Subgroups — C₃×#10592714 #13163053

Order 1728.

1216 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3753	C₂	1	2	yes	maximal, complemented
3	C₃×#64083	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63636	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592714	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5015	—	3	4	no	maximal
8	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃³×C₄	—	2	1	no	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
13	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃³×C₂²⋊C₄	—	2	1	no	—
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#3753	—	2	4	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃²×C₄×D₄	—	2	4	no	—
19	C₃²×C₄×D₄	D₃	1	4	yes	complemented
20	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#3753	C₂×C₃	1	2	yes	complemented
22	#64083	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×#213	—	3	8	no	—
24	C₃×#874	—	3	4	no	—
25	#63636	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#809	—	3	8	no	—
28	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
29	C₂³×C₃³	—	2	1	no	—
30	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
31	C₃³×D₄	—	2	1	no	—
32	C₃³×D₄	D₄	1	2	yes	complemented
33	C₂×C₃³×C₄	—	2	1	no	—
34	C₂×C₃³×C₄	—	4	2	no	—
35	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
36	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
38	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
39	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
40	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
41	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
42	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
43	C₃×#459	—	9	1	no	—
44	#5015	—	3	4	no	—
45	C₂×C₃²×D₄	—	2	4	no	—
46	C₂×C₃²×D₄	C₂×D₃	1	4	yes	—
47	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃²×C₄	—	2	8	no	—
49	C₂²×C₃²×C₄	—	2	4	no	—
50	C₂²×C₃²×C₄	—	2	1	no	—
51	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
52	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
53	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
54	C₂×C₃×D₁₂	—	3	8	no	—
55	C₂×C₃×D₁₂	—	6	4	no	—
56	C₂×C₃×D₁₂	—	6	4	no	—
57	C₃²×C₄⋊C₄	—	2	4	no	—
58	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
59	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₃²×C₂²⋊C₄	—	2	8	no	—
61	C₃²×C₂²⋊C₄	—	2	4	no	—
62	C₃²×C₂²⋊C₄	—	2	1	no	—
63	C₃²×C₄²	—	2	4	no	—
64	C₃²×C₄²	C₂×D₃	1	4	yes	—
65	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
66	#3753	—	2	4	no	—
67	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
68	C₃×#213	—	6	16	no	—
69	C₂×C₃×C₃⋊C₈	—	3	8	no	—
70	C₂²×C₃³	—	2	2	no	—
71	C₂²×C₃³	—	4	1	no	—
72	C₂²×C₃³	—	4	1	no	—
73	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₃⋊D₃	—	4	2	no	—
75	C₂×C₃×C₃⋊D₃	—	4	2	no	—
76	C₂×C₃×C₃⋊D₃	—	4	2	no	—
77	C₃³×C₄	—	2	1	no	—
78	C₃³×C₄	—	4	1	no	—
79	C₃³×C₄	D₈	1	2	yes	complemented
80	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
81	C₃³×C₄	#41	1	1	yes	characteristic
