Subgroups — C₃²×#4871 #13162291

Order 1728.

756 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	C₃²×#893	C₂	1	1	yes	maximal, characteristic
2	C₃²×#890	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#821	C₂	1	1	yes	maximal, characteristic
5	C₃²×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#809	C₂	1	1	yes	maximal, characteristic
7	C₃²×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#478	—	3	1	no	maximal
9	C₃×#4871	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×D₃	—	2	1	no	—
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₃²×C₄⋊Dic₃	—	2	1	no	—
17	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₃²×Dic₃⋊C₄	—	2	1	no	—
19	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×#125	—	3	1	no	—
22	C₃²×C₄×D₄	—	3	1	no	—
23	C₃×#893	C₂×C₃	1	4	yes	—
24	C₃×#890	C₂×C₃	1	4	yes	complemented
25	C₃²×#107	—	3	1	no	—
26	C₃²×#106	—	2	4	no	—
27	C₃²×#106	D₃	1	1	yes	characteristic, complemented
28	C₃²×#106	C₂×C₃	1	4	yes	complemented
29	C₃²×C₄⋊C₈	—	3	1	no	—
30	C₃²×#102	—	3	1	no	—
31	C₃²×#102	—	3	1	no	—
32	C₃×#821	C₂×C₃	1	4	yes	—
33	C₃×Dic₃⋊C₈	C₂×C₃	1	4	yes	complemented
34	C₃×#809	C₂×C₃	1	4	yes	—
35	C₃×#807	C₂×C₃	1	4	yes	complemented
36	C₂²×C₃²×D₃	—	2	1	no	—
37	C₂×C₃³×C₄	—	2	1	no	—
38	C₂×C₃³×C₄	—	2	1	no	—
39	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₃²×Dic₃	—	2	1	no	—
41	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
42	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
43	C₃²×D₁₂	—	2	1	no	—
44	C₃²×D₁₂	Q₈	1	2	yes	—
45	C₃²×C₄×D₃	—	4	2	no	—
46	C₃³×C₈	—	2	1	no	—
47	C₃²×C₃⋊C₈	—	2	1	no	—
48	C₃×#478	—	3	4	no	—
49	#4871	C₃²	1	1	yes	characteristic, direct product factor
50	C₂×C₃²×D₄	—	3	1	no	—
51	C₂²×C₃²×C₄	—	6	1	no	—
52	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
53	C₂×C₃×C₄×D₃	—	2	4	no	—
54	C₂×C₃²×C₈	—	2	4	no	—
55	C₂×C₃²×C₈	—	3	1	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
58	C₃²×C₄⋊C₄	—	2	4	no	—
59	C₃²×C₄⋊C₄	—	2	4	no	—
60	C₃²×C₄⋊C₄	—	3	1	no	—
61	C₃²×C₄⋊C₄	—	6	1	no	—
62	C₃²×C₄⋊C₄	—	6	1	no	—
63	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
64	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
66	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
67	C₃²×C₂²⋊C₄	—	6	1	no	—
68	C₃²×C₄²	—	3	1	no	—
69	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
70	C₃×C₄⋊Dic₃	—	2	4	no	—
71	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
72	C₃×Dic₃⋊C₄	—	2	4	no	—
73	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
74	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
75	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃²×D₃	—	2	2	no	—
77	C₂×C₃²×D₃	—	4	1	no	—
78	C₂×C₃²×D₃	—	4	1	no	—
79	C₃³×C₄	—	4	1	no	—
80	C₃³×C₄	—	4	1	no	—
81	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₃³×C₄	#41	1	1	yes	characteristic
83	C₃²×Dic₃	—	2	1	no	—
84	C₃²×Dic₃	—	2	1	no	—
85	C₃²×Dic₃	—	4	1	no	—
86	C₃×#125	—	3	4	no	—
87	C₃×C₄×D₄	—	3	4	no	—
88	#893	C₂×C₃²	1	1	yes	characteristic
89	#890	C₂×C₃²	1	1	yes	characteristic, complemented
90	C₃×#107	—	3	4	no	—
91	C₃×#106	—	2	4	no	—
92	C₃×#106	C₃×D₃	1	4	yes	complemented
93	C₃×#106	C₂×C₃²	1	1	yes	characteristic, complemented
94	C₃×C₄⋊C₈	—	3	4	no	—
95	C₃×#102	—	3	4	no	—
96	C₃×#102	—	3	4	no	—
97	#821	C₂×C₃²	1	1	yes	characteristic
98	Dic₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
99	#809	C₂×C₃²	1	1	yes	characteristic
100	#807	C₂×C₃²	1	1	yes	characteristic, complemented
101	C₂³×C₃²	—	6	1	no	—
102	C₂²×C₃×D₃	—	2	4	no	—
103	C₃²×D₄	—	3	2	no	—
104	C₃²×D₄	—	6	1	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	4	no	—
108	C₂×C₃²×C₄	—	2	4	no	—
109	C₂×C₃²×C₄	—	2	4	no	—
110	C₂×C₃²×C₄	—	3	1	no	—
111	C₂×C₃²×C₄	—	3	1	no	—
112	C₂×C₃²×C₄	—	4	4	no	—
113	C₂×C₃²×C₄	—	4	4	no	—
114	C₂×C₃²×C₄	—	6	1	no	—
115	C₂×C₃²×C₄	—	12	2	no	—
116	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
118	C₂×C₃×Dic₃	—	2	4	no	—
119	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
120	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
121	C₃×D₁₂	—	2	4	no	—
122	C₃×D₁₂	C₃×Q₈	1	8	yes	—
123	C₃×C₄×D₃	—	4	8	no	—
124	C₃²×C₈	—	2	8	no	—
125	C₃²×C₈	—	2	1	no	—
126	C₃²×C₈	—	2	4	no	—
127	C₃²×C₈	—	6	1	no	—
128	C₃×C₃⋊C₈	—	2	4	no	—
129	#478	—	3	1	no	Sylow
130	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₃³	#135	1	1	yes	characteristic
132	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₃²×D₃	—	4	2	no	—
134	C₂×C₃×D₄	—	3	4	no	—
135	C₂²×C₃×C₄	—	6	4	no	—
136	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
137	C₂×C₄×D₃	—	2	1	no	—
138	C₂×C₃×C₈	—	2	4	no	—
139	C₂×C₃×C₈	—	3	4	no	—
140	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
141	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	2	4	no	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	3	4	no	—
145	C₃×C₄⋊C₄	—	6	4	no	—
146	C₃×C₄⋊C₄	—	6	4	no	—
147	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
148	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
149	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
