Subgroups — C₃²×#5721 #13179099

Order 1728.

1472 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×Q₈×Dic₃	C₂	1	2	yes	maximal
3	C₃²×#944	C₂	1	2	yes	maximal
4	C₃²×#890	C₂	1	2	yes	maximal
5	C₃²×#889	C₂	1	4	yes	maximal
6	C₃²×#887	C₂	1	2	yes	maximal
7	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
8	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
9	C₃²×#557	—	3	1	no	maximal
10	C₃×#5721	C₃	1	4	yes	maximal, direct product factor
11	C₂×C₃³×Q₈	C₂²	1	2	yes	—
12	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃³×C₄²	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊Dic₃	C₂²	1	4	yes	—
19	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	—
22	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
23	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
25	C₃²×C₄×Dic₃	C₂²	1	4	yes	—
26	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
27	C₃²×C₄⋊Q₈	—	2	4	no	—
28	C₃²×C₄⋊Q₈	—	3	2	no	—
29	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
30	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	—
31	C₃²×#125	—	3	4	no	—
32	C₃²×#125	—	3	2	no	—
33	C₃²×C₄×Q₈	—	3	2	no	—
34	C₃²×C₄×Q₈	—	3	2	no	—
35	C₃²×C₄×Q₈	—	3	1	no	—
36	C₃²×C₄×Q₈	—	3	1	no	—
37	C₃×Q₈×Dic₃	C₂×C₃	1	8	yes	—
38	C₃×#944	C₂×C₃	1	8	yes	—
39	C₃×#890	C₂×C₃	1	8	yes	—
40	C₃×#889	C₂×C₃	1	16	yes	—
41	C₃×#887	C₂×C₃	1	8	yes	—
42	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
43	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
44	C₃³×Q₈	—	2	4	no	—
45	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
46	C₂×C₃³×C₄	C₂³	1	4	yes	—
47	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
50	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
51	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
52	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
53	C₃²×Dic₆	—	2	2	no	—
54	C₃²×Dic₆	Q₈	1	4	yes	complemented
55	C₃×#557	—	3	4	no	—
56	#5721	C₃²	1	1	yes	characteristic, direct product factor
57	C₂×C₃²×Q₈	—	2	8	no	—
58	C₂×C₃²×Q₈	—	3	1	no	—
59	C₂×C₃²×Q₈	—	3	1	no	—
60	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
61	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
62	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
63	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
64	C₃²×C₄⋊C₄	—	2	8	no	—
65	C₃²×C₄⋊C₄	—	2	8	no	—
66	C₃²×C₄⋊C₄	—	3	1	no	—
67	C₃²×C₄⋊C₄	—	3	2	no	—
68	C₃²×C₄⋊C₄	—	3	2	no	—
69	C₃²×C₄⋊C₄	—	3	4	no	—
70	C₃²×C₄⋊C₄	—	3	1	no	—
71	C₃²×C₄⋊C₄	—	3	4	no	—
72	C₃²×C₄⋊C₄	—	3	4	no	—
73	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
74	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
75	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
76	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
77	C₃²×C₄²	—	2	4	no	—
78	C₃²×C₄²	—	3	4	no	—
79	C₃²×C₄²	—	3	2	no	—
80	C₃²×C₄²	—	3	2	no	—
81	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
82	C₃²×C₄²	C₂²×C₃	1	4	yes	—
83	C₃×C₄⋊Dic₃	C₂²×C₃	1	16	yes	—
84	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
85	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
86	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
87	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
88	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
89	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
90	C₃×C₄×Dic₃	C₂²×C₃	1	16	yes	—
91	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
92	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
93	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
94	C₃³×C₄	—	2	4	no	—
95	C₃³×C₄	—	2	1	no	—
96	C₃³×C₄	C₂×Q₈	1	2	yes	—
97	C₃³×C₄	#41	1	2	yes	—
98	C₃²×Dic₃	—	2	2	no	—
99	C₃²×Dic₃	—	2	2	no	—
100	C₃²×Dic₃	—	2	2	no	—
101	C₃²×Dic₃	C₂×Q₈	1	4	yes	—
102	C₃×C₄⋊Q₈	—	2	4	no	—
103	C₃×C₄⋊Q₈	—	3	8	no	—
104	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	—
105	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
106	C₃×#125	—	3	16	no	—
107	C₃×#125	—	3	8	no	—
108	C₃×C₄×Q₈	—	3	8	no	—
109	C₃×C₄×Q₈	—	3	8	no	—
110	C₃×C₄×Q₈	—	3	4	no	—
111	C₃×C₄×Q₈	—	3	4	no	—
