Subgroups — C₃×#10588822 #13143247

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63512	C₂	1	1	yes	maximal, characteristic
3	C₃×#63511	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3664	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63383	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63372	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5141	—	3	1	no	maximal
9	#10588822	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃×C₉×#41	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
13	C₃×C₉×M₁₆	—	2	1	no	—
14	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₈×C₉	—	2	1	no	—
16	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
17	C₃×#3675	—	2	1	no	—
18	C₃×#3674	—	2	1	no	—
19	C₃×#3673	—	2	1	no	—
20	C₃×#3672	—	2	1	no	—
21	C₃×#3667	C₂²	1	1	yes	characteristic
22	C₃×#3667	C₂²	1	1	yes	characteristic
23	C₃×#3664	—	2	1	no	—
24	C₃×#3664	—	2	1	no	—
25	C₃²×#131	D₃	1	1	yes	characteristic
26	C₃×#951	—	3	1	no	—
27	C₃×#949	—	3	1	no	—
28	C₂×C₃×#204	—	3	1	no	—
29	C₃×#824	—	3	1	no	—
30	C₃×#823	—	3	1	no	—
31	C₃×#819	—	3	1	no	—
32	C₉×#131	—	2	1	no	—
33	C₉×#131	C₂×C₃	1	1	yes	characteristic, complemented
34	#63512	C₂×C₃	1	1	yes	characteristic
35	#63511	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#3664	C₂×C₃	1	1	yes	characteristic, complemented
37	#63383	C₂×C₃	1	1	yes	characteristic, complemented
38	#63382	C₂×C₃	1	1	yes	characteristic, complemented
39	#63372	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂²×C₃×D₉	—	2	1	no	—
41	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
42	C₃×D₄×C₉	—	2	1	no	—
43	C₃×D₄×C₉	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	—	2	1	no	—
45	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₃×Dic₉	—	2	1	no	—
47	C₃×D₃₆	—	2	1	no	—
48	C₃×D₃₆	—	2	1	no	—
49	C₃×Dic₁₈	—	2	1	no	—
50	C₃×Dic₁₈	—	2	1	no	—
51	C₃×C₈×C₉	—	2	1	no	—
52	C₃×C₈×C₉	D₄	1	1	yes	characteristic
53	C₃×C₈×C₉	D₄	1	1	yes	characteristic
54	C₃×C₉⋊C₈	—	2	1	no	—
55	C₃×C₉⋊C₈	—	2	1	no	—
56	C₃×#471	—	9	1	no	—
57	#5141	—	3	1	no	—
58	C₃²×#41	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×D₁₂	—	3	1	no	—
60	C₂×C₃×Dic₆	—	3	1	no	—
61	C₃²×M₁₆	—	2	1	no	—
62	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃²×C₈	—	2	1	no	—
64	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×#216	—	6	1	no	—
66	C₃×#215	—	6	1	no	—
67	C₃×#214	—	6	1	no	—
68	C₃×#213	—	6	1	no	—
69	C₃×#208	—	3	1	no	—
70	C₃×#208	—	3	1	no	—
71	C₃×#204	—	6	1	no	—
72	C₃×#204	—	6	1	no	—
73	C₉×#41	—	2	1	no	—
74	C₉×#41	C₂²×C₃	1	1	yes	characteristic
75	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
77	C₉×M₁₆	—	2	2	no	—
78	C₉×M₁₆	—	2	1	no	—
79	C₉×M₁₆	—	2	1	no	—
80	C₉×M₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₈×C₉	—	2	2	no	—
82	C₂×C₈×C₉	—	2	1	no	—
83	C₂×C₈×C₉	—	2	1	no	—
84	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
