Subgroups — C₃×#10592523 #13162862

Order 1728.

812 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64159	C₂	1	1	yes	maximal, characteristic
2	C₃×#64157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64116	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63649	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63632	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592523	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4824	—	3	4	no	maximal
10	C₃³×#41	C₂²	1	1	yes	characteristic
11	C₃×#3828	C₂²	1	1	yes	characteristic
12	C₃³×M₁₆	C₂²	1	1	yes	characteristic
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#3756	C₄	1	1	yes	characteristic, complemented
15	C₃×#3755	C₄	1	1	yes	characteristic, complemented
16	C₃×#3754	C₄	1	1	yes	characteristic, complemented
17	C₃×#3753	C₄	1	1	yes	characteristic, complemented
18	C₃×#3748	—	2	1	no	—
19	C₃×#3748	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
21	C₂×C₃×C₃⋊(C₃⋊C₈)	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	#64159	C₂×C₃	1	1	yes	characteristic
26	#64157	C₂×C₃	1	1	yes	characteristic, complemented
27	#64116	C₂×C₃	1	1	yes	characteristic, complemented
28	C₃×#950	—	3	4	no	—
29	C₃×#948	—	3	4	no	—
30	C₃×#907	—	3	4	no	—
31	C₃²×#104	—	2	4	no	—
32	C₃²×#104	D₃	1	4	yes	complemented
33	C₃²×#104	C₂×C₃	1	1	yes	characteristic, complemented
34	#63649	C₂×C₃	1	1	yes	characteristic, complemented
35	#63632	C₂×C₃	1	1	yes	characteristic, complemented
36	C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
37	C₃×#822	—	3	4	no	—
38	C₃×#805	—	3	4	no	—
39	C₃×C₄×C₃⋊C₈	—	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	—	2	1	no	—
45	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
46	C₃×#638	—	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	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
52	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
53	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
54	C₃×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
55	C₃×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
56	C₃×#443	—	9	1	no	—
57	#4824	—	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	4	no	—
64	C₃²×M₁₆	C₂×D₃	1	4	yes	—
65	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
66	C₃²×C₄²	—	2	4	no	—
67	C₃²×C₄²	C₂×D₃	1	4	yes	—
68	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
69	#3756	C₃×C₄	1	1	yes	characteristic, complemented
70	#3755	C₃×C₄	1	1	yes	characteristic, complemented
71	#3754	C₃×C₄	1	1	yes	characteristic, complemented
72	#3753	C₃×C₄	1	1	yes	characteristic, complemented
73	#3748	—	2	1	no	—
74	#3748	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
76	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
78	#3743	—	2	1	no	—
79	C₈×C₃⋊D₃	—	2	1	no	—
80	C₃×#216	—	3	4	no	—
81	C₃×#215	—	3	4	no	—
82	C₃×#214	—	3	4	no	—
83	C₃×#213	—	3	4	no	—
84	C₃×#208	—	3	4	no	—
85	C₃×#208	—	6	4	no	—
86	C₂×C₃×C₃⋊C₈	—	3	4	no	—
87	C₂×C₃×C₃⋊C₈	—	3	4	no	—
88	C₂×C₃×C₃⋊C₈	—	6	4	no	—
