Subgroups — C₃²×#4798 #13162234

Order 1728.

840 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#947	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#204	C₂	1	1	yes	maximal, characteristic
4	C₃²×#887	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃³×#103	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#809	C₂	1	1	yes	maximal, characteristic
8	C₃²×#464	—	3	1	no	maximal
9	C₃×#4798	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃²×#216	—	2	2	no	—
16	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
18	C₃²×Dic₃⋊C₄	—	2	1	no	—
19	C₃²×C₄×Dic₃	—	2	1	no	—
20	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
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	C₃×#947	C₂×C₃	1	4	yes	—
28	C₂×C₃×#216	C₂×C₃	1	4	yes	complemented
29	C₂×C₃×#204	C₂×C₃	1	4	yes	—
30	C₃×#887	C₂×C₃	1	4	yes	complemented
31	C₃²×C₄⋊C₈	—	3	1	no	—
32	C₃²×#103	—	2	4	no	—
33	C₃²×#103	D₃	1	1	yes	characteristic, complemented
34	C₃²×#103	C₂×C₃	1	4	yes	complemented
35	C₃²×#102	—	3	1	no	—
36	C₃×Dic₃⋊C₈	C₂×C₃	1	4	yes	complemented
37	C₃×#809	C₂×C₃	1	4	yes	—
38	C₂²×C₃²×D₃	—	2	1	no	—
39	C₃³×Q₈	—	2	1	no	—
40	C₃³×Q₈	—	2	1	no	—
41	C₂×C₃³×C₄	—	2	1	no	—
42	C₂×C₃³×C₄	—	2	1	no	—
43	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
44	C₂×C₃²×Dic₃	—	2	1	no	—
45	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
46	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
47	C₃²×D₁₂	—	2	1	no	—
48	C₃²×D₁₂	—	2	1	no	—
49	C₃²×Dic₆	—	2	1	no	—
50	C₃²×Dic₆	D₄	1	2	yes	—
51	C₃³×C₈	—	2	1	no	—
52	C₃²×C₃⋊C₈	—	2	1	no	—
53	C₃×#464	—	3	4	no	—
54	#4798	C₃²	1	1	yes	characteristic, direct product factor
55	C₂×C₃²×Q₈	—	2	4	no	—
56	C₂×C₃²×Q₈	—	3	1	no	—
57	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
59	C₂×C₃²×D₄	—	3	1	no	—
60	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
61	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
62	C₃²×Q₁₆	—	6	2	no	—
63	C₃²×SD₁₆	—	6	2	no	—
64	C₂×C₃²×C₈	—	2	4	no	—
65	C₂×C₃²×C₈	—	3	1	no	—
66	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
68	C₃²×C₄⋊C₄	—	2	4	no	—
69	C₃²×C₄⋊C₄	—	6	1	no	—
70	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
72	C₃²×C₂²⋊C₄	—	6	1	no	—
73	C₃²×C₂²⋊C₄	—	6	1	no	—
74	C₃²×C₄²	—	3	1	no	—
75	C₃²×C₄²	—	6	1	no	—
76	C₃×#216	—	2	8	no	—
77	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
78	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
79	C₃×Dic₃⋊C₄	—	2	4	no	—
80	C₃×C₄×Dic₃	—	2	4	no	—
81	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
82	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
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	—	4	1	no	—
89	C₃³×C₄	—	4	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	—	2	1	no	—
94	C₃²×Dic₃	—	2	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	#947	C₂×C₃²	1	1	yes	characteristic
