Subgroups — C₃²×#5716 #13179094

Order 1728.

2292 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×C₃²×#237	C₂	1	2	yes	maximal, complemented
2	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#938	C₂	1	4	yes	maximal, complemented
4	C₃²×#936	C₂	1	2	yes	maximal, complemented
5	C₃²×D₄×Dic₃	C₂	1	4	yes	maximal, complemented
6	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#869	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#532	—	3	1	no	maximal
9	C₃×#5716	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×D₄	C₂²	1	4	yes	—
11	C₂×C₃³×D₄	C₂²	1	2	yes	—
12	C₂×C₃²×#67	C₂²	1	4	yes	—
13	C₂²×C₃²×Dic₃	C₂²	1	4	yes	—
14	C₃²×#237	—	2	8	no	—
15	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
16	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
18	C₃³×C₄²	C₂²	1	1	yes	characteristic
19	C₃²×#217	C₂²	1	8	yes	—
20	C₃²×C₄⋊Dic₃	C₂²	1	4	yes	—
21	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
22	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
23	C₂×C₃²×#41	—	3	2	no	—
24	C₂×C₃×#237	C₂×C₃	1	8	yes	complemented
25	C₃²×C₄⋊Q₈	—	3	1	no	—
26	C₃²×#127	—	2	4	no	—
27	C₃²×#127	D₃	1	1	yes	characteristic, complemented
28	C₃²×#127	C₂×C₃	1	4	yes	complemented
29	C₃²×#124	—	3	2	no	—
30	C₃²×#121	—	3	4	no	—
31	C₃²×C₄×D₄	—	3	4	no	—
32	C₂×C₃²×C₄²	—	3	1	no	—
33	C₃×#938	C₂×C₃	1	16	yes	complemented
34	C₃×#936	C₂×C₃	1	8	yes	complemented
35	C₃×D₄×Dic₃	C₂×C₃	1	16	yes	complemented
36	C₃×C₄²×D₃	C₂×C₃	1	4	yes	complemented
37	C₃×#869	C₂×C₃	1	4	yes	complemented
38	C₂³×C₃³	C₂³	1	4	yes	—
39	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
40	C₃³×D₄	—	2	4	no	—
41	C₃³×D₄	—	2	8	no	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃³×C₄	C₂³	1	2	yes	—
44	C₃²×#67	—	2	8	no	—
45	C₂×C₃²×Dic₃	—	2	8	no	—
46	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
47	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
48	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
49	C₃²×C₄×D₃	—	2	4	no	—
50	C₃²×C₄×D₃	D₄	1	4	yes	complemented
51	C₃²×Dic₆	—	2	4	no	—
52	C₃×#532	—	3	4	no	—
53	#5716	C₃²	1	1	yes	characteristic, direct product factor
54	C₃²×#41	—	6	8	no	—
55	C₂×C₃²×Q₈	—	3	2	no	—
56	C₂×C₃²×D₄	—	2	16	no	—
57	C₂×C₃²×D₄	—	2	8	no	—
58	C₂×C₃²×D₄	—	3	4	no	—
59	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
60	C₂×C₃²×D₄	C₂×D₃	1	4	yes	—
61	C₂×C₃²×D₄	C₂²×C₃	1	16	yes	—
62	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	—
63	C₂²×C₃²×C₄	—	3	1	no	—
64	C₂²×C₃²×C₄	—	3	4	no	—
65	C₂²×C₃²×C₄	—	3	2	no	—
66	C₂×C₃×#67	C₂²×C₃	1	16	yes	—
67	C₂²×C₃×Dic₃	C₂²×C₃	1	16	yes	—
68	C₃×#237	—	2	32	no	—
69	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
70	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
71	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
72	C₃²×C₄⋊C₄	—	3	4	no	—
73	C₃²×C₂²⋊C₄	—	3	8	no	—
74	C₃²×C₄²	—	2	4	no	—
75	C₃²×C₄²	—	3	2	no	—
76	C₃²×C₄²	—	3	1	no	—
77	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
78	C₃²×C₄²	C₂²×C₃	1	4	yes	—
79	C₃×#217	C₂²×C₃	1	32	yes	—
80	C₃×C₄⋊Dic₃	C₂²×C₃	1	16	yes	—
81	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
82	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