89	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
90	C₃×D₃×C₈	—	6	4	no	—
91	C₂²×C₃³	—	2	1	no	—
92	C₂²×C₃³	#41	1	1	yes	characteristic
93	C₂×C₃×C₃⋊D₃	—	2	1	no	—
94	C₃³×C₄	—	2	2	no	—
95	C₃³×C₄	—	2	1	no	—
96	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
98	C₃×C₃⋊Dic₃	—	2	1	no	—
99	C₃×#135	—	9	1	no	—
100	C₃×#131	—	9	1	no	—
101	C₃×#131	—	9	1	no	—
102	#950	—	3	4	no	—
103	#948	—	3	4	no	—
104	#907	—	3	4	no	—
105	C₃×#108	—	9	1	no	—
106	C₃×#104	—	2	4	no	—
107	C₃×#104	—	9	1	no	—
108	C₃×#104	C₃×D₃	1	4	yes	complemented
109	C₃×#104	C₃⋊D₃	1	1	yes	characteristic, complemented
110	C₃×C₄×C₈	—	9	1	no	—
111	#822	—	3	4	no	—
112	#805	—	3	4	no	—
113	C₄×C₃⋊C₈	—	3	4	no	—
114	C₃²×Q₈	—	2	4	no	—
115	C₃²×Q₈	C₄×D₃	1	4	yes	—
116	C₃²×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
117	C₃²×D₄	—	2	1	no	—
118	C₃²×D₄	—	2	4	no	—
119	C₃²×D₄	—	2	4	no	—
120	C₃²×D₄	—	4	4	no	—
121	C₃²×D₄	C₄×D₃	1	4	yes	—
122	C₃²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×C₃²×C₄	—	2	1	no	—
124	C₂×C₃²×C₄	—	2	1	no	—
125	C₂×C₃²×C₄	—	2	4	no	—
126	C₂×C₃²×C₄	—	2	4	no	—
127	C₂×C₃²×C₄	—	2	4	no	—
128	C₂×C₃²×C₄	—	4	4	no	—
129	C₂×C₃²×C₄	—	4	4	no	—
130	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
131	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
132	#638	—	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	C₃×D₁₂	—	3	4	no	—
138	C₃×C₄×D₃	—	6	4	no	—
139	C₃×Dic₆	—	3	4	no	—
140	C₃²×C₈	—	2	1	no	—
141	C₃²×C₈	—	2	4	no	—
142	C₃²×C₈	—	4	4	no	—
143	C₃⋊(C₃⋊C₈)	—	2	1	no	—
144	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	1	yes	characteristic
145	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	1	yes	characteristic
146	C₃⋊(C₃⋊C₈)	C₃×D₄	1	1	yes	characteristic
147	C₃⋊(C₃⋊C₈)	C₃×D₄	1	1	yes	characteristic
148	C₃×C₃⋊C₈	—	3	4	no	—
149	C₃×C₃⋊C₈	—	3	4	no	—
150	C₃×C₃⋊C₈	—	3	4	no	—
151	C₃×C₃⋊C₈	—	3	4	no	—
152	C₃×C₃⋊C₈	—	6	4	no	—
153	#443	—	9	1	no	Sylow
154	C₂×C₃³	—	2	1	no	—
155	C₂×C₃³	—	4	1	no	—
156	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
157	C₃×C₃⋊D₃	—	4	1	no	—
158	C₃×#41	—	2	4	no	—
159	C₃×#41	—	9	1	no	—
160	C₃×#41	C₂×C₃×D₃	1	4	yes	—
161	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
162	#235	—	3	4	no	—
163	C₃×Q₁₆	—	9	1	no	—
164	C₃×SD₁₆	—	9	1	no	—
165	C₃×SD₁₆	—	9	1	no	—
166	C₃×D₈	—	9	1	no	—
167	C₃×M₁₆	—	2	4	no	—
168	C₃×M₁₆	—	9	1	no	—
169	C₃×M₁₆	—	18	1	no	—
170	C₃×M₁₆	—	18	1	no	—
171	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	—
172	C₃×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
173	C₂×C₃×C₈	—	9	1	no	—
174	C₂×C₃×C₈	—	9	1	no	—
175	C₂×C₃×C₈	—	18	1	no	—
176	C₂×C₃×C₈	—	18	1	no	—
177	C₃×C₄²	—	2	4	no	—
178	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
179	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
180	#216	—	3	4	no	—
181	#215	—	3	4	no	—
182	#214	—	3	4	no	—
183	#213	—	3	4	no	—
184	#208	—	3	4	no	—
185	#208	—	6	4	no	—
186	C₂×C₃⋊C₈	—	3	4	no	—
187	C₂×C₃⋊C₈	—	3	4	no	—
188	C₂×C₃⋊C₈	—	6	4	no	—
189	ΓL(1,𝔽₂₅)	—	6	4	no	—
190	D₃×C₈	—	6	4	no	—
191	C₂²×C₃²	—	2	1	no	—
192	C₂²×C₃²	—	2	4	no	—
