Subgroups — C₃×#81799 #11874291

Order 1152.

1205 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#919	C₂	1	4	yes	maximal, complemented
2	C₃×#5725	C₂	1	2	yes	maximal, complemented
3	C₃²×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5191	C₂	1	4	yes	maximal, complemented
5	C₃×#5085	C₂	1	2	yes	maximal, complemented
6	C₃×#4695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3147	—	3	1	no	maximal
9	#81799	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#239	C₂²	1	4	yes	—
11	C₂×C₃²×Q₁₆	C₂²	1	4	yes	—
12	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×#947	C₂²	1	4	yes	—
14	C₂×C₃×#215	C₂²	1	8	yes	—
15	C₃×#919	—	2	16	no	—
16	C₂×C₃×D₂₄	C₂²	1	4	yes	—
17	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
18	C₃×#895	—	2	4	no	—
19	C₃×#893	—	2	4	no	—
20	C₃×#874	C₂²	1	2	yes	—
21	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₈×Dic₃	C₂²	1	2	yes	—
24	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃×#135	—	3	4	no	—
26	C₃×#532	—	3	2	no	—
27	C₂×#919	C₂×C₃	1	4	yes	complemented
28	#5725	C₂×C₃	1	2	yes	complemented
29	C₃×#495	—	3	4	no	—
30	C₃×#494	—	2	1	no	—
31	C₃×#494	D₃	1	1	yes	characteristic, complemented
32	C₃×#494	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#493	—	3	1	no	—
34	C₃×#492	—	3	2	no	—
35	C₂×C₃×C₄×C₈	—	3	1	no	—
36	#5191	C₂×C₃	1	4	yes	complemented
37	#5085	C₂×C₃	1	2	yes	complemented
38	#4695	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃²×Q₈	C₂³	1	4	yes	—
41	C₃×#239	—	2	16	no	—
42	C₂×C₃×D₁₂	—	2	4	no	—
43	C₂×C₃×D₁₂	—	2	4	no	—
44	C₂×C₃×D₁₂	C₂³	1	4	yes	—
45	C₂×C₃×C₄×D₃	—	2	4	no	—
46	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
47	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
48	C₃²×Q₁₆	—	2	8	no	—
49	C₂×C₃²×C₈	C₂³	1	2	yes	—
50	C₃²×C₄⋊C₄	—	2	4	no	—
51	C₃²×C₄²	C₂³	1	1	yes	characteristic
52	C₃×#215	—	2	16	no	—
53	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
54	C₃×C₄×Dic₃	D₄	1	2	yes	—
55	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
57	C₃×D₂₄	—	2	8	no	—
58	C₃×D₃×C₈	D₄	1	8	yes	complemented
59	#3147	—	3	1	no	Sylow
60	C₂×C₃×#41	—	3	4	no	—
61	C₂×#239	C₂²×C₃	1	4	yes	—
62	C₃×#135	—	6	16	no	—
63	C₂×C₃×Q₁₆	—	2	4	no	—
64	C₂×C₃×Q₁₆	C₂×D₃	1	4	yes	—
65	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
66	C₂×C₃×SD₁₆	—	3	8	no	—
67	C₂×C₃×D₈	—	3	4	no	—
68	C₂²×C₃×C₈	—	3	2	no	—
69	C₃×C₄⋊Q₈	—	2	2	no	—
70	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
71	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
72	C₃×#127	—	3	2	no	—
73	C₃×#124	—	3	4	no	—
74	C₃×#121	—	6	4	no	—
75	C₃×C₄×D₄	—	6	4	no	—
76	C₂×C₃×C₄²	—	3	1	no	—
77	#947	C₂²×C₃	1	4	yes	—
78	C₂×#215	C₂²×C₃	1	8	yes	—
79	#919	—	2	16	no	—
80	C₂×D₂₄	C₂²×C₃	1	4	yes	—
81	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
82	#895	—	2	4	no	—
83	#893	—	2	4	no	—
84	#874	C₂²×C₃	1	2	yes	—
85	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
86	C₃×C₄×C₈	—	2	1	no	—
87	C₃×C₄×C₈	—	3	2	no	—
88	C₃×C₄×C₈	—	3	1	no	—
89	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
90	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
91	C₈×Dic₃	C₂²×C₃	1	2	yes	—
92	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
93	C₂²×C₃×D₃	—	2	4	no	—
94	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
95	C₃²×Q₈	—	2	8	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
98	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
100	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
101	C₃×D₁₂	—	2	8	no	—
102	C₃×D₁₂	—	4	8	no	—
103	C₃×D₁₂	—	4	8	no	—
