Subgroups — C₃×#9838 #10628445

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1957	C₂	1	1	yes	maximal, characteristic
4	C₃×#1957	C₂	1	1	yes	maximal, characteristic
5	C₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	#9838	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×#127	C₂²	1	1	yes	characteristic
11	C₃×#486	—	2	2	no	—
12	C₃×#486	C₄	1	4	yes	complemented
13	C₂×C₃×#103	—	2	2	no	—
14	C₂×C₃×#102	—	2	2	no	—
15	C₂×C₃×#102	—	2	2	no	—
16	C₂×C₃×#102	—	2	2	no	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#384	C₂²	1	1	yes	characteristic
21	C₃×#384	C₂²	1	1	yes	characteristic
22	C₃×#340	—	2	2	no	—
23	C₃×#340	—	2	2	no	—
24	C₃×#322	—	2	2	no	—
25	C₂×#486	C₂×C₃	1	1	yes	characteristic
26	#1990	C₂×C₃	1	1	yes	characteristic, complemented
27	#1957	C₂×C₃	1	1	yes	characteristic
28	#1957	C₂×C₃	1	1	yes	characteristic
29	#1956	C₂×C₃	1	1	yes	characteristic, complemented
30	#1834	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#322	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×Q₈	—	2	1	no	—
33	C₂²×C₃×D₄	—	2	1	no	—
34	C₂²×C₃×D₄	—	2	1	no	—
35	C₂²×C₃×D₄	—	2	1	no	—
36	C₂²×C₃×C₈	D₄	1	2	yes	—
37	C₂²×C₃×C₈	D₄	1	2	yes	—
38	C₂²×C₃×C₈	D₄	1	2	yes	—
39	C₃×C₄⋊Q₈	—	2	1	no	—
40	C₃×C₄⋊Q₈	—	2	1	no	—
41	C₃×C₄⋊Q₈	—	2	1	no	—
42	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
43	C₃×#127	—	2	1	no	—
44	C₃×#127	—	2	1	no	—
45	C₃×#127	—	2	1	no	—
46	C₃×#127	C₂×C₄	1	2	yes	—
47	C₂×C₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₃×C₄⋊C₄	—	2	1	no	—
50	C₂×C₃×C₄⋊C₄	—	2	1	no	—
51	C₂×C₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
56	C₃×#103	—	4	4	no	—
57	C₃×#102	—	2	8	no	—
58	C₃×#102	—	4	4	no	—
59	C₃×#102	—	4	4	no	—
60	C₃×C₄×C₈	—	2	1	no	—
61	C₃×C₄×C₈	—	2	1	no	—
62	C₃×C₄×C₈	—	2	1	no	—
63	C₃×C₄×C₈	—	2	1	no	—
64	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
65	C₃×C₄×C₈	D₄	1	2	yes	—
66	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×#127	C₂²×C₃	1	1	yes	characteristic
68	#486	—	2	2	no	—
69	#486	C₃×C₄	1	4	yes	complemented
70	C₂×#103	—	2	2	no	—
71	C₂×#102	—	2	2	no	—
72	C₂×#102	—	2	2	no	—
73	C₂×#102	—	2	2	no	—
74	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
77	#384	C₂²×C₃	1	1	yes	characteristic
78	#384	C₂²×C₃	1	1	yes	characteristic
79	#340	—	2	2	no	—
80	#340	—	2	2	no	—
81	#322	—	2	2	no	—
82	C₂⁴×C₃	—	4	1	no	—
83	C₂×C₃×Q₈	—	2	2	no	—
84	C₂×C₃×Q₈	—	4	1	no	—
85	C₂×C₃×Q₈	—	4	2	no	—
86	C₂×C₃×Q₈	—	4	2	no	—
87	C₂×C₃×D₄	—	2	2	no	—
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂×C₃×D₄	—	2	2	no	—
90	C₂×C₃×D₄	—	4	1	no	—
91	C₂×C₃×D₄	—	4	1	no	—
92	C₂×C₃×D₄	—	4	1	no	—
93	C₂×C₃×D₄	—	4	2	no	—
94	C₂×C₃×D₄	—	4	2	no	—
95	C₂×C₃×D₄	—	4	2	no	—
96	C₂×C₃×D₄	—	4	2	no	—
97	C₂×C₃×D₄	—	4	2	no	—
98	C₂×C₃×D₄	—	4	2	no	—
99	C₂²×C₃×C₄	—	4	1	no	—
100	C₂²×C₃×C₄	—	4	1	no	—
101	C₂²×C₃×C₄	—	4	1	no	—
