Subgroups — C₃×#10589785 #13153096

Order 1728.

856 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63509	C₂	1	2	yes	maximal, complemented
3	C₃×#63499	C₂	1	2	yes	maximal, complemented
4	C₃×#63447	C₂	1	4	yes	maximal, complemented
5	C₃×#63444	C₂	1	4	yes	maximal, complemented
6	C₃×#63437	C₂	1	2	yes	maximal, complemented
7	C₃×#5683	—	3	1	no	maximal
8	#10589785	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂³×C₃×D₉	C₂²	1	2	yes	complemented
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×#611	C₂²	1	4	yes	—
13	C₂×C₃×D₃₆	C₂²	1	2	yes	—
14	C₂×C₃×D₃₆	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₈	C₂²	1	2	yes	complemented
16	C₃×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
18	C₃×#3676	C₂²	1	2	yes	complemented
19	C₃×(C₂×D₉)⋊C₄	C₂²	1	4	yes	complemented
20	C₃×(C₂×D₉)⋊C₄	C₂²	1	4	yes	complemented
21	C₃×(C₂×D₉)⋊C₄	C₂²	1	4	yes	complemented
22	C₃×C₄×Dic₉	C₂²	1	2	yes	complemented
23	C₃²×#124	D₃	1	1	yes	characteristic
24	C₃×#947	—	3	2	no	—
25	C₃×#937	—	3	2	no	—
26	C₃×#885	—	3	4	no	—
27	C₃×#882	—	3	4	no	—
28	C₃×#875	—	3	2	no	—
29	C₉×#124	—	2	1	no	—
30	C₉×#124	C₂×C₃	1	1	yes	characteristic, complemented
31	#63509	C₂×C₃	1	2	yes	complemented
32	#63499	C₂×C₃	1	2	yes	complemented
33	#63447	C₂×C₃	1	4	yes	complemented
34	#63444	C₂×C₃	1	4	yes	complemented
35	#63437	C₂×C₃	1	2	yes	complemented
36	C₂³×C₃×C₉	C₂³	1	2	yes	—
37	C₂²×C₃×D₉	—	4	2	no	—
38	C₂²×C₃×D₉	—	4	4	no	—
39	C₂²×C₃×D₉	C₂³	1	4	yes	—
40	C₃×Q₈×C₉	—	4	1	no	—
41	C₃×D₄×C₉	—	4	1	no	—
42	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₃×#611	—	4	4	no	—
46	C₂×C₃×Dic₉	C₂³	1	4	yes	—
47	C₃×D₃₆	—	4	2	no	—
48	C₃×D₃₆	—	4	2	no	—
49	C₃×Dic₁₈	—	4	2	no	—
50	C₃×H(3,𝔽₄)	—	9	1	no	—
51	#5683	—	3	1	no	—
52	C₂³×C₃×D₃	—	3	2	no	—
53	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×#67	—	3	4	no	—
56	C₂×C₃×D₁₂	—	3	2	no	—
57	C₂×C₃×D₁₂	—	3	2	no	—
58	C₂×C₃×Dic₆	—	3	2	no	—
59	C₂³×D₉	C₂²×C₃	1	2	yes	complemented
60	C₃²×C₂²⋊C₄	C₂×D₃	1	4	yes	—
61	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₃×#217	—	3	2	no	—
63	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
64	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
65	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
66	C₃×C₄×Dic₃	—	3	2	no	—
67	C₂×Q₈×C₉	—	2	1	no	—
68	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
69	C₂×D₄×C₉	—	2	1	no	—
70	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
71	C₂×#611	C₂²×C₃	1	4	yes	—
72	C₂×D₃₆	C₂²×C₃	1	2	yes	—
73	C₂×D₃₆	C₂²×C₃	1	2	yes	—
74	C₂×Dic₁₈	C₂²×C₃	1	2	yes	complemented
75	C₉×C₂²⋊C₄	—	2	4	no	—
76	C₉×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
77	C₄²×C₉	—	2	1	no	—
78	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
79	#3676	C₂²×C₃	1	2	yes	complemented
80	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	complemented
81	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	complemented
82	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	complemented
83	C₄×Dic₉	C₂²×C₃	1	2	yes	complemented
