Subgroups — C₃²×#4845 #13162286

Order 1728.

824 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#204	C₂	1	1	yes	maximal, characteristic
3	C₃²×#875	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#821	C₂	1	1	yes	maximal, characteristic
7	C₃²×#816	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#464	—	3	1	no	maximal
9	C₃×#4845	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
18	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₃²×Dic₃⋊C₄	—	2	1	no	—
20	C₃²×C₄×Dic₃	—	2	1	no	—
21	C₃²×Dic₁₂	—	2	2	no	—
22	C₃²×#204	—	2	2	no	—
23	C₂×C₃²×Q₁₆	—	3	1	no	—
24	C₂×C₃²×SD₁₆	—	3	1	no	—
25	C₃²×#124	—	3	1	no	—
26	C₃²×C₄×Q₈	—	3	1	no	—
27	C₂×C₃×Dic₁₂	C₂×C₃	1	4	yes	complemented
28	C₂×C₃×#204	C₂×C₃	1	4	yes	—
29	C₃×#875	C₂×C₃	1	4	yes	—
30	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	complemented
31	C₃²×C₄⋊C₈	—	2	4	no	—
32	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
33	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
34	C₃²×#103	—	3	1	no	—
35	C₃²×#102	—	3	1	no	—
36	C₃×#821	C₂×C₃	1	4	yes	—
37	C₃×#816	C₂×C₃	1	4	yes	complemented
38	C₂²×C₃²×D₃	—	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	C₂×C₃²×Dic₃	—	2	1	no	—
44	C₂×C₃²×Dic₃	—	2	1	no	—
45	C₃²×D₁₂	—	2	1	no	—
46	C₃²×D₁₂	—	2	1	no	—
47	C₃²×Dic₆	—	2	1	no	—
48	C₃²×Dic₆	—	2	1	no	—
49	C₃²×Dic₆	—	2	1	no	—
50	C₃²×Dic₆	D₄	1	2	yes	—
51	C₃³×C₈	—	2	1	no	—
52	C₃³×C₈	—	2	1	no	—
53	C₃×#464	—	3	4	no	—
54	#4845	C₃²	1	1	yes	characteristic, direct product factor
55	C₂×C₃²×Q₈	—	3	1	no	—
56	C₂×C₃²×Q₈	—	3	1	no	—
57	C₂×C₃²×D₄	—	3	1	no	—
58	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
59	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
60	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
61	C₃²×Q₁₆	—	6	2	no	—
62	C₃²×SD₁₆	—	6	2	no	—
63	C₂×C₃²×C₈	—	2	4	no	—
64	C₂×C₃²×C₈	—	2	4	no	—
65	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
68	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
69	C₃²×C₄⋊C₄	—	3	1	no	—
70	C₃²×C₄⋊C₄	—	6	1	no	—
71	C₃²×C₂²⋊C₄	—	6	1	no	—
72	C₃²×C₂²⋊C₄	—	6	1	no	—
73	C₃²×C₄²	—	2	4	no	—
74	C₃²×C₄²	—	6	1	no	—
75	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
76	C₃²×C₄²	C₂²×C₃	1	4	yes	—
77	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
78	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
79	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
80	C₃×Dic₃⋊C₄	—	2	4	no	—
81	C₃×C₄×Dic₃	—	2	4	no	—
82	C₃×Dic₁₂	—	2	8	no	—
83	C₃×#204	—	2	8	no	—
84	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×D₃	—	4	1	no	—
86	C₂×C₃²×D₃	—	4	1	no	—
87	C₂×C₃²×D₃	—	4	1	no	—
88	C₃³×C₄	—	2	1	no	—
89	C₃³×C₄	—	2	1	no	—
90	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
91	C₃³×C₄	#41	1	1	yes	characteristic
92	C₃²×Dic₃	—	2	2	no	—
93	C₃²×Dic₃	—	4	1	no	—