82	C₂×C₃×D₈	—	9	2	no	—
83	C₃×#127	—	9	1	no	—
84	C₃×C₄×D₄	—	2	4	no	—
85	C₃×C₄×D₄	C₃×D₃	1	4	yes	complemented
86	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic, complemented
87	C₂×#213	—	3	8	no	—
88	#874	—	3	4	no	—
89	C₃×C₄⋊C₈	—	9	1	no	—
90	C₃×#102	—	9	2	no	—
91	#809	—	3	8	no	—
92	C₄⋊(C₃⋊C₈)	—	3	4	no	—
93	C₂³×C₃²	—	2	8	no	—
94	C₂³×C₃²	—	2	4	no	—
95	C₂³×C₃²	—	2	1	no	—
96	C₂²×C₃⋊D₃	—	2	2	no	—
97	C₂²×C₃×D₃	—	6	8	no	—
98	C₃²×D₄	—	2	8	no	—
99	C₃²×D₄	—	2	4	no	—
100	C₃²×D₄	—	2	1	no	—
101	C₃²×D₄	—	2	8	no	—
102	C₃²×D₄	D₁₂	1	8	yes	complemented
103	C₃²×D₄	C₃×D₄	1	2	yes	complemented
104	C₂×C₃²×C₄	—	2	8	no	—
105	C₂×C₃²×C₄	—	2	4	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	4	no	—
108	C₂×C₃²×C₄	—	2	4	no	—
109	C₂×C₃²×C₄	—	2	4	no	—
110	C₂×C₃²×C₄	—	4	16	no	—
111	C₂×C₃²×C₄	—	4	8	no	—
112	C₂×C₃²×C₄	—	4	2	no	—
113	C₂×C₃²×C₄	#67	1	4	yes	—
114	C₂×C₃²×C₄	#67	1	4	yes	—
115	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
117	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
118	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₄⋊(C₃⋊D₃)	—	2	2	no	—
120	C₄⋊(C₃⋊D₃)	—	2	2	no	—
121	C₄⋊(C₃⋊D₃)	—	4	2	no	—
122	C₄⋊(C₃⋊D₃)	—	4	2	no	—
123	C₃×D₁₂	—	6	8	no	—
124	C₃×D₁₂	—	6	8	no	—
125	C₃×D₁₂	—	12	8	no	—
126	C₃×D₁₂	—	12	8	no	—
127	C₃⋊(C₃⋊C₈)	—	2	2	no	—
128	C₃×C₃⋊C₈	—	6	8	no	—
129	#459	—	9	1	no	Sylow
130	C₂×C₃³	—	4	2	no	—
131	C₂×C₃³	#121	1	1	yes	characteristic
132	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
133	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₃×C₃⋊D₃	—	8	2	no	—
135	C₂×C₃×D₄	—	2	4	no	—
136	C₂×C₃×D₄	—	9	2	no	—
137	C₂×C₃×D₄	—	18	1	no	—
138	C₂×C₃×D₄	—	18	1	no	—
139	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
140	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
141	C₂²×C₃×C₄	—	2	8	no	—
142	C₂²×C₃×C₄	—	2	4	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂×D₁₂	—	3	8	no	—
145	C₂×D₁₂	—	6	4	no	—
146	C₂×D₁₂	—	6	4	no	—
147	C₃×D₈	—	18	4	no	—
148	C₂×C₃×C₈	—	9	2	no	—
149	C₃×C₄⋊C₄	—	2	4	no	—
150	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
151	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
152	C₃×C₂²⋊C₄	—	2	8	no	—
153	C₃×C₂²⋊C₄	—	2	4	no	—
154	C₃×C₂²⋊C₄	—	2	1	no	—
155	C₃×C₄²	—	2	4	no	—
156	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
157	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
158	#213	—	6	16	no	—
159	C₂×C₃⋊C₈	—	3	8	no	—
160	C₂²×C₃²	—	2	16	no	—
161	C₂²×C₃²	—	2	8	no	—
162	C₂²×C₃²	—	2	2	no	—
163	C₂²×C₃²	—	2	4	no	—
164	C₂²×C₃²	—	4	8	no	—
165	C₂²×C₃²	—	4	8	no	—
166	C₂²×C₃²	—	4	4	no	—
167	C₂²×C₃²	—	4	4	no	—
168	C₂²×C₃²	—	4	1	no	—
169	C₂²×C₃²	—	4	1	no	—
170	C₂²×C₃²	C₂×#67	1	4	yes	—
171	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₃⋊D₃	—	4	2	no	—
173	C₂×C₃⋊D₃	—	4	2	no	—
174	C₂×C₃⋊D₃	—	4	2	no	—
175	C₂×C₃×D₃	—	12	8	no	—
176	C₂×C₃×D₃	—	12	8	no	—
177	C₂×C₃×D₃	—	12	8	no	—
178	C₃²×C₄	—	2	1	no	—
179	C₃²×C₄	—	2	8	no	—
180	C₃²×C₄	—	2	4	no	—
181	C₃²×C₄	—	2	4	no	—
182	C₃²×C₄	—	2	4	no	—