104	C₃×C₄×D₃	—	2	4	no	—
105	C₃×C₄×D₃	—	4	8	no	—
106	C₃×C₄×D₃	C₂×D₄	1	4	yes	—
107	C₃²×C₈	C₂×D₄	1	4	yes	—
108	C₃×C₃⋊C₈	C₂×D₄	1	4	yes	—
109	C₂×#135	—	3	4	no	—
110	#532	—	3	2	no	—
111	#495	—	3	4	no	—
112	#494	C₃×D₃	1	1	yes	characteristic, complemented
113	#493	—	3	1	no	—
114	#492	—	3	2	no	—
115	C₂×C₄×C₈	—	3	1	no	—
116	C₃×#41	—	6	16	no	—
117	C₂×C₃×Q₈	—	2	4	no	—
118	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
119	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
120	C₂×C₃×D₄	—	3	4	no	—
121	C₂×C₃×D₄	—	6	4	no	—
122	C₂×C₃×D₄	—	6	4	no	—
123	C₂²×C₃×C₄	—	3	1	no	—
124	C₂²×C₃×C₄	—	3	2	no	—
125	C₂²×C₃×C₄	—	6	4	no	—
126	#239	—	2	16	no	—
127	C₂×D₁₂	—	2	4	no	—
128	C₂×D₁₂	—	2	4	no	—
129	C₂×D₁₂	C₂³×C₃	1	4	yes	—
130	C₂×C₄×D₃	—	2	4	no	—
131	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
132	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
133	C₃×Q₁₆	—	2	8	no	—
134	C₃×Q₁₆	—	2	8	no	—
135	C₃×Q₁₆	—	4	8	no	—
136	C₃×SD₁₆	—	6	16	no	—
137	C₃×D₈	—	6	8	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	—	3	2	no	—
140	C₂×C₃×C₈	—	3	8	no	—
141	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
142	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₄⋊C₄	—	4	4	no	—
146	C₃×C₂²⋊C₄	—	6	8	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	3	2	no	—
149	C₃×C₄²	—	3	1	no	—
150	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
151	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
152	#215	—	2	16	no	—
153	(C₂×D₃)⋊C₄	—	2	8	no	—
154	C₄×Dic₃	C₃×D₄	1	2	yes	—
155	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
157	D₂₄	—	2	8	no	—
158	D₃×C₈	C₃×D₄	1	8	yes	complemented
159	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₃×D₃	—	2	2	no	—
161	C₂×C₃×D₃	—	4	4	no	—
162	C₂×C₃×D₃	—	4	8	no	—
163	C₂×C₃×D₃	#127	1	2	yes	—
164	C₃²×C₄	—	4	4	no	—
165	C₃²×C₄	#135	1	4	yes	complemented
166	C₃²×C₄	C₂²×D₄	1	2	yes	—
167	C₃×Dic₃	—	2	2	no	—
168	C₃×Dic₃	#127	1	2	yes	—
169	C₂×#41	—	3	4	no	—
170	#135	—	6	16	no	—
171	C₂×Q₁₆	C₂×C₃×D₃	1	4	yes	—
172	C₂×SD₁₆	—	3	8	no	—
173	C₂×D₈	—	3	4	no	—
174	C₂²×C₈	—	3	2	no	—
175	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
176	#127	—	3	2	no	—
177	#124	—	3	4	no	—
178	#121	—	6	4	no	—
179	C₄×D₄	—	6	4	no	—
180	C₂×C₄²	—	3	1	no	—
181	C₄×C₈	—	3	2	no	—
182	C₄×C₈	—	3	1	no	—
183	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
184	C₂³×C₃	—	3	1	no	—
185	C₂³×C₃	—	6	4	no	—
186	C₂²×D₃	—	2	4	no	—
187	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
188	C₃×Q₈	—	2	8	no	—
189	C₃×Q₈	—	2	8	no	—
190	C₃×Q₈	—	4	8	no	—
191	C₃×D₄	—	6	8	no	—
192	C₃×D₄	—	12	8	no	—
193	C₃×D₄	—	12	8	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	3	1	no	—
199	C₂×C₃×C₄	—	3	4	no	—
200	C₂×C₃×C₄	—	3	2	no	—
201	C₂×C₃×C₄	—	4	4	no	—
202	C₂×C₃×C₄	—	6	4	no	—
203	C₂×C₃×C₄	—	12	8	no	—
204	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
208	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
209	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
210	D₁₂	—	2	8	no	—
211	D₁₂	—	4	8	no	—
212	D₁₂	—	4	8	no	—
213	C₄×D₃	—	2	4	no	—
214	C₄×D₃	—	4	8	no	—
215	C₄×D₃	C₂×C₃×D₄	1	4	yes	—
216	C₃×C₈	—	2	4	no	—
217	C₃×C₈	—	3	4	no	—
218	C₃×C₈	D₃×D₄	1	4	yes	—
219	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
220	C₃⋊C₈	C₂×C₃×D₄	1	4	yes	—
221	C₂×C₃²	C₂×#127	1	1	yes	characteristic
222	C₂×C₃²	C₂×#135	1	2	yes	—
223	C₃×D₃	—	2	2	no	—
224	C₃×D₃	—	8	4	no	—
225	#41	—	6	16	no	—
226	C₂×Q₈	C₂²×C₃×D₃	1	4	yes	—
227	C₂×D₄	—	3	4	no	—
228	C₂×D₄	—	6	4	no	—
229	C₂×D₄	—	6	4	no	—
