Subgroups — C₃×#10588491 #13141870

Order 1728.

354 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	C₃×#63508	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₄×D₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63395	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×#63378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4811	—	3	1	no	maximal
9	#10588491	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₉	—	2	1	no	—
12	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
16	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₃×Dic₉⋊C₄	—	2	1	no	—
19	C₃×Dic₉⋊C₄	—	2	1	no	—
20	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#946	—	3	1	no	—
22	C₃×D₃×C₄⋊C₄	—	3	1	no	—
23	C₃²×#103	D₃	1	1	yes	characteristic
24	C₃×#835	—	3	1	no	—
25	C₃×(C₂×D₃)⋊C₈	—	3	1	no	—
26	C₃×#818	—	3	1	no	—
27	C₃×#807	—	3	1	no	—
28	#63508	C₂×C₃	1	1	yes	characteristic
29	C₄⋊C₄×D₉	C₂×C₃	1	1	yes	characteristic
30	C₉×#103	—	2	1	no	—
31	C₉×#103	C₂×C₃	1	1	yes	characteristic, complemented
32	#63395	C₂×C₃	1	1	yes	characteristic, complemented
33	(C₂×D₉)⋊C₈	C₂×C₃	1	1	yes	characteristic
34	#63378	C₂×C₃	1	1	yes	characteristic, complemented
35	#63366	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×D₉	D₄	1	1	yes	characteristic
37	C₃×Q₈×C₉	—	2	1	no	—
38	C₃×Q₈×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₉	—	2	1	no	—
43	C₂×C₃×Dic₉	—	2	1	no	—
44	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
45	C₃×C₄×D₉	—	2	1	no	—
46	C₃×C₄×D₉	—	2	1	no	—
47	C₃×C₄×D₉	—	4	2	no	—
48	C₃×C₈×C₉	—	2	1	no	—
49	C₃×C₉⋊C₈	—	2	1	no	—
50	C₃×#484	—	9	1	no	—
51	#4811	—	3	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄×D₃	—	3	1	no	—
54	C₂×C₃×C₄×D₃	—	6	1	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
57	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
58	C₃×C₄⋊Dic₃	—	3	1	no	—
59	C₃×C₄⋊Dic₃	—	3	1	no	—
60	C₃×Dic₃⋊C₄	—	6	1	no	—
61	C₃×Dic₃⋊C₄	—	6	1	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×Q₈×C₉	—	2	1	no	—
64	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄×D₉	—	2	1	no	—
66	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₈×C₉	—	2	1	no	—
68	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
69	C₉×C₄⋊C₄	—	2	1	no	—
70	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	(C₂×D₉)⋊C₄	—	2	1	no	—
72	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
73	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
74	Dic₉⋊C₄	—	2	1	no	—
75	Dic₉⋊C₄	—	2	1	no	—
76	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×D₉	—	2	1	no	—
79	C₂×C₃×D₉	—	2	1	no	—
80	C₂×C₃×D₉	Q₁₆	1	2	yes	—
81	C₃×C₄×C₉	—	4	1	no	—
82	C₃×C₄×C₉	—	4	1	no	—
83	C₃×C₄×C₉	#41	1	1	yes	characteristic
84	C₃×C₄×C₉	#41	1	1	yes	characteristic
85	C₃×Dic₉	—	2	1	no	—
86	C₃×Dic₉	—	4	1	no	—
87	C₃×Dic₉	—	4	1	no	—
88	C₃×C₂²⋊Q₈	—	9	1	no	—
89	C₂×C₃×C₄⋊C₄	—	9	1	no	—
90	#946	—	3	1	no	—
91	D₃×C₄⋊C₄	—	3	1	no	—
92	C₃×#107	—	9	1	no	—
93	C₃×#107	—	9	1	no	—
94	C₃×#103	—	2	1	no	—
95	C₃×#103	—	9	1	no	—
96	C₃×#103	D₉	1	1	yes	characteristic, complemented
97	C₃×#103	C₃×D₃	1	1	yes	characteristic
98	C₃×C₂²⋊C₈	—	9	1	no	—
99	#835	—	3	1	no	—
100	(C₂×D₃)⋊C₈	—	3	1	no	—
101	#818	—	3	1	no	—
102	#807	—	3	1	no	—
103	C₂²×C₃×D₃	—	3	1	no	—
104	C₃²×Q₈	—	2	1	no	—
105	C₃²×Q₈	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	1	no	—
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	—	3	1	no	—