94	C₃²×Dic₃	—	4	1	no	—
95	C₂×C₃×Q₁₆	—	3	4	no	—
96	C₂×C₃×SD₁₆	—	3	4	no	—
97	C₃×#124	—	3	4	no	—
98	C₃×C₄×Q₈	—	3	4	no	—
99	C₂×Dic₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
100	C₂×#204	C₂×C₃²	1	1	yes	characteristic
101	#875	C₂×C₃²	1	1	yes	characteristic
102	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic, complemented
103	C₃×C₄⋊C₈	—	2	4	no	—
104	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
105	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
106	C₃×#103	—	3	4	no	—
107	C₃×#102	—	3	4	no	—
108	#821	C₂×C₃²	1	1	yes	characteristic
109	#816	C₂×C₃²	1	1	yes	characteristic, complemented
110	C₂³×C₃²	—	6	1	no	—
111	C₂²×C₃×D₃	—	2	4	no	—
112	C₃²×Q₈	—	3	2	no	—
113	C₃²×Q₈	—	6	1	no	—
114	C₃²×Q₈	—	6	1	no	—
115	C₃²×Q₈	—	6	1	no	—
116	C₃²×D₄	—	6	1	no	—
117	C₃²×D₄	—	6	1	no	—
118	C₂×C₃²×C₄	—	2	4	no	—
119	C₂×C₃²×C₄	—	2	4	no	—
120	C₂×C₃²×C₄	—	2	4	no	—
121	C₂×C₃²×C₄	—	6	1	no	—
122	C₂×C₃²×C₄	—	6	1	no	—
123	C₂×C₃²×C₄	—	6	1	no	—
124	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
125	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
126	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
127	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
128	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
130	C₂×C₃×Dic₃	—	2	4	no	—
131	C₂×C₃×Dic₃	—	2	4	no	—
132	C₂×C₃×Dic₃	—	2	4	no	—
133	C₃×D₁₂	—	2	4	no	—
134	C₃×D₁₂	—	2	4	no	—
135	C₃×Dic₆	—	2	4	no	—
136	C₃×Dic₆	—	2	4	no	—
137	C₃×Dic₆	—	2	4	no	—
138	C₃×Dic₆	C₃×D₄	1	8	yes	—
139	C₃²×C₈	—	2	1	no	—
140	C₃²×C₈	—	2	1	no	—
141	C₃²×C₈	—	2	4	no	—
142	C₃²×C₈	—	2	4	no	—
143	C₃²×C₈	—	4	4	no	—
144	C₃²×C₈	—	4	4	no	—
145	#464	—	3	1	no	Sylow
146	C₂×C₃³	#121	1	1	yes	characteristic
147	C₂×C₃³	#135	1	1	yes	characteristic
148	C₂×C₃³	#137	1	1	yes	characteristic
149	C₃²×D₃	—	8	1	no	—
150	C₂×C₃×Q₈	—	3	4	no	—
151	C₂×C₃×Q₈	—	3	4	no	—
152	C₂×C₃×D₄	—	3	4	no	—
153	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
154	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
155	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
156	C₃×Q₁₆	—	6	8	no	—
157	C₃×SD₁₆	—	6	8	no	—
158	C₂×C₃×C₈	—	2	4	no	—
159	C₂×C₃×C₈	—	2	4	no	—
160	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
161	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
162	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
163	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	3	4	no	—
165	C₃×C₄⋊C₄	—	6	4	no	—
166	C₃×C₂²⋊C₄	—	6	4	no	—
167	C₃×C₂²⋊C₄	—	6	4	no	—
168	C₃×C₄²	—	2	4	no	—
169	C₃×C₄²	—	6	4	no	—
170	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
171	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
172	(C₂×D₃)⋊C₄	—	2	1	no	—
173	(C₂×D₃)⋊C₄	—	2	1	no	—
174	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
175	Dic₃⋊C₄	—	2	1	no	—
