Subgroups — C₃×#10589936 #13153700

Order 1728.

1100 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3701	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63502	C₂	1	3	yes	maximal, complemented
5	C₃×#63500	C₂	1	3	yes	maximal, complemented
6	C₃×#63492	C₂	1	3	yes	maximal, complemented
7	C₃×C₄×#611	C₂	1	3	yes	maximal, complemented
8	C₃×#5834	—	3	1	no	maximal
9	#10589936	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃×C₉×#41	—	2	4	no	—
11	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₄×C₉	C₂²	1	3	yes	—
13	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
14	C₂×C₃×#611	C₂²	1	3	yes	—
15	C₂×C₃×#611	C₂²	1	6	yes	—
16	C₃×#3701	—	2	4	no	—
17	C₂×C₃×D₃₆	C₂²	1	3	yes	—
18	C₂×C₃×C₄×D₉	C₂²	1	3	yes	—
19	C₃×#3676	C₂²	1	3	yes	—
20	C₃×(C₂×D₉)⋊C₄	C₂²	1	3	yes	—
21	C₃×C₄⋊Dic₉	C₂²	1	3	yes	—
22	C₃×Dic₉⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₄×Dic₉	C₂²	1	3	yes	—
24	C₂×C₃²×#41	D₃	1	1	yes	characteristic
25	C₂×C₃×#239	—	3	1	no	—
26	C₃×Q₈×Dic₃	—	3	1	no	—
27	C₃×#940	—	3	3	no	—
28	C₃×#938	—	3	3	no	—
29	C₃×#930	—	3	3	no	—
30	C₃×C₄×#67	—	3	3	no	—
31	C₂×C₉×#41	—	2	1	no	—
32	C₂×C₉×#41	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#3701	C₂×C₃	1	1	yes	characteristic, complemented
34	Q₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
35	#63502	C₂×C₃	1	3	yes	complemented
36	#63500	C₂×C₃	1	3	yes	complemented
37	#63492	C₂×C₃	1	3	yes	complemented
38	C₄×#611	C₂×C₃	1	3	yes	complemented
39	C₂³×C₃×C₉	C₂³	1	3	yes	—
40	C₂²×C₃×D₉	C₂³	1	3	yes	—
41	C₃×Q₈×C₉	D₄	1	4	yes	complemented
42	C₃×D₄×C₉	—	2	6	no	—
43	C₂×C₃×C₄×C₉	—	2	6	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
46	C₃×#611	—	2	6	no	—
47	C₃×#611	—	4	6	no	—
48	C₂×C₃×Dic₉	C₂³	1	1	yes	characteristic
49	C₂×C₃×Dic₉	C₂³	1	3	yes	—
50	C₂×C₃×Dic₉	C₂³	1	1	yes	characteristic
51	C₃×D₃₆	—	2	6	no	—
52	C₃×C₄×D₉	—	2	6	no	—
53	C₃×#550	—	9	1	no	—
54	#5834	—	3	1	no	—
55	C₃²×#41	—	2	4	no	—
56	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×D₄	C₂×D₃	1	3	yes	—
58	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
59	C₂×C₃×#67	—	3	3	no	—
60	C₂×C₃×#67	—	3	6	no	—
61	C₃×#239	—	6	4	no	—
62	C₂×C₃×D₁₂	—	3	3	no	—
63	C₂×C₃×C₄×D₃	—	3	3	no	—
64	C₃×#217	—	3	3	no	—
65	C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
66	C₃×C₄⋊Dic₃	—	3	3	no	—
67	C₃×Dic₃⋊C₄	—	3	1	no	—
68	C₃×C₄×Dic₃	—	3	3	no	—
69	C₉×#41	—	2	8	no	—