110	C₂×C₃×Dic₃	—	6	1	no	—
111	C₂×C₃×Dic₃	—	6	1	no	—
112	C₃×C₄×D₃	—	6	1	no	—
113	C₃×C₄×D₃	—	6	1	no	—
114	C₃×C₄×D₃	—	12	2	no	—
115	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
116	C₃²×C₈	—	2	1	no	—
117	C₃×C₃⋊C₈	—	6	1	no	—
118	Q₈×C₉	—	2	2	no	—
119	Q₈×C₉	—	2	1	no	—
120	Q₈×C₉	—	2	1	no	—
121	Q₈×C₉	—	4	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	4	1	no	—
126	C₂×C₄×C₉	—	4	1	no	—
127	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₉	—	2	1	no	—
129	C₂×Dic₉	—	2	1	no	—
130	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
131	C₄×D₉	—	2	1	no	—
132	C₄×D₉	—	2	1	no	—
133	C₄×D₉	—	4	2	no	—
134	C₈×C₉	—	2	1	no	—
135	C₈×C₉	—	4	1	no	—
136	C₉⋊C₈	—	2	1	no	—
137	#484	—	9	1	no	Sylow
138	C₂×C₃×C₉	#123	1	1	yes	characteristic
139	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×D₉	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
144	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
145	C₂²×C₃×C₄	—	9	1	no	—
146	C₂²×C₃×C₄	—	18	1	no	—
147	C₂×C₄×D₃	—	3	1	no	—
148	C₂×C₄×D₃	—	6	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	9	1	no	—
151	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
152	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	—	9	1	no	—
155	C₃×C₄⋊C₄	—	9	1	no	—
156	C₃×C₄⋊C₄	—	18	1	no	—
157	C₃×C₄⋊C₄	—	18	1	no	—
158	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	18	1	no	—
161	(C₂×D₃)⋊C₄	—	6	1	no	—
162	C₄⋊Dic₃	—	3	1	no	—
163	C₄⋊Dic₃	—	3	1	no	—
164	Dic₃⋊C₄	—	6	1	no	—
165	Dic₃⋊C₄	—	6	1	no	—
166	C₂×C₃⋊C₈	—	3	1	no	—
167	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
168	C₂×C₃×D₃	—	3	2	no	—
169	C₂×C₃×D₃	—	6	1	no	—
170	C₂×C₃×D₃	—	6	1	no	—
171	C₃²×C₄	—	4	1	no	—
172	C₃²×C₄	—	4	1	no	—
173	C₃²×C₄	#235	1	1	yes	characteristic
174	C₃²×C₄	#237	1	1	yes	characteristic
175	C₃×Dic₃	—	6	1	no	—
176	C₃×Dic₃	—	12	1	no	—
177	C₃×Dic₃	—	12	1	no	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×D₉	—	2	1	no	—
181	C₂×D₉	—	2	1	no	—
182	C₂×D₉	C₃×Q₁₆	1	2	yes	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	4	2	no	—
186	C₄×C₉	—	4	1	no	—
187	C₄×C₉	—	4	1	no	—
188	C₄×C₉	—	8	1	no	—
189	C₄×C₉	C₃×#41	1	1	yes	characteristic
190	C₄×C₉	C₃×#41	1	1	yes	characteristic
191	Dic₉	—	2	1	no	—
192	Dic₉	—	4	1	no	—
193	Dic₉	—	4	1	no	—
194	C₂²⋊Q₈	—	9	1	no	—
195	C₂×C₄⋊C₄	—	9	1	no	—
196	#107	—	9	1	no	—
197	#107	—	9	1	no	—
198	#103	—	9	1	no	—
199	#103	C₃×D₉	1	1	yes	characteristic, complemented
200	C₂²⋊C₈	—	9	1	no	—
201	C₃×C₉	#484	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	9	1	no	—
203	C₂²×D₃	—	3	1	no	—
204	C₃×Q₈	—	2	2	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	2	1	no	—
207	C₃×Q₈	—	2	1	no	—
208	C₃×Q₈	—	2	1	no	—
209	C₃×Q₈	—	4	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	4	1	no	—
217	C₂×C₃×C₄	—	9	1	no	—
218	C₂×C₃×C₄	—	18	1	no	—
219	C₂×C₃×C₄	—	18	1	no	—
220	C₂×C₃×C₄	—	18	1	no	—
221	C₂×C₃×C₄	—	18	1	no	—
222	C₂×C₃×C₄	—	36	2	no	—
223	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂×Dic₃	—	3	1	no	—
226	C₂×Dic₃	—	6	1	no	—
227	C₂×Dic₃	—	6	1	no	—
228	C₄×D₃	—	6	1	no	—
229	C₄×D₃	—	6	1	no	—
230	C₄×D₃	—	12	2	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃×C₈	—	18	1	no	—
235	C₃⋊C₈	—	6	1	no	—
236	C₂×C₃²	#883	1	1	yes	characteristic
237	C₂×C₃²	#914	1	1	yes	characteristic
238	C₂×C₃²	D₃×Q₁₆	1	1	yes	characteristic
