Subgroups — C₃²×#5066 #13162502

Order 1728.

836 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	C₃³×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×#809	C₂	1	1	yes	maximal, characteristic
4	C₃²×#809	C₂	1	1	yes	maximal, characteristic
5	C₃²×#808	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#807	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#478	—	3	1	no	maximal
9	C₃×#5066	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	C₃³×C₄⋊C₄	—	2	1	no	—
13	C₃³×C₄⋊C₄	—	2	1	no	—
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃³×C₄²	C₂²	1	1	yes	characteristic
17	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
18	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×#125	—	2	4	no	—
22	C₃²×#125	D₃	1	1	yes	characteristic, complemented
23	C₃²×#125	C₂×C₃	1	4	yes	complemented
24	C₃²×C₄×D₄	—	3	1	no	—
25	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
26	C₃²×#107	—	3	1	no	—
27	C₃²×#106	—	3	1	no	—
28	C₃²×C₄⋊C₈	—	3	1	no	—
29	C₃²×#102	—	3	1	no	—
30	C₃²×#102	—	3	1	no	—
31	C₃×#809	C₂×C₃	1	4	yes	—
32	C₃×#809	C₂×C₃	1	4	yes	—
33	C₃×#808	C₂×C₃	1	4	yes	complemented
34	C₃×#807	C₂×C₃	1	4	yes	complemented
35	C₃×C₄⋊(C₃⋊C₈)	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	D₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃²×Dic₃	—	2	1	no	—
43	C₃²×D₁₂	—	2	1	no	—
44	C₃²×D₁₂	Q₈	1	2	yes	—
45	C₃²×C₄×D₃	—	4	2	no	—
46	C₃²×C₃⋊C₈	—	2	1	no	—
47	C₃²×C₃⋊C₈	—	2	1	no	—
48	C₃×#478	—	3	4	no	—
49	#5066	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	—	3	1	no	—
55	C₂×C₃²×C₈	—	3	1	no	—
56	C₃²×C₄⋊C₄	—	2	8	no	—
57	C₃²×C₄⋊C₄	—	2	8	no	—
58	C₃²×C₄⋊C₄	—	2	4	no	—
59	C₃²×C₄⋊C₄	—	2	4	no	—
60	C₃²×C₄⋊C₄	—	2	1	no	—
61	C₃²×C₄⋊C₄	—	2	1	no	—
62	C₃²×C₄⋊C₄	—	2	4	no	—
63	C₃²×C₄⋊C₄	—	2	4	no	—
64	C₃²×C₄⋊C₄	—	3	1	no	—
65	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
69	C₃²×C₂²⋊C₄	—	6	1	no	—
70	C₃²×C₄²	—	2	4	no	—
71	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
72	C₃²×C₄²	C₂²×C₃	1	4	yes	—
73	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
74	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
75	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
76	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
77	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×D₃	—	2	2	no	—
79	C₂×C₃²×D₃	—	4	1	no	—
80	C₂×C₃²×D₃	—	4	1	no	—
81	C₃³×C₄	—	2	1	no	—
82	C₃³×C₄	—	2	1	no	—
83	C₃³×C₄	—	4	1	no	—
84	C₃³×C₄	—	4	1	no	—
85	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₃³×C₄	#41	1	1	yes	characteristic
87	C₃²×Dic₃	—	4	1	no	—
88	C₃×#125	—	2	4	no	—
89	C₃×#125	C₃×D₃	1	4	yes	complemented
90	C₃×#125	C₂×C₃²	1	1	yes	characteristic, complemented
91	C₃×C₄×D₄	—	3	4	no	—
92	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
93	C₃×#107	—	3	4	no	—
94	C₃×#106	—	3	4	no	—
95	C₃×C₄⋊C₈	—	3	4	no	—
96	C₃×#102	—	3	4	no	—
97	C₃×#102	—	3	4	no	—
98	#809	C₂×C₃²	1	1	yes	characteristic
99	#809	C₂×C₃²	1	1	yes	characteristic
100	#808	C₂×C₃²	1	1	yes	characteristic, complemented
101	#807	C₂×C₃²	1	1	yes	characteristic, complemented
102	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic, complemented
103	C₂³×C₃²	—	6	1	no	—
104	C₂²×C₃×D₃	—	2	4	no	—
105	C₃²×D₄	—	3	2	no	—
106	C₃²×D₄	—	6	1	no	—
107	C₂×C₃²×C₄	—	2	8	no	—
108	C₂×C₃²×C₄	—	2	8	no	—
109	C₂×C₃²×C₄	—	2	1	no	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	4	no	—
112	C₂×C₃²×C₄	—	2	4	no	—
113	C₂×C₃²×C₄	—	2	4	no	—
114	C₂×C₃²×C₄	—	2	4	no	—
115	C₂×C₃²×C₄	—	2	4	no	—
116	C₂×C₃²×C₄	—	6	1	no	—
117	C₂×C₃²×C₄	—	12	2	no	—
118	C₂×C₃²×C₄	#67	1	1	yes	characteristic
119	C₂×C₃²×C₄	#67	1	1	yes	characteristic
120	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
121	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
124	C₂×C₃×Dic₃	—	2	4	no	—
125	C₃×D₁₂	—	2	4	no	—
126	C₃×D₁₂	C₃×Q₈	1	8	yes	—
127	C₃×C₄×D₃	—	4	8	no	—
128	C₃²×C₈	—	6	1	no	—
129	C₃²×C₈	—	6	1	no	—
130	C₃×C₃⋊C₈	—	2	4	no	—
131	C₃×C₃⋊C₈	—	2	4	no	—
132	#478	—	3	1	no	Sylow
133	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₃³	#135	1	1	yes	characteristic
135	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₃²×D₃	—	4	2	no	—
137	C₂×C₃×D₄	—	3	4	no	—
138	C₂²×C₃×C₄	—	6	4	no	—
139	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
140	C₂×C₄×D₃	—	2	1	no	—
141	C₂×C₃×C₈	—	3	4	no	—
142	C₂×C₃×C₈	—	3	4	no	—
143	C₃×C₄⋊C₄	—	2	8	no	—
144	C₃×C₄⋊C₄	—	2	8	no	—
145	C₃×C₄⋊C₄	—	2	4	no	—
146	C₃×C₄⋊C₄	—	2	4	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	4	no	—
150	C₃×C₄⋊C₄	—	2	4	no	—
151	C₃×C₄⋊C₄	—	3	4	no	—
152	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
153	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
154	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
155	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
156	C₃×C₂²⋊C₄	—	6	4	no	—
157	C₃×C₄²	—	2	4	no	—
158	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
