Subgroups — C₃²×#4805 #13162240

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#946	C₂	1	1	yes	maximal, characteristic
2	C₃²×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃³×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃²×#817	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#483	—	3	1	no	maximal
9	C₃×#4805	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×D₃	—	2	1	no	—
12	C₂×C₃²×C₄×D₃	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×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₃²×Dic₃⋊C₄	—	2	1	no	—
19	C₃²×Dic₃⋊C₄	—	2	1	no	—
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×C₂²⋊Q₈	—	3	1	no	—
22	C₂×C₃²×C₄⋊C₄	—	3	1	no	—
23	C₃×#946	C₂×C₃	1	4	yes	—
24	C₃×D₃×C₄⋊C₄	C₂×C₃	1	4	yes	—
25	C₃²×#106	—	3	1	no	—
26	C₃²×#106	—	3	1	no	—
27	C₃²×#103	—	2	4	no	—
28	C₃²×#103	—	3	1	no	—
29	C₃²×#103	D₃	1	1	yes	characteristic, complemented
30	C₃²×#103	C₂×C₃	1	4	yes	complemented
31	C₃²×C₂²⋊C₈	—	3	1	no	—
32	C₃×#835	C₂×C₃	1	4	yes	complemented
33	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	4	yes	—
34	C₃×#817	C₂×C₃	1	4	yes	complemented
35	C₃×#808	C₂×C₃	1	4	yes	complemented
36	C₂²×C₃²×D₃	D₄	1	1	yes	characteristic
37	C₃³×Q₈	—	2	1	no	—
38	C₃³×Q₈	—	2	1	no	—
39	C₂×C₃³×C₄	—	2	1	no	—
40	C₂×C₃³×C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
45	C₃²×C₄×D₃	—	2	1	no	—
46	C₃²×C₄×D₃	—	2	1	no	—
47	C₃²×C₄×D₃	—	4	2	no	—
48	C₃³×C₈	—	2	1	no	—
49	C₃²×C₃⋊C₈	—	2	1	no	—
50	C₃×#483	—	3	4	no	—
51	#4805	C₃²	1	1	yes	characteristic, direct product factor
52	C₂×C₃²×Q₈	—	2	4	no	—
53	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
55	C₂²×C₃²×C₄	—	3	1	no	—
56	C₂²×C₃²×C₄	—	6	1	no	—
57	C₂×C₃×C₄×D₃	—	2	4	no	—
58	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
59	C₂×C₃²×C₈	—	2	4	no	—
60	C₂×C₃²×C₈	—	3	1	no	—
61	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
63	C₃²×C₄⋊C₄	—	2	4	no	—
64	C₃²×C₄⋊C₄	—	3	1	no	—
65	C₃²×C₄⋊C₄	—	3	1	no	—
66	C₃²×C₄⋊C₄	—	6	1	no	—
67	C₃²×C₄⋊C₄	—	6	1	no	—
68	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
69	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
70	C₃²×C₂²⋊C₄	—	6	1	no	—
71	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
72	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
73	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
74	C₃×Dic₃⋊C₄	—	2	4	no	—
75	C₃×Dic₃⋊C₄	—	2	4	no	—
76	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
77	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×D₃	—	2	1	no	—
79	C₂×C₃²×D₃	—	2	1	no	—
80	C₂×C₃²×D₃	SD₁₆	1	2	yes	—
81	C₃³×C₄	—	4	1	no	—
82	C₃³×C₄	—	4	1	no	—
83	C₃³×C₄	#41	1	1	yes	characteristic
84	C₃³×C₄	#41	1	1	yes	characteristic
