Subgroups — C₃²×#4814 #13162247

Order 1728.

744 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	C₃²×#946	C₂	1	1	yes	maximal, characteristic
2	C₃²×#892	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×#818	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#485	—	3	1	no	maximal
9	C₃×#4814	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
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×C₄×Dic₃	—	2	1	no	—
20	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×C₂²⋊Q₈	—	3	1	no	—
22	C₃²×#117	—	3	1	no	—
23	C₃×#946	C₂×C₃	1	4	yes	—
24	C₃×#892	C₂×C₃	1	4	yes	—
25	C₃²×#107	—	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×#818	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	C₃³×C₈	—	2	1	no	—
48	C₃²×C₃⋊C₈	—	2	1	no	—
49	C₃×#485	—	3	4	no	—
50	#4814	C₃²	1	1	yes	characteristic, direct product factor
51	C₂×C₃²×Q₈	—	2	4	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
54	C₂²×C₃²×C₄	—	3	1	no	—
55	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
56	C₂×C₃²×C₈	—	2	4	no	—
57	C₂×C₃²×C₈	—	3	1	no	—
58	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
60	C₃²×C₄⋊C₄	—	2	4	no	—
61	C₃²×C₄⋊C₄	—	3	1	no	—
62	C₃²×C₄⋊C₄	—	3	1	no	—
63	C₃²×C₄⋊C₄	—	6	1	no	—
64	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
66	C₃²×C₂²⋊C₄	—	6	1	no	—
67	C₃²×C₂²⋊C₄	—	6	1	no	—
68	C₃²×C₄²	—	6	1	no	—
69	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
70	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
71	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
72	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
73	C₃×Dic₃⋊C₄	—	2	4	no	—
74	C₃×C₄×Dic₃	—	2	4	no	—
75	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
76	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×D₃	—	2	1	no	—
78	C₂×C₃²×D₃	—	2	1	no	—
79	C₂×C₃²×D₃	—	2	1	no	—
80	C₃³×C₄	—	4	1	no	—
81	C₃³×C₄	—	4	1	no	—
82	C₃³×C₄	#41	1	1	yes	characteristic
83	C₃³×C₄	#41	1	1	yes	characteristic
84	C₃²×Dic₃	—	2	1	no	—
85	C₃²×Dic₃	—	4	1	no	—
86	C₃²×Dic₃	—	4	1	no	—
87	C₃×C₂²⋊Q₈	—	3	4	no	—
88	C₃×#117	—	3	4	no	—
89	#946	C₂×C₃²	1	1	yes	characteristic
90	#892	C₂×C₃²	1	1	yes	characteristic
91	C₃×#107	—	3	4	no	—
92	C₃×#106	—	3	4	no	—
93	C₃×#103	—	2	4	no	—
94	C₃×#103	—	3	4	no	—
95	C₃×#103	C₃×D₃	1	4	yes	complemented
96	C₃×#103	C₂×C₃²	1	1	yes	characteristic, complemented
97	C₃×C₂²⋊C₈	—	3	4	no	—
98	#835	C₂×C₃²	1	1	yes	characteristic, complemented
99	(C₂×D₃)⋊C₈	C₂×C₃²	1	1	yes	characteristic
100	#818	C₂×C₃²	1	1	yes	characteristic, complemented
101	#808	C₂×C₃²	1	1	yes	characteristic, complemented
102	C₂³×C₃²	—	3	1	no	—
103	C₂²×C₃×D₃	C₃×D₄	1	4	yes	—
104	C₃²×Q₈	—	2	8	no	—
105	C₃²×Q₈	—	2	1	no	—
106	C₃²×Q₈	—	2	1	no	—
107	C₃²×Q₈	—	2	4	no	—
108	C₃²×Q₈	—	2	4	no	—
109	C₃²×Q₈	—	4	4	no	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	1	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	—	3	1	no	—
116	C₂×C₃²×C₄	—	4	4	no	—
117	C₂×C₃²×C₄	—	4	4	no	—
118	C₂×C₃²×C₄	—	6	1	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	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
124	C₂×C₃×Dic₃	—	2	4	no	—
125	C₂×C₃×Dic₃	—	2	4	no	—
126	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
127	C₃×C₄×D₃	—	2	4	no	—
128	C₃×C₄×D₃	—	2	4	no	—
