Subgroups — C₃×#10592844 #13163183

Order 1728.

872 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	C₃³×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64159	C₂	1	1	yes	maximal, characteristic
3	C₃×#64110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63664	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63652	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63649	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63642	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592844	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#5145	—	3	4	no	maximal
10	C₃³×#41	C₂²	1	1	yes	characteristic
11	C₃×#3828	C₂²	1	1	yes	characteristic
12	C₃³×M₁₆	—	2	1	no	—
13	C₃³×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃³×C₈	—	2	1	no	—
15	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
16	C₃×#3756	C₄	1	1	yes	characteristic, complemented
17	C₃×#3755	C₄	1	1	yes	characteristic, complemented
18	C₃×#3754	C₄	1	1	yes	characteristic, complemented
19	C₃×#3753	C₄	1	1	yes	characteristic, complemented
20	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₃×#3748	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×#3743	—	2	1	no	—
24	C₃×C₈×C₃⋊D₃	—	2	1	no	—
25	C₃²×#131	—	2	4	no	—
26	C₃²×#131	D₃	1	4	yes	complemented
27	C₃²×#131	C₂×C₃	1	1	yes	characteristic, complemented
28	#64159	C₂×C₃	1	1	yes	characteristic
29	#64110	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×#950	—	3	4	no	—
31	C₃×#901	—	3	4	no	—
32	#63664	C₂×C₃	1	1	yes	characteristic, complemented
33	#63652	C₂×C₃	1	1	yes	characteristic, complemented
34	#63649	C₂×C₃	1	1	yes	characteristic, complemented
35	#63642	C₂×C₃	1	1	yes	characteristic, complemented
36	C₃×#837	—	3	4	no	—
37	C₃×#825	—	3	4	no	—
38	C₃×#822	—	3	4	no	—
39	C₃×#815	—	3	4	no	—
40	C₃³×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₃³×D₄	—	2	1	no	—
42	C₃³×D₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃³×C₄	—	2	1	no	—
44	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
45	C₃×#638	—	2	1	no	—
46	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
47	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₃⋊D₃	—	2	1	no	—
49	C₃×#634	C₂×C₄	1	1	yes	characteristic
50	C₃³×C₈	—	2	1	no	—
51	C₃³×C₈	D₄	1	1	yes	characteristic
52	C₃³×C₈	D₄	1	1	yes	characteristic
53	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
54	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
55	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
56	C₃×#444	—	9	1	no	—
57	#5145	—	3	4	no	—
58	C₃²×#41	—	2	4	no	—
59	C₃²×#41	C₂×D₃	1	4	yes	—
60	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
61	#3828	C₂²×C₃	1	1	yes	characteristic
62	C₃×#235	—	3	4	no	—
63	C₃²×M₁₆	—	2	8	no	—
64	C₃²×M₁₆	—	2	4	no	—
65	C₃²×M₁₆	—	2	4	no	—
66	C₃²×M₁₆	—	2	1	no	—
67	C₃²×M₁₆	C₂×D₃	1	4	yes	—
68	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃²×C₈	—	2	8	no	—
70	C₂×C₃²×C₈	—	2	4	no	—
71	C₂×C₃²×C₈	—	2	4	no	—
72	C₂×C₃²×C₈	—	2	1	no	—
73	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
74	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
75	#3756	C₃×C₄	1	1	yes	characteristic, complemented
76	#3755	C₃×C₄	1	1	yes	characteristic, complemented
