₃²	1	4	yes	—
188	C₂³×C₃	—	2	8	no	—
189	C₂³×C₃	—	2	8	no	—
190	C₂³×C₃	—	2	8	no	—
191	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
192	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
193	C₃×Q₈	—	2	8	no	—
194	C₃×Q₈	—	2	8	no	—
195	C₃×Q₈	—	4	8	no	—
196	C₂×C₃×C₄	—	2	8	no	—
197	C₂×C₃×C₄	—	2	16	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	2	8	no	—
200	C₂×C₃×C₄	—	2	8	no	—
201	C₂×C₃×C₄	—	2	8	no	—
202	C₂×C₃×C₄	—	2	8	no	—
203	C₂×C₃×C₄	—	2	8	no	—
204	C₂×C₃×C₄	—	4	8	no	—
205	C₂×C₃×C₄	—	4	8	no	—
206	C₂×C₃×C₄	—	4	8	no	—
207	C₂×C₃×C₄	—	4	8	no	—
208	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
210	C₂×C₃×C₄	C₃×#41	1	8	yes	—
211	C₂×C₃×C₄	C₃×#41	1	8	yes	—
212	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
213	C₃×C₈	—	2	8	no	—
214	C₃×C₈	—	2	8	no	—
215	C₂×C₃²	—	4	2	no	—
216	C₂×C₃²	#485	1	4	yes	direct product factor
217	C₂×C₃²	C₂×#123	1	1	yes	characteristic
218	C₂×C₃²	C₂×#135	1	1	yes	characteristic
219	C₂×C₃²	C₂×#137	1	1	yes	characteristic
220	C₂⁴	C₃²×D₄	1	1	yes	characteristic
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	C₂³×C₃²	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₃²×D₄	1	2	yes	—
233	C₂²×C₄	C₂³×C₃²	1	2	yes	—
234	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂³×C₃²	1	2	yes	—
238	C₂×C₈	C₂³×C₃²	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
244	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
245	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₂²×C₃	—	2	8	no	—
250	C₂²×C₃	—	2	8	no	—
251	C₂²×C₃	—	2	8	no	—
252	C₂²×C₃	—	4	16	no	—
253	C₂²×C₃	C₃×#123	1	8	yes	—
254	C₂²×C₃	C₃×#135	1	8	yes	—
255	C₂²×C₃	C₃×#137	1	8	yes	—
256	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
257	C₃×C₄	—	2	8	no	—
258	C₃×C₄	—	4	8	no	—
259	C₃×C₄	—	4	8	no	—
260	C₃×C₄	—	4	8	no	—
261	C₃×C₄	—	4	8	no	—
262	C₃×C₄	C₃×#123	1	8	yes	—
263	C₃×C₄	C₂×C₃×#41	1	4	yes	—
264	C₃×C₄	C₂×C₃×#41	1	4	yes	—
265	C₃²	C₂×#485	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂×C₃²×D₄	1	2	yes	—
270	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	Q₈	—	2	2	no	—
273	Q₈	—	4	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
287	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
288	C₂×C₄	C₃²×#41	1	2	yes	—
289	C₂×C₄	C₃²×#41	1	2	yes	—
290	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₂×C₃	—	4	8	no	—
294	C₂×C₃	C₃×#485	1	16	yes	direct product factor
295	C₂×C₃	C₂×C₃×#123	1	4	yes	—
296	C₂×C₃	C₂×C₃×#135	1	4	yes	—
297	C₂×C₃	C₂×C₃×#137	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	C₃²×#123	1	2	yes	—
303	C₂²	C₃²×#135	1	2	yes	—
304	C₂²	C₃²×#137	1	2	yes	—
305	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	C₃²×#123	1	2	yes	—
312	C₄	C₂×C₃²×#41	1	1	yes	characteristic
313	C₄	C₂×C₃²×#41	1	1	yes	characteristic
314	C₃	C₂×C₃×#485	1	4	yes	direct product factor
315	C₂	—	4	2	no	—
316	C₂	C₃²×#485	1	4	yes	direct product factor
317	C₂	C₂×C₃²×#123	1	1	yes	characteristic
318	C₂	C₂×C₃²×#135	1	1	yes	characteristic
319	C₂	C₂×C₃²×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
prime radical	C₂⁴×C₃²	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂³×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#485	C₂	direct (N × H)
4. C₂×C₃²×#107	C₂	semidirect
5. C₂×C₃²×#106	C₂	semidirect
6. C₂×C₃²×#103	C₂	semidirect
7. C₂×C₃²×#103	C₂	semidirect
9. C₂×C₃×#485	C₃	direct (N × H)
25. C₃²×#107	C₂²	semidirect
26. C₃²×#106	C₂²	semidirect
27. C₃²×#103	C₂²	semidirect
28. C₃²×#103	C₂²	semidirect
32. C₃×#485	C₂×C₃	direct (N × H)
33. C₂×C₃×#107	C₂×C₃	semidirect
34. C₂×C₃×#106	C₂×C₃	semidirect
35. C₂×C₃×#103	C₂×C₃	semidirect
36. C₂×C₃×#103	C₂×C₃	semidirect
67. C₂×#485	C₃²	direct (N × H)
83. C₃×#107	C₂²×C₃	semidirect
84. C₃×#106	C₂²×C₃	semidirect
85. C₃×#103	C₂²×C₃	semidirect
86. C₃×#103	C₂²×C₃	semidirect
117. #485	C₂×C₃²	direct (N × H)
118. C₂×#107	C₂×C₃²	semidirect
119. C₂×#106	C₂×C₃²	semidirect
120. C₂×#103	C₂×C₃²	semidirect
121. C₂×#103	C₂×C₃²	semidirect
183. #107	C₂²×C₃²	semidirect
184. #106	C₂²×C₃²	semidirect
185. #103	C₂²×C₃²	semidirect
186. #103	C₂²×C₃²	semidirect
216. C₂×C₃²	#485	direct (N × H)
265. C₃²	C₂×#485	direct (N × H)
294. C₂×C₃	C₃×#485	direct (N × H)
314. C₃	C₂×C₃×#485	direct (N × H)
316. C₂	C₃²×#485	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×C₂²⋊Q₈	2
C₂×C₃²×#117	2
C₃²×#485	2
C₂×C₃²×#107	2
C₂×C₃²×#106	2
C₂×C₃²×#103	2
C₂×C₃²×C₂²⋊C₈	2
C₂×C₃×#485	3

Subgroups — C2×C32×#485 #11770137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#485 #11770137