85	C₃²×Dic₃	—	2	1	no	—
86	C₃²×Dic₃	—	4	1	no	—
87	C₃²×Dic₃	—	4	1	no	—
88	C₃×C₂²⋊Q₈	—	3	4	no	—
89	C₂×C₃×C₄⋊C₄	—	3	4	no	—
90	#946	C₂×C₃²	1	1	yes	characteristic
91	D₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
92	C₃×#106	—	3	4	no	—
93	C₃×#106	—	3	4	no	—
94	C₃×#103	—	2	4	no	—
95	C₃×#103	—	3	4	no	—
96	C₃×#103	C₃×D₃	1	4	yes	complemented
97	C₃×#103	C₂×C₃²	1	1	yes	characteristic, complemented
98	C₃×C₂²⋊C₈	—	3	4	no	—
99	#835	C₂×C₃²	1	1	yes	characteristic, complemented
100	(C₂×D₃)⋊C₈	C₂×C₃²	1	1	yes	characteristic
101	#817	C₂×C₃²	1	1	yes	characteristic, complemented
102	#808	C₂×C₃²	1	1	yes	characteristic, complemented
103	C₂³×C₃²	—	3	1	no	—
104	C₂²×C₃×D₃	C₃×D₄	1	4	yes	—
105	C₃²×Q₈	—	2	8	no	—
106	C₃²×Q₈	—	2	1	no	—
107	C₃²×Q₈	—	2	1	no	—
108	C₃²×Q₈	—	2	4	no	—
109	C₃²×Q₈	—	2	4	no	—
110	C₃²×Q₈	—	4	4	no	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	—	2	1	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	—	3	1	no	—
117	C₂×C₃²×C₄	—	4	4	no	—
118	C₂×C₃²×C₄	—	4	4	no	—
119	C₂×C₃²×C₄	—	6	1	no	—
120	C₂×C₃²×C₄	—	6	1	no	—
121	C₂×C₃²×C₄	—	6	1	no	—
122	C₂×C₃²×C₄	—	6	1	no	—
123	C₂×C₃²×C₄	—	12	2	no	—
124	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
126	C₂×C₃×Dic₃	—	2	4	no	—
127	C₂×C₃×Dic₃	—	2	4	no	—
128	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
129	C₃×C₄×D₃	—	2	4	no	—
130	C₃×C₄×D₃	—	2	4	no	—
131	C₃×C₄×D₃	—	4	8	no	—
132	C₃²×C₈	—	2	1	no	—
133	C₃²×C₈	—	2	4	no	—
134	C₃²×C₈	—	4	4	no	—
135	C₃²×C₈	—	6	1	no	—
136	C₃×C₃⋊C₈	—	2	4	no	—
137	#483	—	3	1	no	Sylow
138	C₂×C₃³	#123	1	1	yes	characteristic
139	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₃³	#137	1	1	yes	characteristic
141	C₃²×D₃	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	4	no	—
143	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
144	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
145	C₂²×C₃×C₄	—	3	4	no	—
146	C₂²×C₃×C₄	—	6	4	no	—
147	C₂×C₄×D₃	—	2	1	no	—
148	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
149	C₂×C₃×C₈	—	2	4	no	—
150	C₂×C₃×C₈	—	3	4	no	—
151	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
152	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	3	4	no	—
155	C₃×C₄⋊C₄	—	3	4	no	—
156	C₃×C₄⋊C₄	—	6	4	no	—
157	C₃×C₄⋊C₄	—	6	4	no	—
158	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
159	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	6	4	no	—
161	(C₂×D₃)⋊C₄	—	2	1	no	—
162	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
163	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
164	Dic₃⋊C₄	—	2	1	no	—
165	Dic₃⋊C₄	—	2	1	no	—
166	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
167	C₂²×C₃²	—	2	4	no	—