77	#3754	C₃×C₄	1	1	yes	characteristic, complemented
78	#3753	C₃×C₄	1	1	yes	characteristic, complemented
79	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
80	#3748	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
82	#3743	—	2	1	no	—
83	C₈×C₃⋊D₃	—	2	1	no	—
84	C₃×#216	—	3	4	no	—
85	C₃×#215	—	3	4	no	—
86	C₃×#214	—	3	4	no	—
87	C₃×#213	—	3	4	no	—
88	C₃×C₄×Dic₃	—	3	4	no	—
89	C₃×#208	—	3	4	no	—
90	C₂×C₃×C₃⋊C₈	—	3	4	no	—
91	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
92	C₃×D₃×C₈	—	6	4	no	—
93	C₂²×C₃³	—	2	1	no	—
94	C₂²×C₃³	#41	1	1	yes	characteristic
95	C₂×C₃×C₃⋊D₃	—	2	1	no	—
96	C₃³×C₄	—	2	1	no	—
97	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
99	C₃×C₃⋊Dic₃	—	2	1	no	—
100	C₃×C₃⋊Dic₃	—	4	1	no	—
101	C₃×#135	—	9	1	no	—
102	C₃×#131	—	2	4	no	—
103	C₃×#131	—	9	1	no	—
104	C₃×#131	C₃×D₃	1	4	yes	complemented
105	C₃×#131	C₃⋊D₃	1	1	yes	characteristic, complemented
106	#950	—	3	4	no	—
107	#901	—	3	4	no	—
108	C₃×#108	—	9	1	no	—
109	C₃×#104	—	9	1	no	—
110	C₃×#104	—	9	1	no	—
111	C₃×#97	—	9	1	no	—
112	#837	—	3	4	no	—
113	#825	—	3	4	no	—
114	#822	—	3	4	no	—
115	#815	—	3	4	no	—
116	C₃²×Q₈	—	2	4	no	—
117	C₃²×Q₈	C₄×D₃	1	4	yes	—
118	C₃²×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
119	C₃²×D₄	—	2	8	no	—
120	C₃²×D₄	—	2	4	no	—
121	C₃²×D₄	—	2	1	no	—
122	C₃²×D₄	—	2	4	no	—
123	C₃²×D₄	C₄×D₃	1	4	yes	—
124	C₃²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃²×C₄	—	2	8	no	—
126	C₂×C₃²×C₄	—	2	4	no	—
127	C₂×C₃²×C₄	—	2	1	no	—
128	C₂×C₃²×C₄	—	2	4	no	—
129	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
130	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
131	#638	—	2	1	no	—
132	C₂×C₃⋊Dic₃	—	2	1	no	—
133	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	1	yes	characteristic
134	C₄×C₃⋊D₃	—	2	1	no	—
135	#634	C₂×C₃×C₄	1	1	yes	characteristic
136	C₃×#67	—	6	4	no	—
137	C₂×C₃×Dic₃	—	6	4	no	—
138	C₃×D₁₂	—	3	4	no	—
139	C₃×C₄×D₃	—	6	4	no	—
140	C₃×Dic₆	—	3	4	no	—
141	C₃²×C₈	—	2	8	no	—
142	C₃²×C₈	—	2	4	no	—
143	C₃²×C₈	—	2	1	no	—
144	C₃²×C₈	—	2	4	no	—
145	C₃²×C₈	—	2	4	no	—
146	C₃²×C₈	#67	1	4	yes	—
147	C₃²×C₈	#67	1	4	yes	—
148	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
149	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
150	C₃⋊(C₃⋊C₈)	—	2	1	no	—
151	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	1	yes	characteristic
152	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	1	yes	characteristic
153	C₃×C₃⋊C₈	—	3	4	no	—
154	C₃×C₃⋊C₈	—	3	4	no	—
155	C₃×C₃⋊C₈	—	6	4	no	—
156	#444	—	9	1	no	Sylow
157	C₂×C₃³	—	2	1	no	—
158	C₂×C₃³	—	4	1	no	—
159	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
160	C₃×C₃⋊D₃	—	4	1	no	—
161	C₃×#41	—	2	4	no	—
162	C₃×#41	—	9	1	no	—
163	C₃×#41	C₂×C₃×D₃	1	4	yes	—
164	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
165	#235	—	3	4	no	—
166	C₃×Q₁₆	—	9	1	no	—
167	C₃×SD₁₆	—	9	1	no	—
168	C₃×SD₁₆	—	9	1	no	—
169	C₃×D₈	—	9	1	no	—
170	C₃×M₁₆	—	2	8	no	—
171	C₃×M₁₆	—	2	4	no	—
172	C₃×M₁₆	—	2	1	no	—
173	C₃×M₁₆	—	2	4	no	—
174	C₃×M₁₆	—	9	1	no	—
175	C₃×M₁₆	—	18	1	no	—
176	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
177	C₃×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
178	C₂×C₃×C₈	—	2	8	no	—
179	C₂×C₃×C₈	—	2	4	no	—
180	C₂×C₃×C₈	—	2	1	no	—
