Subgroups — C₃×#33637 #10652244

Order 768.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₃×#2231	C₂	1	4	yes	maximal, complemented
4	C₃×#2230	C₂	1	4	yes	maximal, complemented
5	C₃×#2155	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2145	C₂	1	2	yes	maximal
8	#33637	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₃×#131	—	2	1	no	—
11	C₃×#533	—	2	1	no	—
12	C₂×C₃×M₃₂	C₂²	1	2	yes	—
13	C₃×#500	—	2	1	no	—
14	C₃×#499	—	2	1	no	—
15	C₃×#499	C₂²	1	2	yes	complemented
16	C₃×#498	C₂²	1	2	yes	complemented
17	C₃×#443	—	2	1	no	—
18	C₃×#443	—	2	1	no	—
19	C₃×#443	C₂²	1	2	yes	—
20	C₃×#443	C₂²	1	2	yes	—
21	C₂×C₃×#108	C₂²	1	1	yes	characteristic
22	C₃×#427	—	2	1	no	—
23	C₃×#427	C₂²	1	2	yes	—
24	C₂×C₃×#104	—	2	1	no	—
25	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₃×#365	C₂²	1	8	yes	complemented
27	C₃×#364	C₂²	1	2	yes	complemented
28	C₃×#364	C₂²	1	2	yes	complemented
29	C₃×#360	C₂²	1	8	yes	—
30	#3160	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#443	C₂×C₃	1	1	yes	characteristic
32	#2231	C₂×C₃	1	4	yes	complemented
33	#2230	C₂×C₃	1	4	yes	complemented
34	#2155	C₂×C₃	1	2	yes	complemented
35	C₂×#364	C₂×C₃	1	1	yes	characteristic, complemented
36	#2145	C₂×C₃	1	2	yes	—
37	C₂×C₃×#41	—	2	1	no	—
38	C₃×#135	—	2	1	no	—
39	C₃×#135	—	2	1	no	—
40	C₃×#135	Q₈	1	2	yes	complemented
41	C₃×#135	C₂³	1	2	yes	—
42	C₂×C₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
43	C₂×C₃×SD₁₆	—	2	1	no	—
44	C₂×C₃×D₈	Q₈	1	1	yes	characteristic, complemented
45	C₃×#131	—	2	2	no	—
46	C₃×#131	—	2	2	no	—
47	C₃×#131	—	4	1	no	—
48	C₃×#131	—	4	1	no	—
49	C₂×C₃×M₁₆	—	2	1	no	—
50	C₂×C₃×M₁₆	—	4	1	no	—
51	C₂²×C₃×C₈	—	4	1	no	—
52	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	2	no	—
55	C₃×#125	—	2	2	no	—
56	C₃×C₂²⋊Q₈	—	4	2	no	—
57	C₃×C₄×Q₈	—	4	2	no	—
58	C₃×#117	—	2	2	no	—
59	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
60	C₃×M₃₂	—	2	2	no	—
61	C₃×M₃₂	C₂³	1	4	yes	—
62	C₂×C₃×C₁₆	—	2	2	no	—
63	C₃×#108	—	2	1	no	—
64	C₃×#108	C₂³	1	2	yes	—
65	C₃×#107	—	2	2	no	—
66	C₃×#107	—	2	2	no	—
67	C₃×#106	—	2	2	no	—
68	C₃×#106	C₂³	1	4	yes	—
69	C₃×#104	—	2	2	no	—
70	C₃×#104	—	2	2	no	—
71	C₃×C₄×C₈	D₄	1	1	yes	characteristic
72	C₃×C₄×C₈	D₄	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×#135	C₂²×C₃	1	1	yes	characteristic
76	C₂×#131	—	2	1	no	—
77	#533	—	2	1	no	—
78	C₂×M₃₂	C₂²×C₃	1	2	yes	—
79	#500	—	2	1	no	—
80	#499	—	2	1	no	—
81	#499	C₂²×C₃	1	2	yes	complemented
82	#498	C₂²×C₃	1	2	yes	complemented
83	#443	—	2	1	no	—
84	#443	—	2	1	no	—
85	#443	C₂²×C₃	1	2	yes	—
86	#443	C₂²×C₃	1	2	yes	—
87	C₂×#108	C₂²×C₃	1	1	yes	characteristic
88	#427	—	2	1	no	—
89	#427	C₂²×C₃	1	2	yes	—
90	C₂×#104	—	2	1	no	—
91	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
92	#365	C₂²×C₃	1	8	yes	complemented
93	#364	C₂²×C₃	1	2	yes	complemented
94	#364	C₂²×C₃	1	2	yes	complemented
95	#360	C₂²×C₃	1	8	yes	—
96	C₃×#41	—	2	2	no	—
97	C₃×#41	—	2	2	no	—
98	C₃×#41	—	4	1	no	—
99	C₃×#41	—	4	1	no	—
100	C₂×C₃×Q₈	—	2	1	no	—
101	C₂×C₃×Q₈	—	4	2	no	—
102	C₂×C₃×D₄	—	2	1	no	—
103	C₂×C₃×D₄	—	4	1	no	—
104	C₂²×C₃×C₄	—	2	1	no	—
105	C₂²×C₃×C₄	—	4	1	no	—
106	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₃×Q₁₆	—	2	1	no	—
108	C₃×Q₁₆	C₂×Q₈	1	2	yes	—
109	C₃×SD₁₆	—	2	2	no	—
110	C₃×SD₁₆	—	2	2	no	—
111	C₃×D₈	—	2	1	no	—
