Subgroups — C₃×#10588391 #13141504

Order 1728.

400 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3664	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63380	C₂	1	1	yes	maximal, characteristic
6	C₃×#63378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4710	—	3	1	no	maximal
9	#10588391	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₉	—	2	1	no	—
12	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
17	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₃×Dic₉⋊C₄	—	2	1	no	—
20	C₃×C₄×Dic₉	—	2	1	no	—
21	C₃×#3664	C₄	1	4	yes	complemented
22	C₂×C₃×#204	—	3	1	no	—
23	C₃×C₄×D₁₂	—	3	1	no	—
24	C₃×C₄×Dic₆	—	3	1	no	—
25	C₃²×#97	D₃	1	1	yes	characteristic
26	C₃×#821	—	3	1	no	—
27	C₃×#818	—	3	1	no	—
28	C₃×#816	—	3	1	no	—
29	C₂×#3664	C₂×C₃	1	1	yes	characteristic
30	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic
31	C₄×Dic₁₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₉×#97	—	2	1	no	—
33	C₉×#97	C₂×C₃	1	1	yes	characteristic, complemented
34	#63380	C₂×C₃	1	1	yes	characteristic
35	#63378	C₂×C₃	1	1	yes	characteristic, complemented
36	#63376	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×C₃×D₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃×Dic₉	—	2	1	no	—
42	C₂×C₃×Dic₉	—	2	1	no	—
43	C₂×C₃×Dic₉	—	2	1	no	—
44	C₃×D₃₆	—	2	1	no	—
45	C₃×D₃₆	C₂×C₄	1	2	yes	—
46	C₃×C₄×D₉	—	4	2	no	—
47	C₃×Dic₁₈	—	2	1	no	—
48	C₃×Dic₁₈	C₂×C₄	1	2	yes	complemented
49	C₃×C₈×C₉	—	2	1	no	—
50	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	9	1	no	—
52	#4710	—	3	1	no	—
53	C₂×C₃×D₁₂	—	3	1	no	—
54	C₂×C₃×C₄×D₃	—	6	1	no	—
55	C₂×C₃×Dic₆	—	3	1	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
60	C₃×C₄⋊Dic₃	—	3	1	no	—
61	C₃×C₄⋊Dic₃	—	3	1	no	—
62	C₃×Dic₃⋊C₄	—	6	1	no	—
63	C₃×C₄×Dic₃	—	6	1	no	—
64	C₃×#204	—	3	4	no	—
65	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄×D₉	—	2	1	no	—
67	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₈×C₉	—	2	1	no	—
69	C₂×C₈×C₉	—	2	1	no	—
70	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
72	C₄²×C₉	—	2	1	no	—
73	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
74	(C₂×D₉)⋊C₄	—	2	1	no	—
75	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
76	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
77	Dic₉⋊C₄	—	2	1	no	—
78	C₄×Dic₉	—	2	1	no	—
79	#3664	C₃×C₄	1	4	yes	complemented
80	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×D₉	—	2	2	no	—
82	C₂×C₃×D₉	—	4	1	no	—
83	C₂×C₃×D₉	—	4	1	no	—
84	C₃×C₄×C₉	—	2	1	no	—
85	C₃×C₄×C₉	—	2	1	no	—
86	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₄×C₉	#41	1	1	yes	characteristic
88	C₃×Dic₉	—	2	2	no	—
89	C₃×Dic₉	—	4	1	no	—
90	C₃×Dic₉	—	4	1	no	—
91	C₂×C₃×SD₁₆	—	9	1	no	—
92	C₃×C₄×Q₈	—	9	1	no	—
93	C₃×C₄×D₄	—	9	1	no	—
94	C₂×#204	—	3	1	no	—
95	C₄×D₁₂	—	3	1	no	—
96	C₄×Dic₆	—	3	1	no	—
97	C₃×#107	—	9	1	no	—
98	C₃×#103	—	9	1	no	—
99	C₃×#102	—	9	1	no	—
100	C₃×#97	—	2	1	no	—
101	C₃×#97	D₉	1	1	yes	characteristic, complemented
102	C₃×#97	C₃×D₃	1	1	yes	characteristic
103	#821	—	3	1	no	—
104	#818	—	3	1	no	—
105	#816	—	3	1	no	—
106	C₂²×C₃×D₃	—	6	1	no	—
107	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
108	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	—	6	1	no	—
111	C₂×C₃×Dic₃	—	6	1	no	—
112	C₂×C₃×Dic₃	—	6	1	no	—
113	C₃×D₁₂	—	3	2	no	—
114	C₃×D₁₂	—	6	1	no	—
115	C₃×C₄×D₃	—	12	2	no	—
116	C₃×Dic₆	—	3	2	no	—
117	C₃×Dic₆	—	6	1	no	—
118	C₂²×D₉	—	2	1	no	—
119	C₃²×C₈	—	2	1	no	—