181	C₂×C₃×C₈	—	2	4	no	—
182	C₂×C₃×C₈	—	9	1	no	—
183	C₂×C₃×C₈	—	18	1	no	—
184	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
185	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
186	C₃×C₄²	—	9	1	no	—
187	#216	—	3	4	no	—
188	#215	—	3	4	no	—
189	#214	—	3	4	no	—
190	#213	—	3	4	no	—
191	C₄×Dic₃	—	3	4	no	—
192	#208	—	3	4	no	—
193	C₂×C₃⋊C₈	—	3	4	no	—
194	ΓL(1,𝔽₂₅)	—	6	4	no	—
195	D₃×C₈	—	6	4	no	—
196	C₂²×C₃²	—	2	8	no	—
197	C₂²×C₃²	—	2	4	no	—
198	C₂²×C₃²	—	2	1	no	—
199	C₂²×C₃²	—	2	4	no	—
200	C₂²×C₃²	#235	1	4	yes	—
201	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
202	C₂×C₃⋊D₃	—	2	1	no	—
203	C₂×C₃×D₃	—	6	4	no	—
204	C₃²×C₄	—	2	8	no	—
205	C₃²×C₄	—	2	4	no	—
206	C₃²×C₄	—	2	1	no	—
207	C₃²×C₄	—	2	4	no	—
208	C₃²×C₄	—	2	4	no	—
209	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
210	C₃²×C₄	C₂×#67	1	4	yes	—
211	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₃⋊Dic₃	—	2	1	no	—
214	C₃⋊Dic₃	—	4	1	no	—
215	C₃×Dic₃	—	6	4	no	—
216	C₃×Dic₃	—	12	4	no	—
217	#135	—	9	1	no	—
218	#131	—	9	1	no	—
219	#131	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
220	#108	—	9	1	no	—
221	#104	—	9	1	no	—
222	#104	—	9	1	no	—
223	#97	—	9	1	no	—
224	C₃³	#444	1	1	yes	characteristic, complemented, Sylow
225	C₃×Q₈	—	2	4	no	—
226	C₃×Q₈	—	9	1	no	—
227	C₃×Q₈	C₃×C₄×D₃	1	4	yes	—
228	C₃×Q₈	C₄×C₃⋊D₃	1	1	yes	characteristic
229	C₃×D₄	—	2	8	no	—
230	C₃×D₄	—	2	4	no	—
231	C₃×D₄	—	2	1	no	—
232	C₃×D₄	—	2	4	no	—
233	C₃×D₄	—	9	1	no	—
234	C₃×D₄	—	18	1	no	—
235	C₃×D₄	C₃×C₄×D₃	1	4	yes	—
236	C₃×D₄	C₄×C₃⋊D₃	1	1	yes	characteristic
237	C₂×C₃×C₄	—	2	8	no	—
238	C₂×C₃×C₄	—	2	4	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	4	no	—
241	C₂×C₃×C₄	—	18	1	no	—
242	C₂×C₃×C₄	—	18	1	no	—
243	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
244	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
245	#67	—	6	4	no	—
246	C₂×Dic₃	—	6	4	no	—
247	D₁₂	—	3	4	no	—
248	C₄×D₃	—	6	4	no	—
249	Dic₆	—	3	4	no	—
250	C₃×C₈	—	2	8	no	—
251	C₃×C₈	—	2	4	no	—
252	C₃×C₈	—	2	1	no	—
253	C₃×C₈	—	2	4	no	—
254	C₃×C₈	—	2	4	no	—
255	C₃×C₈	—	9	1	no	—
256	C₃×C₈	—	9	1	no	—
257	C₃×C₈	—	18	1	no	—
258	C₃×C₈	C₃×#67	1	4	yes	—
259	C₃×C₈	C₃×#67	1	4	yes	—
260	C₃×C₈	#638	1	1	yes	characteristic
261	C₃×C₈	#638	1	1	yes	characteristic
262	C₃⋊C₈	—	3	4	no	—
263	C₃⋊C₈	—	3	4	no	—
264	C₃⋊C₈	—	6	4	no	—
265	C₂×C₃²	—	2	1	no	—
266	C₂×C₃²	—	2	4	no	—
267	C₂×C₃²	—	2	4	no	—
268	C₂×C₃²	—	4	4	no	—
269	C₂×C₃²	—	4	8	no	—
270	C₂×C₃²	—	4	4	no	—
271	C₂×C₃²	—	4	1	no	—
272	C₂×C₃²	C₄×#67	1	4	yes	—
273	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
274	C₃⋊D₃	—	4	1	no	—
275	C₃×D₃	—	12	4	no	—
276	#41	—	9	1	no	—
277	#41	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
278	Q₁₆	—	9	1	no	—
279	SD₁₆	—	9	1	no	—
280	SD₁₆	—	9	1	no	—
281	D₈	—	9	1	no	—
282	M₁₆	—	2	1	no	—
283	M₁₆	—	9	1	no	—
284	M₁₆	—	18	1	no	—
285	M₁₆	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	9	1	no	—
288	C₂×C₈	—	18	1	no	—
289	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
290	C₄²	—	9	1	no	—
291	C₂²×C₃	—	2	8	no	—
292	C₂²×C₃	—	2	4	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	2	4	no	—
295	C₂²×C₃	—	18	1	no	—
296	C₂²×C₃	C₃×#235	1	4	yes	—
297	C₂²×C₃	#3828	1	1	yes	characteristic