129	C₃²×C₈	—	2	1	no	—
130	C₃²×C₈	—	2	4	no	—
131	C₃²×C₈	—	4	4	no	—
132	C₃²×C₈	—	6	1	no	—
133	C₃×C₃⋊C₈	—	2	4	no	—
134	#485	—	3	1	no	Sylow
135	C₂×C₃³	#123	1	1	yes	characteristic
136	C₂×C₃³	#135	1	1	yes	characteristic
137	C₂×C₃³	#137	1	1	yes	characteristic
138	C₃²×D₃	—	4	1	no	—
139	C₂×C₃×Q₈	—	2	4	no	—
140	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
141	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
142	C₂²×C₃×C₄	—	3	4	no	—
143	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
144	C₂×C₃×C₈	—	2	4	no	—
145	C₂×C₃×C₈	—	3	4	no	—
146	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
147	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
148	C₃×C₄⋊C₄	—	2	4	no	—
149	C₃×C₄⋊C₄	—	3	4	no	—
150	C₃×C₄⋊C₄	—	3	4	no	—
151	C₃×C₄⋊C₄	—	6	4	no	—
152	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
153	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	—	6	4	no	—
155	C₃×C₂²⋊C₄	—	6	4	no	—
156	C₃×C₄²	—	6	4	no	—
157	(C₂×D₃)⋊C₄	—	2	1	no	—
158	(C₂×D₃)⋊C₄	—	2	1	no	—
159	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
160	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
161	Dic₃⋊C₄	—	2	1	no	—
162	C₄×Dic₃	—	2	1	no	—
163	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
164	C₂²×C₃²	—	2	4	no	—
165	C₂²×C₃²	—	6	1	no	—
166	C₂²×C₃²	—	6	1	no	—
167	C₂²×C₃²	—	6	1	no	—
168	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
169	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
170	C₂×C₃×D₃	—	2	4	no	—
171	C₂×C₃×D₃	—	2	4	no	—
172	C₂×C₃×D₃	—	2	4	no	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	—	2	4	no	—
175	C₃²×C₄	—	4	8	no	—
176	C₃²×C₄	—	4	1	no	—
177	C₃²×C₄	—	4	1	no	—
178	C₃²×C₄	—	4	4	no	—
179	C₃²×C₄	—	4	4	no	—
180	C₃²×C₄	—	6	1	no	—
181	C₃²×C₄	—	8	4	no	—
182	C₃²×C₄	—	12	1	no	—
183	C₃²×C₄	—	12	1	no	—
184	C₃²×C₄	#235	1	1	yes	characteristic
185	C₃²×C₄	#237	1	1	yes	characteristic
186	C₃²×C₄	C₃×#41	1	4	yes	—
187	C₃²×C₄	C₃×#41	1	4	yes	—
188	C₃×Dic₃	—	2	4	no	—
189	C₃×Dic₃	—	4	4	no	—
190	C₃×Dic₃	—	4	4	no	—
191	C₂²⋊Q₈	—	3	1	no	—
192	#117	—	3	1	no	—
193	#107	—	3	1	no	—
194	#106	—	3	1	no	—
195	#103	—	3	1	no	—
196	#103	C₃²×D₃	1	1	yes	characteristic, complemented
197	C₂²⋊C₈	—	3	1	no	—
198	C₃³	#485	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	3	4	no	—
200	C₂²×D₃	C₃²×D₄	1	1	yes	characteristic
201	C₃×Q₈	—	2	8	no	—
202	C₃×Q₈	—	2	4	no	—
203	C₃×Q₈	—	2	4	no	—
204	C₃×Q₈	—	2	1	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	4	4	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	—	3	4	no	—
213	C₂×C₃×C₄	—	4	4	no	—
214	C₂×C₃×C₄	—	4	4	no	—
215	C₂×C₃×C₄	—	6	4	no	—
216	C₂×C₃×C₄	—	6	4	no	—
217	C₂×C₃×C₄	—	6	4	no	—
218	C₂×C₃×C₄	—	6	4	no	—
219	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
220	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
221	C₂×Dic₃	—	2	1	no	—
222	C₂×Dic₃	—	2	1	no	—
223	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
224	C₄×D₃	—	2	1	no	—
225	C₄×D₃	—	2	1	no	—
226	C₃×C₈	—	2	4	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	4	4	no	—
229	C₃×C₈	—	6	4	no	—
230	C₃⋊C₈	—	2	1	no	—
231	C₂×C₃²	—	2	4	no	—
232	C₂×C₃²	—	2	4	no	—
233	C₂×C₃²	—	2	4	no	—
234	C₂×C₃²	—	12	1	no	—
235	C₂×C₃²	#883	1	1	yes	characteristic