85	#3675	—	2	1	no	—
86	#3674	—	2	1	no	—
87	#3673	—	2	1	no	—
88	#3672	—	2	1	no	—
89	#3667	C₂²×C₃	1	1	yes	characteristic
90	#3667	C₂²×C₃	1	1	yes	characteristic
91	#3664	—	2	1	no	—
92	#3664	—	2	1	no	—
93	C₂²×C₃×C₉	—	2	1	no	—
94	C₂²×C₃×C₉	#41	1	1	yes	characteristic
95	C₂×C₃×D₉	—	4	1	no	—
96	C₂×C₃×D₉	—	8	1	no	—
97	C₃×C₄×C₉	—	2	1	no	—
98	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₃×Dic₉	—	4	1	no	—
101	C₃×#137	—	9	1	no	—
102	C₃×AGL(1,ℤ/8)	—	9	1	no	—
103	C₂×C₃×SD₁₆	—	9	1	no	—
104	C₃×#131	—	2	1	no	—
105	C₃×#131	D₉	1	1	yes	characteristic, complemented
106	C₃×#131	C₃×D₃	1	1	yes	characteristic
107	#951	—	3	1	no	—
108	#949	—	3	1	no	—
109	C₂×#204	—	3	1	no	—
110	C₃×#108	—	9	1	no	—
111	C₃×#101	—	9	1	no	—
112	C₃×#100	—	9	1	no	—
113	#824	—	3	1	no	—
114	#823	—	3	1	no	—
115	#819	—	3	1	no	—
116	C₂²×C₃×D₃	—	6	1	no	—
117	C₃²×Q₈	D₁₂	1	1	yes	characteristic
118	C₃²×D₄	—	2	1	no	—
119	C₃²×D₄	D₁₂	1	1	yes	characteristic
120	C₂×C₃²×C₄	—	2	1	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₃×Dic₃	—	6	1	no	—
123	C₃×D₁₂	—	6	1	no	—
124	C₃×D₁₂	—	6	1	no	—
125	C₃×Dic₆	—	6	1	no	—
126	C₃×Dic₆	—	6	1	no	—
127	C₂²×D₉	—	2	1	no	—
128	C₃²×C₈	—	2	1	no	—
129	C₃²×C₈	#67	1	1	yes	characteristic
130	C₃²×C₈	#67	1	1	yes	characteristic
131	C₃×C₃⋊C₈	—	6	1	no	—
132	C₃×C₃⋊C₈	—	6	1	no	—
133	Q₈×C₉	—	2	1	no	—
134	Q₈×C₉	C₃×D₄	1	1	yes	characteristic
135	D₄×C₉	—	2	2	no	—
136	D₄×C₉	—	2	1	no	—
137	D₄×C₉	—	2	1	no	—
138	D₄×C₉	C₃×D₄	1	1	yes	characteristic
139	C₂×C₄×C₉	—	2	2	no	—
140	C₂×C₄×C₉	—	2	1	no	—
141	C₂×C₄×C₉	—	2	1	no	—
142	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
143	C₂×Dic₉	—	2	1	no	—
144	D₃₆	—	2	1	no	—
145	D₃₆	—	2	1	no	—
146	Dic₁₈	—	2	1	no	—
147	Dic₁₈	—	2	1	no	—
148	C₈×C₉	—	2	2	no	—
149	C₈×C₉	—	2	1	no	—
150	C₈×C₉	—	2	1	no	—
151	C₈×C₉	—	2	1	no	—
152	C₈×C₉	C₃×D₄	1	1	yes	characteristic
153	C₈×C₉	C₃×D₄	1	1	yes	characteristic
154	C₉⋊C₈	—	2	1	no	—
155	C₉⋊C₈	—	2	1	no	—
156	#471	—	9	1	no	Sylow
157	C₂×C₃×C₉	—	2	1	no	—
158	C₂×C₃×C₉	—	4	1	no	—
159	C₂×C₃×C₉	#121	1	1	yes	characteristic
160	C₃×D₉	—	8	1	no	—
161	C₃×#41	—	2	1	no	—
162	C₃×#41	C₂×D₉	1	1	yes	characteristic
163	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
164	C₂×C₃×Q₈	—	9	1	no	—
165	C₂×C₃×D₄	—	9	1	no	—
166	C₂×D₁₂	—	3	1	no	—
167	C₂×Dic₆	—	3	1	no	—
168	C₃×Q₁₆	—	18	1	no	—
169	C₃×SD₁₆	—	18	1	no	—
170	C₃×SD₁₆	—	18	1	no	—
171	C₃×SD₁₆	—	18	1	no	—
172	C₃×SD₁₆	—	18	1	no	—
173	C₃×D₈	—	18	1	no	—
174	C₃×M₁₆	—	2	2	no	—
175	C₃×M₁₆	—	2	1	no	—
176	C₃×M₁₆	—	2	1	no	—
177	C₃×M₁₆	—	2	1	no	—
178	C₃×M₁₆	—	9	1	no	—
179	C₃×M₁₆	—	9	1	no	—
180	C₃×M₁₆	C₂×D₉	1	1	yes	characteristic
181	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
182	C₂×C₃×C₈	—	2	2	no	—
183	C₂×C₃×C₈	—	2	1	no	—
184	C₂×C₃×C₈	—	2	1	no	—
185	C₂×C₃×C₈	—	2	1	no	—
186	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