70	C₉×#41	—	2	4	no	—
71	C₂×Q₈×C₉	—	2	1	no	—
72	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
73	C₂×D₄×C₉	—	2	3	no	—
74	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
75	C₂²×C₄×C₉	—	2	3	no	—
76	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
77	C₂×#611	C₂²×C₃	1	3	yes	—
78	C₂×#611	C₂²×C₃	1	6	yes	—
79	#3701	—	2	4	no	—
80	C₂×D₃₆	C₂²×C₃	1	3	yes	—
81	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
82	#3676	C₂²×C₃	1	3	yes	—
83	(C₂×D₉)⋊C₄	C₂²×C₃	1	3	yes	—
84	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
85	Dic₉⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₄×Dic₉	C₂²×C₃	1	3	yes	—
87	C₂²×C₃×C₉	—	2	3	no	—
88	C₂²×C₃×C₉	—	2	3	no	—
89	C₂²×C₃×C₉	—	2	3	no	—
90	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₃×D₉	—	2	3	no	—
92	C₂×C₃×D₉	—	2	3	no	—
93	C₂×C₃×D₉	—	2	3	no	—
94	C₃×C₄×C₉	—	2	1	no	—
95	C₃×C₄×C₉	C₂×D₄	1	6	yes	—
96	C₃×Dic₉	—	2	1	no	—
97	C₃×Dic₉	—	2	3	no	—
98	C₃×Dic₉	#41	1	2	yes	complemented
99	C₂×C₃×#41	—	2	1	no	—
100	C₂×C₃×#41	—	9	1	no	—
101	C₂×C₃×#41	D₉	1	1	yes	characteristic, complemented
102	C₂×C₃×#41	C₃×D₃	1	1	yes	characteristic
103	C₂×#239	—	3	1	no	—
104	C₃×#127	—	9	3	no	—
105	C₃×#121	—	9	3	no	—
106	C₃×#121	—	9	3	no	—
107	C₃×C₄×Q₈	—	9	1	no	—
108	C₃×C₄×D₄	—	9	3	no	—
109	Q₈×Dic₃	—	3	1	no	—
110	#940	—	3	3	no	—
111	#938	—	3	3	no	—
112	#930	—	3	3	no	—
113	C₄×#67	—	3	3	no	—
114	C₂³×C₃²	C₂²×D₃	1	3	yes	—
115	C₂²×C₃×D₃	—	3	3	no	—
116	C₃²×Q₈	#67	1	4	yes	—
117	C₃²×D₄	—	2	6	no	—
118	C₂×C₃²×C₄	—	2	6	no	—
119	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
121	C₃×#67	—	6	6	no	—
122	C₃×#67	—	12	6	no	—
123	C₂×C₃×Dic₃	—	3	1	no	—
124	C₂×C₃×Dic₃	—	3	3	no	—
125	C₂×C₃×Dic₃	—	3	1	no	—
126	C₃×D₁₂	—	6	6	no	—
127	C₃×C₄×D₃	—	6	6	no	—
128	C₂³×C₉	—	2	3	no	—
129	C₂³×C₉	C₂³×C₃	1	3	yes	—
130	C₂²×D₉	C₂³×C₃	1	3	yes	—
131	Q₈×C₉	—	2	4	no	—
132	Q₈×C₉	C₃×D₄	1	4	yes	complemented
133	D₄×C₉	—	2	12	no	—
134	D₄×C₉	—	2	6	no	—
135	C₂×C₄×C₉	—	2	12	no	—
136	C₂×C₄×C₉	—	2	6	no	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	—	2	3	no	—
139	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
141	#611	—	2	6	no	—
142	#611	—	4	6	no	—
143	C₂×Dic₉	C₂³×C₃	1	3	yes	—
144	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
145	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
146	D₃₆	—	2	6	no	—
147	C₄×D₉	—	2	6	no	—
148	#550	—	9	1	no	Sylow
149	C₂×C₃×C₉	—	4	3	no	—