239	C₃×D₃	—	6	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	C₃×#123	1	1	yes	characteristic
244	C₂×C₉	C₂×C₃×Q₁₆	1	1	yes	characteristic
245	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
246	D₉	—	2	2	no	—
247	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
248	C₂²×C₄	—	9	1	no	—
249	C₂²×C₄	—	18	1	no	—
250	C₂×C₈	—	9	1	no	—
251	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
252	C₄⋊C₄	—	9	1	no	—
253	C₄⋊C₄	—	9	1	no	—
254	C₄⋊C₄	—	18	1	no	—
255	C₄⋊C₄	—	18	1	no	—
256	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	18	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	9	2	no	—
260	C₂²×C₃	—	18	1	no	—
261	C₂²×C₃	—	18	1	no	—
262	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
263	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
264	C₂×D₃	—	3	2	no	—
265	C₂×D₃	—	6	1	no	—
266	C₂×D₃	—	6	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	4	2	no	—
270	C₃×C₄	—	4	1	no	—
271	C₃×C₄	—	4	1	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	8	1	no	—
275	C₃×C₄	—	18	1	no	—
276	C₃×C₄	—	36	1	no	—
277	C₃×C₄	—	36	1	no	—
278	C₃×C₄	#3697	1	1	yes	characteristic
279	C₃×C₄	#3699	1	1	yes	characteristic
280	C₃×C₄	C₃×#235	1	1	yes	characteristic
281	C₃×C₄	C₃×#237	1	1	yes	characteristic
282	Dic₃	—	6	1	no	—
283	Dic₃	—	12	1	no	—
284	Dic₃	—	12	1	no	—
285	C₃²	#4811	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	C₃×#484	1	1	yes	characteristic, complemented
288	C₂³	—	9	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	—	9	1	no	—
294	C₂×C₄	—	18	1	no	—
295	C₂×C₄	—	18	1	no	—
296	C₂×C₄	—	18	1	no	—
297	C₂×C₄	—	18	1	no	—
298	C₂×C₄	—	36	2	no	—
299	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	18	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	18	2	no	—
306	C₂×C₃	#63445	1	1	yes	characteristic
307	C₂×C₃	#63476	1	1	yes	characteristic
308	C₂×C₃	Q₁₆×D₉	1	1	yes	characteristic
309	C₂×C₃	C₃×#883	1	1	yes	characteristic
310	C₂×C₃	C₃×#914	1	1	yes	characteristic
311	C₂×C₃	C₃×D₃×Q₁₆	1	1	yes	characteristic
312	D₃	—	6	2	no	—
313	C₂²	—	9	2	no	—
314	C₂²	—	18	1	no	—
315	C₂²	—	18	1	no	—
316	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	18	1	no	—
320	C₄	—	36	1	no	—
321	C₄	—	36	1	no	—
322	C₄	C₃×#3697	1	1	yes	characteristic
323	C₄	C₃×#3699	1	1	yes	characteristic
324	C₃	—	2	1	no	—
325	C₃	#10588491	1	1	yes	characteristic, direct product factor
326	C₃	C₃×#4811	1	1	yes	characteristic
327	C₂	—	18	2	no	—
328	C₂	C₃×#63445	1	1	yes	characteristic
329	C₂	C₃×#63476	1	1	yes	characteristic
330	C₂	C₃×Q₁₆×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#103	C₂
socle	C₂²×C₃²	D₃×D₄
prime radical	C₂²×C₃×D₉	D₄
hypercenter	C₃×#103	D₉
hypocenter	C₉	C₃×#484
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#103	C₂	semidirect
4. C₃×#63395	C₂	semidirect
6. C₃×#63378	C₂	semidirect
7. C₃×#63366	C₂	semidirect
9. #10588491	C₃	direct (N × H)
31. C₉×#103	C₂×C₃	semidirect
32. #63395	C₂×C₃	semidirect
34. #63378	C₂×C₃	semidirect
35. #63366	C₂×C₃	semidirect
96. C₃×#103	D₉	semidirect
199. #103	C₃×D₉	semidirect
201. C₃×C₉	#484	semidirect
287. C₉	C₃×#484	semidirect
325. C₃	#10588491	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#63508	2
C₃×C₄⋊C₄×D₉	2
C₃×C₉×#103	2
C₃×#63395	2
C₃×(C₂×D₉)⋊C₈	2
C₃×#63378	2
C₃×#63366	2
C₃×#4811	3
#10588491	3

Subgroups — C3×#10588491 #13141870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588491 #13141870