112	Q₈×Dic₃	C₂×C₃²	1	2	yes	—
113	#944	C₂×C₃²	1	2	yes	—
114	#890	C₂×C₃²	1	2	yes	—
115	#889	C₂×C₃²	1	4	yes	—
116	#887	C₂×C₃²	1	2	yes	—
117	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
118	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
119	C₃²×Q₈	—	2	4	no	—
120	C₃²×Q₈	—	2	16	no	—
121	C₃²×Q₈	—	3	4	no	—
122	C₃²×Q₈	—	4	16	no	—
123	C₃²×Q₈	—	6	2	no	—
124	C₂×C₃²×C₄	—	2	4	no	—
125	C₂×C₃²×C₄	—	2	4	no	—
126	C₂×C₃²×C₄	—	2	4	no	—
127	C₂×C₃²×C₄	—	2	16	no	—
128	C₂×C₃²×C₄	—	3	2	no	—
129	C₂×C₃²×C₄	—	3	2	no	—
130	C₂×C₃²×C₄	—	3	2	no	—
131	C₂×C₃²×C₄	—	3	2	no	—
132	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
133	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₃²×C₄	C₂³×C₃	1	16	yes	—
137	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
138	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
139	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
140	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
141	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
142	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
143	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
144	C₃×Dic₆	—	2	8	no	—
145	C₃×Dic₆	C₃×Q₈	1	16	yes	complemented
146	#557	—	3	1	no	Sylow
147	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₃³	C₂×#41	1	1	yes	characteristic
149	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×C₃×Q₈	—	2	8	no	—
151	C₂×C₃×Q₈	—	3	4	no	—
152	C₂×C₃×Q₈	—	3	4	no	—
153	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
154	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
155	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
156	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	8	no	—
158	C₃×C₄⋊C₄	—	2	8	no	—
159	C₃×C₄⋊C₄	—	3	8	no	—
160	C₃×C₄⋊C₄	—	3	8	no	—
161	C₃×C₄⋊C₄	—	3	16	no	—
162	C₃×C₄⋊C₄	—	3	16	no	—
163	C₃×C₄⋊C₄	—	3	4	no	—
164	C₃×C₄⋊C₄	—	3	16	no	—
165	C₃×C₄⋊C₄	—	3	4	no	—
166	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
167	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
168	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
169	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
170	C₃×C₄²	—	2	4	no	—
171	C₃×C₄²	—	3	16	no	—
172	C₃×C₄²	—	3	8	no	—
173	C₃×C₄²	—	3	8	no	—
174	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
175	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
176	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
177	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
178	C₄⋊Dic₃	C₂²×C₃²	1	4	yes	—
179	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
180	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
181	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	—
182	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	—
183	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
184	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
185	C₄×Dic₃	C₂²×C₃²	1	4	yes	—
186	C₂²×C₃²	—	2	4	no	—
187	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
188	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
189	C₃²×C₄	—	2	8	no	—
190	C₃²×C₄	—	2	4	no	—
191	C₃²×C₄	—	2	1	no	—
192	C₃²×C₄	—	2	4	no	—
193	C₃²×C₄	—	2	8	no	—
194	C₃²×C₄	—	2	8	no	—
195	C₃²×C₄	—	2	16	no	—
196	C₃²×C₄	—	3	4	no	—
197	C₃²×C₄	—	4	16	no	—
198	C₃²×C₄	—	6	2	no	—
199	C₃²×C₄	—	6	2	no	—
200	C₃²×C₄	—	6	2	no	—
201	C₃²×C₄	#237	1	2	yes	—
202	C₃²×C₄	D₃×Q₈	1	2	yes	—
203	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	—
204	C₃²×C₄	C₃×#41	1	8	yes	—
205	C₃×Dic₃	—	2	8	no	—
206	C₃×Dic₃	—	2	8	no	—
207	C₃×Dic₃	—	2	8	no	—
208	C₃×Dic₃	C₂×C₃×Q₈	1	16	yes	—
209	C₄⋊Q₈	—	3	2	no	—
210	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic
211	#125	—	3	4	no	—
212	#125	—	3	2	no	—
213	C₄×Q₈	—	3	2	no	—
214	C₄×Q₈	—	3	2	no	—
215	C₄×Q₈	—	3	1	no	—
216	C₄×Q₈	—	3	1	no	—
217	C₃³	#557	1	1	yes	characteristic, complemented, Sylow
218	C₃×Q₈	—	2	16	no	—
219	C₃×Q₈	—	2	4	no	—
220	C₃×Q₈	—	3	16	no	—
221	C₃×Q₈	—	4	16	no	—
222	C₃×Q₈	—	6	8	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	2	16	no	—
