Subgroups — C₃×#27691 #10646298

Order 768.

640 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2103	C₂	1	2	yes	maximal, complemented
4	C₃×#2091	C₂	1	2	yes	maximal, complemented
5	C₃×#1953	C₂	1	2	yes	maximal, complemented
6	C₃×#1891	C₂	1	2	yes	maximal
7	C₃×#1858	C₂	1	2	yes	maximal, complemented
8	C₃×#1813	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#1810	C₂	1	2	yes	maximal, complemented
10	#27691	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#568	C₂²	1	2	yes	—
12	C₂×C₃×#131	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₃×#525	C₂²	1	1	yes	characteristic
15	C₃×#435	—	2	2	no	—
16	C₃×C₈×D₄	—	2	2	no	—
17	C₃×#430	—	2	2	no	—
18	C₃×#430	C₂²	1	2	yes	—
19	C₃×#430	C₂²	1	4	yes	—
20	C₂×C₃×#108	—	2	2	no	—
21	C₂×C₃×#108	C₂²	1	2	yes	—
22	C₂×C₃×#108	C₂²	1	2	yes	—
23	C₃×#424	C₂²	1	1	yes	characteristic
24	C₃×#413	C₂²	1	2	yes	—
25	C₃×#408	C₂²	1	1	yes	characteristic
26	C₃×#407	C₂²	1	1	yes	characteristic
27	C₃×#407	C₂²	1	4	yes	—
28	C₃×#405	C₂²	1	1	yes	characteristic
29	C₃×#344	C₂²	1	4	yes	—
30	C₃×#341	C₂²	1	4	yes	—
31	C₃×#338	C₂²	1	2	yes	—
32	#2929	C₂×C₃	1	1	yes	characteristic
33	#2916	C₂×C₃	1	1	yes	characteristic, complemented
34	#2103	C₂×C₃	1	2	yes	complemented
35	#2091	C₂×C₃	1	2	yes	complemented
36	#1953	C₂×C₃	1	2	yes	complemented
37	#1891	C₂×C₃	1	2	yes	—
38	#1858	C₂×C₃	1	2	yes	complemented
39	#1813	C₂×C₃	1	1	yes	characteristic, complemented
40	#1810	C₂×C₃	1	2	yes	complemented
41	C₂×C₃×#41	C₂³	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₃×#131	—	2	4	no	—
44	C₃×#131	—	4	4	no	—
45	C₂×C₃×M₁₆	—	2	4	no	—
46	C₂×C₃×M₁₆	D₄	1	2	yes	—
47	C₂×C₃×M₁₆	C₂³	1	4	yes	—
48	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₃×M₁₆	C₂³	1	4	yes	—
50	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	—	2	4	no	—
54	C₂²×C₃×C₈	D₄	1	2	yes	—
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₃×#123	Q₈	1	2	yes	—
58	C₃×C₂²⋊Q₈	—	2	1	no	—
59	C₃×#121	—	2	1	no	—
60	C₃×#120	Q₈	1	2	yes	—
61	C₃×C₄×D₄	—	2	2	no	—
62	C₃×#117	C₂³	1	1	yes	characteristic
63	C₃×#108	—	2	2	no	—
64	C₃×#108	—	2	4	no	—
65	C₃×#108	—	2	2	no	—
66	C₃×#108	—	2	4	no	—
67	C₃×#108	—	4	4	no	—
68	C₃×C₄⋊C₈	—	2	2	no	—
69	C₃×#101	—	2	2	no	—
70	C₃×#100	—	2	2	no	—
71	C₃×C₂²⋊C₈	—	2	2	no	—
72	C₃×C₂²⋊C₈	—	2	2	no	—
73	C₃×C₂²⋊C₈	—	4	4	no	—
74	C₃×#97	—	2	1	no	—
75	C₃×C₄×C₈	—	2	1	no	—
76	#568	C₂²×C₃	1	2	yes	—
77	C₂×#131	C₂²×C₃	1	1	yes	characteristic
78	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
79	#525	C₂²×C₃	1	1	yes	characteristic
80	#435	—	2	2	no	—
81	C₈×D₄	—	2	2	no	—
82	#430	—	2	2	no	—
83	#430	C₂²×C₃	1	2	yes	—
84	#430	C₂²×C₃	1	4	yes	—
85	C₂×#108	—	2	2	no	—
86	C₂×#108	C₂²×C₃	1	2	yes	—
87	C₂×#108	C₂²×C₃	1	2	yes	—
88	#424	C₂²×C₃	1	1	yes	characteristic
89	#413	C₂²×C₃	1	2	yes	—
90	#408	C₂²×C₃	1	1	yes	characteristic
91	#407	C₂²×C₃	1	1	yes	characteristic
92	#407	C₂²×C₃	1	4	yes	—
93	#405	C₂²×C₃	1	1	yes	characteristic
94	#344	C₂²×C₃	1	4	yes	—
95	#341	C₂²×C₃	1	4	yes	—
96	#338	C₂²×C₃	1	2	yes	—
97	C₂⁴×C₃	C₂×Q₈	1	1	yes	characteristic
98	C₃×#41	—	2	2	no	—
99	C₃×#41	—	2	2	no	—
100	C₂×C₃×Q₈	#41	1	1	yes	characteristic
101	C₂×C₃×D₄	—	2	2	no	—
102	C₂×C₃×D₄	C₂×Q₈	1	2	yes	—
103	C₂×C₃×D₄	#41	1	1	yes	characteristic
104	C₂²×C₃×C₄	—	2	1	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
109	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
111	C₃×M₁₆	—	2	2	no	—
112	C₃×M₁₆	—	2	4	no	—
113	C₃×M₁₆	—	2	4	no	—
114	C₃×M₁₆	—	2	4	no	—
115	C₃×M₁₆	—	2	4	no	—
116	C₃×M₁₆	—	2	2	no	—
117	C₃×M₁₆	—	2	4	no	—
118	C₃×M₁₆	—	4	4	no	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	4	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₂×C₃×C₈	—	4	4	no	—
126	C₂×C₃×C₈	—	4	2	no	—
127	C₂×C₃×C₈	—	4	2	no	—
128	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
129	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₈	#41	1	4	yes	—
132	C₃×C₄⋊C₄	—	2	2	no	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₂²⋊C₄	—	2	2	no	—
135	C₃×C₂²⋊C₄	—	2	2	no	—
136	C₃×C₂²⋊C₄	C₂×Q₈	1	2	yes	—
137	C₃×C₄²	—	2	1	no	—
138	C₂×#41	C₂³×C₃	1	1	yes	characteristic
139	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
140	#131	—	2	4	no	—