150	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₃×D₉	—	4	3	no	—
154	C₃×#41	—	2	8	no	—
155	C₃×#41	—	2	4	no	—
156	C₃×#41	—	2	4	no	—
157	C₃×#41	—	18	4	no	—
158	C₂×C₃×Q₈	—	2	1	no	—
159	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
160	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
161	C₂×C₃×D₄	—	2	3	no	—
162	C₂×C₃×D₄	—	9	3	no	—
163	C₂×C₃×D₄	—	9	6	no	—
164	C₂×C₃×D₄	—	9	3	no	—
165	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
166	C₂×C₃×D₄	C₂×C₃×D₃	1	3	yes	—
167	C₂²×C₃×C₄	—	2	3	no	—
168	C₂²×C₃×C₄	—	9	3	no	—
169	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
170	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
171	C₂×#67	—	3	3	no	—
172	C₂×#67	—	3	6	no	—
173	#239	—	6	4	no	—
174	C₂×D₁₂	—	3	3	no	—
175	C₂×C₄×D₃	—	3	3	no	—
176	C₃×C₄⋊C₄	—	9	1	no	—
177	C₃×C₄⋊C₄	—	9	3	no	—
178	C₃×C₂²⋊C₄	—	9	3	no	—
179	C₃×C₂²⋊C₄	—	9	3	no	—
180	C₃×C₄²	—	9	3	no	—
181	#217	—	3	3	no	—
182	(C₂×D₃)⋊C₄	—	3	3	no	—
183	C₄⋊Dic₃	—	3	3	no	—
184	Dic₃⋊C₄	—	3	1	no	—
185	C₄×Dic₃	—	3	3	no	—
186	C₂²×C₃²	—	2	3	no	—
187	C₂²×C₃²	—	2	3	no	—
188	C₂²×C₃²	—	2	3	no	—
189	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
190	C₂×C₃×D₃	—	6	3	no	—
191	C₂×C₃×D₃	—	6	3	no	—
192	C₂×C₃×D₃	—	6	3	no	—
193	C₃²×C₄	—	2	1	no	—
194	C₃²×C₄	C₂×#67	1	6	yes	—
195	C₃×Dic₃	—	3	2	no	—
196	C₃×Dic₃	—	6	3	no	—
197	C₃×Dic₃	—	6	1	no	—
198	C₂²×C₉	—	2	6	no	—
199	C₂²×C₉	—	2	3	no	—
200	C₂²×C₉	—	2	3	no	—
201	C₂²×C₉	—	2	3	no	—
202	C₂²×C₉	—	2	1	no	—
203	C₂²×C₉	—	4	3	no	—
204	C₂²×C₉	—	4	3	no	—
205	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
206	C₂×D₉	—	2	3	no	—
207	C₂×D₉	—	2	3	no	—
208	C₂×D₉	—	2	3	no	—
209	C₄×C₉	—	2	2	no	—
210	C₄×C₉	—	2	1	no	—
211	C₄×C₉	—	2	6	no	—
212	C₄×C₉	C₂×C₃×D₄	1	6	yes	—
213	Dic₉	—	2	1	no	—
214	Dic₉	—	2	3	no	—
215	Dic₉	C₃×#41	1	2	yes	complemented
216	C₂×#41	—	9	1	no	—
217	C₂×#41	C₃×D₉	1	1	yes	characteristic, complemented
218	#127	—	9	3	no	—
219	#121	—	9	3	no	—
220	#121	—	9	3	no	—
221	C₄×Q₈	—	9	1	no	—
222	C₄×D₄	—	9	3	no	—
223	C₃×C₉	#550	1	1	yes	characteristic, complemented, Sylow
224	C₂³×C₃	—	2	3	no	—
225	C₂³×C₃	—	9	3	no	—
226	C₂³×C₃	C₂²×D₉	1	3	yes	—
227	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
228	C₂²×D₃	—	3	3	no	—
229	C₃×Q₈	—	2	4	no	—
230	C₃×Q₈	#611	1	4	yes	complemented
231	C₃×Q₈	C₃×#67	1	4	yes	—
232	C₃×D₄	—	2	12	no	—
233	C₃×D₄	—	2	6	no	—
234	C₃×D₄	—	2	6	no	—
