Subgroups — C₃×#27673 #10646280

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3093	C₂	1	2	yes	maximal
2	C₃×#2929	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2089	C₂	1	2	yes	maximal, complemented
5	C₃×#2078	C₂	1	2	yes	maximal, complemented
6	C₃×#1951	C₂	1	2	yes	maximal, complemented
7	C₃×#1822	C₂	1	2	yes	maximal, complemented
8	C₃×#1815	C₂	1	2	yes	maximal, complemented
9	C₃×#1810	C₂	1	1	yes	maximal, characteristic, complemented
10	#27673	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#131	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₃×#525	C₂²	1	1	yes	characteristic
14	C₂×C₃×#117	C₂²	1	2	yes	—
15	C₃×#485	—	2	4	no	—
16	C₃×#482	—	2	4	no	—
17	C₃×#435	—	2	2	no	—
18	C₃×C₈×D₄	—	2	2	no	—
19	C₂×C₃×#108	C₂²	1	2	yes	—
20	C₃×#428	C₂²	1	4	yes	—
21	C₃×#427	—	2	2	no	—
22	C₃×#427	—	2	2	no	—
23	C₂×C₃×#107	C₂²	1	2	yes	—
24	C₂×C₃×#106	C₂²	1	2	yes	—
25	C₃×#424	C₂²	1	1	yes	characteristic
26	C₃×#417	C₂²	1	4	yes	—
27	C₃×#408	C₂²	1	1	yes	characteristic
28	C₃×#407	C₂²	1	1	yes	characteristic
29	C₃×#407	C₂²	1	2	yes	—
30	C₃×#405	C₂²	1	1	yes	characteristic
31	C₃×#343	C₂²	1	4	yes	—
32	C₃×#343	C₂²	1	4	yes	—
33	C₃×#341	C₂²	1	2	yes	—
34	#3093	C₂×C₃	1	2	yes	—
35	#2929	C₂×C₃	1	1	yes	characteristic
36	C₂×#427	C₂×C₃	1	1	yes	characteristic, complemented
37	#2089	C₂×C₃	1	2	yes	complemented
38	#2078	C₂×C₃	1	2	yes	complemented
39	#1951	C₂×C₃	1	2	yes	complemented
40	#1822	C₂×C₃	1	2	yes	complemented
41	#1815	C₂×C₃	1	2	yes	complemented
42	#1810	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂×C₃×#41	C₂³	1	1	yes	characteristic
44	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₃×#131	—	2	4	no	—
46	C₂×C₃×M₁₆	—	2	1	no	—
47	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₃×M₁₆	C₂³	1	4	yes	—
50	C₂²×C₃×C₈	—	2	4	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	D₄	1	2	yes	—
53	C₂²×C₃×C₈	D₄	1	2	yes	—
54	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×#123	—	2	1	no	—
57	C₃×C₂²⋊Q₈	D₄	1	2	yes	—
58	C₃×#121	D₄	1	2	yes	—
59	C₃×#120	—	2	1	no	—
60	C₃×C₄×D₄	—	2	2	no	—
61	C₃×#117	—	2	4	no	—
62	C₃×#117	—	2	4	no	—
63	C₃×#117	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
65	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
66	C₂×C₃×C₄²	—	2	2	no	—
67	C₃×#108	—	2	4	no	—
68	C₃×#107	—	2	4	no	—
69	C₃×#107	—	2	4	no	—
70	C₃×#106	—	2	2	no	—
71	C₃×#106	—	2	2	no	—
72	C₃×#106	—	2	4	no	—
73	C₃×C₄⋊C₈	—	2	2	no	—
74	C₃×#103	—	2	4	no	—
75	C₃×#102	—	2	4	no	—
76	C₃×C₂²⋊C₈	—	2	2	no	—
77	C₃×C₂²⋊C₈	—	2	4	no	—
78	C₃×C₂²⋊C₈	—	2	2	no	—
79	C₃×#97	—	2	1	no	—
80	C₃×C₄×C₈	—	2	1	no	—
81	C₂×#131	C₂²×C₃	1	1	yes	characteristic
82	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
83	#525	C₂²×C₃	1	1	yes	characteristic
84	C₂×#117	C₂²×C₃	1	2	yes	—
85	#485	—	2	4	no	—
86	#482	—	2	4	no	—
87	#435	—	2	2	no	—
88	C₈×D₄	—	2	2	no	—
89	C₂×#108	C₂²×C₃	1	2	yes	—
90	#428	C₂²×C₃	1	4	yes	—
91	#427	—	2	2	no	—
92	#427	—	2	2	no	—
93	C₂×#107	C₂²×C₃	1	2	yes	—
94	C₂×#106	C₂²×C₃	1	2	yes	—
95	#424	C₂²×C₃	1	1	yes	characteristic
96	#417	C₂²×C₃	1	4	yes	—
97	#408	C₂²×C₃	1	1	yes	characteristic
98	#407	C₂²×C₃	1	2	yes	—
99	#407	C₂²×C₃	1	1	yes	characteristic
100	#405	C₂²×C₃	1	1	yes	characteristic
101	#343	C₂²×C₃	1	4	yes	—
102	#343	C₂²×C₃	1	4	yes	—
103	#341	C₂²×C₃	1	2	yes	—
104	C₂⁴×C₃	#41	1	1	yes	characteristic
105	C₃×#41	—	2	2	no	—
106	C₃×#41	—	2	2	no	—
107	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	—	2	1	no	—
110	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	2	2	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	2	4	no	—
115	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
116	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
117	C₂²×C₃×C₄	#41	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
119	C₃×M₁₆	—	2	4	no	—
120	C₃×M₁₆	—	2	2	no	—
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	2	4	no	—
123	C₃×M₁₆	—	4	2	no	—
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	—	2	4	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	4	2	no	—
131	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	#41	1	4	yes	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	2	4	no	—
138	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
139	C₃×C₂²⋊C₄	—	2	2	no	—
