Subgroups — C₃×#12980 #10631587

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	#12980	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#135	C₄	1	1	yes	characteristic
10	C₂×C₃×#131	C₂²	1	1	yes	characteristic
11	C₂×C₃×#131	C₂²	1	1	yes	characteristic
12	C₃×#504	—	2	2	no	—
13	C₃×#504	—	2	2	no	—
14	C₂×C₃×M₃₂	—	2	1	no	—
15	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	—	2	1	no	—
18	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
20	C₃×#443	—	2	2	no	—
21	C₃×#443	—	2	2	no	—
22	C₂×C₃×#108	C₄	1	1	yes	characteristic
23	C₂×C₃×#104	C₄	1	1	yes	characteristic
24	C₂×C₃×#104	C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₃×#350	—	2	2	no	—
27	C₃×#350	—	2	2	no	—
28	C₃×C₂²⋊C₁₆	—	2	1	no	—
29	C₃×C₂²⋊C₁₆	—	2	1	no	—
30	C₂×#504	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#443	C₂×C₃	1	1	yes	characteristic
32	C₂×#350	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#350	C₂×C₃	1	1	yes	characteristic, complemented
34	#2113	C₂×C₃	1	1	yes	characteristic, complemented
35	#1376	C₂×C₃	1	1	yes	characteristic, complemented
36	#1374	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
39	C₃×#135	—	2	2	no	—
40	C₃×#135	—	2	2	no	—
41	C₂×C₃×Q₁₆	C₈	1	1	yes	characteristic
42	C₂×C₃×SD₁₆	C₈	1	1	yes	characteristic
43	C₂×C₃×SD₁₆	C₈	1	1	yes	characteristic
44	C₂×C₃×D₈	C₈	1	1	yes	characteristic
45	C₃×#131	—	2	1	no	—
46	C₃×#131	—	2	1	no	—
47	C₃×#131	—	2	1	no	—
48	C₃×#131	—	2	1	no	—
49	C₃×#131	—	2	1	no	—
50	C₃×#131	—	2	1	no	—
51	C₃×#131	D₄	1	2	yes	—
52	C₃×#131	D₄	1	2	yes	—
53	C₂×C₃×M₁₆	—	2	1	no	—
54	C₂×C₃×M₁₆	—	2	1	no	—
55	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₈	—	2	1	no	—
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₃×M₃₂	—	2	4	no	—
63	C₃×M₃₂	—	2	1	no	—
64	C₃×M₃₂	—	2	1	no	—
65	C₃×M₃₂	—	2	1	no	—
66	C₃×M₃₂	D₄	1	2	yes	—
67	C₂×C₃×C₁₆	—	2	4	no	—
68	C₂×C₃×C₁₆	—	2	1	no	—
69	C₂×C₃×C₁₆	—	2	1	no	—
70	C₂×C₃×C₁₆	—	2	1	no	—
71	C₂×C₃×C₁₆	—	2	1	no	—
72	C₂×C₃×C₁₆	—	2	1	no	—
73	C₂×C₃×C₁₆	—	2	1	no	—
74	C₂×C₃×C₁₆	—	2	1	no	—
75	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
76	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
77	C₃×#108	—	2	2	no	—
78	C₃×#104	—	2	2	no	—
79	C₃×#104	—	2	2	no	—
80	C₃×C₄×C₈	—	2	1	no	—
81	C₃×C₄×C₈	—	2	1	no	—
82	C₂×#135	C₃×C₄	1	1	yes	characteristic
83	C₂×#131	C₂²×C₃	1	1	yes	characteristic
84	C₂×#131	C₂²×C₃	1	1	yes	characteristic
85	#504	—	2	2	no	—
86	#504	—	2	2	no	—
87	C₂×M₃₂	—	2	1	no	—
88	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
89	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
90	C₂²×C₁₆	—	2	1	no	—
91	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
93	#443	—	2	2	no	—
94	#443	—	2	2	no	—
95	C₂×#108	C₃×C₄	1	1	yes	characteristic
96	C₂×#104	C₃×C₄	1	1	yes	characteristic
97	C₂×#104	C₃×C₄	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	#350	—	2	2	no	—
100	#350	—	2	2	no	—
101	C₂²⋊C₁₆	—	2	1	no	—
102	C₂²⋊C₁₆	—	2	1	no	—
103	C₃×#41	—	2	1	no	—
104	C₃×#41	—	2	1	no	—
105	C₃×#41	—	2	1	no	—
106	C₃×#41	—	2	1	no	—
107	C₃×#41	—	2	1	no	—
108	C₃×#41	—	2	1	no	—
109	C₃×#41	C₂²⋊C₄	1	2	yes	—
110	C₃×#41	M₁₆	1	2	yes	—
111	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
112	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
113	C₂×C₃×D₄	—	2	1	no	—
114	C₂×C₃×D₄	—	2	1	no	—