84	C₂²×C₃×C₉	—	2	4	no	—
85	C₂²×C₃×C₉	—	2	2	no	—
86	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
87	C₂×C₃×D₉	—	2	4	no	—
88	C₂×C₃×D₉	—	2	4	no	—
89	C₂×C₃×D₉	—	2	4	no	—
90	C₂×C₃×D₉	—	4	8	no	—
91	C₃×C₄×C₉	—	2	1	no	—
92	C₃×C₄×C₉	—	2	2	no	—
93	C₃×C₄×C₉	—	2	2	no	—
94	C₃×Dic₉	—	2	4	no	—
95	C₃×#124	—	2	1	no	—
96	C₃×#124	—	9	2	no	—
97	C₃×#124	—	9	4	no	—
98	C₃×#124	—	9	2	no	—
99	C₃×#124	D₉	1	1	yes	characteristic, complemented
100	C₃×#124	C₃×D₃	1	1	yes	characteristic
101	C₃×#120	—	9	2	no	—
102	C₃×#120	—	9	4	no	—
103	#947	—	3	2	no	—
104	#937	—	3	2	no	—
105	#885	—	3	4	no	—
106	#882	—	3	4	no	—
107	#875	—	3	2	no	—
108	C₂³×C₃²	C₂²×D₃	1	2	yes	—
109	C₂²×C₃×D₃	—	3	4	no	—
110	C₂²×C₃×D₃	—	12	2	no	—
111	C₂²×C₃×D₃	—	12	4	no	—
112	C₃²×Q₈	—	4	1	no	—
113	C₃²×D₄	—	4	1	no	—
114	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
115	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
117	C₃×#67	—	12	4	no	—
118	C₂×C₃×Dic₃	—	3	4	no	—
119	C₃×D₁₂	—	12	2	no	—
120	C₃×D₁₂	—	12	2	no	—
121	C₃×Dic₆	—	12	2	no	—
122	C₂³×C₉	—	2	2	no	—
123	C₂³×C₉	C₂³×C₃	1	2	yes	—
124	C₂²×D₉	—	4	2	no	—
125	C₂²×D₉	—	4	4	no	—
126	C₂²×D₉	C₂³×C₃	1	4	yes	—
127	Q₈×C₉	—	4	1	no	—
128	Q₈×C₉	—	4	2	no	—
129	D₄×C₉	—	4	1	no	—
130	D₄×C₉	—	4	2	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	2	2	no	—
133	C₂×C₄×C₉	—	2	2	no	—
134	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
135	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
137	#611	—	4	4	no	—
138	C₂×Dic₉	C₂³×C₃	1	4	yes	—
139	D₃₆	—	4	2	no	—
140	D₃₆	—	4	2	no	—
141	Dic₁₈	—	4	2	no	—
142	H(3,𝔽₄)	—	9	1	no	Sylow
143	C₂×C₃×C₉	—	4	2	no	—
144	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
146	C₃×D₉	—	4	4	no	—
147	C₂⁴×C₃	—	9	2	no	—
148	C₂³×D₃	—	3	2	no	—
149	C₂×C₃×Q₈	—	2	1	no	—
150	C₂×C₃×Q₈	—	9	2	no	—
151	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic, complemented
152	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂×C₃×D₄	—	2	1	no	—
154	C₂×C₃×D₄	—	9	4	no	—
155	C₂×C₃×D₄	—	9	2	no	—
156	C₂×C₃×D₄	—	9	2	no	—
157	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
158	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
159	C₂×#67	—	3	4	no	—
160	C₂×D₁₂	—	3	2	no	—
161	C₂×D₁₂	—	3	2	no	—
162	C₂×Dic₆	—	3	2	no	—
163	C₃×C₂²⋊C₄	—	2	4	no	—
164	C₃×C₂²⋊C₄	—	9	2	no	—
165	C₃×C₂²⋊C₄	—	9	4	no	—
166	C₃×C₂²⋊C₄	—	9	4	no	—
167	C₃×C₂²⋊C₄	—	9	4	no	—
168	C₃×C₂²⋊C₄	C₂×D₉	1	4	yes	complemented
169	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	4	yes	—
170	C₃×C₄²	—	2	1	no	—
171	C₃×C₄²	—	9	2	no	—
172	C₃×C₄²	C₂×D₉	1	1	yes	characteristic, complemented
173	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
174	#217	—	3	2	no	—
175	(C₂×D₃)⋊C₄	—	3	4	no	—
176	(C₂×D₃)⋊C₄	—	3	4	no	—
177	(C₂×D₃)⋊C₄	—	3	4	no	—
178	C₄×Dic₃	—	3	2	no	—
179	C₂²×C₃²	—	2	2	no	—
180	C₂²×C₃²	—	2	4	no	—
181	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
182	C₂×C₃×D₃	—	6	4	no	—
183	C₂×C₃×D₃	—	6	4	no	—
184	C₂×C₃×D₃	—	6	4	no	—
185	C₂×C₃×D₃	—	12	8	no	—
186	C₃²×C₄	—	2	1	no	—