176	C₄×Dic₃	—	2	1	no	—
177	Dic₁₂	—	2	2	no	—
178	#204	—	2	2	no	—
179	C₂²×C₃²	—	2	4	no	—
180	C₂²×C₃²	—	12	1	no	—
181	C₂²×C₃²	—	12	1	no	—
182	C₂²×C₃²	—	12	1	no	—
183	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
184	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
185	C₂×C₃×D₃	—	4	4	no	—
186	C₂×C₃×D₃	—	4	4	no	—
187	C₂×C₃×D₃	—	4	4	no	—
188	C₃²×C₄	—	2	8	no	—
189	C₃²×C₄	—	2	1	no	—
190	C₃²×C₄	—	2	1	no	—
191	C₃²×C₄	—	2	4	no	—
192	C₃²×C₄	—	2	4	no	—
193	C₃²×C₄	—	2	4	no	—
194	C₃²×C₄	—	2	4	no	—
195	C₃²×C₄	—	4	4	no	—
196	C₃²×C₄	—	6	2	no	—
197	C₃²×C₄	—	12	1	no	—
198	C₃²×C₄	—	12	1	no	—
199	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
200	C₃²×C₄	#239	1	1	yes	characteristic
201	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
202	C₃²×C₄	C₃×#41	1	4	yes	—
203	C₃×Dic₃	—	2	8	no	—
204	C₃×Dic₃	—	4	4	no	—
205	C₃×Dic₃	—	4	4	no	—
206	C₂×Q₁₆	—	3	1	no	—
207	C₂×SD₁₆	—	3	1	no	—
208	#124	—	3	1	no	—
209	C₄×Q₈	—	3	1	no	—
210	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
211	#103	—	3	1	no	—
212	#102	—	3	1	no	—
213	C₃³	#464	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₃	—	6	4	no	—
215	C₂²×D₃	—	2	1	no	—
216	C₃×Q₈	—	3	8	no	—
217	C₃×Q₈	—	6	4	no	—
218	C₃×Q₈	—	6	4	no	—
219	C₃×Q₈	—	6	4	no	—
220	C₃×D₄	—	6	4	no	—
221	C₃×D₄	—	6	4	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	6	4	no	—
226	C₂×C₃×C₄	—	6	4	no	—
227	C₂×C₃×C₄	—	6	4	no	—
228	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
229	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
230	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
233	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
234	C₂×Dic₃	—	2	1	no	—
235	C₂×Dic₃	—	2	1	no	—
236	C₂×Dic₃	—	2	1	no	—
237	D₁₂	—	2	1	no	—
238	D₁₂	—	2	1	no	—
239	Dic₆	—	2	1	no	—
240	Dic₆	—	2	1	no	—
241	Dic₆	—	2	1	no	—
242	Dic₆	C₃²×D₄	1	2	yes	—
243	C₃×C₈	—	2	4	no	—
244	C₃×C₈	—	2	4	no	—
245	C₃×C₈	—	2	1	no	—
246	C₃×C₈	—	2	1	no	—
247	C₃×C₈	—	4	4	no	—
248	C₃×C₈	—	4	4	no	—
249	C₂×C₃²	—	2	4	no	—
250	C₂×C₃²	—	2	4	no	—
251	C₂×C₃²	—	2	4	no	—
252	C₂×C₃²	—	24	1	no	—
253	C₂×C₃²	#895	1	1	yes	characteristic
254	C₂×C₃²	#904	1	1	yes	characteristic
255	C₂×C₃²	#909	1	1	yes	characteristic
256	C₂×C₃²	C₃×#121	1	4	yes	—
257	C₂×C₃²	C₃×#135	1	4	yes	—
258	C₂×C₃²	C₃×#137	1	4	yes	—
259	C₃×D₃	—	8	4	no	—
260	C₂×Q₈	—	3	1	no	—
261	C₂×Q₈	—	3	1	no	—
262	C₂×D₄	—	3	1	no	—
263	Q₁₆	—	6	2	no	—
264	SD₁₆	—	6	2	no	—
265	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
267	C₄⋊C₄	—	3	1	no	—
268	C₄⋊C₄	—	6	1	no	—
269	C₂²⋊C₄	—	6	1	no	—
270	C₂²⋊C₄	—	6	1	no	—
271	C₄²	—	6	1	no	—
272	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
273	C₂²×C₃	—	2	4	no	—
274	C₂²×C₃	—	12	4	no	—
275	C₂²×C₃	—	12	4	no	—
276	C₂²×C₃	—	12	4	no	—
277	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