187	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
188	#216	—	6	1	no	—
189	#215	—	6	1	no	—
190	#214	—	6	1	no	—
191	#213	—	6	1	no	—
192	#208	—	3	1	no	—
193	#208	—	3	1	no	—
194	#204	—	6	1	no	—
195	#204	—	6	1	no	—
196	C₂²×C₃²	—	2	1	no	—
197	C₂²×C₃²	#235	1	1	yes	characteristic
198	C₂×C₃×D₃	—	12	1	no	—
199	C₂×C₃×D₃	—	24	1	no	—
200	C₃²×C₄	—	2	1	no	—
201	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
202	C₃²×C₄	C₂×#67	1	1	yes	characteristic
203	C₃×Dic₃	—	12	1	no	—
204	C₂²×C₉	—	2	2	no	—
205	C₂²×C₉	—	2	1	no	—
206	C₂²×C₉	—	2	1	no	—
207	C₂²×C₉	C₃×#41	1	1	yes	characteristic
208	C₂×D₉	—	4	1	no	—
209	C₂×D₉	—	8	1	no	—
210	C₄×C₉	—	2	2	no	—
211	C₄×C₉	—	2	1	no	—
212	C₄×C₉	—	2	1	no	—
213	C₄×C₉	—	2	1	no	—
214	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
215	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
216	Dic₉	—	4	1	no	—
217	#137	—	9	1	no	—
218	AGL(1,ℤ/8)	—	9	1	no	—
219	C₂×SD₁₆	—	9	1	no	—
220	#131	C₃×D₉	1	1	yes	characteristic, complemented
221	#108	—	9	1	no	—
222	#101	—	9	1	no	—
223	#100	—	9	1	no	—
224	C₃×C₉	#471	1	1	yes	characteristic, complemented, Sylow
225	C₂³×C₃	—	18	1	no	—
226	C₂²×D₃	—	6	1	no	—
227	C₃×Q₈	—	2	1	no	—
228	C₃×Q₈	—	18	1	no	—
229	C₃×Q₈	—	18	1	no	—
230	C₃×Q₈	D₃₆	1	1	yes	characteristic
231	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
232	C₃×D₄	—	2	2	no	—
233	C₃×D₄	—	2	1	no	—
234	C₃×D₄	—	2	1	no	—
235	C₃×D₄	—	2	1	no	—
236	C₃×D₄	—	18	1	no	—
237	C₃×D₄	—	18	1	no	—
238	C₃×D₄	D₃₆	1	1	yes	characteristic
239	C₃×D₄	C₃×D₁₂	1	1	yes	characteristic
240	C₂×C₃×C₄	—	2	2	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	18	1	no	—
245	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
247	C₂×Dic₃	—	6	1	no	—
248	D₁₂	—	6	1	no	—
249	D₁₂	—	6	1	no	—
250	Dic₆	—	6	1	no	—
251	Dic₆	—	6	1	no	—
252	C₃×C₈	—	2	2	no	—
253	C₃×C₈	—	2	1	no	—
254	C₃×C₈	—	2	1	no	—
255	C₃×C₈	—	2	1	no	—
256	C₃×C₈	—	2	1	no	—
257	C₃×C₈	—	18	1	no	—
258	C₃×C₈	—	18	1	no	—
259	C₃×C₈	#611	1	1	yes	characteristic
260	C₃×C₈	#611	1	1	yes	characteristic
261	C₃×C₈	C₃×#67	1	1	yes	characteristic
262	C₃×C₈	C₃×#67	1	1	yes	characteristic
263	C₃⋊C₈	—	6	1	no	—
264	C₃⋊C₈	—	6	1	no	—
265	C₂×C₃²	—	2	1	no	—
266	C₂×C₃²	—	4	1	no	—
267	C₂×C₃²	#930	1	1	yes	characteristic
268	C₃×D₃	—	24	1	no	—
269	C₂×C₉	—	2	1	no	—
270	C₂×C₉	—	2	1	no	—
271	C₂×C₉	—	4	1	no	—
272	C₂×C₉	—	4	2	no	—
273	C₂×C₉	—	4	1	no	—
274	C₂×C₉	C₃×#121	1	1	yes	characteristic
275	D₉	—	8	1	no	—
276	#41	C₂×C₃×D₉	1	1	yes	characteristic
277	C₂×Q₈	—	9	1	no	—
278	C₂×D₄	—	9	1	no	—
279	Q₁₆	—	18	1	no	—
280	SD₁₆	—	18	1	no	—
281	SD₁₆	—	18	1	no	—
282	SD₁₆	—	18	1	no	—
283	SD₁₆	—	18	1	no	—
284	D₈	—	18	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	9	1	no	—
287	M₁₆	—	9	1	no	—
288	M₁₆	C₂×C₃×D₉	1	1	yes	characteristic
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
291	C₂²×C₃	—	2	2	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	2	1	no	—