187	C₃²×C₄	—	2	2	no	—
188	C₃²×C₄	—	2	2	no	—
189	C₃×Dic₃	—	6	4	no	—
190	C₂²×C₉	—	2	4	no	—
191	C₂²×C₉	—	2	2	no	—
192	C₂²×C₉	—	2	1	no	—
193	C₂²×C₉	—	4	2	no	—
194	C₂²×C₉	—	4	4	no	—
195	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
196	C₂×D₉	—	2	4	no	—
197	C₂×D₉	—	2	4	no	—
198	C₂×D₉	—	2	4	no	—
199	C₂×D₉	—	4	8	no	—
200	C₄×C₉	—	2	2	no	—
201	C₄×C₉	—	2	2	no	—
202	C₄×C₉	—	2	2	no	—
203	C₄×C₉	—	2	1	no	—
204	C₄×C₉	—	4	2	no	—
205	C₄×C₉	—	4	2	no	—
206	Dic₉	—	2	4	no	—
207	#124	—	9	2	no	—
208	#124	—	9	4	no	—
209	#124	—	9	2	no	—
210	#124	C₃×D₉	1	1	yes	characteristic, complemented
211	#120	—	9	2	no	—
212	#120	—	9	4	no	—
213	C₃×C₉	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₃	—	2	2	no	—
215	C₂³×C₃	—	9	4	no	—
216	C₂³×C₃	—	36	2	no	—
217	C₂³×C₃	—	36	4	no	—
218	C₂³×C₃	C₂²×D₉	1	2	yes	—
219	C₂³×C₃	C₂²×C₃×D₃	1	2	yes	—
220	C₂²×D₃	—	3	4	no	—
221	C₂²×D₃	—	12	2	no	—
222	C₂²×D₃	—	12	4	no	—
223	C₃×Q₈	—	4	1	no	—
224	C₃×Q₈	—	4	1	no	—
225	C₃×Q₈	—	4	2	no	—
226	C₃×Q₈	—	36	2	no	—
227	C₃×D₄	—	4	1	no	—
228	C₃×D₄	—	4	1	no	—
229	C₃×D₄	—	4	2	no	—
230	C₃×D₄	—	36	2	no	—
231	C₃×D₄	—	36	2	no	—
232	C₃×D₄	—	36	4	no	—
233	C₂×C₃×C₄	—	2	1	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	—	2	2	no	—
236	C₂×C₃×C₄	—	9	4	no	—
237	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
238	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
240	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
241	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
243	#67	—	12	4	no	—
244	C₂×Dic₃	—	3	4	no	—
245	D₁₂	—	12	2	no	—
246	D₁₂	—	12	2	no	—
247	Dic₆	—	12	2	no	—
248	C₂×C₃²	—	4	2	no	—
249	C₂×C₃²	#1004	1	1	yes	characteristic
250	C₂×C₃²	#1009	1	2	yes	—
251	C₃×D₃	—	12	4	no	—
252	C₂×C₉	—	2	1	no	—
253	C₂×C₉	—	2	2	no	—
254	C₂×C₉	—	4	2	no	—
255	C₂×C₉	—	8	2	no	—
256	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
257	C₂×C₉	C₃×H(5,𝔽₂)	1	2	yes	—
258	D₉	—	4	4	no	—
259	C₂⁴	—	9	2	no	—
260	C₂×Q₈	—	9	2	no	—
261	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic, complemented
262	C₂×D₄	—	9	4	no	—
263	C₂×D₄	—	9	2	no	—
264	C₂×D₄	—	9	2	no	—
265	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	9	2	no	—
267	C₂²⋊C₄	—	9	4	no	—
268	C₂²⋊C₄	—	9	4	no	—
269	C₂²⋊C₄	—	9	4	no	—
270	C₂²⋊C₄	C₂×C₃×D₉	1	4	yes	complemented
271	C₄²	—	9	2	no	—
272	C₄²	C₂×C₃×D₉	1	1	yes	characteristic, complemented
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	4	no	—
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	2	4	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	—	4	4	no	—
280	C₂²×C₃	—	18	4	no	—
281	C₂²×C₃	—	18	4	no	—
282	C₂²×C₃	—	18	4	no	—
283	C₂²×C₃	—	36	8	no	—
284	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
285	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
286	C₂×D₃	—	6	4	no	—
287	C₂×D₃	—	6	4	no	—
288	C₂×D₃	—	6	4	no	—
289	C₂×D₃	—	12	8	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	2	no	—
296	C₃×C₄	—	2	2	no	—
297	C₃×C₄	—	4	2	no	—
298	C₃×C₄	—	4	2	no	—
299	C₃×C₄	—	18	4	no	—