100	C₂×#216	C₂×C₃²	1	1	yes	characteristic, complemented
101	C₂×#204	C₂×C₃²	1	1	yes	characteristic
102	#887	C₂×C₃²	1	1	yes	characteristic, complemented
103	C₃×C₄⋊C₈	—	3	4	no	—
104	C₃×#103	—	2	4	no	—
105	C₃×#103	C₃×D₃	1	4	yes	complemented
106	C₃×#103	C₂×C₃²	1	1	yes	characteristic, complemented
107	C₃×#102	—	3	4	no	—
108	Dic₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
109	#809	C₂×C₃²	1	1	yes	characteristic
110	C₂³×C₃²	—	6	1	no	—
111	C₂²×C₃×D₃	—	2	4	no	—
112	C₃²×Q₈	—	2	8	no	—
113	C₃²×Q₈	—	2	1	no	—
114	C₃²×Q₈	—	2	1	no	—
115	C₃²×Q₈	—	2	4	no	—
116	C₃²×Q₈	—	2	4	no	—
117	C₃²×Q₈	—	3	2	no	—
118	C₃²×Q₈	—	4	4	no	—
119	C₃²×Q₈	—	6	1	no	—
120	C₃²×D₄	—	6	1	no	—
121	C₃²×D₄	—	6	1	no	—
122	C₂×C₃²×C₄	—	2	1	no	—
123	C₂×C₃²×C₄	—	2	1	no	—
124	C₂×C₃²×C₄	—	2	4	no	—
125	C₂×C₃²×C₄	—	2	4	no	—
126	C₂×C₃²×C₄	—	2	4	no	—
127	C₂×C₃²×C₄	—	3	1	no	—
128	C₂×C₃²×C₄	—	3	1	no	—
129	C₂×C₃²×C₄	—	4	4	no	—
130	C₂×C₃²×C₄	—	4	4	no	—
131	C₂×C₃²×C₄	—	6	1	no	—
132	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
134	C₂×C₃×Dic₃	—	2	4	no	—
135	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
136	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
137	C₃×D₁₂	—	2	4	no	—
138	C₃×D₁₂	—	2	4	no	—
139	C₃×Dic₆	—	2	4	no	—
140	C₃×Dic₆	C₃×D₄	1	8	yes	—
141	C₃²×C₈	—	2	1	no	—
142	C₃²×C₈	—	2	4	no	—
143	C₃²×C₈	—	4	4	no	—
144	C₃²×C₈	—	6	1	no	—
145	C₃×C₃⋊C₈	—	2	4	no	—
146	#464	—	3	1	no	Sylow
147	C₂×C₃³	#121	1	1	yes	characteristic
148	C₂×C₃³	#135	1	1	yes	characteristic
149	C₂×C₃³	#137	1	1	yes	characteristic
150	C₃²×D₃	—	8	1	no	—
151	C₂×C₃×Q₈	—	2	4	no	—
152	C₂×C₃×Q₈	—	3	4	no	—
153	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
154	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
155	C₂×C₃×D₄	—	3	4	no	—
156	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
157	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
158	C₃×Q₁₆	—	6	8	no	—
159	C₃×SD₁₆	—	6	8	no	—
160	C₂×C₃×C₈	—	2	4	no	—
161	C₂×C₃×C₈	—	3	4	no	—
162	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
163	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	2	4	no	—
165	C₃×C₄⋊C₄	—	6	4	no	—
166	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
167	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
168	C₃×C₂²⋊C₄	—	6	4	no	—
169	C₃×C₂²⋊C₄	—	6	4	no	—
170	C₃×C₄²	—	3	4	no	—
171	C₃×C₄²	—	6	4	no	—
172	#216	—	2	2	no	—
173	(C₂×D₃)⋊C₄	—	2	1	no	—
174	(C₂×D₃)⋊C₄	—	2	1	no	—
175	Dic₃⋊C₄	—	2	1	no	—
176	C₄×Dic₃	—	2	1	no	—
177	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
178	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
179	#204	—	2	2	no	—
180	C₂²×C₃²	—	2	4	no	—
181	C₂²×C₃²	—	12	1	no	—
182	C₂²×C₃²	—	12	1	no	—
183	C₂²×C₃²	—	12	1	no	—
184	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
185	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