141	#131	—	4	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	C₃×D₄	1	2	yes	—
144	C₂×M₁₆	C₂³×C₃	1	4	yes	—
145	C₂×M₁₆	C₂³×C₃	1	4	yes	—
146	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
147	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	C₃×D₄	1	2	yes	—
152	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
154	#123	C₃×Q₈	1	2	yes	—
155	C₂²⋊Q₈	—	2	1	no	—
156	#121	—	2	1	no	—
157	#120	C₃×Q₈	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	#117	C₂³×C₃	1	1	yes	characteristic
160	#108	—	2	2	no	—
161	#108	—	2	4	no	—
162	#108	—	2	2	no	—
163	#108	—	2	4	no	—
164	#108	—	4	4	no	—
165	C₄⋊C₈	—	2	2	no	—
166	#101	—	2	2	no	—
167	#100	—	2	2	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	4	4	no	—
171	#97	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	—	2	2	no	—
175	C₂³×C₃	—	4	2	no	—
176	C₂³×C₃	C₂²⋊Q₈	1	2	yes	—
177	C₂³×C₃	C₂²×Q₈	1	1	yes	characteristic
178	C₃×Q₈	—	2	2	no	—
179	C₃×D₄	—	2	4	no	—
180	C₃×D₄	—	2	2	no	—
181	C₃×D₄	—	4	4	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
191	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×#41	1	2	yes	—
193	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
194	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	4	1	no	—
197	C₃×C₈	—	4	4	no	—
198	C₃×C₈	—	4	1	no	—
199	C₃×C₈	—	4	4	no	—
200	C₃×C₈	#123	1	2	yes	—
201	C₃×C₈	#123	1	2	yes	—
202	C₂⁴	C₂×C₃×Q₈	1	1	yes	characteristic
203	#41	—	2	2	no	—
204	#41	—	2	2	no	—
205	C₂×Q₈	C₃×#41	1	1	yes	characteristic
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₂×C₃×Q₈	1	2	yes	—
208	C₂×D₄	C₃×#41	1	1	yes	characteristic
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
234	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
236	C₂×C₈	C₃×#41	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	C₂×C₃×Q₈	1	2	yes	—
242	C₄²	—	2	1	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	4	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	4	1	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	—	4	1	no	—
251	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂²×C₃	#554	1	2	yes	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	4	2	no	—
255	C₃×C₄	—	4	1	no	—
256	C₃×C₄	C₂×#123	1	1	yes	characteristic
257	C₃×C₄	#535	1	1	yes	characteristic
258	C₃×C₄	#553	1	2	yes	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	C₃×C₂²⋊Q₈	1	2	yes	—
263	C₂³	C₂²×C₃×Q₈	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
277	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×#41	1	2	yes	—
279	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
280	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
281	C₈	—	2	2	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	4	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	4	no	—
286	C₈	C₃×#123	1	2	yes	—
287	C₈	C₃×#123	1	2	yes	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	4	2	no	—
291	C₂×C₃	—	4	2	no	—
292	C₂×C₃	#2487	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
302	C₂²	C₃×#554	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	C₂×C₃×#123	1	1	yes	characteristic
307	C₄	C₃×#535	1	1	yes	characteristic
308	C₄	C₃×#553	1	2	yes	—
309	C₃	#27691	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	—	2	1	no	—
311	C₂	—	2	2	no	—
312	C₂	—	4	2	no	—
313	C₂	—	4	2	no	—
314	C₂	C₃×#2487	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	#2487
prime radical	C₃×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2916	C₂	semidirect
3. C₃×#2103	C₂	semidirect
4. C₃×#2091	C₂	semidirect
5. C₃×#1953	C₂	semidirect
7. C₃×#1858	C₂	semidirect
8. C₃×#1813	C₂	semidirect
9. C₃×#1810	C₂	semidirect
10. #27691	C₃	direct (N × H)
33. #2916	C₂×C₃	semidirect
34. #2103	C₂×C₃	semidirect
35. #2091	C₂×C₃	semidirect
36. #1953	C₂×C₃	semidirect
38. #1858	C₂×C₃	semidirect
39. #1813	C₂×C₃	semidirect
40. #1810	C₂×C₃	semidirect
309. C₃	#27691	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×#2929	2
C₃×#2916	2
C₃×#2103	2
C₃×#2091	2
C₃×#1953	2
C₃×#1891	2
C₃×#1858	2
C₃×#1813	2
C₃×#1810	2
#27691	3

Subgroups — C3×#27691 #10646298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#27691 #10646298