235	C₃×D₄	—	18	6	no	—
236	C₃×D₄	—	18	6	no	—
237	C₃×D₄	—	36	6	no	—
238	C₂×C₃×C₄	—	2	12	no	—
239	C₂×C₃×C₄	—	2	6	no	—
240	C₂×C₃×C₄	—	2	6	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	3	no	—
243	C₂×C₃×C₄	—	9	1	no	—
244	C₂×C₃×C₄	—	9	3	no	—
245	C₂×C₃×C₄	—	9	1	no	—
246	C₂×C₃×C₄	—	18	6	no	—
247	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
248	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
249	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
251	#67	—	6	6	no	—
252	#67	—	12	6	no	—
253	C₂×Dic₃	—	3	1	no	—
254	C₂×Dic₃	—	3	3	no	—
255	C₂×Dic₃	—	3	1	no	—
256	D₁₂	—	6	6	no	—
257	C₄×D₃	—	6	6	no	—
258	C₂×C₃²	—	4	3	no	—
259	C₂×C₃²	D₃×#41	1	1	yes	characteristic
260	C₂×C₃²	#1009	1	1	yes	characteristic
261	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
262	C₃×D₃	—	12	3	no	—
263	C₂×C₉	—	2	1	no	—
264	C₂×C₉	—	2	1	no	—
265	C₂×C₉	—	2	1	no	—
266	C₂×C₉	—	4	6	no	—
267	C₂×C₉	—	4	3	no	—
268	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
269	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
270	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
271	D₉	—	4	3	no	—
272	#41	—	2	4	no	—
273	#41	—	18	4	no	—
274	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
275	C₂×D₄	—	9	3	no	—
276	C₂×D₄	—	9	6	no	—
277	C₂×D₄	—	9	3	no	—
278	C₂×D₄	C₂×C₃×D₉	1	3	yes	—
279	C₂²×C₄	—	9	3	no	—
280	C₂²×C₄	C₂×C₃×D₉	1	3	yes	—
281	C₄⋊C₄	—	9	1	no	—
282	C₄⋊C₄	—	9	3	no	—
283	C₂²⋊C₄	—	9	3	no	—
284	C₂²⋊C₄	—	9	3	no	—
285	C₄²	—	9	3	no	—
286	C₂²×C₃	—	2	6	no	—
287	C₂²×C₃	—	2	3	no	—
288	C₂²×C₃	—	2	3	no	—
289	C₂²×C₃	—	2	3	no	—
290	C₂²×C₃	—	2	3	no	—
291	C₂²×C₃	—	2	3	no	—
292	C₂²×C₃	—	2	3	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	4	3	no	—
295	C₂²×C₃	—	4	3	no	—
296	C₂²×C₃	—	18	3	no	—
297	C₂²×C₃	—	18	3	no	—
298	C₂²×C₃	—	18	3	no	—
299	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
300	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
301	C₂×D₃	—	6	3	no	—
302	C₂×D₃	—	6	3	no	—
303	C₂×D₃	—	6	3	no	—
304	C₃×C₄	—	2	2	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	1	no	—
307	C₃×C₄	—	2	6	no	—
308	C₃×C₄	—	9	2	no	—
309	C₃×C₄	—	18	3	no	—
310	C₃×C₄	—	18	1	no	—
311	C₃×C₄	C₂×#611	1	6	yes	—
312	C₃×C₄	C₂×C₃×#67	1	6	yes	—
313	Dic₃	—	3	2	no	—
314	Dic₃	—	6	3	no	—
315	Dic₃	—	6	1	no	—
316	C₃²	#5834	1	1	yes	characteristic
317	C₉	—	2	1	no	—
318	C₉	C₃×#550	1	1	yes	characteristic, complemented