102	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	—	4	1	no	—
117	C₃×C₄⋊C₄	—	4	1	no	—
118	C₃×C₄⋊C₄	—	4	2	no	—
119	C₃×C₄⋊C₄	—	4	2	no	—
120	C₃×C₄⋊C₄	—	4	2	no	—
121	C₃×C₄⋊C₄	—	4	2	no	—
122	C₃×C₄⋊C₄	—	4	2	no	—
123	C₃×C₄⋊C₄	—	4	2	no	—
124	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
125	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
126	C₃×C₄²	#41	1	1	yes	characteristic
127	C₃×C₄²	#41	1	1	yes	characteristic
128	C₂²×Q₈	—	2	1	no	—
129	C₂²×D₄	—	2	1	no	—
130	C₂²×D₄	—	2	1	no	—
131	C₂²×D₄	—	2	1	no	—
132	C₂²×C₈	C₃×D₄	1	2	yes	—
133	C₂²×C₈	C₃×D₄	1	2	yes	—
134	C₂²×C₈	C₃×D₄	1	2	yes	—
135	C₄⋊Q₈	—	2	1	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
139	#127	—	2	1	no	—
140	#127	—	2	1	no	—
141	#127	—	2	1	no	—
142	#127	C₂×C₃×C₄	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
152	#103	—	4	4	no	—
153	#102	—	2	8	no	—
154	#102	—	4	4	no	—
155	#102	—	4	4	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
161	C₄×C₈	C₃×D₄	1	2	yes	—
162	C₂³×C₃	—	4	2	no	—
163	C₂³×C₃	—	8	1	no	—
164	C₂³×C₃	—	8	1	no	—
165	C₂³×C₃	—	8	1	no	—
166	C₂³×C₃	—	8	1	no	—
167	C₂³×C₃	—	8	1	no	—
168	C₂³×C₃	—	8	1	no	—
169	C₂³×C₃	#127	1	1	yes	characteristic
170	C₃×Q₈	—	4	4	no	—
171	C₃×Q₈	—	8	2	no	—
172	C₃×D₄	—	4	4	no	—
173	C₃×D₄	—	4	4	no	—
174	C₃×D₄	—	4	4	no	—
175	C₃×D₄	—	8	2	no	—
176	C₃×D₄	—	8	2	no	—
177	C₃×D₄	—	8	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	8	1	no	—
180	C₂×C₃×C₄	—	8	1	no	—
181	C₂×C₃×C₄	—	8	1	no	—
182	C₂×C₃×C₄	—	8	1	no	—
183	C₂×C₃×C₄	—	8	1	no	—
184	C₂×C₃×C₄	—	8	1	no	—
185	C₂×C₃×C₄	—	8	1	no	—
186	C₂×C₃×C₄	—	8	1	no	—
187	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	#121	1	1	yes	characteristic
191	C₂×C₃×C₄	#121	1	1	yes	characteristic
192	C₂×C₃×C₄	#124	1	1	yes	characteristic
193	C₂×C₃×C₄	#135	1	2	yes	—
194	C₂×C₃×C₄	#135	1	2	yes	—
195	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
196	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
198	C₂×C₃×C₄	#137	1	2	yes	—
199	C₃×C₈	—	2	4	no	—
200	C₃×C₈	—	4	2	no	—
201	C₃×C₈	—	4	2	no	—
202	C₂⁴	—	4	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₃×C₂²⋊C₄	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
245	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
246	C₄²	C₃×#41	1	1	yes	characteristic
247	C₄²	C₃×#41	1	1	yes	characteristic
248	C₂²×C₃	—	8	2	no	—
249	C₂²×C₃	—	8	2	no	—
250	C₂²×C₃	—	8	2	no	—
251	C₂²×C₃	—	16	1	no	—
252	C₂²×C₃	—	16	1	no	—
253	C₂²×C₃	—	16	1	no	—
254	C₂²×C₃	—	16	1	no	—
255	C₂²×C₃	#390	1	1	yes	characteristic
256	C₂²×C₃	#495	1	1	yes	characteristic
257	C₂²×C₃	#495	1	1	yes	characteristic
258	C₂²×C₃	#496	1	1	yes	characteristic
259	C₂²×C₃	#496	1	1	yes	characteristic
260	C₂²×C₃	#496	1	1	yes	characteristic
261	C₂²×C₃	#497	1	1	yes	characteristic
262	C₃×C₄	—	8	2	no	—
263	C₃×C₄	—	16	1	no	—
264	C₃×C₄	—	16	1	no	—
265	C₃×C₄	#416	1	2	yes	—
266	C₃×C₄	#440	1	2	yes	—