150	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
151	C₃×C₂²⋊C₄	—	6	4	no	—
152	C₃×C₄²	—	3	4	no	—
153	(C₂×D₃)⋊C₄	—	2	1	no	—
154	C₄⋊Dic₃	—	2	1	no	—
155	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
156	Dic₃⋊C₄	—	2	1	no	—
157	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
159	C₂²×C₃²	—	2	4	no	—
160	C₂²×C₃²	—	6	2	no	—
161	C₂²×C₃²	—	12	1	no	—
162	C₂²×C₃²	—	12	1	no	—
163	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
164	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
165	C₂×C₃×D₃	—	2	8	no	—
166	C₂×C₃×D₃	—	4	4	no	—
167	C₂×C₃×D₃	—	4	4	no	—
168	C₃²×C₄	—	2	4	no	—
169	C₃²×C₄	—	2	4	no	—
170	C₃²×C₄	—	4	1	no	—
171	C₃²×C₄	—	4	1	no	—
172	C₃²×C₄	—	4	4	no	—
173	C₃²×C₄	—	4	4	no	—
174	C₃²×C₄	—	6	1	no	—
175	C₃²×C₄	—	6	1	no	—
176	C₃²×C₄	—	8	4	no	—
177	C₃²×C₄	—	8	4	no	—
178	C₃²×C₄	—	12	1	no	—
179	C₃²×C₄	#235	1	1	yes	characteristic
180	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
181	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
182	C₃²×C₄	C₃×#41	1	4	yes	—
183	C₃×Dic₃	—	2	4	no	—
184	C₃×Dic₃	—	2	4	no	—
185	C₃×Dic₃	—	4	4	no	—
186	#125	—	3	1	no	—
187	C₄×D₄	—	3	1	no	—
188	#107	—	3	1	no	—
189	#106	C₃²×D₃	1	1	yes	characteristic, complemented
190	C₄⋊C₈	—	3	1	no	—
191	#102	—	3	1	no	—
192	#102	—	3	1	no	—
193	C₃³	#478	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₃	—	6	4	no	—
195	C₂²×D₃	—	2	1	no	—
196	C₃×D₄	—	3	8	no	—
197	C₃×D₄	—	6	4	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	3	4	no	—
204	C₂×C₃×C₄	—	3	4	no	—
205	C₂×C₃×C₄	—	4	4	no	—
206	C₂×C₃×C₄	—	4	4	no	—
207	C₂×C₃×C₄	—	6	4	no	—
208	C₂×C₃×C₄	—	12	8	no	—
209	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
210	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
211	C₂×Dic₃	—	2	1	no	—
212	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
213	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
214	D₁₂	—	2	1	no	—
215	D₁₂	C₃²×Q₈	1	2	yes	—
216	C₄×D₃	—	4	2	no	—
217	C₃×C₈	—	2	8	no	—
218	C₃×C₈	—	2	4	no	—
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	6	4	no	—
221	C₃⋊C₈	—	2	1	no	—
222	C₂×C₃²	—	2	4	no	—
223	C₂×C₃²	—	2	4	no	—
224	C₂×C₃²	—	2	4	no	—
225	C₂×C₃²	—	12	2	no	—
226	C₂×C₃²	#896	1	1	yes	characteristic
227	C₂×C₃²	#914	1	1	yes	characteristic
228	C₂×C₃²	#916	1	1	yes	characteristic
229	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
230	C₂×C₃²	C₃×#135	1	4	yes	—
231	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
232	C₃×D₃	—	4	8	no	—
233	C₂×D₄	—	3	1	no	—
234	C₂²×C₄	—	6	1	no	—
235	C₂×C₈	—	3	1	no	—
236	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
237	C₄⋊C₄	—	3	1	no	—
238	C₄⋊C₄	—	6	1	no	—
239	C₄⋊C₄	—	6	1	no	—
240	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
241	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
242	C₂²⋊C₄	—	6	1	no	—
243	C₄²	—	3	1	no	—
244	C₂²×C₃	—	2	4	no	—
245	C₂²×C₃	—	6	8	no	—
246	C₂²×C₃	—	12	4	no	—
247	C₂²×C₃	—	12	4	no	—
248	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
249	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
250	C₂×D₃	—	2	2	no	—
251	C₂×D₃	—	4	1	no	—
252	C₂×D₃	—	4	1	no	—
253	C₃×C₄	—	2	4	no	—
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	4	4	no	—
256	C₃×C₄	—	4	4	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	6	4	no	—
260	C₃×C₄	—	6	4	no	—
261	C₃×C₄	—	8	4	no	—
262	C₃×C₄	—	8	4	no	—
263	C₃×C₄	—	12	4	no	—
264	C₃×C₄	C₃×#235	1	4	yes	—
265	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
266	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
267	C₃×C₄	C₃²×#41	1	1	yes	characteristic
268	Dic₃	—	2	1	no	—
269	Dic₃	—	2	1	no	—
270	Dic₃	—	4	1	no	—
271	C₃²	—	2	4	no	—
272	C₃²	#4871	1	1	yes	characteristic, direct product factor
273	C₃²	C₃×#478	1	4	yes	complemented
274	C₂³	—	6	1	no	—
275	D₄	—	3	2	no	—
276	D₄	—	6	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	6	1	no	—
286	C₂×C₃	—	2	4	no	—
287	C₂×C₃	—	2	4	no	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	12	8	no	—
290	C₂×C₃	C₃×#896	1	4	yes	—
291	C₂×C₃	C₃×#914	1	4	yes	—
292	C₂×C₃	C₃×#916	1	4	yes	—
293	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
294	C₂×C₃	C₃²×#135	1	1	yes	characteristic
295	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
296	D₃	—	4	2	no	—
297	C₂²	—	6	2	no	—
298	C₂²	—	12	1	no	—
299	C₂²	—	12	1	no	—
300	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	6	1	no	—
304	C₄	—	6	1	no	—
305	C₄	—	12	1	no	—
306	C₄	C₃²×#235	1	1	yes	characteristic
307	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
308	C₃	—	2	4	no	—
309	C₃	C₃×#4871	1	4	yes	direct product factor
310	C₃	C₃²×#478	1	1	yes	characteristic, complemented
311	C₂	—	12	2	no	—
312	C₂	C₃²×#896	1	1	yes	characteristic
313	C₂	C₃²×#914	1	1	yes	characteristic
314	C₂	C₃²×#916	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#106	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×#106	D₃
hypocenter	C₃	C₃²×#478
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#106