193	C₂²×C₃²	—	2	4	no	—
194	C₂²×C₃²	—	4	4	no	—
195	C₂²×C₃²	#237	1	4	yes	—
196	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
197	C₂×C₃⋊D₃	—	2	1	no	—
198	C₂×C₃×D₃	—	6	4	no	—
199	C₃²×C₄	—	2	2	no	—
200	C₃²×C₄	—	2	1	no	—
201	C₃²×C₄	—	2	4	no	—
202	C₃²×C₄	—	2	4	no	—
203	C₃²×C₄	—	2	8	no	—
204	C₃²×C₄	—	2	4	no	—
205	C₃²×C₄	—	4	4	no	—
206	C₃²×C₄	—	4	8	no	—
207	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
208	C₃²×C₄	D₃×D₄	1	4	yes	—
209	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
210	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₃⋊Dic₃	—	2	1	no	—
212	C₃×Dic₃	—	6	4	no	—
213	#135	—	9	1	no	—
214	#131	—	9	1	no	—
215	#131	—	9	1	no	—
216	#108	—	9	1	no	—
217	#104	—	9	1	no	—
218	#104	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
219	C₄×C₈	—	9	1	no	—
220	C₃³	#443	1	1	yes	characteristic, complemented, Sylow
221	C₃×Q₈	—	2	4	no	—
222	C₃×Q₈	—	9	1	no	—
223	C₃×Q₈	C₃×C₄×D₃	1	4	yes	—
224	C₃×Q₈	C₄×C₃⋊D₃	1	1	yes	characteristic
225	C₃×D₄	—	2	4	no	—
226	C₃×D₄	—	2	1	no	—
227	C₃×D₄	—	2	4	no	—
228	C₃×D₄	—	4	4	no	—
229	C₃×D₄	—	9	1	no	—
230	C₃×D₄	—	18	1	no	—
231	C₃×D₄	C₃×C₄×D₃	1	4	yes	—
232	C₃×D₄	C₄×C₃⋊D₃	1	1	yes	characteristic
233	C₂×C₃×C₄	—	2	4	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	4	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	2	4	no	—
238	C₂×C₃×C₄	—	4	4	no	—
239	C₂×C₃×C₄	—	4	4	no	—
240	C₂×C₃×C₄	—	18	1	no	—
241	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
242	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
243	#67	—	6	4	no	—
244	D₁₂	—	3	4	no	—
245	C₄×D₃	—	6	4	no	—
246	Dic₆	—	3	4	no	—
247	C₃×C₈	—	2	4	no	—
248	C₃×C₈	—	2	1	no	—
249	C₃×C₈	—	4	4	no	—
250	C₃×C₈	—	9	1	no	—
251	C₃×C₈	—	9	1	no	—
252	C₃×C₈	—	9	1	no	—
253	C₃×C₈	—	9	1	no	—
254	C₃×C₈	—	18	1	no	—
255	C₃⋊C₈	—	3	4	no	—
256	C₃⋊C₈	—	3	4	no	—
257	C₃⋊C₈	—	3	4	no	—
258	C₃⋊C₈	—	3	4	no	—
259	C₃⋊C₈	—	6	4	no	—
260	C₂×C₃²	—	2	1	no	—
261	C₂×C₃²	—	2	4	no	—
262	C₂×C₃²	—	2	4	no	—
263	C₂×C₃²	—	4	4	no	—
264	C₂×C₃²	—	4	1	no	—
265	C₂×C₃²	—	4	4	no	—
266	C₂×C₃²	—	8	4	no	—
267	C₂×C₃²	#67⋊C₄	1	4	yes	—
268	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
269	C₃⋊D₃	—	4	1	no	—
270	C₃×D₃	—	12	4	no	—
271	#41	—	9	1	no	—
272	#41	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
273	Q₁₆	—	9	1	no	—
274	SD₁₆	—	9	1	no	—
275	SD₁₆	—	9	1	no	—
276	D₈	—	9	1	no	—
277	M₁₆	—	9	1	no	—
278	M₁₆	—	18	1	no	—
279	M₁₆	—	18	1	no	—
280	M₁₆	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
281	C₂×C₈	—	9	1	no	—
282	C₂×C₈	—	9	1	no	—
283	C₂×C₈	—	18	1	no	—
284	C₂×C₈	—	18	1	no	—
285	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
286	C₂²×C₃	—	2	4	no	—
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	2	4	no	—
289	C₂²×C₃	—	4	4	no	—
290	C₂²×C₃	—	18	1	no	—
291	C₂²×C₃	C₃×#237	1	4	yes	—
292	C₂²×C₃	#3830	1	1	yes	characteristic
293	C₂×D₃	—	6	4	no	—
294	C₃×C₄	—	2	2	no	—
295	C₃×C₄	—	2	8	no	—
296	C₃×C₄	—	2	4	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	4	no	—
299	C₃×C₄	—	2	4	no	—
300	C₃×C₄	—	4	4	no	—
301	C₃×C₄	—	4	8	no	—