112	C₃×D₈	C₂×Q₈	1	2	yes	—
113	C₃×M₁₆	—	2	2	no	—
114	C₃×M₁₆	—	4	1	no	—
115	C₃×M₁₆	—	4	2	no	—
116	C₃×M₁₆	—	4	2	no	—
117	C₂×C₃×C₈	—	4	1	no	—
118	C₂×C₃×C₈	—	4	2	no	—
119	C₂×C₃×C₈	—	4	2	no	—
120	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
121	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
122	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
123	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
129	C₂×C₃×C₈	#41	1	1	yes	characteristic
130	C₂×C₃×C₈	#41	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₄⋊C₄	—	4	4	no	—
134	C₃×C₄⋊C₄	—	4	2	no	—
135	C₃×C₂²⋊C₄	—	4	2	no	—
136	C₃×C₄²	—	2	1	no	—
137	C₃×C₄²	—	4	2	no	—
138	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
139	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
140	C₃×C₁₆	—	2	4	no	—
141	C₂×#41	—	2	1	no	—
142	#135	—	2	1	no	—
143	#135	—	2	1	no	—
144	#135	C₃×Q₈	1	2	yes	complemented
145	#135	C₂³×C₃	1	2	yes	—
146	C₂×Q₁₆	C₃×Q₈	1	1	yes	characteristic, complemented
147	C₂×SD₁₆	—	2	1	no	—
148	C₂×D₈	C₃×Q₈	1	1	yes	characteristic, complemented
149	#131	—	2	2	no	—
150	#131	—	2	2	no	—
151	#131	—	4	1	no	—
152	#131	—	4	1	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	4	1	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
158	C₄⋊Q₈	—	2	2	no	—
159	#125	—	2	2	no	—
160	C₂²⋊Q₈	—	4	2	no	—
161	C₄×Q₈	—	4	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
164	M₃₂	—	2	2	no	—
165	M₃₂	C₂³×C₃	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	#108	—	2	1	no	—
168	#108	C₂³×C₃	1	2	yes	—
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#106	—	2	2	no	—
172	#106	C₂³×C₃	1	4	yes	—
173	#104	—	2	2	no	—
174	#104	—	2	2	no	—
175	C₄×C₈	C₃×D₄	1	1	yes	characteristic
176	C₄×C₈	C₃×D₄	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
179	C₂³×C₃	—	4	1	no	—
180	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₃×Q₈	—	2	2	no	—
182	C₃×Q₈	—	4	1	no	—
183	C₃×Q₈	—	8	4	no	—
184	C₃×D₄	—	2	2	no	—
185	C₃×D₄	—	4	2	no	—
186	C₃×D₄	—	4	2	no	—
187	C₃×D₄	—	4	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	1	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	4	4	no	—
195	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
199	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
201	C₃×C₈	—	4	2	no	—
202	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
203	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
204	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
205	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
206	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
207	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
208	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
209	C₃×C₈	C₂×#41	1	1	yes	characteristic
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	#41	—	4	1	no	—
213	#41	—	4	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	Q₁₆	—	2	1	no	—
222	Q₁₆	C₂×C₃×Q₈	1	2	yes	—
223	SD₁₆	—	2	2	no	—
224	SD₁₆	—	2	2	no	—
225	D₈	—	2	1	no	—
226	D₈	C₂×C₃×Q₈	1	2	yes	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	1	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
235	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
236	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
237	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