120	C₃²×C₈	C₄×D₃	1	2	yes	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₉	—	2	1	no	—
128	C₂×Dic₉	—	2	1	no	—
129	C₂×Dic₉	—	2	1	no	—
130	D₃₆	—	2	1	no	—
131	D₃₆	C₂×C₃×C₄	1	2	yes	—
132	C₄×D₉	—	4	2	no	—
133	Dic₁₈	—	2	1	no	—
134	Dic₁₈	C₂×C₃×C₄	1	2	yes	complemented
135	C₈×C₉	—	2	2	no	—
136	C₈×C₉	—	2	1	no	—
137	C₈×C₉	—	2	2	no	—
138	C₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
139	#440	—	9	1	no	Sylow
140	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₃×C₉	#137	1	1	yes	characteristic
143	C₃×D₉	—	4	2	no	—
144	C₂×C₃×Q₈	—	9	1	no	—
145	C₂×C₃×D₄	—	9	1	no	—
146	C₂²×C₃×C₄	—	18	1	no	—
147	C₂×D₁₂	—	3	1	no	—
148	C₂×C₄×D₃	—	6	1	no	—
149	C₂×Dic₆	—	3	1	no	—
150	C₃×SD₁₆	—	9	4	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
154	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	9	1	no	—
158	C₃×C₄⋊C₄	—	9	1	no	—
159	C₃×C₄⋊C₄	—	18	1	no	—
160	C₃×C₂²⋊C₄	—	18	1	no	—
161	C₃×C₄²	—	2	1	no	—
162	C₃×C₄²	—	18	1	no	—
163	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
164	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
165	(C₂×D₃)⋊C₄	—	6	1	no	—
166	C₄⋊Dic₃	—	3	1	no	—
167	C₄⋊Dic₃	—	3	1	no	—
168	Dic₃⋊C₄	—	6	1	no	—
169	C₄×Dic₃	—	6	1	no	—
170	#204	—	3	4	no	—
171	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₃×D₃	—	6	2	no	—
173	C₂×C₃×D₃	—	12	1	no	—
174	C₂×C₃×D₃	—	12	1	no	—
175	C₃²×C₄	—	2	1	no	—
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₃²×C₄	#235	1	1	yes	characteristic
179	C₃×Dic₃	—	6	2	no	—
180	C₃×Dic₃	—	12	1	no	—
181	C₃×Dic₃	—	12	1	no	—
182	C₂²×C₉	—	2	1	no	—
183	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×D₉	—	2	2	no	—
185	C₂×D₉	—	4	1	no	—
186	C₂×D₉	—	4	1	no	—
187	C₄×C₉	—	2	1	no	—
188	C₄×C₉	—	2	1	no	—
189	C₄×C₉	—	2	1	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	4	1	no	—
192	C₄×C₉	—	4	1	no	—
193	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
194	C₄×C₉	C₃×#41	1	1	yes	characteristic
195	Dic₉	—	2	2	no	—
196	Dic₉	—	4	1	no	—
197	Dic₉	—	4	1	no	—
198	C₂×SD₁₆	—	9	1	no	—
199	C₄×Q₈	—	9	1	no	—
200	C₄×D₄	—	9	1	no	—
201	#107	—	9	1	no	—
202	#103	—	9	1	no	—
203	#102	—	9	1	no	—
204	#97	C₃×D₉	1	1	yes	characteristic, complemented
205	C₃×C₉	#440	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	18	1	no	—
207	C₂²×D₃	—	6	1	no	—
208	C₃×Q₈	—	9	2	no	—
209	C₃×Q₈	—	18	1	no	—
210	C₃×D₄	—	9	2	no	—
211	C₃×D₄	—	18	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	—	18	1	no	—
216	C₂×C₃×C₄	—	18	1	no	—
217	C₂×C₃×C₄	—	18	1	no	—
218	C₂×C₃×C₄	—	36	2	no	—
219	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
220	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂×Dic₃	—	6	1	no	—
226	C₂×Dic₃	—	6	1	no	—
227	C₂×Dic₃	—	6	1	no	—
228	D₁₂	—	3	2	no	—
229	D₁₂	—	6	1	no	—
230	C₄×D₃	—	12	2	no	—
231	Dic₆	—	3	2	no	—
232	Dic₆	—	6	1	no	—
233	C₃×C₈	—	2	2	no	—
234	C₃×C₈	—	2	1	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	—	2	2	no	—
237	C₃×C₈	C₄×D₉	1	2	yes	complemented
238	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
239	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
240	C₂×C₃²	#908	1	1	yes	characteristic
241	C₂×C₃²	#909	1	1	yes	characteristic
242	C₃×D₃	—	12	2	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
247	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂×C₉	C₃×#137	1	1	yes	characteristic
249	D₉	—	4	2	no	—
250	C₂×Q₈	—	9	1	no	—
251	C₂×D₄	—	9	1	no	—
252	C₂²×C₄	—	18	1	no	—
253	SD₁₆	—	9	4	no	—