183	C₃²×C₄	—	4	4	no	—
184	C₃²×C₄	—	4	8	no	—
185	C₃²×C₄	—	4	4	no	—
186	C₃²×C₄	—	4	1	no	—
187	C₃²×C₄	#213	1	8	yes	complemented
188	C₃²×C₄	C₃×D₈	1	2	yes	complemented
189	C₃²×C₄	C₂×D₁₂	1	4	yes	—
190	C₃²×C₄	#235	1	4	yes	—
191	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₃²×C₄	C₃×#41	1	1	yes	characteristic
193	C₂×D₈	—	9	2	no	—
194	#127	—	9	1	no	—
195	C₄×D₄	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
196	C₄⋊C₈	—	9	1	no	—
197	#102	—	9	2	no	—
198	C₃³	#459	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	2	8	no	—
200	C₂³×C₃	—	2	4	no	—
201	C₂³×C₃	—	2	1	no	—
202	C₂³×C₃	—	18	2	no	—
203	C₂²×D₃	—	6	8	no	—
204	C₃×D₄	—	2	8	no	—
205	C₃×D₄	—	2	4	no	—
206	C₃×D₄	—	2	1	no	—
207	C₃×D₄	—	2	8	no	—
208	C₃×D₄	—	18	2	no	—
209	C₃×D₄	—	18	2	no	—
210	C₃×D₄	—	36	2	no	—
211	C₃×D₄	—	36	2	no	—
212	C₃×D₄	C₃×D₁₂	1	8	yes	complemented
213	C₃×D₄	C₄⋊(C₃⋊D₃)	1	2	yes	complemented
214	C₂×C₃×C₄	—	2	8	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	4	16	no	—
221	C₂×C₃×C₄	—	4	8	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	C₃×#67	1	4	yes	—
224	C₂×C₃×C₄	C₃×#67	1	4	yes	—
225	C₂×C₃×C₄	#638	1	1	yes	characteristic
226	C₂×C₃×C₄	#638	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
228	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
229	D₁₂	—	6	8	no	—
230	D₁₂	—	6	8	no	—
231	D₁₂	—	12	8	no	—
232	D₁₂	—	12	8	no	—
233	C₃×C₈	—	18	2	no	—
234	C₃⋊C₈	—	6	8	no	—
235	C₂×C₃²	—	2	4	no	—
236	C₂×C₃²	—	2	4	no	—
237	C₂×C₃²	—	2	4	no	—
238	C₂×C₃²	—	4	16	no	—
239	C₂×C₃²	—	4	8	no	—
240	C₂×C₃²	—	4	2	no	—
241	C₂×C₃²	#930	1	4	yes	—
242	C₂×C₃²	C₂×#213	1	4	yes	—
243	C₂×C₃²	#949	1	4	yes	—
244	C₂×C₃²	C₃×#121	1	1	yes	characteristic
245	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
246	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₃⋊D₃	—	8	2	no	—
248	C₃×D₃	—	24	8	no	—
249	C₂×D₄	—	9	2	no	—
250	C₂×D₄	—	18	1	no	—
251	C₂×D₄	—	18	1	no	—
252	C₂×D₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	D₈	—	18	4	no	—
255	C₂×C₈	—	9	2	no	—
256	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
259	C₂²×C₃	—	2	16	no	—
260	C₂²×C₃	—	2	8	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	4	no	—
263	C₂²×C₃	—	4	8	no	—
264	C₂²×C₃	—	4	8	no	—
265	C₂²×C₃	—	4	4	no	—
266	C₂²×C₃	—	4	4	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	36	2	no	—
270	C₂²×C₃	—	36	2	no	—
271	C₂²×C₃	—	36	2	no	—
272	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
273	C₂²×C₃	C₂×#638	1	1	yes	characteristic
274	C₂×D₃	—	12	8	no	—
275	C₂×D₃	—	12	8	no	—
276	C₂×D₃	—	12	8	no	—
277	C₃×C₄	—	2	4	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	8	no	—
280	C₃×C₄	—	2	4	no	—
281	C₃×C₄	—	2	4	no	—
282	C₃×C₄	—	4	4	no	—
283	C₃×C₄	—	4	8	no	—
284	C₃×C₄	—	4	4	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	C₃×#213	1	8	yes	complemented
287	C₃×C₄	#3753	1	2	yes	complemented
288	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
289	C₃×C₄	C₃×#235	1	4	yes	—
290	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