83	C₂²×C₃³	—	2	4	no	—
84	C₂²×C₃³	—	2	8	no	—
85	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
86	C₂×C₃²×D₃	—	2	2	no	—
87	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
88	C₃³×C₄	C₂×D₄	1	2	yes	—
89	C₃³×C₄	#41	1	4	yes	complemented
90	C₃²×Dic₃	—	2	4	no	—
91	C₃²×Dic₃	—	2	2	no	—
92	C₃²×Dic₃	C₂×D₄	1	2	yes	—
93	C₂×C₃×#41	—	3	8	no	—
94	C₂×#237	C₂×C₃²	1	2	yes	complemented
95	C₃×C₄⋊Q₈	—	3	4	no	—
96	C₃×#127	—	2	4	no	—
97	C₃×#127	C₃×D₃	1	4	yes	complemented
98	C₃×#127	C₂×C₃²	1	1	yes	characteristic, complemented
99	C₃×#124	—	3	8	no	—
100	C₃×#121	—	3	16	no	—
101	C₃×C₄×D₄	—	3	16	no	—
102	C₂×C₃×C₄²	—	3	4	no	—
103	#938	C₂×C₃²	1	4	yes	complemented
104	#936	C₂×C₃²	1	2	yes	complemented
105	D₄×Dic₃	C₂×C₃²	1	4	yes	complemented
106	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic, complemented
107	#869	C₂×C₃²	1	1	yes	characteristic, complemented
108	C₂³×C₃²	—	2	16	no	—
109	C₂³×C₃²	—	3	1	no	—
110	C₂³×C₃²	C₂²×D₃	1	4	yes	—
111	C₂³×C₃²	C₂³×C₃	1	16	yes	—
112	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
113	C₃²×Q₈	—	6	4	no	—
114	C₃²×D₄	—	2	4	no	—
115	C₃²×D₄	—	2	8	no	—
116	C₃²×D₄	—	2	16	no	—
117	C₃²×D₄	—	2	32	no	—
118	C₃²×D₄	—	4	16	no	—
119	C₃²×D₄	—	4	32	no	—
120	C₃²×D₄	—	6	8	no	—
121	C₂×C₃²×C₄	—	2	4	no	—
122	C₂×C₃²×C₄	—	2	8	no	—
123	C₂×C₃²×C₄	—	3	1	no	—
124	C₂×C₃²×C₄	—	3	4	no	—
125	C₂×C₃²×C₄	—	3	2	no	—
126	C₂×C₃²×C₄	—	3	4	no	—
127	C₂×C₃²×C₄	—	6	8	no	—
128	C₂×C₃²×C₄	—	6	4	no	—
129	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
130	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
131	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
132	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
133	C₃×#67	—	2	32	no	—
134	C₂×C₃×Dic₃	—	2	32	no	—
135	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
136	C₂×C₃×Dic₃	C₂³×C₃	1	16	yes	—
137	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
138	C₃×C₄×D₃	—	2	16	no	—
139	C₃×C₄×D₃	C₃×D₄	1	16	yes	complemented
140	C₃×Dic₆	—	2	16	no	—
141	#532	—	3	1	no	Sylow
142	C₂×C₃³	—	4	4	no	—
143	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃³	C₂×#41	1	2	yes	—
145	C₃²×D₃	—	2	2	no	—
146	C₃×#41	—	6	32	no	—
147	C₂×C₃×Q₈	—	3	8	no	—
148	C₂×C₃×D₄	—	2	16	no	—
149	C₂×C₃×D₄	—	2	8	no	—
150	C₂×C₃×D₄	—	3	16	no	—
151	C₂×C₃×D₄	C₂×C₃×D₃	1	16	yes	—
152	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
153	C₂×C₃×D₄	C₂²×C₃²	1	4	yes	—
154	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
155	C₂²×C₃×C₄	—	3	4	no	—
156	C₂²×C₃×C₄	—	3	16	no	—
157	C₂²×C₃×C₄	—	3	8	no	—
158	C₂×#67	C₂²×C₃²	1	4	yes	—
159	C₂²×Dic₃	C₂²×C₃²	1	4	yes	—
160	#237	—	2	8	no	—
161	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
162	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
163	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
164	C₃×C₄⋊C₄	—	3	16	no	—
165	C₃×C₂²⋊C₄	—	3	32	no	—
166	C₃×C₄²	—	2	4	no	—
167	C₃×C₄²	—	3	8	no	—
168	C₃×C₄²	—	3	4	no	—
169	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
170	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
171	#217	C₂²×C₃²	1	8	yes	—
172	C₄⋊Dic₃	C₂²×C₃²	1	4	yes	—
173	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
