Subgroups — C₃×#9851 #10628458

Order 768.

492 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1962	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1492	C₂	1	1	yes	maximal, characteristic
8	#9851	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×#498	C₄	1	1	yes	characteristic, complemented
12	C₃×#492	C₄	1	1	yes	characteristic, complemented
13	C₃×#487	C₄	1	1	yes	characteristic, complemented
14	C₃×#486	C₄	1	1	yes	characteristic, complemented
15	C₃×#465	—	2	2	no	—
16	C₃×C₄×SD₁₆	—	2	2	no	—
17	C₃×#423	C₂²	1	1	yes	characteristic
18	C₃×#423	C₂²	1	1	yes	characteristic
19	C₂×C₃×#104	—	2	2	no	—
20	C₂×C₃×#97	—	2	2	no	—
21	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄³	C₂²	1	1	yes	characteristic
24	C₃×#339	—	2	2	no	—
25	C₃×#335	—	2	2	no	—
26	C₃×Q₈⋊C₈	—	2	2	no	—
27	C₃×D₄⋊C₈	—	2	2	no	—
28	#3167	C₂×C₃	1	1	yes	characteristic
29	#1994	C₂×C₃	1	1	yes	characteristic, complemented
30	#1962	C₂×C₃	1	1	yes	characteristic
31	C₄×#97	C₂×C₃	1	1	yes	characteristic, complemented
32	#1567	C₂×C₃	1	1	yes	characteristic, complemented
33	#1493	C₂×C₃	1	1	yes	characteristic, complemented
34	#1492	C₂×C₃	1	1	yes	characteristic
35	C₂×C₃×#41	—	2	1	no	—
36	C₂×C₃×SD₁₆	—	2	2	no	—
37	C₂×C₃×M₁₆	—	2	1	no	—
38	C₂×C₃×M₁₆	—	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	—	2	1	no	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	D₄	1	2	yes	—
59	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
60	C₃×#106	—	2	2	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	—	2	1	no	—
65	C₃×#104	—	4	4	no	—
66	C₃×#103	—	2	2	no	—
67	C₃×#102	—	2	2	no	—
68	C₃×#97	—	2	8	no	—
69	C₃×C₄×C₈	—	2	1	no	—
70	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₃×C₄×C₈	D₄	1	2	yes	—
73	C₃×C₄×C₈	D₄	1	2	yes	—
74	#533	C₂²×C₃	1	1	yes	characteristic
75	#532	C₂²×C₃	1	1	yes	characteristic
76	#498	C₃×C₄	1	1	yes	characteristic, complemented
77	#492	C₃×C₄	1	1	yes	characteristic, complemented
78	#487	C₃×C₄	1	1	yes	characteristic, complemented
79	#486	C₃×C₄	1	1	yes	characteristic, complemented
80	#465	—	2	2	no	—
81	C₄×SD₁₆	—	2	2	no	—
82	#423	C₂²×C₃	1	1	yes	characteristic
83	#423	C₂²×C₃	1	1	yes	characteristic
84	C₂×#104	—	2	2	no	—
85	C₂×#97	—	2	2	no	—
86	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
88	C₄³	C₂²×C₃	1	1	yes	characteristic
89	#339	—	2	2	no	—
90	#335	—	2	2	no	—
91	Q₈⋊C₈	—	2	2	no	—
92	D₄⋊C₈	—	2	2	no	—
93	C₃×#41	—	4	2	no	—
94	C₃×#41	—	4	2	no	—
95	C₂×C₃×Q₈	—	2	1	no	—
96	C₂×C₃×Q₈	—	2	1	no	—
97	C₂×C₃×Q₈	—	2	1	no	—
98	C₂×C₃×D₄	—	2	1	no	—
99	C₂×C₃×D₄	—	2	1	no	—
100	C₂×C₃×D₄	—	2	1	no	—
101	C₂×C₃×D₄	—	4	1	no	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	—	4	1	no	—
104	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₄	#41	1	1	yes	characteristic
107	C₃×SD₁₆	—	4	4	no	—
108	C₃×M₁₆	—	4	2	no	—
109	C₃×M₁₆	—	4	2	no	—
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	4	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	—	2	4	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	4	2	no	—
132	C₃×C₄²	—	4	1	no	—
133	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
134	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
135	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
136	C₃×C₄²	#41	1	1	yes	characteristic
137	C₂×#41	—	2	1	no	—
138	C₂×SD₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	1	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	C₃×D₄	1	2	yes	—
143	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
144	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
145	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
146	#127	C₂×C₃×C₄	1	1	yes	characteristic
147	#125	—	2	1	no	—
148	#124	—	2	1	no	—
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	#121	—	2	1	no	—
152	#121	—	2	1	no	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	#117	—	2	1	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	C₃×D₄	1	2	yes	—
161	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
162	#106	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₄⋊C₈	—	2	1	no	—