302	C₃×C₄	—	18	1	no	—
303	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
304	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
305	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
306	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
307	Dic₃	—	6	4	no	—
308	C₃²	—	2	4	no	—
309	C₃²	#4824	1	4	yes	complemented
310	C₃²	C₃×#443	1	1	yes	characteristic, complemented
311	Q₈	—	9	1	no	—
312	Q₈	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
313	D₄	—	2	1	no	—
314	D₄	—	9	1	no	—
315	D₄	—	18	1	no	—
316	D₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	18	1	no	—
320	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	9	1	no	—
323	C₈	—	9	1	no	—
324	C₈	—	9	1	no	—
325	C₈	—	9	1	no	—
326	C₈	—	18	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	4	no	—
329	C₂×C₃	—	2	4	no	—
330	C₂×C₃	—	4	4	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	—	4	4	no	—
333	C₂×C₃	—	8	4	no	—
334	C₂×C₃	—	36	1	no	—
335	C₂×C₃	C₃×#67⋊C₄	1	4	yes	—
336	C₂×C₃	#64090	1	1	yes	characteristic
337	D₃	—	12	4	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	18	1	no	—
340	C₂²	C₃×#3830	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	18	1	no	—
344	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
345	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
346	C₃	—	2	4	no	—
347	C₃	C₃×#4824	1	4	yes	complemented
348	C₃	#10592523	1	1	yes	characteristic, direct product factor
349	C₂	—	2	1	no	—
350	C₂	—	4	1	no	—
351	C₂	—	36	1	no	—
352	C₂	C₃×#64090	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#104	C₂
socle	C₂×C₃³	C₄×D₄
prime radical	C₃×#64159	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#104	C₃⋊D₃
hypocenter	C₃²	C₃×#443
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#104

Complements

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

N	H ≅ G/N	type
2. C₃×#64157	C₂	semidirect
3. C₃×#64116	C₂	semidirect
4. C₃³×#104	C₂	semidirect
5. C₃×#63649	C₂	semidirect
6. C₃×#63632	C₂	semidirect
7. C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
8. #10592523	C₃	direct (N × H)
14. C₃×#3756	C₄	semidirect
15. C₃×#3755	C₄	semidirect
16. C₃×#3754	C₄	semidirect
17. C₃×#3753	C₄	semidirect
26. #64157	C₂×C₃	semidirect
27. #64116	C₂×C₃	semidirect
32. C₃²×#104	D₃	semidirect
33. C₃²×#104	C₂×C₃	semidirect
34. #63649	C₂×C₃	semidirect
35. #63632	C₂×C₃	semidirect
36. C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
69. #3756	C₃×C₄	semidirect
70. #3755	C₃×C₄	semidirect
71. #3754	C₃×C₄	semidirect
72. #3753	C₃×C₄	semidirect
108. C₃×#104	C₃×D₃	semidirect
109. C₃×#104	C₃⋊D₃	semidirect
218. #104	C₃×C₃⋊D₃	semidirect
220. C₃³	#443	semidirect
309. C₃²	#4824	semidirect
310. C₃²	C₃×#443	semidirect
347. C₃	C₃×#4824	semidirect
348. C₃	#10592523	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×#64159	2
C₃×#64157	2
C₃×#64116	2
C₃³×#104	2
C₃×#63649	2
C₃×#63632	2
C₃×C₄×C₃⋊(C₃⋊C₈)	2
#10592523	3
C₃×#4824	3

Subgroups — C3×#10592523 #13162862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592523 #13162862