Subgroups — C₃×#10588698 #13142729

Order 1728.

514 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3673	C₂	1	2	yes	maximal, complemented
3	C₃×#63436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63369	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5015	—	3	1	no	maximal
7	#10588698	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×C₃×D₃₆	—	2	1	no	—
11	C₂×C₃×D₃₆	—	2	1	no	—
12	C₂×C₃×D₃₆	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₃×#3673	—	2	4	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
18	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
19	C₂×C₃×#213	—	3	2	no	—
20	C₃×#874	—	3	1	no	—
21	C₃×#809	—	3	2	no	—
22	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₄×D₄×C₉	—	2	1	no	—
24	C₄×D₄×C₉	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#3673	C₂×C₃	1	2	yes	complemented
26	#63436	C₂×C₃	1	1	yes	characteristic, complemented
27	#63369	C₂×C₃	1	2	yes	complemented
28	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×C₉	—	2	1	no	—
30	C₂²×C₃×D₉	—	2	2	no	—
31	C₃×D₄×C₉	—	2	1	no	—
32	C₃×D₄×C₉	D₄	1	2	yes	complemented
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	—	4	2	no	—
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₃×D₃₆	—	2	2	no	—
39	C₃×D₃₆	—	2	2	no	—
40	C₃×D₃₆	—	4	2	no	—
41	C₃×D₃₆	—	4	2	no	—
42	C₃×C₉⋊C₈	—	2	2	no	—
43	C₃×#459	—	9	1	no	—
44	#5015	—	3	1	no	—
45	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	—	2	1	no	—
47	C₂×C₃×D₁₂	—	3	2	no	—
48	C₂×C₃×D₁₂	—	6	1	no	—
49	C₂×C₃×D₁₂	—	6	1	no	—
50	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
51	C₃²×C₂²⋊C₄	—	2	1	no	—
52	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₃×#213	—	6	4	no	—
54	C₂×C₃×C₃⋊C₈	—	3	2	no	—
55	C₂×D₄×C₉	—	2	1	no	—
56	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₄×C₉	—	2	2	no	—
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂×D₃₆	—	2	1	no	—
60	C₂×D₃₆	—	2	1	no	—
61	C₂×D₃₆	C₂²×C₃	1	2	yes	—
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₉×C₂²⋊C₄	—	2	2	no	—
65	C₉×C₂²⋊C₄	—	2	1	no	—
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
68	#3673	—	2	4	no	—
69	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
70	C₂²×C₃×C₉	—	2	2	no	—
71	C₂²×C₃×C₉	—	4	1	no	—
72	C₂²×C₃×C₉	—	4	1	no	—
73	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×D₉	—	4	2	no	—
75	C₂×C₃×D₉	—	4	2	no	—
76	C₂×C₃×D₉	—	4	2	no	—
77	C₃×C₄×C₉	—	2	1	no	—
78	C₃×C₄×C₉	—	4	1	no	—
79	C₃×C₄×C₉	D₈	1	2	yes	complemented
80	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄×C₉	#41	1	1	yes	characteristic
82	C₂×C₃×D₈	—	9	2	no	—
83	C₃×#127	—	9	1	no	—
84	C₃×C₄×D₄	—	2	1	no	—
85	C₃×C₄×D₄	D₉	1	1	yes	characteristic, complemented
86	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
87	C₂×#213	—	3	2	no	—
88	#874	—	3	1	no	—
89	C₃×C₄⋊C₈	—	9	1	no	—
90	C₃×#102	—	9	2	no	—
91	#809	—	3	2	no	—
92	C₄⋊(C₃⋊C₈)	—	3	1	no	—
93	C₂³×C₃²	—	2	1	no	—
94	C₂²×C₃×D₃	—	6	2	no	—
95	C₃²×D₄	—	2	1	no	—
96	C₃²×D₄	D₁₂	1	2	yes	—
97	C₂×C₃²×C₄	—	2	1	no	—
98	C₂×C₃²×C₄	—	4	2	no	—
99	C₂×C₃²×C₄	#67	1	1	yes	characteristic
100	C₂×C₃²×C₄	#67	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₃×D₁₂	—	6	2	no	—
103	C₃×D₁₂	—	6	2	no	—
104	C₃×D₁₂	—	12	2	no	—
105	C₃×D₁₂	—	12	2	no	—
106	C₂³×C₉	—	2	2	no	—
107	C₂³×C₉	—	2	1	no	—
108	C₂²×D₉	—	2	2	no	—
109	C₃×C₃⋊C₈	—	6	2	no	—
110	D₄×C₉	—	2	2	no	—
111	D₄×C₉	—	2	1	no	—
112	D₄×C₉	—	2	2	no	—
113	D₄×C₉	C₃×D₄	1	2	yes	complemented
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	1	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	4	4	no	—