319	C₂³	—	9	3	no	—
320	C₂³	C₂²×C₃×D₉	1	3	yes	—
321	Q₈	C₃×#611	1	4	yes	complemented
322	D₄	—	2	6	no	—
323	D₄	—	18	6	no	—
324	D₄	—	18	6	no	—
325	D₄	—	36	6	no	—
326	C₂×C₄	—	2	6	no	—
327	C₂×C₄	—	9	1	no	—
328	C₂×C₄	—	9	3	no	—
329	C₂×C₄	—	9	1	no	—
330	C₂×C₄	—	18	6	no	—
331	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	4	6	no	—
337	C₂×C₃	—	4	3	no	—
338	C₂×C₃	—	4	3	no	—
339	C₂×C₃	—	36	3	no	—
340	C₂×C₃	#41×D₉	1	1	yes	characteristic
341	C₂×C₃	#63715	1	1	yes	characteristic
342	C₂×C₃	C₂²×#611	1	1	yes	characteristic
343	C₂×C₃	C₃×D₃×#41	1	1	yes	characteristic
344	C₂×C₃	C₃×#1009	1	1	yes	characteristic
345	C₂×C₃	C₂²×C₃×#67	1	1	yes	characteristic
346	D₃	—	12	3	no	—
347	C₂²	—	2	3	no	—
348	C₂²	—	2	3	no	—
349	C₂²	—	2	3	no	—
350	C₂²	—	18	3	no	—
351	C₂²	—	18	3	no	—
352	C₂²	—	18	3	no	—
353	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	9	2	no	—
356	C₄	—	18	3	no	—
357	C₄	—	18	1	no	—
358	C₄	C₂×C₃×#611	1	6	yes	—
359	C₃	—	2	1	no	—
360	C₃	#10589936	1	1	yes	characteristic, direct product factor
361	C₃	C₃×#5834	1	1	yes	characteristic
362	C₂	—	4	3	no	—
363	C₂	—	36	3	no	—
364	C₂	C₃×#41×D₉	1	1	yes	characteristic
365	C₂	C₃×#63715	1	1	yes	characteristic
366	C₂	C₂²×C₃×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₂×C₃×#41	D₉
hypocenter	C₉	C₃×#550
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×#41	C₂	semidirect
2. C₂×C₃×#3701	C₂	semidirect
3. C₃×Q₈×Dic₉	C₂	semidirect
4. C₃×#63502	C₂	semidirect
5. C₃×#63500	C₂	semidirect
6. C₃×#63492	C₂	semidirect
7. C₃×C₄×#611	C₂	semidirect
9. #10589936	C₃	direct (N × H)
32. C₂×C₉×#41	C₂×C₃	semidirect
33. C₂×#3701	C₂×C₃	semidirect
34. Q₈×Dic₉	C₂×C₃	semidirect
35. #63502	C₂×C₃	semidirect
36. #63500	C₂×C₃	semidirect
37. #63492	C₂×C₃	semidirect
38. C₄×#611	C₂×C₃	semidirect
41. C₃×Q₈×C₉	D₄	semidirect
98. C₃×Dic₉	#41	semidirect
101. C₂×C₃×#41	D₉	semidirect
132. Q₈×C₉	C₃×D₄	semidirect
215. Dic₉	C₃×#41	semidirect
217. C₂×#41	C₃×D₉	semidirect
223. C₃×C₉	#550	semidirect
230. C₃×Q₈	#611	semidirect
318. C₉	C₃×#550	semidirect
321. Q₈	C₃×#611	semidirect
360. C₃	#10589936	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×#41	2
C₂×C₃×#3701	2
C₃×Q₈×Dic₉	2
C₃×#63502	2
C₃×#63500	2
C₃×#63492	2
C₃×C₄×#611	2
C₃×#5834	3
#10589936	3

Subgroups — C3×#10589936 #13153700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589936 #13153700