227	C₂×C₃×C₄	—	3	8	no	—
228	C₂×C₃×C₄	—	3	8	no	—
229	C₂×C₃×C₄	—	3	8	no	—
230	C₂×C₃×C₄	—	3	8	no	—
231	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
232	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
233	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
234	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
235	C₂×C₃×C₄	C₂³×C₃²	1	4	yes	—
236	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
239	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
240	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
241	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
242	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
243	Dic₆	—	2	2	no	—
244	Dic₆	C₃²×Q₈	1	4	yes	complemented
245	C₂×C₃²	—	2	4	no	—
246	C₂×C₃²	—	2	4	no	—
247	C₂×C₃²	—	2	4	no	—
248	C₂×C₃²	C₂×#237	1	1	yes	characteristic
249	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
250	C₂×C₃²	#1007	1	1	yes	characteristic
251	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
252	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
253	C₂×C₃²	C₃×2₋¹⁺⁴	1	4	yes	—
254	C₂×Q₈	—	3	1	no	—
255	C₂×Q₈	—	3	1	no	—
256	C₂×Q₈	C₂×C₃²×D₃	1	2	yes	—
257	C₄⋊C₄	—	3	1	no	—
258	C₄⋊C₄	—	3	2	no	—
259	C₄⋊C₄	—	3	2	no	—
260	C₄⋊C₄	—	3	4	no	—
261	C₄⋊C₄	—	3	1	no	—
262	C₄⋊C₄	—	3	4	no	—
263	C₄⋊C₄	—	3	4	no	—
264	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
265	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
266	C₄²	—	3	4	no	—
267	C₄²	—	3	2	no	—
268	C₄²	—	3	2	no	—
269	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
270	C₂²×C₃	—	2	4	no	—
271	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
272	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
273	C₃×C₄	—	2	8	no	—
274	C₃×C₄	—	2	16	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	8	no	—
279	C₃×C₄	—	2	8	no	—
280	C₃×C₄	—	3	16	no	—
281	C₃×C₄	—	4	16	no	—
282	C₃×C₄	—	6	8	no	—
283	C₃×C₄	—	6	8	no	—
284	C₃×C₄	—	6	8	no	—
285	C₃×C₄	C₃×#237	1	8	yes	—
286	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
287	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	—
288	C₃×C₄	C₃²×#41	1	2	yes	—
289	Dic₃	—	2	2	no	—
290	Dic₃	—	2	2	no	—
291	Dic₃	—	2	2	no	—
292	Dic₃	C₂×C₃²×Q₈	1	4	yes	—
293	C₃²	—	2	4	no	—
294	C₃²	#5721	1	1	yes	characteristic, direct product factor
295	C₃²	C₃×#557	1	4	yes	complemented
296	Q₈	—	2	4	no	—
297	Q₈	—	3	4	no	—
298	Q₈	—	6	2	no	—
299	C₂×C₄	—	3	2	no	—
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	3	2	no	—
302	C₂×C₄	—	3	2	no	—
303	C₂×C₄	C₂²×C₃²×D₃	1	4	yes	—
304	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
307	C₂×C₃	—	2	4	no	—
308	C₂×C₃	—	2	4	no	—
309	C₂×C₃	—	2	4	no	—
310	C₂×C₃	C₂×C₃×#237	1	4	yes	—
311	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
312	C₂×C₃	C₃×#1007	1	4	yes	—
313	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
314	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
315	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
316	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	2	1	no	—
319	C₄	—	3	4	no	—
320	C₄	—	6	2	no	—
321	C₄	—	6	2	no	—
322	C₄	—	6	2	no	—
323	C₄	C₃²×#237	1	2	yes	—
324	C₄	C₃²×D₃×Q₈	1	2	yes	—
325	C₃	—	2	4	no	—
326	C₃	C₃×#5721	1	4	yes	direct product factor
327	C₃	C₃²×#557	1	1	yes	characteristic, complemented
328	C₂	C₂×C₃²×#237	1	1	yes	characteristic
329	C₂	C₂×C₃²×D₃×Q₈	1	1	yes	characteristic
330	C₂	C₃²×#1007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂⁴
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
10. C₃×#5721	C₃	direct (N × H)
54. C₃²×Dic₆	Q₈	semidirect
56. #5721	C₃²	direct (N × H)
145. C₃×Dic₆	C₃×Q₈	semidirect
217. C₃³	#557	semidirect
244. Dic₆	C₃²×Q₈	semidirect
294. C₃²	#5721	direct (N × H)
295. C₃²	C₃×#557	semidirect
326. C₃	C₃×#5721	direct (N × H)
327. C₃	C₃²×#557	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₃²×Q₈×Dic₃	2
C₃²×#944	2
C₃²×#890	2
C₃²×#889	2
C₃²×#887	2
C₃²×C₄×Dic₆	2
C₃²×#557	3
C₃×#5721	3

Subgroups — C32×#5721 #13179099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5721 #13179099