120	C₂×C₄×C₉	—	4	2	no	—
121	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
122	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
123	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
124	D₃₆	—	2	2	no	—
125	D₃₆	—	2	2	no	—
126	D₃₆	—	4	2	no	—
127	D₃₆	—	4	2	no	—
128	C₉⋊C₈	—	2	2	no	—
129	#459	—	9	1	no	Sylow
130	C₂×C₃×C₉	—	4	2	no	—
131	C₂×C₃×C₉	#121	1	1	yes	characteristic
132	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₃×D₉	—	8	2	no	—
135	C₂×C₃×D₄	—	2	1	no	—
136	C₂×C₃×D₄	—	9	2	no	—
137	C₂×C₃×D₄	—	18	1	no	—
138	C₂×C₃×D₄	—	18	1	no	—
139	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
140	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	—	2	1	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂×D₁₂	—	3	2	no	—
145	C₂×D₁₂	—	6	1	no	—
146	C₂×D₁₂	—	6	1	no	—
147	C₃×D₈	—	18	4	no	—
148	C₂×C₃×C₈	—	9	2	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
151	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₂²⋊C₄	—	2	1	no	—
154	C₃×C₂²⋊C₄	—	2	1	no	—
155	C₃×C₄²	—	2	1	no	—
156	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
157	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
158	#213	—	6	4	no	—
159	C₂×C₃⋊C₈	—	3	2	no	—
160	C₂²×C₃²	—	2	2	no	—
161	C₂²×C₃²	—	4	1	no	—
162	C₂²×C₃²	—	4	1	no	—
163	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
164	C₂×C₃×D₃	—	12	2	no	—
165	C₂×C₃×D₃	—	12	2	no	—
166	C₂×C₃×D₃	—	12	2	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	4	1	no	—
169	C₃²×C₄	#213	1	2	yes	—
170	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
171	C₃²×C₄	#235	1	1	yes	characteristic
172	C₂²×C₉	—	2	4	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	4	2	no	—
176	C₂²×C₉	—	4	2	no	—
177	C₂²×C₉	—	4	1	no	—
178	C₂²×C₉	—	4	1	no	—
179	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×D₉	—	4	2	no	—
181	C₂×D₉	—	4	2	no	—
182	C₂×D₉	—	4	2	no	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	4	1	no	—
188	C₄×C₉	—	4	2	no	—
189	C₄×C₉	—	4	1	no	—
190	C₄×C₉	C₃×D₈	1	2	yes	complemented
191	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
192	C₄×C₉	C₃×#41	1	1	yes	characteristic
193	C₂×D₈	—	9	2	no	—
194	#127	—	9	1	no	—
195	C₄×D₄	C₃×D₉	1	1	yes	characteristic, complemented
196	C₄⋊C₈	—	9	1	no	—
197	#102	—	9	2	no	—
198	C₃×C₉	#459	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	—	2	1	no	—
202	C₂³×C₃	—	18	2	no	—
203	C₂²×D₃	—	6	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	2	1	no	—
206	C₃×D₄	—	2	1	no	—
207	C₃×D₄	—	2	2	no	—
208	C₃×D₄	—	18	2	no	—
209	C₃×D₄	—	18	2	no	—
210	C₃×D₄	—	36	2	no	—
211	C₃×D₄	—	36	2	no	—
212	C₃×D₄	D₃₆	1	2	yes	complemented
213	C₃×D₄	C₃×D₁₂	1	2	yes	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	#611	1	1	yes	characteristic
224	C₂×C₃×C₄	#611	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
226	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
227	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
229	D₁₂	—	6	2	no	—
230	D₁₂	—	6	2	no	—
231	D₁₂	—	12	2	no	—
232	D₁₂	—	12	2	no	—
233	C₃×C₈	—	18	2	no	—
234	C₃⋊C₈	—	6	2	no	—
235	C₂×C₃²	—	4	2	no	—
236	C₂×C₃²	#930	1	1	yes	characteristic
237	C₂×C₃²	C₂×#213	1	1	yes	characteristic
238	C₂×C₃²	#949	1	1	yes	characteristic
239	C₃×D₃	—	24	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	4	4	no	—
244	C₂×C₉	—	4	2	no	—
245	C₂×C₉	C₃×#121	1	1	yes	characteristic
246	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
247	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	D₉	—	8	2	no	—