291	C₃×C₄	#3828	1	1	yes	characteristic
292	C₃²	—	2	4	no	—
293	C₃²	#5015	1	4	yes	complemented
294	C₃²	C₃×#459	1	1	yes	characteristic, complemented
295	C₂³	—	2	1	no	—
296	C₂³	—	18	2	no	—
297	D₄	—	2	1	no	—
298	D₄	—	18	2	no	—
299	D₄	—	18	2	no	—
300	D₄	—	36	2	no	—
301	D₄	—	36	2	no	—
302	D₄	C₃×C₄⋊(C₃⋊D₃)	1	2	yes	complemented
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₃×#638	1	1	yes	characteristic
306	C₂×C₄	C₃×#638	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
308	C₈	—	18	2	no	—
309	C₂×C₃	—	2	4	no	—
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	—	2	4	no	—
312	C₂×C₃	—	4	16	no	—
313	C₂×C₃	—	4	8	no	—
314	C₂×C₃	—	4	2	no	—
315	C₂×C₃	—	72	2	no	—
316	C₂×C₃	C₃×#930	1	4	yes	—
317	C₂×C₃	C₂×C₃×#213	1	4	yes	—
318	C₂×C₃	C₃×#949	1	4	yes	—
319	C₂×C₃	#64139	1	1	yes	characteristic
320	C₂×C₃	C₂×#3753	1	1	yes	characteristic
321	C₂×C₃	#64158	1	1	yes	characteristic
322	D₃	—	24	8	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	36	2	no	—
327	C₂²	—	36	2	no	—
328	C₂²	—	36	2	no	—
329	C₂²	C₂×C₃×#638	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	C₃×#3753	1	2	yes	complemented
333	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
334	C₄	C₃×#3828	1	1	yes	characteristic
335	C₃	—	2	4	no	—
336	C₃	C₃×#5015	1	4	yes	complemented
337	C₃	#10592714	1	1	yes	characteristic, direct product factor
338	C₂	—	4	2	no	—
339	C₂	—	72	2	no	—
340	C₂	C₃×#64139	1	1	yes	characteristic
341	C₂	C₂×C₃×#3753	1	1	yes	characteristic
342	C₂	C₃×#64158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×D₄	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₃×C₄×D₄	C₃⋊D₃
hypocenter	C₃²	C₃×#459
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄×D₄	C₂	semidirect
2. C₂×C₃×#3753	C₂	semidirect
3. C₃×#64083	C₂	semidirect
4. C₃×#63636	C₂	semidirect
5. C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
6. #10592714	C₃	direct (N × H)
19. C₃²×C₄×D₄	D₃	semidirect
20. C₃²×C₄×D₄	C₂×C₃	semidirect
21. C₂×#3753	C₂×C₃	semidirect
22. #64083	C₂×C₃	semidirect
25. #63636	C₂×C₃	semidirect
26. C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	semidirect
32. C₃³×D₄	D₄	semidirect
79. C₃³×C₄	D₈	semidirect
85. C₃×C₄×D₄	C₃×D₃	semidirect
86. C₃×C₄×D₄	C₃⋊D₃	semidirect
102. C₃²×D₄	D₁₂	semidirect
103. C₃²×D₄	C₃×D₄	semidirect
187. C₃²×C₄	#213	semidirect
188. C₃²×C₄	C₃×D₈	semidirect
195. C₄×D₄	C₃×C₃⋊D₃	semidirect
198. C₃³	#459	semidirect
212. C₃×D₄	C₃×D₁₂	semidirect
213. C₃×D₄	C₄⋊(C₃⋊D₃)	semidirect
286. C₃×C₄	C₃×#213	semidirect
287. C₃×C₄	#3753	semidirect
293. C₃²	#5015	semidirect
294. C₃²	C₃×#459	semidirect
302. D₄	C₃×C₄⋊(C₃⋊D₃)	semidirect
332. C₄	C₃×#3753	semidirect
336. C₃	C₃×#5015	semidirect
337. C₃	#10592714	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×C₄×D₄	2
C₂×C₃×#3753	2
C₃×#64083	2
C₃×#63636	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10592714	3
C₃×#5015	3

Subgroups — C3×#10592714 #13163053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592714 #13163053