254	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
255	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
256	C₄⋊C₄	—	9	1	no	—
257	C₄⋊C₄	—	9	1	no	—
258	C₄⋊C₄	—	18	1	no	—
259	C₂²⋊C₄	—	18	1	no	—
260	C₄²	—	18	1	no	—
261	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	18	2	no	—
264	C₂²×C₃	—	36	1	no	—
265	C₂²×C₃	—	36	1	no	—
266	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
267	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
268	C₂×D₃	—	6	2	no	—
269	C₂×D₃	—	12	1	no	—
270	C₂×D₃	—	12	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	—	4	1	no	—
279	C₃×C₄	—	18	2	no	—
280	C₃×C₄	—	36	1	no	—
281	C₃×C₄	—	36	1	no	—
282	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
283	C₃×C₄	#3697	1	1	yes	characteristic
284	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
285	C₃×C₄	C₃×#235	1	1	yes	characteristic
286	Dic₃	—	6	2	no	—
287	Dic₃	—	12	1	no	—
288	Dic₃	—	12	1	no	—
289	C₃²	#4710	1	1	yes	characteristic
290	C₉	—	2	1	no	—
291	C₉	C₃×#440	1	1	yes	characteristic, complemented
292	C₂³	—	18	1	no	—
293	Q₈	—	9	2	no	—
294	Q₈	—	18	1	no	—
295	D₄	—	9	2	no	—
296	D₄	—	18	1	no	—
297	C₂×C₄	—	18	1	no	—
298	C₂×C₄	—	18	1	no	—
299	C₂×C₄	—	18	1	no	—
300	C₂×C₄	—	36	2	no	—
301	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
302	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	C₃×C₄×D₉	1	2	yes	complemented
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	36	2	no	—
310	C₂×C₃	C₄×D₃₆	1	1	yes	characteristic
311	C₂×C₃	#63470	1	1	yes	characteristic
312	C₂×C₃	#63471	1	1	yes	characteristic
313	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
314	C₂×C₃	C₃×#908	1	1	yes	characteristic
315	C₂×C₃	C₃×#909	1	1	yes	characteristic
316	D₃	—	12	2	no	—
317	C₂²	—	18	2	no	—
318	C₂²	—	36	1	no	—
319	C₂²	—	36	1	no	—
320	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	18	2	no	—
324	C₄	—	36	1	no	—
325	C₄	—	36	1	no	—
326	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
327	C₄	C₃×#3697	1	1	yes	characteristic
328	C₃	—	2	1	no	—
329	C₃	#10588391	1	1	yes	characteristic, direct product factor
330	C₃	C₃×#4710	1	1	yes	characteristic
331	C₂	—	36	2	no	—
332	C₂	C₃×C₄×D₃₆	1	1	yes	characteristic
333	C₂	C₃×#63470	1	1	yes	characteristic
334	C₂	C₃×#63471	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃₆
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂²×C₃²	C₂×D₁₂
prime radical	C₂×C₃×D₃₆	C₂²
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×#97	D₉
hypocenter	C₉	C₃×#440
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×Dic₁₈	C₂	semidirect
4. C₃×C₉×#97	C₂	semidirect
6. C₃×#63378	C₂	semidirect
7. C₃×#63376	C₂	semidirect
9. #10588391	C₃	direct (N × H)
21. C₃×#3664	C₄	semidirect
31. C₄×Dic₁₈	C₂×C₃	semidirect
33. C₉×#97	C₂×C₃	semidirect
35. #63378	C₂×C₃	semidirect
36. #63376	C₂×C₃	semidirect
48. C₃×Dic₁₈	C₂×C₄	semidirect
50. C₃×C₈×C₉	C₂×C₄	semidirect
79. #3664	C₃×C₄	semidirect
101. C₃×#97	D₉	semidirect
134. Dic₁₈	C₂×C₃×C₄	semidirect
138. C₈×C₉	C₂×C₃×C₄	semidirect
204. #97	C₃×D₉	semidirect
205. C₃×C₉	#440	semidirect
237. C₃×C₈	C₄×D₉	semidirect
291. C₉	C₃×#440	semidirect
305. C₈	C₃×C₄×D₉	semidirect
329. C₃	#10588391	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×#3664	2
C₃×C₄×D₃₆	2
C₃×C₄×Dic₁₈	2
C₃×C₉×#97	2
C₃×#63380	2
C₃×#63378	2
C₃×#63376	2
C₃×#4710	3
#10588391	3

Subgroups — C3×#10588391 #13141504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588391 #13141504