238	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
242	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
243	C₂×C₈	C₃×#41	1	1	yes	characteristic
244	C₂×C₈	C₃×#41	1	1	yes	characteristic
245	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	4	2	no	—
252	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
253	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
254	C₁₆	—	2	4	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	—	4	2	no	—
257	C₂²×C₃	—	8	2	no	—
258	C₂²×C₃	#476	1	1	yes	characteristic
259	C₂²×C₃	#476	1	1	yes	characteristic
260	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	4	2	no	—
264	C₃×C₄	—	8	4	no	—
265	C₃×C₄	#476	1	1	yes	characteristic
266	C₃×C₄	#476	1	1	yes	characteristic
267	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
268	C₃×C₄	C₂²×SD₁₆	1	1	yes	characteristic
269	C₂³	—	4	1	no	—
270	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	8	4	no	—
274	D₄	—	2	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
286	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
288	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
289	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
293	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
294	C₈	C₂×C₃×SD₁₆	1	1	yes	characteristic
295	C₈	C₂×C₃×SD₁₆	1	1	yes	characteristic
296	C₈	C₂×C₃×SD₁₆	1	1	yes	characteristic
297	C₈	C₂×C₃×SD₁₆	1	1	yes	characteristic
298	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
299	C₈	C₂×C₃×#41	1	1	yes	characteristic
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	8	2	no	—
304	C₂×C₃	C₂×#476	1	1	yes	characteristic
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₃×#476	1	1	yes	characteristic
309	C₂²	C₃×#476	1	1	yes	characteristic
310	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	8	4	no	—
315	C₄	C₃×#476	1	1	yes	characteristic
316	C₄	C₃×#476	1	1	yes	characteristic
317	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
318	C₄	C₂²×C₃×SD₁₆	1	1	yes	characteristic
319	C₃	#33637	1	1	yes	characteristic, direct product factor, Sylow
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×C₃×#476	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂×C₃	C₂×#476
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3160	C₂	semidirect
3. C₃×#2231	C₂	semidirect
4. C₃×#2230	C₂	semidirect
5. C₃×#2155	C₂	semidirect
6. C₂×C₃×#364	C₂	semidirect
8. #33637	C₃	direct (N × H)
15. C₃×#499	C₂²	semidirect
16. C₃×#498	C₂²	semidirect
26. C₃×#365	C₂²	semidirect
27. C₃×#364	C₂²	semidirect
28. C₃×#364	C₂²	semidirect
30. #3160	C₂×C₃	semidirect
32. #2231	C₂×C₃	semidirect
33. #2230	C₂×C₃	semidirect
34. #2155	C₂×C₃	semidirect
35. C₂×#364	C₂×C₃	semidirect
40. C₃×#135	Q₈	semidirect
42. C₂×C₃×Q₁₆	Q₈	semidirect
44. C₂×C₃×D₈	Q₈	semidirect
81. #499	C₂²×C₃	semidirect
82. #498	C₂²×C₃	semidirect
92. #365	C₂²×C₃	semidirect
93. #364	C₂²×C₃	semidirect
94. #364	C₂²×C₃	semidirect
144. #135	C₃×Q₈	semidirect
146. C₂×Q₁₆	C₃×Q₈	semidirect
148. C₂×D₈	C₃×Q₈	semidirect
319. C₃	#33637	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×#3160	2
C₂×C₃×#443	2
C₃×#2231	2
C₃×#2230	2
C₃×#2155	2
C₂×C₃×#364	2
C₃×#2145	2
#33637	3

Subgroups — C3×#33637 #10652244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33637 #10652244