115	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
116	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₃×Q₁₆	—	2	2	no	—
122	C₃×SD₁₆	—	2	2	no	—
123	C₃×SD₁₆	—	2	2	no	—
124	C₃×D₈	—	2	2	no	—
125	C₃×M₁₆	—	2	1	no	—
126	C₃×M₁₆	—	2	2	no	—
127	C₃×M₁₆	—	2	2	no	—
128	C₃×M₁₆	—	2	1	no	—
129	C₃×M₁₆	—	2	1	no	—
130	C₃×M₁₆	—	4	1	no	—
131	C₃×M₁₆	—	4	1	no	—
132	C₃×M₁₆	C₂²⋊C₄	1	2	yes	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	—	4	1	no	—
140	C₂×C₃×C₈	—	4	1	no	—
141	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₈	#41	1	1	yes	characteristic
148	C₂×C₃×C₈	#41	1	1	yes	characteristic
149	C₃×C₄²	—	2	1	no	—
150	C₃×C₄²	—	2	1	no	—
151	C₃×C₁₆	—	2	1	no	—
152	C₃×C₁₆	—	2	2	no	—
153	C₃×C₁₆	—	2	1	no	—
154	C₃×C₁₆	—	4	1	no	—
155	C₃×C₁₆	—	4	1	no	—
156	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
157	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
158	#135	—	2	2	no	—
159	#135	—	2	2	no	—
160	C₂×Q₁₆	C₃×C₈	1	1	yes	characteristic
161	C₂×SD₁₆	C₃×C₈	1	1	yes	characteristic
162	C₂×SD₁₆	C₃×C₈	1	1	yes	characteristic
163	C₂×D₈	C₃×C₈	1	1	yes	characteristic
164	#131	—	2	1	no	—
165	#131	—	2	1	no	—
166	#131	—	2	1	no	—
167	#131	—	2	1	no	—
168	#131	—	2	1	no	—
169	#131	—	2	1	no	—
170	#131	C₃×D₄	1	2	yes	—
171	#131	C₃×D₄	1	2	yes	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
175	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
181	M₃₂	—	2	4	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	C₃×D₄	1	2	yes	—
186	C₂×C₁₆	—	2	4	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	C₃×D₄	1	1	yes	characteristic
195	C₂×C₁₆	C₃×D₄	1	1	yes	characteristic
196	#108	—	2	2	no	—
197	#104	—	2	2	no	—
198	#104	—	2	2	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₂³×C₃	—	2	1	no	—
202	C₂³×C₃	—	2	1	no	—
203	C₂³×C₃	#131	1	1	yes	characteristic
204	C₃×Q₈	—	2	1	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	2	1	no	—
207	C₃×Q₈	C₂²⋊C₈	1	2	yes	—
208	C₃×D₄	—	2	2	no	—
209	C₃×D₄	—	2	2	no	—
210	C₃×D₄	—	2	1	no	—
211	C₃×D₄	—	2	1	no	—
212	C₃×D₄	—	2	1	no	—
213	C₃×D₄	—	4	1	no	—
214	C₃×D₄	—	4	1	no	—
215	C₃×D₄	C₂²⋊C₈	1	2	yes	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	4	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
230	C₂×C₃×C₄	#131	1	1	yes	characteristic
231	C₃×C₈	—	2	2	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	2	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	#120	1	1	yes	characteristic
236	C₃×C₈	#121	1	1	yes	characteristic
237	C₃×C₈	#121	1	1	yes	characteristic
238	C₃×C₈	#123	1	1	yes	characteristic
239	#41	—	2	1	no	—
240	#41	—	2	1	no	—
241	#41	—	2	1	no	—
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	—	2	1	no	—
245	#41	C₃×C₂²⋊C₄	1	2	yes	—
246	#41	C₃×M₁₆	1	2	yes	—
247	C₂×Q₈	C₂×C₃×C₈	1	1	yes	characteristic
248	C₂×Q₈	C₂×C₃×C₈	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	C₂×C₃×C₈	1	1	yes	characteristic
252	C₂×D₄	C₂×C₃×C₈	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	2	2	no	—
260	D₈	—	2	2	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	2	1	no	—
265	M₁₆	—	2	1	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	—	4	1	no	—
268	M₁₆	C₃×C₂²⋊C₄	1	2	yes	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	C₂×C₃×C₈	1	1	yes	characteristic
278	C₂×C₈	C₂×C₃×C₈	1	1	yes	characteristic
279	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