168	C₂²×C₃²	—	3	2	no	—
169	C₂²×C₃²	—	6	1	no	—
170	C₂²×C₃²	—	6	1	no	—
171	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
172	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
173	C₂×C₃×D₃	—	2	4	no	—
174	C₂×C₃×D₃	—	2	4	no	—
175	C₂×C₃×D₃	C₃×SD₁₆	1	8	yes	—
176	C₃²×C₄	—	2	4	no	—
177	C₃²×C₄	—	2	4	no	—
178	C₃²×C₄	—	4	8	no	—
179	C₃²×C₄	—	4	1	no	—
180	C₃²×C₄	—	4	1	no	—
181	C₃²×C₄	—	4	4	no	—
182	C₃²×C₄	—	4	4	no	—
183	C₃²×C₄	—	6	1	no	—
184	C₃²×C₄	—	8	4	no	—
185	C₃²×C₄	—	12	1	no	—
186	C₃²×C₄	—	12	1	no	—
187	C₃²×C₄	#235	1	1	yes	characteristic
188	C₃²×C₄	#237	1	1	yes	characteristic
189	C₃²×C₄	C₃×#41	1	4	yes	—
190	C₃²×C₄	C₃×#41	1	4	yes	—
191	C₃×Dic₃	—	2	4	no	—
192	C₃×Dic₃	—	4	4	no	—
193	C₃×Dic₃	—	4	4	no	—
194	C₂²⋊Q₈	—	3	1	no	—
195	C₂×C₄⋊C₄	—	3	1	no	—
196	#106	—	3	1	no	—
197	#106	—	3	1	no	—
198	#103	—	3	1	no	—
199	#103	C₃²×D₃	1	1	yes	characteristic, complemented
200	C₂²⋊C₈	—	3	1	no	—
201	C₃³	#483	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	3	4	no	—
203	C₂²×D₃	C₃²×D₄	1	1	yes	characteristic
204	C₃×Q₈	—	2	8	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₃×Q₈	—	2	4	no	—
207	C₃×Q₈	—	2	1	no	—
208	C₃×Q₈	—	2	1	no	—
209	C₃×Q₈	—	4	4	no	—
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	3	4	no	—
216	C₂×C₃×C₄	—	4	4	no	—
217	C₂×C₃×C₄	—	4	4	no	—
218	C₂×C₃×C₄	—	6	4	no	—
219	C₂×C₃×C₄	—	6	4	no	—
220	C₂×C₃×C₄	—	6	4	no	—
221	C₂×C₃×C₄	—	6	4	no	—
222	C₂×C₃×C₄	—	12	8	no	—
223	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
224	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
225	C₂×Dic₃	—	2	1	no	—
226	C₂×Dic₃	—	2	1	no	—
227	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
228	C₄×D₃	—	2	1	no	—
229	C₄×D₃	—	2	1	no	—
230	C₄×D₃	—	4	2	no	—
231	C₃×C₈	—	2	4	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	4	4	no	—
234	C₃×C₈	—	6	4	no	—
235	C₃⋊C₈	—	2	1	no	—
236	C₂×C₃²	—	2	4	no	—
237	C₂×C₃²	—	2	4	no	—
238	C₂×C₃²	—	2	4	no	—
239	C₂×C₃²	—	6	2	no	—
240	C₂×C₃²	#883	1	1	yes	characteristic
241	C₂×C₃²	D₃×SD₁₆	1	1	yes	characteristic
242	C₂×C₃²	#918	1	1	yes	characteristic
243	C₂×C₃²	C₃×#123	1	4	yes	—
244	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
245	C₂×C₃²	C₃×#137	1	4	yes	—
246	C₃×D₃	—	2	8	no	—
247	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
248	C₂²×C₄	—	3	1	no	—
249	C₂²×C₄	—	6	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₄⋊C₄	—	6	1	no	—
255	C₄⋊C₄	—	6	1	no	—
256	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	6	1	no	—
258	C₂²×C₃	—	2	4	no	—
259	C₂²×C₃	—	3	8	no	—
260	C₂²×C₃	—	6	4	no	—
261	C₂²×C₃	—	6	4	no	—
262	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
263	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	—	2	1	no	—