140	C₃×C₂²⋊C₄	—	2	1	no	—
141	C₃×C₂²⋊C₄	—	2	2	no	—
142	C₃×C₂²⋊C₄	—	4	4	no	—
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	—	4	4	no	—
145	C₂×#41	C₂³×C₃	1	1	yes	characteristic
146	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
147	#131	—	2	4	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
150	C₂×M₁₆	C₂³×C₃	1	4	yes	—
151	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₃×D₄	1	2	yes	—
155	C₂²×C₈	C₃×D₄	1	2	yes	—
156	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
158	#123	—	2	1	no	—
159	C₂²⋊Q₈	C₃×D₄	1	2	yes	—
160	#121	C₃×D₄	1	2	yes	—
161	#120	—	2	1	no	—
162	C₄×D₄	—	2	2	no	—
163	#117	—	2	4	no	—
164	#117	—	2	4	no	—
165	#117	C₂³×C₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₄²	—	2	2	no	—
169	#108	—	2	4	no	—
170	#107	—	2	4	no	—
171	#107	—	2	4	no	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	#106	—	2	4	no	—
175	C₄⋊C₈	—	2	2	no	—
176	#103	—	2	4	no	—
177	#102	—	2	4	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	4	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	#97	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	1	no	—
185	C₂³×C₃	—	4	2	no	—
186	C₂³×C₃	#121	1	2	yes	—
187	C₂³×C₃	C₂×#41	1	1	yes	characteristic
188	C₃×Q₈	—	2	2	no	—
189	C₃×D₄	—	2	2	no	—
190	C₃×D₄	—	4	4	no	—
191	C₃×D₄	—	4	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	—	4	1	no	—
200	C₂×C₃×C₄	—	4	4	no	—
201	C₂×C₃×C₄	—	8	4	no	—
202	C₂×C₃×C₄	#121	1	2	yes	—
203	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×#41	1	2	yes	—
206	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃×C₈	—	4	1	no	—
210	C₃×C₈	—	4	4	no	—
211	C₃×C₈	#123	1	2	yes	—
212	C₃×C₈	#123	1	2	yes	—
213	C₂⁴	C₃×#41	1	1	yes	characteristic
214	#41	—	2	2	no	—
215	#41	—	2	2	no	—
216	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
225	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
226	C₂²×C₄	C₃×#41	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×C₈	C₃×#41	1	4	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	4	4	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	4	4	no	—
259	C₂²×C₃	—	4	1	no	—
260	C₂²×C₃	—	4	2	no	—
261	C₂²×C₃	C₂×#121	1	1	yes	characteristic
262	C₂²×C₃	#552	1	2	yes	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	8	4	no	—
267	C₃×C₄	C₂×#123	1	1	yes	characteristic
268	C₃×C₄	#536	1	1	yes	characteristic
269	C₃×C₄	#547	1	2	yes	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	4	2	no	—
273	C₂³	C₃×#121	1	2	yes	—
274	C₂³	C₂×C₃×#41	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₃×#121	1	2	yes	—
290	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×#41	1	2	yes	—
293	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
294	C₈	—	2	2	no	—
295	C₈	—	4	1	no	—
296	C₈	—	4	1	no	—
297	C₈	—	4	4	no	—
298	C₈	C₃×#123	1	2	yes	—
299	C₈	C₃×#123	1	2	yes	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	4	2	no	—
302	C₂×C₃	—	8	1	no	—
303	C₂×C₃	#2486	1	1	yes	characteristic
304	C₂×C₃	#3067	1	2	yes	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	C₂×C₃×#121	1	1	yes	characteristic
313	C₂²	C₃×#552	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	8	4	no	—
318	C₄	C₂×C₃×#123	1	1	yes	characteristic
319	C₄	C₃×#536	1	1	yes	characteristic
320	C₄	C₃×#547	1	2	yes	—
321	C₃	#27673	1	1	yes	characteristic, direct product factor, Sylow
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	8	1	no	—
325	C₂	C₃×#2486	1	1	yes	characteristic
326	C₂	C₃×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₃×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#427	C₂	semidirect
4. C₃×#2089	C₂	semidirect
5. C₃×#2078	C₂	semidirect
6. C₃×#1951	C₂	semidirect
7. C₃×#1822	C₂	semidirect
8. C₃×#1815	C₂	semidirect
9. C₃×#1810	C₂	semidirect
10. #27673	C₃	direct (N × H)
36. C₂×#427	C₂×C₃	semidirect
37. #2089	C₂×C₃	semidirect
38. #2078	C₂×C₃	semidirect
39. #1951	C₂×C₃	semidirect
40. #1822	C₂×C₃	semidirect
41. #1815	C₂×C₃	semidirect
42. #1810	C₂×C₃	semidirect
321. C₃	#27673	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×#3093	2
C₃×#2929	2
C₂×C₃×#427	2
C₃×#2089	2
C₃×#2078	2
C₃×#1951	2
C₃×#1822	2
C₃×#1815	2
C₃×#1810	2
#27673	3

Subgroups — C3×#27673 #10646280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#27673 #10646280