236	C₂×C₃²	#915	1	1	yes	characteristic
237	C₂×C₃²	#919	1	1	yes	characteristic
238	C₂×C₃²	C₃×#123	1	4	yes	—
239	C₂×C₃²	C₃×#135	1	4	yes	—
240	C₂×C₃²	C₃×#137	1	4	yes	—
241	C₃×D₃	—	4	4	no	—
242	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
243	C₂²×C₄	—	3	1	no	—
244	C₂×C₈	—	3	1	no	—
245	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
246	C₄⋊C₄	—	3	1	no	—
247	C₄⋊C₄	—	3	1	no	—
248	C₄⋊C₄	—	6	1	no	—
249	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	6	1	no	—
251	C₂²⋊C₄	—	6	1	no	—
252	C₄²	—	6	1	no	—
253	C₂²×C₃	—	2	4	no	—
254	C₂²×C₃	—	6	4	no	—
255	C₂²×C₃	—	6	4	no	—
256	C₂²×C₃	—	6	4	no	—
257	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
258	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
259	C₂×D₃	—	2	1	no	—
260	C₂×D₃	—	2	1	no	—
261	C₂×D₃	—	2	1	no	—
262	C₃×C₄	—	2	4	no	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	4	8	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	—	4	4	no	—
267	C₃×C₄	—	4	1	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	6	4	no	—
270	C₃×C₄	—	8	4	no	—
271	C₃×C₄	—	12	4	no	—
272	C₃×C₄	—	12	4	no	—
273	C₃×C₄	C₃×#235	1	4	yes	—
274	C₃×C₄	C₃×#237	1	4	yes	—
275	C₃×C₄	C₃²×#41	1	1	yes	characteristic
276	C₃×C₄	C₃²×#41	1	1	yes	characteristic
277	Dic₃	—	2	1	no	—
278	Dic₃	—	4	1	no	—
279	Dic₃	—	4	1	no	—
280	C₃²	—	2	4	no	—
281	C₃²	#4814	1	1	yes	characteristic, direct product factor
282	C₃²	C₃×#485	1	4	yes	complemented
283	C₂³	—	3	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
294	C₈	—	2	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	4	no	—
300	C₂×C₃	C₃×#883	1	4	yes	—
301	C₂×C₃	C₃×#915	1	4	yes	—
302	C₂×C₃	C₃×#919	1	4	yes	—
303	C₂×C₃	C₃²×#123	1	1	yes	characteristic
304	C₂×C₃	C₃²×#135	1	1	yes	characteristic
305	C₂×C₃	C₃²×#137	1	1	yes	characteristic
306	D₃	—	4	1	no	—
307	C₂²	—	6	1	no	—
308	C₂²	—	6	1	no	—
309	C₂²	—	6	1	no	—
310	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	6	1	no	—
314	C₄	—	12	1	no	—
315	C₄	—	12	1	no	—
316	C₄	C₃²×#235	1	1	yes	characteristic
317	C₄	C₃²×#237	1	1	yes	characteristic
318	C₃	—	2	4	no	—
319	C₃	C₃×#4814	1	4	yes	direct product factor
320	C₃	C₃²×#485	1	1	yes	characteristic, complemented
321	C₂	—	12	1	no	—
322	C₂	C₃²×#883	1	1	yes	characteristic
323	C₂	C₃²×#915	1	1	yes	characteristic
324	C₂	C₃²×#919	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×#485
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×#818	C₂	semidirect
7. C₃²×#808	C₂	semidirect
9. C₃×#4814	C₃	direct (N × H)
29. C₃²×#103	D₃	semidirect
30. C₃²×#103	C₂×C₃	semidirect
32. C₃×#835	C₂×C₃	semidirect
34. C₃×#818	C₂×C₃	semidirect
35. C₃×#808	C₂×C₃	semidirect
50. #4814	C₃²	direct (N × H)
95. C₃×#103	C₃×D₃	semidirect
96. C₃×#103	C₂×C₃²	semidirect
98. #835	C₂×C₃²	semidirect
100. #818	C₂×C₃²	semidirect
101. #808	C₂×C₃²	semidirect
196. #103	C₃²×D₃	semidirect
198. C₃³	#485	semidirect
281. C₃²	#4814	direct (N × H)
282. C₃²	C₃×#485	semidirect
319. C₃	C₃×#4814	direct (N × H)
320. C₃	C₃²×#485	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×#892	2
C₃³×#103	2
C₃²×#835	2
C₃²×(C₂×D₃)⋊C₈	2
C₃²×#818	2
C₃²×#808	2
C₃²×#485	3
C₃×#4814	3

Subgroups — C32×#4814 #13162247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4814 #13162247