298	C₂×D₃	—	6	4	no	—
299	C₃×C₄	—	2	8	no	—
300	C₃×C₄	—	2	4	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	4	no	—
303	C₃×C₄	—	2	4	no	—
304	C₃×C₄	—	18	1	no	—
305	C₃×C₄	—	36	1	no	—
306	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
307	C₃×C₄	C₂×C₃×#67	1	4	yes	—
308	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
309	C₃×C₄	C₂×#638	1	1	yes	characteristic
310	Dic₃	—	6	4	no	—
311	Dic₃	—	12	4	no	—
312	C₃²	—	2	4	no	—
313	C₃²	#5145	1	4	yes	complemented
314	C₃²	C₃×#444	1	1	yes	characteristic, complemented
315	Q₈	—	9	1	no	—
316	Q₈	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
317	D₄	—	2	1	no	—
318	D₄	—	9	1	no	—
319	D₄	—	18	1	no	—
320	D₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	18	1	no	—
323	C₂×C₄	—	18	1	no	—
324	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	9	1	no	—
327	C₈	—	9	1	no	—
328	C₈	—	18	1	no	—
329	C₈	C₃×#638	1	1	yes	characteristic
330	C₈	C₃×#638	1	1	yes	characteristic
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	4	no	—
333	C₂×C₃	—	2	4	no	—
334	C₂×C₃	—	4	4	no	—
335	C₂×C₃	—	4	8	no	—
336	C₂×C₃	—	4	1	no	—
337	C₂×C₃	—	4	4	no	—
338	C₂×C₃	—	36	1	no	—
339	C₂×C₃	C₃×C₄×#67	1	4	yes	—
340	C₂×C₃	C₄×#638	1	1	yes	characteristic
341	D₃	—	12	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	18	1	no	—
344	C₂²	C₃×#3828	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	18	1	no	—
347	C₄	—	36	1	no	—
348	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
349	C₄	C₂×C₃×#638	1	1	yes	characteristic
350	C₃	—	2	4	no	—
351	C₃	C₃×#5145	1	4	yes	complemented
352	C₃	#10592844	1	1	yes	characteristic, direct product factor
353	C₂	—	2	1	no	—
354	C₂	—	4	1	no	—
355	C₂	—	36	1	no	—
356	C₂	C₃×C₄×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×#638
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#131	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃×#64159	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#131	C₃⋊D₃
hypocenter	C₃²	C₃×#444
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#131

Complements

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

N	H ≅ G/N	type
1. C₃³×#131	C₂	semidirect
3. C₃×#64110	C₂	semidirect
4. C₃×#63664	C₂	semidirect
5. C₃×#63652	C₂	semidirect
6. C₃×#63649	C₂	semidirect
7. C₃×#63642	C₂	semidirect
8. #10592844	C₃	direct (N × H)
16. C₃×#3756	C₄	semidirect
17. C₃×#3755	C₄	semidirect
18. C₃×#3754	C₄	semidirect
19. C₃×#3753	C₄	semidirect
26. C₃²×#131	D₃	semidirect
27. C₃²×#131	C₂×C₃	semidirect
29. #64110	C₂×C₃	semidirect
32. #63664	C₂×C₃	semidirect
33. #63652	C₂×C₃	semidirect
34. #63649	C₂×C₃	semidirect
35. #63642	C₂×C₃	semidirect
75. #3756	C₃×C₄	semidirect
76. #3755	C₃×C₄	semidirect
77. #3754	C₃×C₄	semidirect
78. #3753	C₃×C₄	semidirect
104. C₃×#131	C₃×D₃	semidirect
105. C₃×#131	C₃⋊D₃	semidirect
219. #131	C₃×C₃⋊D₃	semidirect
224. C₃³	#444	semidirect
313. C₃²	#5145	semidirect
314. C₃²	C₃×#444	semidirect
351. C₃	C₃×#5145	semidirect
352. C₃	#10592844	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#131	2
C₃×#64159	2
C₃×#64110	2
C₃×#63664	2
C₃×#63652	2
C₃×#63649	2
C₃×#63642	2
#10592844	3
C₃×#5145	3

Subgroups — C3×#10592844 #13163183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592844 #13163183