166	C₄⋊C₈	—	2	1	no	—
167	#104	—	4	4	no	—
168	#103	—	2	2	no	—
169	#102	—	2	2	no	—
170	#97	—	2	8	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
173	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
174	C₄×C₈	C₃×D₄	1	2	yes	—
175	C₄×C₈	C₃×D₄	1	2	yes	—
176	C₂³×C₃	—	4	1	no	—
177	C₂³×C₃	#121	1	1	yes	characteristic
178	C₃×Q₈	—	4	2	no	—
179	C₃×Q₈	—	4	2	no	—
180	C₃×Q₈	—	4	2	no	—
181	C₃×D₄	—	4	2	no	—
182	C₃×D₄	—	4	2	no	—
183	C₃×D₄	—	4	2	no	—
184	C₃×D₄	—	8	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	4	2	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	—	4	1	no	—
192	C₂×C₃×C₄	—	4	1	no	—
193	C₂×C₃×C₄	—	8	1	no	—
194	C₂×C₃×C₄	—	8	1	no	—
195	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	#121	1	1	yes	characteristic
199	C₂×C₃×C₄	#124	1	1	yes	characteristic
200	C₂×C₃×C₄	#127	1	1	yes	characteristic
201	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
202	C₂×C₃×C₄	#137	1	2	yes	—
203	C₃×C₈	—	2	4	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	8	1	no	—
206	C₃×C₈	—	8	1	no	—
207	C₃×C₈	#104	1	4	yes	complemented
208	#41	—	4	2	no	—
209	#41	—	4	2	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₃×#41	1	1	yes	characteristic
222	SD₁₆	—	4	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
231	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	1	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
249	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
250	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
251	C₄²	C₃×#41	1	1	yes	characteristic
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	8	1	no	—
255	C₂²×C₃	—	8	1	no	—
256	C₂²×C₃	—	8	1	no	—
257	C₂²×C₃	#390	1	1	yes	characteristic
258	C₂²×C₃	#468	1	2	yes	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	4	4	no	—
262	C₃×C₄	—	8	1	no	—
263	C₃×C₄	—	8	1	no	—
264	C₃×C₄	—	8	1	no	—
265	C₃×C₄	C₂×#104	1	2	yes	—
266	C₃×C₄	#440	1	2	yes	—
267	C₃×C₄	#444	1	2	yes	—
268	C₃×C₄	#496	1	1	yes	characteristic
269	C₃×C₄	#497	1	1	yes	characteristic
270	C₂³	—	4	1	no	—
271	C₂³	C₃×#121	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
291	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
292	C₂×C₄	C₃×#121	1	1	yes	characteristic
293	C₂×C₄	C₃×#124	1	1	yes	characteristic
294	C₂×C₄	C₃×#127	1	1	yes	characteristic
295	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
296	C₂×C₄	C₃×#137	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	2	2	no	—
299	C₈	—	8	1	no	—
300	C₈	—	8	1	no	—
301	C₈	C₃×#104	1	4	yes	complemented
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	16	1	no	—
304	C₂×C₃	#1962	1	1	yes	characteristic
305	C₂×C₃	#1973	1	1	yes	characteristic
306	C₂×C₃	#1975	1	1	yes	characteristic
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	C₃×#390	1	1	yes	characteristic
313	C₂²	C₃×#468	1	2	yes	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	C₂×C₃×#104	1	2	yes	—
321	C₄	C₃×#440	1	2	yes	—
322	C₄	C₃×#444	1	2	yes	—
323	C₄	C₃×#496	1	1	yes	characteristic
324	C₄	C₃×#497	1	1	yes	characteristic
325	C₃	#9851	1	1	yes	characteristic, direct product factor, Sylow
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	C₃×#1962	1	1	yes	characteristic
329	C₂	C₃×#1973	1	1	yes	characteristic
330	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×#1994	C₂	semidirect
4. C₃×C₄×#97	C₂	semidirect
5. C₃×#1567	C₂	semidirect
6. C₃×#1493	C₂	semidirect
8. #9851	C₃	direct (N × H)
11. C₃×#498	C₄	semidirect
12. C₃×#492	C₄	semidirect
13. C₃×#487	C₄	semidirect
14. C₃×#486	C₄	semidirect
29. #1994	C₂×C₃	semidirect
31. C₄×#97	C₂×C₃	semidirect
32. #1567	C₂×C₃	semidirect
33. #1493	C₂×C₃	semidirect
76. #498	C₃×C₄	semidirect
77. #492	C₃×C₄	semidirect
78. #487	C₃×C₄	semidirect
79. #486	C₃×C₄	semidirect
207. C₃×C₈	#104	semidirect
301. C₈	C₃×#104	semidirect
325. C₃	#9851	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×#3167	2
C₃×#1994	2
C₃×#1962	2
C₃×C₄×#97	2
C₃×#1567	2
C₃×#1493	2
C₃×#1492	2
#9851	3

Subgroups — C3×#9851 #10628458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9851 #10628458