186	C₂×C₃×D₃	—	4	4	no	—
187	C₂×C₃×D₃	—	4	4	no	—
188	C₂×C₃×D₃	—	4	4	no	—
189	C₃²×C₄	—	2	4	no	—
190	C₃²×C₄	—	2	4	no	—
191	C₃²×C₄	—	4	8	no	—
192	C₃²×C₄	—	4	1	no	—
193	C₃²×C₄	—	4	1	no	—
194	C₃²×C₄	—	4	4	no	—
195	C₃²×C₄	—	4	4	no	—
196	C₃²×C₄	—	6	2	no	—
197	C₃²×C₄	—	6	1	no	—
198	C₃²×C₄	—	6	1	no	—
199	C₃²×C₄	—	8	4	no	—
200	C₃²×C₄	#235	1	1	yes	characteristic
201	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
202	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
203	C₃²×C₄	C₃×#41	1	4	yes	—
204	C₃×Dic₃	—	2	8	no	—
205	C₃×Dic₃	—	2	4	no	—
206	C₃×Dic₃	—	2	4	no	—
207	C₂×Q₁₆	—	3	1	no	—
208	C₂×SD₁₆	—	3	1	no	—
209	#124	—	3	1	no	—
210	C₄×Q₈	—	3	1	no	—
211	C₄⋊C₈	—	3	1	no	—
212	#103	C₃²×D₃	1	1	yes	characteristic, complemented
213	#102	—	3	1	no	—
214	C₃³	#464	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₃	—	6	4	no	—
216	C₂²×D₃	—	2	1	no	—
217	C₃×Q₈	—	2	8	no	—
218	C₃×Q₈	—	2	4	no	—
219	C₃×Q₈	—	2	4	no	—
220	C₃×Q₈	—	2	1	no	—
221	C₃×Q₈	—	2	1	no	—
222	C₃×Q₈	—	3	8	no	—
223	C₃×Q₈	—	4	4	no	—
224	C₃×Q₈	—	6	4	no	—
225	C₃×D₄	—	6	4	no	—
226	C₃×D₄	—	6	4	no	—
227	C₂×C₃×C₄	—	2	4	no	—
228	C₂×C₃×C₄	—	2	4	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	4	no	—
232	C₂×C₃×C₄	—	3	4	no	—
233	C₂×C₃×C₄	—	3	4	no	—
234	C₂×C₃×C₄	—	4	4	no	—
235	C₂×C₃×C₄	—	4	4	no	—
236	C₂×C₃×C₄	—	6	4	no	—
237	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
238	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
239	C₂×Dic₃	—	2	1	no	—
240	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
241	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
242	D₁₂	—	2	1	no	—
243	D₁₂	—	2	1	no	—
244	Dic₆	—	2	1	no	—
245	Dic₆	C₃²×D₄	1	2	yes	—
246	C₃×C₈	—	2	4	no	—
247	C₃×C₈	—	2	1	no	—
248	C₃×C₈	—	4	4	no	—
249	C₃×C₈	—	6	4	no	—
250	C₃⋊C₈	—	2	1	no	—
251	C₂×C₃²	—	2	4	no	—
252	C₂×C₃²	—	2	4	no	—
253	C₂×C₃²	—	2	4	no	—
254	C₂×C₃²	—	24	1	no	—
255	C₂×C₃²	#884	1	1	yes	characteristic
256	C₂×C₃²	#916	1	1	yes	characteristic
257	C₂×C₃²	#918	1	1	yes	characteristic
258	C₂×C₃²	C₃×#121	1	4	yes	—
259	C₂×C₃²	C₃×#135	1	4	yes	—
260	C₂×C₃²	C₃×#137	1	4	yes	—
261	C₃×D₃	—	8	4	no	—
262	C₂×Q₈	—	3	1	no	—
263	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
264	C₂×D₄	—	3	1	no	—
265	Q₁₆	—	6	2	no	—
266	SD₁₆	—	6	2	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
269	C₄⋊C₄	—	6	1	no	—
270	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
271	C₂²⋊C₄	—	6	1	no	—
272	C₂²⋊C₄	—	6	1	no	—
273	C₄²	—	3	1	no	—
274	C₄²	—	6	1	no	—
275	C₂²×C₃	—	2	4	no	—
276	C₂²×C₃	—	12	4	no	—
277	C₂²×C₃	—	12	4	no	—
278	C₂²×C₃	—	12	4	no	—
279	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
280	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
281	C₂×D₃	—	4	1	no	—
282	C₂×D₃	—	4	1	no	—
283	C₂×D₃	—	4	1	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	4	no	—