281	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
282	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
283	C₂×C₈	C₃×#41	1	1	yes	characteristic
284	C₂×C₈	C₃×#41	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	4	1	no	—
291	C₁₆	—	4	1	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	2	2	no	—
295	C₂²×C₃	—	4	1	no	—
296	C₂²×C₃	—	4	1	no	—
297	C₂²×C₃	—	4	1	no	—
298	C₂²×C₃	—	4	1	no	—
299	C₂²×C₃	—	4	1	no	—
300	C₂²×C₃	#408	1	1	yes	characteristic
301	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
302	C₂²×C₃	#435	1	1	yes	characteristic
303	C₃×C₄	—	2	2	no	—
304	C₃×C₄	—	4	2	no	—
305	C₃×C₄	—	4	1	no	—
306	C₃×C₄	#386	1	1	yes	characteristic
307	C₃×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
308	C₃×C₄	C₈×D₄	1	1	yes	characteristic
309	C₃×C₄	#435	1	1	yes	characteristic
310	C₂³	—	2	1	no	—
311	C₂³	—	2	1	no	—
312	C₂³	C₃×#131	1	1	yes	characteristic
313	Q₈	—	2	1	no	—
314	Q₈	—	2	1	no	—
315	Q₈	—	2	1	no	—
316	Q₈	C₃×C₂²⋊C₈	1	2	yes	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	2	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	4	1	no	—
323	D₄	—	4	1	no	—
324	D₄	C₃×C₂²⋊C₈	1	2	yes	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
335	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
336	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
338	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
339	C₂×C₄	C₃×#131	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	4	1	no	—
344	C₈	C₃×#120	1	1	yes	characteristic
345	C₈	C₃×#121	1	1	yes	characteristic
346	C₈	C₃×#121	1	1	yes	characteristic
347	C₈	C₃×#123	1	1	yes	characteristic
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	—	2	1	no	—
350	C₂×C₃	—	4	2	no	—
351	C₂×C₃	—	8	1	no	—
352	C₂×C₃	#1869	1	1	yes	characteristic
353	C₂×C₃	#2170	1	1	yes	characteristic
354	C₂×C₃	#2171	1	1	yes	characteristic
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	C₃×#408	1	1	yes	characteristic
364	C₂²	C₃×C₈×D₄	1	1	yes	characteristic
365	C₂²	C₃×#435	1	1	yes	characteristic
366	C₄	—	2	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	1	no	—
369	C₄	C₃×#386	1	1	yes	characteristic
370	C₄	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
371	C₄	C₃×C₈×D₄	1	1	yes	characteristic
372	C₄	C₃×#435	1	1	yes	characteristic
373	C₃	#12980	1	1	yes	characteristic, direct product factor, Sylow
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	4	2	no	—
377	C₂	—	8	1	no	—
378	C₂	C₃×#1869	1	1	yes	characteristic
379	C₂	C₃×#2170	1	1	yes	characteristic
380	C₂	C₃×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₈×D₄
prime radical	C₂×C₃×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#504	C₂	semidirect
3. C₂×C₃×#350	C₂	semidirect
4. C₂×C₃×#350	C₂	semidirect
5. C₃×#2113	C₂	semidirect
6. C₃×#1376	C₂	semidirect
7. C₃×#1374	C₂	semidirect
8. #12980	C₃	direct (N × H)
30. C₂×#504	C₂×C₃	semidirect
32. C₂×#350	C₂×C₃	semidirect
33. C₂×#350	C₂×C₃	semidirect
34. #2113	C₂×C₃	semidirect
35. #1376	C₂×C₃	semidirect
36. #1374	C₂×C₃	semidirect
373. C₃	#12980	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×#504	2
C₂×C₃×#443	2
C₂×C₃×#350	2
C₃×#2113	2
C₃×#1376	2
C₃×#1374	2
#12980	3

Subgroups — C3×#12980 #10631587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12980 #10631587