266	C₂×D₃	C₃²×SD₁₆	1	2	yes	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	4	no	—
269	C₃×C₄	—	4	8	no	—
270	C₃×C₄	—	4	4	no	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	6	4	no	—
275	C₃×C₄	—	8	4	no	—
276	C₃×C₄	—	12	4	no	—
277	C₃×C₄	—	12	4	no	—
278	C₃×C₄	C₃×#235	1	4	yes	—
279	C₃×C₄	C₃×#237	1	4	yes	—
280	C₃×C₄	C₃²×#41	1	1	yes	characteristic
281	C₃×C₄	C₃²×#41	1	1	yes	characteristic
282	Dic₃	—	2	1	no	—
283	Dic₃	—	4	1	no	—
284	Dic₃	—	4	1	no	—
285	C₃²	—	2	4	no	—
286	C₃²	#4805	1	1	yes	characteristic, direct product factor
287	C₃²	C₃×#483	1	4	yes	complemented
288	C₂³	—	3	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	12	2	no	—
299	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	6	1	no	—
302	C₂×C₃	—	2	4	no	—
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	—	2	4	no	—
305	C₂×C₃	—	6	8	no	—
306	C₂×C₃	C₃×#883	1	4	yes	—
307	C₂×C₃	C₃×D₃×SD₁₆	1	4	yes	—
308	C₂×C₃	C₃×#918	1	4	yes	—
309	C₂×C₃	C₃²×#123	1	1	yes	characteristic
310	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
311	C₂×C₃	C₃²×#137	1	1	yes	characteristic
312	D₃	—	2	2	no	—
313	C₂²	—	3	2	no	—
314	C₂²	—	6	1	no	—
315	C₂²	—	6	1	no	—
316	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	6	1	no	—
320	C₄	—	12	1	no	—
321	C₄	—	12	1	no	—
322	C₄	C₃²×#235	1	1	yes	characteristic
323	C₄	C₃²×#237	1	1	yes	characteristic
324	C₃	—	2	4	no	—
325	C₃	C₃×#4805	1	4	yes	direct product factor
326	C₃	C₃²×#483	1	1	yes	characteristic, complemented
327	C₂	—	6	2	no	—
328	C₂	C₃²×#883	1	1	yes	characteristic
329	C₂	C₃²×D₃×SD₁₆	1	1	yes	characteristic
330	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	C₂²×C₃²×D₃	D₄
hypercenter	C₃²×#103	D₃
hypocenter	C₃	C₃²×#483
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
3. C₃³×#103	C₂	semidirect
4. C₃²×#835	C₂	semidirect
6. C₃²×#817	C₂	semidirect
7. C₃²×#808	C₂	semidirect
9. C₃×#4805	C₃	direct (N × H)
29. C₃²×#103	D₃	semidirect
30. C₃²×#103	C₂×C₃	semidirect
32. C₃×#835	C₂×C₃	semidirect
34. C₃×#817	C₂×C₃	semidirect
35. C₃×#808	C₂×C₃	semidirect
51. #4805	C₃²	direct (N × H)
96. C₃×#103	C₃×D₃	semidirect
97. C₃×#103	C₂×C₃²	semidirect
99. #835	C₂×C₃²	semidirect
101. #817	C₂×C₃²	semidirect
102. #808	C₂×C₃²	semidirect
199. #103	C₃²×D₃	semidirect
201. C₃³	#483	semidirect
286. C₃²	#4805	direct (N × H)
287. C₃²	C₃×#483	semidirect
325. C₃	C₃×#4805	direct (N × H)
326. C₃	C₃²×#483	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#946	2
C₃²×D₃×C₄⋊C₄	2
C₃³×#103	2
C₃²×#835	2
C₃²×(C₂×D₃)⋊C₈	2
C₃²×#817	2
C₃²×#808	2
C₃²×#483	3
C₃×#4805	3

Subgroups — C32×#4805 #13162240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4805 #13162240