286	C₃×C₄	—	4	8	no	—
287	C₃×C₄	—	4	4	no	—
288	C₃×C₄	—	4	4	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	6	8	no	—
292	C₃×C₄	—	6	4	no	—
293	C₃×C₄	—	6	4	no	—
294	C₃×C₄	—	8	4	no	—
295	C₃×C₄	C₃×#235	1	4	yes	—
296	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
297	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
298	C₃×C₄	C₃²×#41	1	1	yes	characteristic
299	Dic₃	—	2	2	no	—
300	Dic₃	—	2	1	no	—
301	Dic₃	—	2	1	no	—
302	C₃²	—	2	4	no	—
303	C₃²	#4798	1	1	yes	characteristic, direct product factor
304	C₃²	C₃×#464	1	4	yes	complemented
305	C₂³	—	6	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	3	2	no	—
309	Q₈	—	6	1	no	—
310	D₄	—	6	1	no	—
311	D₄	—	6	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	3	1	no	—
315	C₂×C₄	—	3	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	6	1	no	—
320	C₂×C₃	—	2	4	no	—
321	C₂×C₃	—	2	4	no	—
322	C₂×C₃	—	2	4	no	—
323	C₂×C₃	—	24	4	no	—
324	C₂×C₃	C₃×#884	1	4	yes	—
325	C₂×C₃	C₃×#916	1	4	yes	—
326	C₂×C₃	C₃×#918	1	4	yes	—
327	C₂×C₃	C₃²×#121	1	1	yes	characteristic
328	C₂×C₃	C₃²×#135	1	1	yes	characteristic
329	C₂×C₃	C₃²×#137	1	1	yes	characteristic
330	D₃	—	8	1	no	—
331	C₂²	—	12	1	no	—
332	C₂²	—	12	1	no	—
333	C₂²	—	12	1	no	—
334	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	6	1	no	—
338	C₄	—	6	1	no	—
339	C₄	—	6	2	no	—
340	C₄	C₃²×#235	1	1	yes	characteristic
341	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
342	C₃	—	2	4	no	—
343	C₃	C₃×#4798	1	4	yes	direct product factor
344	C₃	C₃²×#464	1	1	yes	characteristic, complemented
345	C₂	—	24	1	no	—
346	C₂	C₃²×#884	1	1	yes	characteristic
347	C₂	C₃²×#916	1	1	yes	characteristic
348	C₂	C₃²×#918	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×#103	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×#103	D₃
hypocenter	C₃	C₃²×#464
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#103

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#216	C₂	semidirect
4. C₃²×#887	C₂	semidirect
5. C₃³×#103	C₂	semidirect
6. C₃²×Dic₃⋊C₈	C₂	semidirect
9. C₃×#4798	C₃	direct (N × H)
28. C₂×C₃×#216	C₂×C₃	semidirect
30. C₃×#887	C₂×C₃	semidirect
33. C₃²×#103	D₃	semidirect
34. C₃²×#103	C₂×C₃	semidirect
36. C₃×Dic₃⋊C₈	C₂×C₃	semidirect
54. #4798	C₃²	direct (N × H)
100. C₂×#216	C₂×C₃²	semidirect
102. #887	C₂×C₃²	semidirect
105. C₃×#103	C₃×D₃	semidirect
106. C₃×#103	C₂×C₃²	semidirect
108. Dic₃⋊C₈	C₂×C₃²	semidirect
212. #103	C₃²×D₃	semidirect
214. C₃³	#464	semidirect
303. C₃²	#4798	direct (N × H)
304. C₃²	C₃×#464	semidirect
343. C₃	C₃×#4798	direct (N × H)
344. 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
C₃²×#947	2
C₂×C₃²×#216	2
C₂×C₃²×#204	2
C₃²×#887	2
C₃³×#103	2
C₃²×Dic₃⋊C₈	2
C₃²×#809	2
C₃²×#464	3
C₃×#4798	3

Subgroups — C32×#4798 #13162234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4798 #13162234