174	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
175	C₂²×C₃²	—	2	4	no	—
176	C₂²×C₃²	—	2	8	no	—
177	C₂²×C₃²	—	2	4	no	—
178	C₂²×C₃²	—	2	16	no	—
179	C₂²×C₃²	—	2	32	no	—
180	C₂²×C₃²	—	3	2	no	—
181	C₂²×C₃²	—	4	16	no	—
182	C₂²×C₃²	—	4	32	no	—
183	C₂²×C₃²	—	6	2	no	—
184	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
185	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
186	C₂×C₃×D₃	—	2	8	no	—
187	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
188	C₃²×C₄	—	2	8	no	—
189	C₃²×C₄	—	2	16	no	—
190	C₃²×C₄	—	3	2	no	—
191	C₃²×C₄	—	6	4	no	—
192	C₃²×C₄	—	6	2	no	—
193	C₃²×C₄	D₃×D₄	1	2	yes	—
194	C₃²×C₄	#237	1	4	yes	complemented
195	C₃²×C₄	C₂×C₃×D₄	1	8	yes	—
196	C₃²×C₄	C₃×#41	1	16	yes	complemented
197	C₃×Dic₃	—	2	16	no	—
198	C₃×Dic₃	—	2	8	no	—
199	C₃×Dic₃	C₂×C₃×D₄	1	8	yes	—
200	C₂×#41	—	3	2	no	—
201	C₄⋊Q₈	—	3	1	no	—
202	#127	C₃²×D₃	1	1	yes	characteristic, complemented
203	#124	—	3	2	no	—
204	#121	—	3	4	no	—
205	C₄×D₄	—	3	4	no	—
206	C₂×C₄²	—	3	1	no	—
207	C₃³	#532	1	1	yes	characteristic, complemented, Sylow
208	C₂³×C₃	—	2	16	no	—
209	C₂³×C₃	—	3	4	no	—
210	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
211	C₂³×C₃	C₂³×C₃²	1	4	yes	—
212	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
213	C₃×Q₈	—	6	16	no	—
214	C₃×D₄	—	2	16	no	—
215	C₃×D₄	—	2	32	no	—
216	C₃×D₄	—	2	4	no	—
217	C₃×D₄	—	2	8	no	—
218	C₃×D₄	—	4	16	no	—
219	C₃×D₄	—	4	32	no	—
220	C₃×D₄	—	6	32	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	2	8	no	—
223	C₂×C₃×C₄	—	3	4	no	—
224	C₂×C₃×C₄	—	3	16	no	—
225	C₂×C₃×C₄	—	3	8	no	—
226	C₂×C₃×C₄	—	3	16	no	—
227	C₂×C₃×C₄	—	6	32	no	—
228	C₂×C₃×C₄	—	6	16	no	—
229	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
230	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
231	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
233	#67	—	2	8	no	—
234	C₂×Dic₃	—	2	8	no	—
235	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
236	C₂×Dic₃	C₂³×C₃²	1	4	yes	—
237	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
238	C₄×D₃	—	2	4	no	—
239	C₄×D₃	C₃²×D₄	1	4	yes	complemented
240	Dic₆	—	2	4	no	—
241	C₂×C₃²	—	2	4	no	—
242	C₂×C₃²	—	2	8	no	—
243	C₂×C₃²	—	4	4	no	—
244	C₂×C₃²	—	4	16	no	—
245	C₂×C₃²	—	6	2	no	—
246	C₂×C₃²	—	8	16	no	—
247	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
248	C₂×C₃²	C₂×#237	1	2	yes	—
249	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
250	C₂×C₃²	C₂×C₃×#41	1	8	yes	—
251	C₃×D₃	—	2	8	no	—
252	#41	—	6	8	no	—
253	C₂×Q₈	—	3	2	no	—
254	C₂×D₄	—	3	4	no	—
255	C₂×D₄	C₂×C₃²×D₃	1	4	yes	—
256	C₂×D₄	C₂×C₃²×D₃	1	2	yes	—
257	C₂²×C₄	—	3	1	no	—
258	C₂²×C₄	—	3	4	no	—
259	C₂²×C₄	—	3	2	no	—
260	C₄⋊C₄	—	3	4	no	—
261	C₂²⋊C₄	—	3	8	no	—
262	C₄²	—	3	2	no	—
263	C₄²	—	3	1	no	—
264	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
265	C₂²×C₃	—	2	16	no	—
266	C₂²×C₃	—	2	32	no	—
267	C₂²×C₃	—	2	4	no	—
268	C₂²×C₃	—	2	8	no	—
269	C₂²×C₃	—	2	4	no	—
270	C₂²×C₃	—	3	8	no	—
271	C₂²×C₃	—	4	16	no	—
272	C₂²×C₃	—	4	32	no	—
273	C₂²×C₃	—	6	8	no	—
274	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
275	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
276	C₂×D₃	—	2	2	no	—
277	C₂×D₃	C₂×C₃²×D₄	1	2	yes	—
278	C₃×C₄	—	2	8	no	—