278	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
279	C₂×D₃	—	4	1	no	—
280	C₂×D₃	—	4	1	no	—
281	C₂×D₃	—	4	1	no	—
282	C₃×C₄	—	2	8	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	4	no	—
288	C₃×C₄	—	2	4	no	—
289	C₃×C₄	—	4	4	no	—
290	C₃×C₄	—	6	8	no	—
291	C₃×C₄	—	12	4	no	—
292	C₃×C₄	—	12	4	no	—
293	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
294	C₃×C₄	C₃×#239	1	4	yes	—
295	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
296	C₃×C₄	C₃²×#41	1	1	yes	characteristic
297	Dic₃	—	2	2	no	—
298	Dic₃	—	4	1	no	—
299	Dic₃	—	4	1	no	—
300	C₃²	—	2	4	no	—
301	C₃²	#4845	1	1	yes	characteristic, direct product factor
302	C₃²	C₃×#464	1	4	yes	complemented
303	C₂³	—	6	1	no	—
304	Q₈	—	3	2	no	—
305	Q₈	—	6	1	no	—
306	Q₈	—	6	1	no	—
307	Q₈	—	6	1	no	—
308	D₄	—	6	1	no	—
309	D₄	—	6	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
314	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₂×C₃	—	2	4	no	—
319	C₂×C₃	—	2	4	no	—
320	C₂×C₃	—	2	4	no	—
321	C₂×C₃	—	24	4	no	—
322	C₂×C₃	C₃×#895	1	4	yes	—
323	C₂×C₃	C₃×#904	1	4	yes	—
324	C₂×C₃	C₃×#909	1	4	yes	—
325	C₂×C₃	C₃²×#121	1	1	yes	characteristic
326	C₂×C₃	C₃²×#135	1	1	yes	characteristic
327	C₂×C₃	C₃²×#137	1	1	yes	characteristic
328	D₃	—	8	1	no	—
329	C₂²	—	12	1	no	—
330	C₂²	—	12	1	no	—
331	C₂²	—	12	1	no	—
332	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	6	2	no	—
336	C₄	—	12	1	no	—
337	C₄	—	12	1	no	—
338	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
339	C₄	C₃²×#239	1	1	yes	characteristic
340	C₃	—	2	4	no	—
341	C₃	C₃×#4845	1	4	yes	direct product factor
342	C₃	C₃²×#464	1	1	yes	characteristic, complemented
343	C₂	—	24	1	no	—
344	C₂	C₃²×#895	1	1	yes	characteristic
345	C₂	C₃²×#904	1	1	yes	characteristic
346	C₂	C₃²×#909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#464
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×Dic₁₂	C₂	semidirect
4. C₃²×C₄×Dic₆	C₂	semidirect
5. C₃³×C₄⋊C₈	C₂	semidirect
7. C₃²×#816	C₂	semidirect
9. C₃×#4845	C₃	direct (N × H)
27. C₂×C₃×Dic₁₂	C₂×C₃	semidirect
30. C₃×C₄×Dic₆	C₂×C₃	semidirect
32. C₃²×C₄⋊C₈	D₃	semidirect
33. C₃²×C₄⋊C₈	C₂×C₃	semidirect
37. C₃×#816	C₂×C₃	semidirect
54. #4845	C₃²	direct (N × H)
99. C₂×Dic₁₂	C₂×C₃²	semidirect
102. C₄×Dic₆	C₂×C₃²	semidirect
104. C₃×C₄⋊C₈	C₃×D₃	semidirect
105. C₃×C₄⋊C₈	C₂×C₃²	semidirect
109. #816	C₂×C₃²	semidirect
210. C₄⋊C₈	C₃²×D₃	semidirect
213. C₃³	#464	semidirect
301. C₃²	#4845	direct (N × H)
302. C₃²	C₃×#464	semidirect
341. C₃	C₃×#4845	direct (N × H)
342. C₃	C₃²×#464	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Dic₁₂	2
C₂×C₃²×#204	2
C₃²×#875	2
C₃²×C₄×Dic₆	2
C₃³×C₄⋊C₈	2
C₃²×#821	2
C₃²×#816	2
C₃²×#464	3
C₃×#4845	3

Subgroups — C32×#4845 #13162286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4845 #13162286