295	C₂²×C₃	—	36	1	no	—
296	C₂²×C₃	—	72	1	no	—
297	C₂²×C₃	#3697	1	1	yes	characteristic
298	C₂²×C₃	C₃×#235	1	1	yes	characteristic
299	C₂×D₃	—	12	1	no	—
300	C₂×D₃	—	24	1	no	—
301	C₃×C₄	—	2	2	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	36	1	no	—
307	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
308	C₃×C₄	C₂×#611	1	1	yes	characteristic
309	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
310	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
311	Dic₃	—	12	1	no	—
312	C₃²	#5141	1	1	yes	characteristic
313	C₉	—	2	1	no	—
314	C₉	C₃×#471	1	1	yes	characteristic, complemented
315	C₂³	—	18	1	no	—
316	Q₈	—	18	1	no	—
317	Q₈	—	18	1	no	—
318	Q₈	C₃×D₃₆	1	1	yes	characteristic
319	D₄	—	2	1	no	—
320	D₄	—	18	1	no	—
321	D₄	—	18	1	no	—
322	D₄	C₃×D₃₆	1	1	yes	characteristic
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	18	1	no	—
325	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	18	1	no	—
328	C₈	—	18	1	no	—
329	C₈	C₃×#611	1	1	yes	characteristic
330	C₈	C₃×#611	1	1	yes	characteristic
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	4	1	no	—
335	C₂×C₃	—	4	2	no	—
336	C₂×C₃	—	4	1	no	—
337	C₂×C₃	—	4	1	no	—
338	C₂×C₃	—	72	1	no	—
339	C₂×C₃	#63492	1	1	yes	characteristic
340	C₂×C₃	C₃×#930	1	1	yes	characteristic
341	D₃	—	24	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	36	1	no	—
344	C₂²	—	72	1	no	—
345	C₂²	C₃×#3697	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	36	1	no	—
348	C₄	C₂×C₃×D₃₆	1	1	yes	characteristic
349	C₄	C₂×C₃×#611	1	1	yes	characteristic
350	C₃	—	2	1	no	—
351	C₃	#10588822	1	1	yes	characteristic, direct product factor
352	C₃	C₃×#5141	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	4	1	no	—
355	C₂	—	72	1	no	—
356	C₂	C₃×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#63492
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#131	C₂
socle	C₂×C₃²	#930
prime radical	C₃×#63512	C₂
hypercenter	C₃×#131	D₉
hypocenter	C₉	C₃×#471
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#131

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#131	C₂	semidirect
3. C₃×#63511	C₂	semidirect
4. C₂×C₃×#3664	C₂	semidirect
5. C₃×#63383	C₂	semidirect
6. C₃×#63382	C₂	semidirect
7. C₃×#63372	C₂	semidirect
9. #10588822	C₃	direct (N × H)
33. C₉×#131	C₂×C₃	semidirect
35. #63511	C₂×C₃	semidirect
36. C₂×#3664	C₂×C₃	semidirect
37. #63383	C₂×C₃	semidirect
38. #63382	C₂×C₃	semidirect
39. #63372	C₂×C₃	semidirect
105. C₃×#131	D₉	semidirect
220. #131	C₃×D₉	semidirect
224. C₃×C₉	#471	semidirect
314. C₉	C₃×#471	semidirect
351. C₃	#10588822	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×C₉×#131	2
C₃×#63512	2
C₃×#63511	2
C₂×C₃×#3664	2
C₃×#63383	2
C₃×#63382	2
C₃×#63372	2
C₃×#5141	3
#10588822	3

Subgroups — C3×#10588822 #13143247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588822 #13143247