159	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
160	(C₂×D₃)⋊C₄	—	2	1	no	—
161	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
162	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
163	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
164	C₂²×C₃²	—	2	4	no	—
165	C₂²×C₃²	—	6	2	no	—
166	C₂²×C₃²	—	12	1	no	—
167	C₂²×C₃²	—	12	1	no	—
168	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
169	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
170	C₂×C₃×D₃	—	2	8	no	—
171	C₂×C₃×D₃	—	4	4	no	—
172	C₂×C₃×D₃	—	4	4	no	—
173	C₃²×C₄	—	2	1	no	—
174	C₃²×C₄	—	2	1	no	—
175	C₃²×C₄	—	2	4	no	—
176	C₃²×C₄	—	2	4	no	—
177	C₃²×C₄	—	2	4	no	—
178	C₃²×C₄	—	2	4	no	—
179	C₃²×C₄	—	4	4	no	—
180	C₃²×C₄	—	4	4	no	—
181	C₃²×C₄	—	4	8	no	—
182	C₃²×C₄	—	4	8	no	—
183	C₃²×C₄	—	4	1	no	—
184	C₃²×C₄	—	4	1	no	—
185	C₃²×C₄	—	4	4	no	—
186	C₃²×C₄	—	4	4	no	—
187	C₃²×C₄	—	12	1	no	—
188	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
189	C₃²×C₄	#239	1	1	yes	characteristic
190	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
191	C₃²×C₄	C₃×#41	1	4	yes	—
192	C₃×Dic₃	—	4	4	no	—
193	#125	C₃²×D₃	1	1	yes	characteristic, complemented
194	C₄×D₄	—	3	1	no	—
195	#107	—	3	1	no	—
196	#106	—	3	1	no	—
197	C₄⋊C₈	—	3	1	no	—
198	#102	—	3	1	no	—
199	#102	—	3	1	no	—
200	C₃³	#478	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₃	—	6	4	no	—
202	C₂²×D₃	—	2	1	no	—
203	C₃×D₄	—	3	8	no	—
204	C₃×D₄	—	6	4	no	—
205	C₂×C₃×C₄	—	2	8	no	—
206	C₂×C₃×C₄	—	2	8	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	2	4	no	—
213	C₂×C₃×C₄	—	2	4	no	—
214	C₂×C₃×C₄	—	6	4	no	—
215	C₂×C₃×C₄	—	12	8	no	—
216	C₂×C₃×C₄	C₃×#67	1	4	yes	—
217	C₂×C₃×C₄	C₃×#67	1	4	yes	—
218	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
221	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
222	C₂×Dic₃	—	2	1	no	—
223	D₁₂	—	2	1	no	—
224	D₁₂	C₃²×Q₈	1	2	yes	—
225	C₄×D₃	—	4	2	no	—
226	C₃×C₈	—	6	4	no	—
227	C₃×C₈	—	6	4	no	—
228	C₃⋊C₈	—	2	1	no	—
229	C₃⋊C₈	—	2	1	no	—
230	C₂×C₃²	—	2	4	no	—
231	C₂×C₃²	—	2	4	no	—
232	C₂×C₃²	—	2	4	no	—
233	C₂×C₃²	—	12	2	no	—
234	C₂×C₃²	#946	1	1	yes	characteristic
235	C₂×C₃²	#949	1	1	yes	characteristic
236	C₂×C₃²	#950	1	1	yes	characteristic
237	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
238	C₂×C₃²	C₃×#135	1	4	yes	—
239	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
240	C₃×D₃	—	4	8	no	—
241	C₂×D₄	—	3	1	no	—
242	C₂²×C₄	—	6	1	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	3	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	3	1	no	—
248	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	6	1	no	—
251	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	—	6	8	no	—
254	C₂²×C₃	—	12	4	no	—
255	C₂²×C₃	—	12	4	no	—
256	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
257	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
258	C₂×D₃	—	2	2	no	—
259	C₂×D₃	—	4	1	no	—
260	C₂×D₃	—	4	1	no	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	4	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	4	no	—
266	C₃×C₄	—	2	4	no	—
267	C₃×C₄	—	4	4	no	—
268	C₃×C₄	—	4	4	no	—
269	C₃×C₄	—	4	8	no	—
270	C₃×C₄	—	4	8	no	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	4	4	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	4	1	no	—
275	C₃×C₄	—	12	4	no	—
276	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
277	C₃×C₄	C₃×#239	1	4	yes	—
278	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
279	C₃×C₄	C₃²×#41	1	1	yes	characteristic
280	Dic₃	—	4	1	no	—
281	C₃²	—	2	4	no	—
282	C₃²	#5066	1	1	yes	characteristic, direct product factor
283	C₃²	C₃×#478	1	4	yes	complemented
284	C₂³	—	6	1	no	—
285	D₄	—	3	2	no	—
286	D₄	—	6	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	—	12	2	no	—
291	C₂×C₄	C₃²×#67	1	1	yes	characteristic
292	C₂×C₄	C₃²×#67	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
294	C₈	—	6	1	no	—
295	C₈	—	6	1	no	—
296	C₂×C₃	—	2	4	no	—
297	C₂×C₃	—	2	4	no	—
298	C₂×C₃	—	2	4	no	—
299	C₂×C₃	—	12	8	no	—
300	C₂×C₃	C₃×#946	1	4	yes	—
301	C₂×C₃	C₃×#949	1	4	yes	—
302	C₂×C₃	C₃×#950	1	4	yes	—
303	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₃	C₃²×#135	1	1	yes	characteristic
305	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
306	D₃	—	4	2	no	—
307	C₂²	—	6	2	no	—
308	C₂²	—	12	1	no	—
309	C₂²	—	12	1	no	—
310	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	12	1	no	—
316	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
317	C₄	C₃²×#239	1	1	yes	characteristic
318	C₃	—	2	4	no	—
319	C₃	C₃×#5066	1	4	yes	direct product factor
320	C₃	C₃²×#478	1	1	yes	characteristic, complemented
321	C₂	—	12	2	no	—
322	C₂	C₃²×#946	1	1	yes	characteristic
323	C₂	C₃²×#949	1	1	yes	characteristic
324	C₂	C₃²×#950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#125	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×#125	D₃
hypocenter	C₃	C₃²×#478
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×#125

