Subgroups — C₃×#9858 #10628465

Order 768.

428 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3131	C₂	1	1	yes	maximal, characteristic
2	C₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1492	C₂	1	1	yes	maximal, characteristic
6	C₃×#1492	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	#9858	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#533	C₂²	1	1	yes	characteristic
10	C₃×#532	C₂²	1	1	yes	characteristic
11	C₃×#488	—	2	2	no	—
12	C₃×#487	C₄	1	1	yes	characteristic
13	C₃×#487	C₄	1	1	yes	characteristic
14	C₃×#486	C₄	1	1	yes	characteristic
15	C₃×#486	C₄	1	1	yes	characteristic
16	C₂×C₃×#108	—	2	2	no	—
17	C₃×#423	C₂²	1	1	yes	characteristic
18	C₃×#423	C₂²	1	1	yes	characteristic
19	C₃×#416	—	2	2	no	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×Q₈⋊C₈	—	2	2	no	—
24	C₃×Q₈⋊C₈	—	2	2	no	—
25	C₃×D₄⋊C₈	—	2	2	no	—
26	C₃×D₄⋊C₈	—	2	2	no	—
27	C₃×#322	—	2	2	no	—
28	#3131	C₂×C₃	1	1	yes	characteristic
29	#1848	C₂×C₃	1	1	yes	characteristic, complemented
30	#1493	C₂×C₃	1	1	yes	characteristic, complemented
31	#1493	C₂×C₃	1	1	yes	characteristic, complemented
32	#1492	C₂×C₃	1	1	yes	characteristic
33	#1492	C₂×C₃	1	1	yes	characteristic
34	C₂×#322	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₃×#41	—	2	1	no	—
36	C₂×C₃×M₁₆	—	2	1	no	—
37	C₂×C₃×M₁₆	—	2	1	no	—
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	D₄	1	2	yes	—
41	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
44	C₃×#127	C₂×C₄	1	1	yes	characteristic
45	C₃×#125	—	2	1	no	—
46	C₃×#124	—	2	1	no	—
47	C₃×C₂²⋊Q₈	—	2	1	no	—
48	C₃×C₂²⋊Q₈	—	2	1	no	—
49	C₃×#121	—	2	1	no	—
50	C₃×#121	—	2	1	no	—
51	C₃×C₄×Q₈	—	2	1	no	—
52	C₃×C₄×Q₈	—	2	1	no	—
53	C₃×C₄×D₄	—	2	1	no	—
54	C₃×C₄×D₄	—	2	1	no	—
55	C₃×#117	—	2	1	no	—
56	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₃×#108	—	4	4	no	—
58	C₃×C₄⋊C₈	—	2	1	no	—
59	C₃×C₄⋊C₈	—	2	1	no	—
60	C₃×C₄⋊C₈	—	2	1	no	—
61	C₃×C₄⋊C₈	—	2	1	no	—
62	C₃×#103	—	2	2	no	—
63	C₃×#103	—	2	2	no	—
64	C₃×#102	—	2	2	no	—
65	C₃×#102	—	2	2	no	—
66	C₃×C₄×C₈	—	2	1	no	—
67	C₃×C₄×C₈	—	2	1	no	—
68	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₃×C₄×C₈	D₄	1	2	yes	—
71	C₃×C₄×C₈	D₄	1	2	yes	—
72	C₃×C₄×C₈	D₄	1	2	yes	—
73	#533	C₂²×C₃	1	1	yes	characteristic
74	#532	C₂²×C₃	1	1	yes	characteristic
75	#488	—	2	2	no	—
76	#487	C₃×C₄	1	1	yes	characteristic
77	#487	C₃×C₄	1	1	yes	characteristic
78	#486	C₃×C₄	1	1	yes	characteristic
79	#486	C₃×C₄	1	1	yes	characteristic
80	C₂×#108	—	2	2	no	—
81	#423	C₂²×C₃	1	1	yes	characteristic
82	#423	C₂²×C₃	1	1	yes	characteristic
83	#416	—	2	2	no	—
84	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
87	Q₈⋊C₈	—	2	2	no	—
88	Q₈⋊C₈	—	2	2	no	—
89	D₄⋊C₈	—	2	2	no	—
90	D₄⋊C₈	—	2	2	no	—
91	#322	—	2	2	no	—
92	C₃×#41	—	4	2	no	—
93	C₃×#41	—	4	2	no	—
94	C₂×C₃×Q₈	—	2	1	no	—
95	C₂×C₃×Q₈	—	2	1	no	—
96	C₂×C₃×Q₈	—	2	1	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	—	2	1	no	—
99	C₂×C₃×D₄	—	2	1	no	—
100	C₂×C₃×D₄	—	4	1	no	—
101	C₂²×C₃×C₄	—	4	1	no	—
102	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	#41	1	1	yes	characteristic
104	C₂²×C₃×C₄	#41	1	1	yes	characteristic
105	C₃×M₁₆	—	4	2	no	—
106	C₃×M₁₆	—	4	2	no	—
107	C₂×C₃×C₈	—	2	4	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	—	4	2	no	—
113	C₂×C₃×C₈	—	4	1	no	—
114	C₂×C₃×C₈	—	4	1	no	—
115	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
116	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
117	C₃×C₄⋊C₄	—	2	1	no	—
118	C₃×C₄⋊C₄	—	2	1	no	—
119	C₃×C₄⋊C₄	—	2	1	no	—
120	C₃×C₄⋊C₄	—	2	1	no	—
121	C₃×C₄⋊C₄	—	2	1	no	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	—	4	1	no	—
124	C₃×C₄⋊C₄	—	4	1	no	—
125	C₃×C₂²⋊C₄	—	4	1	no	—
126	C₃×C₂²⋊C₄	—	4	1	no	—
127	C₃×C₂²⋊C₄	—	4	1	no	—
128	C₃×C₄²	—	4	1	no	—
129	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
130	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
131	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₂×#41	—	2	1	no	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₃×D₄	1	2	yes	—
139	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
140	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