279	C₃×C₄	—	2	16	no	—
280	C₃×C₄	—	3	8	no	—
281	C₃×C₄	—	6	16	no	—
282	C₃×C₄	—	6	8	no	—
283	C₃×C₄	C₃×D₃×D₄	1	8	yes	—
284	C₃×C₄	C₃×#237	1	16	yes	complemented
285	C₃×C₄	C₂×C₃²×D₄	1	2	yes	—
286	C₃×C₄	C₃²×#41	1	4	yes	complemented
287	Dic₃	—	2	4	no	—
288	Dic₃	—	2	2	no	—
289	Dic₃	C₂×C₃²×D₄	1	2	yes	—
290	C₃²	—	2	4	no	—
291	C₃²	#5716	1	1	yes	characteristic, direct product factor
292	C₃²	C₃×#532	1	4	yes	complemented
293	C₂³	—	3	1	no	—
294	C₂³	C₂²×C₃²×D₃	1	4	yes	—
295	Q₈	—	6	4	no	—
296	D₄	—	2	4	no	—
297	D₄	—	2	8	no	—
298	D₄	—	6	8	no	—
299	C₂×C₄	—	3	1	no	—
300	C₂×C₄	—	3	4	no	—
301	C₂×C₄	—	3	2	no	—
302	C₂×C₄	—	3	4	no	—
303	C₂×C₄	—	6	8	no	—
304	C₂×C₄	—	6	4	no	—
305	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
307	C₂×C₃	—	2	4	no	—
308	C₂×C₃	—	2	8	no	—
309	C₂×C₃	—	4	16	no	—
310	C₂×C₃	—	4	4	no	—
311	C₂×C₃	—	6	8	no	—
312	C₂×C₃	—	8	16	no	—
313	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
314	C₂×C₃	C₂×C₃×#237	1	8	yes	—
315	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
316	C₂×C₃	C₂×C₃²×#41	1	2	yes	—
317	D₃	—	2	2	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	8	no	—
320	C₂²	—	3	2	no	—
321	C₂²	—	6	2	no	—
322	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
323	C₄	—	3	2	no	—
324	C₄	—	6	4	no	—
325	C₄	—	6	2	no	—
326	C₄	C₃²×D₃×D₄	1	2	yes	—
327	C₄	C₃²×#237	1	4	yes	complemented
328	C₃	—	2	4	no	—
329	C₃	C₃×#5716	1	4	yes	direct product factor
330	C₃	C₃²×#532	1	1	yes	characteristic, complemented
331	C₂	—	4	4	no	—
332	C₂	—	6	2	no	—
333	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic
334	C₂	C₂×C₃²×#237	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#127	D₃
hypocenter	C₃	C₃²×#532
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#237	C₂	semidirect
2. C₃³×#127	C₂	semidirect
3. C₃²×#938	C₂	semidirect
4. C₃²×#936	C₂	semidirect
5. C₃²×D₄×Dic₃	C₂	semidirect
6. C₃²×C₄²×D₃	C₂	semidirect
7. C₃²×#869	C₂	semidirect
9. C₃×#5716	C₃	direct (N × H)
24. C₂×C₃×#237	C₂×C₃	semidirect
27. C₃²×#127	D₃	semidirect
28. C₃²×#127	C₂×C₃	semidirect
33. C₃×#938	C₂×C₃	semidirect
34. C₃×#936	C₂×C₃	semidirect
35. C₃×D₄×Dic₃	C₂×C₃	semidirect
36. C₃×C₄²×D₃	C₂×C₃	semidirect
37. C₃×#869	C₂×C₃	semidirect
50. C₃²×C₄×D₃	D₄	semidirect
53. #5716	C₃²	direct (N × H)
89. C₃³×C₄	#41	semidirect
94. C₂×#237	C₂×C₃²	semidirect
97. C₃×#127	C₃×D₃	semidirect
98. C₃×#127	C₂×C₃²	semidirect
103. #938	C₂×C₃²	semidirect
104. #936	C₂×C₃²	semidirect
105. D₄×Dic₃	C₂×C₃²	semidirect
106. C₄²×D₃	C₂×C₃²	semidirect
107. #869	C₂×C₃²	semidirect
139. C₃×C₄×D₃	C₃×D₄	semidirect
194. C₃²×C₄	#237	semidirect
196. C₃²×C₄	C₃×#41	semidirect
202. #127	C₃²×D₃	semidirect
207. C₃³	#532	semidirect
239. C₄×D₃	C₃²×D₄	semidirect
284. C₃×C₄	C₃×#237	semidirect
286. C₃×C₄	C₃²×#41	semidirect
291. C₃²	#5716	direct (N × H)
292. C₃²	C₃×#532	semidirect
327. C₄	C₃²×#237	semidirect
329. C₃	C₃×#5716	direct (N × H)
330. C₃	C₃²×#532	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#237	2
C₃³×#127	2
C₃²×#938	2
C₃²×#936	2
C₃²×D₄×Dic₃	2
C₃²×C₄²×D₃	2
C₃²×#869	2
C₃²×#532	3
C₃×#5716	3

Subgroups — C32×#5716 #13179094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5716 #13179094