Complements

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

N	H ≅ G/N	type
1. C₃³×#125	C₂	semidirect
5. C₃²×#808	C₂	semidirect
6. C₃²×#807	C₂	semidirect
7. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
9. C₃×#5066	C₃	direct (N × H)
22. C₃²×#125	D₃	semidirect
23. C₃²×#125	C₂×C₃	semidirect
33. C₃×#808	C₂×C₃	semidirect
34. C₃×#807	C₂×C₃	semidirect
35. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
49. #5066	C₃²	direct (N × H)
89. C₃×#125	C₃×D₃	semidirect
90. C₃×#125	C₂×C₃²	semidirect
100. #808	C₂×C₃²	semidirect
101. #807	C₂×C₃²	semidirect
102. C₄⋊(C₃⋊C₈)	C₂×C₃²	semidirect
193. #125	C₃²×D₃	semidirect
200. C₃³	#478	semidirect
282. C₃²	#5066	direct (N × H)
283. C₃²	C₃×#478	semidirect
319. C₃	C₃×#5066	direct (N × H)
320. 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
C₃³×#125	2
C₃²×C₄×D₁₂	2
C₃²×#809	2
C₃²×#808	2
C₃²×#807	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#478	3
C₃×#5066	3

Subgroups — C32×#5066 #13162502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5066 #13162502