249	C₂×D₄	—	9	2	no	—
250	C₂×D₄	—	18	1	no	—
251	C₂×D₄	—	18	1	no	—
252	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	D₈	—	18	4	no	—
255	C₂×C₈	—	9	2	no	—
256	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	36	2	no	—
270	C₂²×C₃	—	36	2	no	—
271	C₂²×C₃	—	36	2	no	—
272	C₂²×C₃	C₂×#611	1	1	yes	characteristic
273	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
274	C₂×D₃	—	12	2	no	—
275	C₂×D₃	—	12	2	no	—
276	C₂×D₃	—	12	2	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	4	1	no	—
283	C₃×C₄	—	4	2	no	—
284	C₃×C₄	—	4	1	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	#3673	1	2	yes	complemented
287	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
288	C₃×C₄	#3697	1	1	yes	characteristic
289	C₃×C₄	C₃×#213	1	2	yes	—
290	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
291	C₃×C₄	C₃×#235	1	1	yes	characteristic
292	C₃²	#5015	1	1	yes	characteristic
293	C₉	—	2	1	no	—
294	C₉	C₃×#459	1	1	yes	characteristic, complemented
295	C₂³	—	2	1	no	—
296	C₂³	—	18	2	no	—
297	D₄	—	2	1	no	—
298	D₄	—	18	2	no	—
299	D₄	—	18	2	no	—
300	D₄	—	36	2	no	—
301	D₄	—	36	2	no	—
302	D₄	C₃×D₃₆	1	2	yes	complemented
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₃×#611	1	1	yes	characteristic
306	C₂×C₄	C₃×#611	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
308	C₈	—	18	2	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	4	4	no	—
313	C₂×C₃	—	4	2	no	—
314	C₂×C₃	—	4	2	no	—
315	C₂×C₃	—	72	2	no	—
316	C₂×C₃	#63492	1	1	yes	characteristic
317	C₂×C₃	C₂×#3673	1	1	yes	characteristic
318	C₂×C₃	#63512	1	1	yes	characteristic
319	C₂×C₃	C₃×#930	1	1	yes	characteristic
320	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
321	C₂×C₃	C₃×#949	1	1	yes	characteristic
322	D₃	—	24	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	36	2	no	—
327	C₂²	—	36	2	no	—
328	C₂²	—	36	2	no	—
329	C₂²	C₂×C₃×#611	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	C₃×#3673	1	2	yes	complemented
333	C₄	C₂×C₃×D₃₆	1	1	yes	characteristic
334	C₄	C₃×#3697	1	1	yes	characteristic
335	C₃	—	2	1	no	—
336	C₃	#10588698	1	1	yes	characteristic, direct product factor
337	C₃	C₃×#5015	1	1	yes	characteristic
338	C₂	—	4	2	no	—
339	C₂	—	72	2	no	—
340	C₂	C₃×#63492	1	1	yes	characteristic
341	C₂	C₂×C₃×#3673	1	1	yes	characteristic
342	C₂	C₃×#63512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×#67
hypercenter	C₃×C₄×D₄	D₉
hypocenter	C₉	C₃×#459
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×D₄×C₉	C₂	semidirect
2. C₂×C₃×#3673	C₂	semidirect
3. C₃×#63436	C₂	semidirect
4. C₃×#63369	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588698	C₃	direct (N × H)
24. C₄×D₄×C₉	C₂×C₃	semidirect
25. C₂×#3673	C₂×C₃	semidirect
26. #63436	C₂×C₃	semidirect
27. #63369	C₂×C₃	semidirect
28. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
32. C₃×D₄×C₉	D₄	semidirect
79. C₃×C₄×C₉	D₈	semidirect
85. C₃×C₄×D₄	D₉	semidirect
113. D₄×C₉	C₃×D₄	semidirect
190. C₄×C₉	C₃×D₈	semidirect
195. C₄×D₄	C₃×D₉	semidirect
198. C₃×C₉	#459	semidirect
212. C₃×D₄	D₃₆	semidirect
286. C₃×C₄	#3673	semidirect
294. C₉	C₃×#459	semidirect
302. D₄	C₃×D₃₆	semidirect
332. C₄	C₃×#3673	semidirect
336. C₃	#10588698	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×C₄×D₄×C₉	2
C₂×C₃×#3673	2
C₃×#63436	2
C₃×#63369	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#5015	3
#10588698	3

Subgroups — C3×#10588698 #13142729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588698 #13142729