141	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
142	#127	C₂×C₃×C₄	1	1	yes	characteristic
143	#125	—	2	1	no	—
144	#124	—	2	1	no	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	#121	—	2	1	no	—
148	#121	—	2	1	no	—
149	C₄×Q₈	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	—	2	1	no	—
153	#117	—	2	1	no	—
154	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
155	#108	—	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	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	#102	—	2	2	no	—
163	#102	—	2	2	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
167	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
168	C₄×C₈	C₃×D₄	1	2	yes	—
169	C₄×C₈	C₃×D₄	1	2	yes	—
170	C₄×C₈	C₃×D₄	1	2	yes	—
171	C₂³×C₃	—	4	1	no	—
172	C₂³×C₃	#124	1	1	yes	characteristic
173	C₃×Q₈	—	4	2	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₃×Q₈	—	4	2	no	—
176	C₃×D₄	—	4	2	no	—
177	C₃×D₄	—	4	2	no	—
178	C₃×D₄	—	4	2	no	—
179	C₃×D₄	—	8	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	4	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	—	4	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	#127	1	1	yes	characteristic
193	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₃×C₄	#137	1	2	yes	—
195	C₃×C₈	—	2	4	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	4	2	no	—
199	C₃×C₈	—	8	1	no	—
200	C₃×C₈	—	8	1	no	—
201	#41	—	4	2	no	—
202	#41	—	4	2	no	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₃×#41	1	1	yes	characteristic
213	C₂²×C₄	C₃×#41	1	1	yes	characteristic
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
225	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
239	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
240	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
241	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	8	1	no	—
245	C₂²×C₃	—	8	1	no	—
246	C₂²×C₃	—	8	1	no	—
247	C₂²×C₃	#390	1	1	yes	characteristic
248	C₂²×C₃	#489	1	2	yes	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	8	1	no	—
252	C₃×C₄	—	8	1	no	—
253	C₃×C₄	—	8	1	no	—
254	C₃×C₄	#417	1	2	yes	—
255	C₃×C₄	#443	1	2	yes	—
256	C₃×C₄	#444	1	2	yes	—
257	C₃×C₄	#496	1	1	yes	characteristic
258	C₃×C₄	#497	1	1	yes	characteristic
259	C₂³	—	4	1	no	—
260	C₂³	C₃×#124	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
277	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
278	C₂×C₄	C₃×#121	1	1	yes	characteristic
279	C₂×C₄	C₃×#121	1	1	yes	characteristic
280	C₂×C₄	C₃×#127	1	1	yes	characteristic
281	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
282	C₂×C₄	C₃×#137	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	4	2	no	—
287	C₈	—	8	1	no	—
288	C₈	—	8	1	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	16	1	no	—
291	C₂×C₃	#1959	1	1	yes	characteristic
292	C₂×C₃	#1974	1	1	yes	characteristic
293	C₂×C₃	#1975	1	1	yes	characteristic
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	—	8	1	no	—
299	C₂²	C₃×#390	1	1	yes	characteristic
300	C₂²	C₃×#489	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	8	1	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	C₃×#417	1	2	yes	—
307	C₄	C₃×#443	1	2	yes	—
308	C₄	C₃×#444	1	2	yes	—
309	C₄	C₃×#496	1	1	yes	characteristic
310	C₄	C₃×#497	1	1	yes	characteristic
311	C₃	#9858	1	1	yes	characteristic, direct product factor, Sylow
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	C₃×#1959	1	1	yes	characteristic
315	C₂	C₃×#1974	1	1	yes	characteristic
316	C₂	C₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#390
prime radical	C₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#1848	C₂	semidirect
3. C₃×#1493	C₂	semidirect
4. C₃×#1493	C₂	semidirect
7. C₂×C₃×#322	C₂	semidirect
8. #9858	C₃	direct (N × H)
29. #1848	C₂×C₃	semidirect
30. #1493	C₂×C₃	semidirect
31. #1493	C₂×C₃	semidirect
34. C₂×#322	C₂×C₃	semidirect
311. C₃	#9858	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#3131	2
C₃×#1848	2
C₃×#1493	2
C₃×#1492	2
C₂×C₃×#322	2
#9858	3

Subgroups — C3×#9858 #10628465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9858 #10628465