Complements

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

N	H ≅ G/N	type
2. C₃²×#890	C₂	semidirect
3. C₃³×#106	C₂	semidirect
5. C₃²×Dic₃⋊C₈	C₂	semidirect
7. C₃²×#807	C₂	semidirect
9. C₃×#4871	C₃	direct (N × H)
24. C₃×#890	C₂×C₃	semidirect
27. C₃²×#106	D₃	semidirect
28. C₃²×#106	C₂×C₃	semidirect
33. C₃×Dic₃⋊C₈	C₂×C₃	semidirect
35. C₃×#807	C₂×C₃	semidirect
49. #4871	C₃²	direct (N × H)
89. #890	C₂×C₃²	semidirect
92. C₃×#106	C₃×D₃	semidirect
93. C₃×#106	C₂×C₃²	semidirect
98. Dic₃⋊C₈	C₂×C₃²	semidirect
100. #807	C₂×C₃²	semidirect
189. #106	C₃²×D₃	semidirect
193. C₃³	#478	semidirect
272. C₃²	#4871	direct (N × H)
273. C₃²	C₃×#478	semidirect
309. C₃	C₃×#4871	direct (N × H)
310. C₃	C₃²×#478	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#893	2
C₃²×#890	2
C₃³×#106	2
C₃²×#821	2
C₃²×Dic₃⋊C₈	2
C₃²×#809	2
C₃²×#807	2
C₃²×#478	3
C₃×#4871	3

Subgroups — C32×#4871 #13162291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4871 #13162291