300	Dic₃	—	6	4	no	—
301	C₃²	#5683	1	1	yes	characteristic
302	C₉	—	2	1	no	—
303	C₉	C₃×H(3,𝔽₄)	1	1	yes	characteristic, complemented
304	C₂³	—	9	4	no	—
305	C₂³	—	36	2	no	—
306	C₂³	—	36	4	no	—
307	C₂³	C₂²×C₃×D₉	1	2	yes	—
308	Q₈	—	4	1	no	—
309	Q₈	—	36	2	no	—
310	D₄	—	4	1	no	—
311	D₄	—	36	2	no	—
312	D₄	—	36	2	no	—
313	D₄	—	36	4	no	—
314	C₂×C₄	—	9	4	no	—
315	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
316	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	4	2	no	—
321	C₂×C₃	—	4	2	no	—
322	C₂×C₃	—	8	2	no	—
323	C₂×C₃	—	36	4	no	—
324	C₂×C₃	#63710	1	1	yes	characteristic
325	C₂×C₃	#63715	1	2	yes	—
326	C₂×C₃	C₃×#1004	1	1	yes	characteristic
327	C₂×C₃	C₃×#1009	1	2	yes	—
328	D₃	—	12	4	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	18	4	no	—
332	C₂²	—	18	4	no	—
333	C₂²	—	18	4	no	—
334	C₂²	—	36	8	no	—
335	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	18	4	no	—
340	C₃	—	2	1	no	—
341	C₃	#10589785	1	1	yes	characteristic, direct product factor
342	C₃	C₃×#5683	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	36	4	no	—
345	C₂	C₃×#63710	1	1	yes	characteristic
346	C₂	C₃×#63715	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#124	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₃×#124	D₉
hypocenter	C₉	C₃×H(3,𝔽₄)
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#124	C₂	semidirect
2. C₃×#63509	C₂	semidirect
3. C₃×#63499	C₂	semidirect
4. C₃×#63447	C₂	semidirect
5. C₃×#63444	C₂	semidirect
6. C₃×#63437	C₂	semidirect
8. #10589785	C₃	direct (N × H)
9. C₂³×C₃×D₉	C₂²	semidirect
10. C₂×C₃×Q₈×C₉	C₂²	semidirect
15. C₂×C₃×Dic₁₈	C₂²	semidirect
16. C₃×C₉×C₂²⋊C₄	C₂²	semidirect
17. C₃×C₄²×C₉	C₂²	semidirect
18. C₃×#3676	C₂²	semidirect
19. C₃×(C₂×D₉)⋊C₄	C₂²	semidirect
20. C₃×(C₂×D₉)⋊C₄	C₂²	semidirect
21. C₃×(C₂×D₉)⋊C₄	C₂²	semidirect
22. C₃×C₄×Dic₉	C₂²	semidirect
30. C₉×#124	C₂×C₃	semidirect
31. #63509	C₂×C₃	semidirect
32. #63499	C₂×C₃	semidirect
33. #63447	C₂×C₃	semidirect
34. #63444	C₂×C₃	semidirect
35. #63437	C₂×C₃	semidirect
59. C₂³×D₉	C₂²×C₃	semidirect
68. C₂×Q₈×C₉	C₂²×C₃	semidirect
74. C₂×Dic₁₈	C₂²×C₃	semidirect
76. C₉×C₂²⋊C₄	C₂²×C₃	semidirect
78. C₄²×C₉	C₂²×C₃	semidirect
79. #3676	C₂²×C₃	semidirect
80. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
81. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
82. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
83. C₄×Dic₉	C₂²×C₃	semidirect
99. C₃×#124	D₉	semidirect
151. C₂×C₃×Q₈	C₂×D₉	semidirect
168. C₃×C₂²⋊C₄	C₂×D₉	semidirect
172. C₃×C₄²	C₂×D₉	semidirect
210. #124	C₃×D₉	semidirect
213. C₃×C₉	H(3,𝔽₄)	semidirect
261. C₂×Q₈	C₂×C₃×D₉	semidirect
270. C₂²⋊C₄	C₂×C₃×D₉	semidirect
272. C₄²	C₂×C₃×D₉	semidirect
303. C₉	C₃×H(3,𝔽₄)	semidirect
341. C₃	#10589785	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×#124	2
C₃×#63509	2
C₃×#63499	2
C₃×#63447	2
C₃×#63444	2
C₃×#63437	2
C₃×#5683	3
#10589785	3

Subgroups — C3×#10589785 #13153096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589785 #13153096