267	C₃×C₄	#442	1	2	yes	—
268	C₃×C₄	#468	1	2	yes	—
269	C₃×C₄	#469	1	2	yes	—
270	C₃×C₄	#488	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	—	8	1	no	—
277	C₂³	—	8	1	no	—
278	C₂³	C₃×#127	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	8	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	2	no	—
285	D₄	—	8	2	no	—
286	D₄	—	8	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
298	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
299	C₂×C₄	C₃×#121	1	1	yes	characteristic
300	C₂×C₄	C₃×#121	1	1	yes	characteristic
301	C₂×C₄	C₃×#124	1	1	yes	characteristic
302	C₂×C₄	C₃×#135	1	2	yes	—
303	C₂×C₄	C₃×#135	1	2	yes	—
304	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
305	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
306	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
307	C₂×C₄	C₃×#137	1	2	yes	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₂×C₃	—	16	2	no	—
312	C₂×C₃	#1717	1	1	yes	characteristic
313	C₂×C₃	#1717	1	1	yes	characteristic
314	C₂×C₃	#1719	1	1	yes	characteristic
315	C₂×C₃	#1720	1	1	yes	characteristic
316	C₂×C₃	#1958	1	1	yes	characteristic
317	C₂×C₃	#1972	1	1	yes	characteristic
318	C₂×C₃	#1973	1	1	yes	characteristic
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	1	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	—	16	1	no	—
326	C₂²	C₃×#390	1	1	yes	characteristic
327	C₂²	C₃×#495	1	1	yes	characteristic
328	C₂²	C₃×#495	1	1	yes	characteristic
329	C₂²	C₃×#496	1	1	yes	characteristic
330	C₂²	C₃×#496	1	1	yes	characteristic
331	C₂²	C₃×#496	1	1	yes	characteristic
332	C₂²	C₃×#497	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	16	1	no	—
335	C₄	—	16	1	no	—
336	C₄	C₃×#416	1	2	yes	—
337	C₄	C₃×#440	1	2	yes	—
338	C₄	C₃×#442	1	2	yes	—
339	C₄	C₃×#468	1	2	yes	—
340	C₄	C₃×#469	1	2	yes	—
341	C₄	C₃×#488	1	2	yes	—
342	C₃	#9838	1	1	yes	characteristic, direct product factor, Sylow
343	C₂	—	16	2	no	—
344	C₂	C₃×#1717	1	1	yes	characteristic
345	C₂	C₃×#1717	1	1	yes	characteristic
346	C₂	C₃×#1719	1	1	yes	characteristic
347	C₂	C₃×#1720	1	1	yes	characteristic
348	C₂	C₃×#1958	1	1	yes	characteristic
349	C₂	C₃×#1972	1	1	yes	characteristic
350	C₂	C₃×#1973	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂×C₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#1990	C₂	semidirect
5. C₃×#1956	C₂	semidirect
6. C₃×#1834	C₂	semidirect
7. C₂×C₃×#322	C₂	semidirect
8. #9838	C₃	direct (N × H)
12. C₃×#486	C₄	semidirect
26. #1990	C₂×C₃	semidirect
29. #1956	C₂×C₃	semidirect
30. #1834	C₂×C₃	semidirect
31. C₂×#322	C₂×C₃	semidirect
69. #486	C₃×C₄	semidirect
342. C₃	#9838	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×#486	2
C₃×#1990	2
C₃×#1957	2
C₃×#1956	2
C₃×#1834	2
C₂×C₃×#322	2
#9838	3

Subgroups — C3×#9838 #10628445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9838 #10628445