230	C₂²×C₄	—	3	1	no	—
231	C₂²×C₄	—	3	2	no	—
232	C₂²×C₄	—	6	4	no	—
233	Q₁₆	—	2	8	no	—
234	SD₁₆	—	6	16	no	—
235	D₈	—	6	8	no	—
236	C₂×C₈	—	3	2	no	—
237	C₂×C₈	—	3	8	no	—
238	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	6	8	no	—
241	C₄²	—	3	1	no	—
242	C₄²	—	3	2	no	—
243	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	3	2	no	—
246	C₂²×C₃	—	6	2	no	—
247	C₂²×C₃	—	12	4	no	—
248	C₂²×C₃	—	12	8	no	—
249	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
250	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
251	C₂×D₃	—	2	2	no	—
252	C₂×D₃	—	4	8	no	—
253	C₂×D₃	—	4	4	no	—
254	C₂×D₃	C₃×#127	1	2	yes	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	3	2	no	—
258	C₃×C₄	—	4	4	no	—
259	C₃×C₄	—	4	4	no	—
260	C₃×C₄	—	6	2	no	—
261	C₃×C₄	—	8	4	no	—
262	C₃×C₄	#919	1	4	yes	complemented
263	C₃×C₄	C₃×#135	1	4	yes	complemented
264	C₃×C₄	C₂×D₃×D₄	1	2	yes	—
265	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
266	Dic₃	—	2	2	no	—
267	Dic₃	C₃×#127	1	2	yes	—
268	C₃²	#3147	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	3	1	no	—
270	C₂³	—	6	4	no	—
271	Q₈	—	2	8	no	—
272	D₄	—	6	8	no	—
273	D₄	—	12	8	no	—
274	D₄	—	12	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	4	no	—
278	C₂×C₄	—	3	2	no	—
279	C₂×C₄	—	6	4	no	—
280	C₂×C₄	—	12	8	no	—
281	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
282	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
283	C₈	—	3	4	no	—
284	C₈	C₃×D₃×D₄	1	4	yes	—
285	C₂×C₃	—	2	2	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	6	2	no	—
288	C₂×C₃	—	24	4	no	—
289	C₂×C₃	D₃×#127	1	1	yes	characteristic
290	C₂×C₃	C₂×#919	1	2	yes	—
291	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
292	C₂×C₃	C₂×C₃×#135	1	2	yes	—
293	D₃	—	2	2	no	—
294	D₃	—	8	4	no	—
295	C₂²	—	3	2	no	—
296	C₂²	—	6	2	no	—
297	C₂²	—	12	8	no	—
298	C₂²	—	12	4	no	—
299	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
300	C₄	—	3	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	6	2	no	—
303	C₄	C₃×#919	1	4	yes	complemented
304	C₄	C₂×C₃×D₃×D₄	1	2	yes	—
305	C₃	—	2	1	no	—
306	C₃	#81799	1	1	yes	characteristic, direct product factor
307	C₃	C₃×#3147	1	1	yes	characteristic, complemented
308	C₂	—	6	2	no	—
309	C₂	—	24	4	no	—
310	C₂	C₃×D₃×#127	1	1	yes	characteristic
311	C₂	C₂×C₃×#919	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#494	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₃×#494	D₃
hypocenter	C₃	C₃×#3147
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#494

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#919	C₂	semidirect
2. C₃×#5725	C₂	semidirect
3. C₃²×#494	C₂	semidirect
4. C₃×#5191	C₂	semidirect
5. C₃×#5085	C₂	semidirect
6. C₃×#4695	C₂	semidirect
7. C₃×C₄×D₃×C₈	C₂	semidirect
9. #81799	C₃	direct (N × H)
27. C₂×#919	C₂×C₃	semidirect
28. #5725	C₂×C₃	semidirect
31. C₃×#494	D₃	semidirect
32. C₃×#494	C₂×C₃	semidirect
36. #5191	C₂×C₃	semidirect
37. #5085	C₂×C₃	semidirect
38. #4695	C₂×C₃	semidirect
39. C₄×D₃×C₈	C₂×C₃	semidirect
58. C₃×D₃×C₈	D₄	semidirect
112. #494	C₃×D₃	semidirect
158. D₃×C₈	C₃×D₄	semidirect
165. C₃²×C₄	#135	semidirect
262. C₃×C₄	#919	semidirect
263. C₃×C₄	C₃×#135	semidirect
268. C₃²	#3147	semidirect
303. C₄	C₃×#919	semidirect
306. C₃	#81799	direct (N × H)
307. C₃	C₃×#3147	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#919	2
C₃×#5725	2
C₃²×#494	2
C₃×#5191	2
C₃×#5085	2
C₃×#4695	2
C₃×C₄×D₃×C₈	2
C₃×#3147	3
#81799	3

Subgroups — C3×#81799 #11874291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#81799 #11874291