Subgroups — C₃²×#5567 #13178944

Order 1728.

1480 conjugacy classes of subgroups.

The subgroup lattice has 338 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×#944	C₂	1	2	yes	maximal
3	C₃²×#889	C₂	1	4	yes	maximal
4	C₃²×#887	C₂	1	2	yes	maximal
5	C₃²×#870	C₂	1	2	yes	maximal
6	C₃²×#869	C₂	1	1	yes	maximal, characteristic
7	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
8	C₃²×C₄×Dic₆	C₂	1	2	yes	maximal
9	C₃²×#557	—	3	1	no	maximal
10	C₃×#5567	C₃	1	4	yes	maximal, direct product factor
11	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
13	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃³×C₄²	C₂²	1	1	yes	characteristic
17	C₃³×C₄²	C₂²	1	2	yes	—
18	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
21	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	—
22	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	—
23	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	—
25	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
26	C₃²×C₄×Dic₃	C₂²	1	4	yes	—
27	C₃²×C₄⋊Q₈	—	3	2	no	—
28	C₃²×C₄⋊Q₈	—	3	1	no	—
29	C₃²×#125	—	3	4	no	—
30	C₃²×#125	—	3	2	no	—
31	C₃²×C₄×Q₈	—	2	4	no	—
32	C₃²×C₄×Q₈	—	3	2	no	—
33	C₃²×C₄×Q₈	—	3	2	no	—
34	C₃²×C₄×Q₈	—	3	1	no	—
35	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
36	C₃²×C₄×Q₈	C₂×C₃	1	4	yes	—
37	C₃×#944	C₂×C₃	1	8	yes	—
38	C₃×#889	C₂×C₃	1	16	yes	—
39	C₃×#887	C₂×C₃	1	8	yes	—
40	C₃×#870	C₂×C₃	1	8	yes	—
41	C₃×#869	C₂×C₃	1	4	yes	—
42	C₃×C₄×Dic₆	C₂×C₃	1	8	yes	—
43	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
44	C₃³×Q₈	—	2	2	no	—
45	C₂×C₃³×C₄	C₂³	1	2	yes	—
46	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
47	C₂×C₃³×C₄	C₂³	1	2	yes	—
48	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
51	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
52	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
53	C₃²×Dic₆	—	2	4	no	—
54	C₃²×Dic₆	Q₈	1	4	yes	complemented
55	C₃×#557	—	3	4	no	—
56	#5567	C₃²	1	1	yes	characteristic, direct product factor
57	C₂×C₃²×Q₈	—	2	4	no	—
58	C₂×C₃²×Q₈	—	3	1	no	—
59	C₂×C₃²×Q₈	—	3	2	no	—
60	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
62	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
63	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
64	C₃²×C₄⋊C₄	—	2	4	no	—
65	C₃²×C₄⋊C₄	—	2	8	no	—
66	C₃²×C₄⋊C₄	—	3	2	no	—
67	C₃²×C₄⋊C₄	—	3	4	no	—
68	C₃²×C₄⋊C₄	—	3	1	no	—
69	C₃²×C₄⋊C₄	—	3	4	no	—
70	C₃²×C₄⋊C₄	—	3	2	no	—
71	C₃²×C₄⋊C₄	—	3	4	no	—
72	C₃²×C₄⋊C₄	—	3	2	no	—
73	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
75	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
76	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
77	C₃²×C₄²	—	2	8	no	—
78	C₃²×C₄²	—	2	4	no	—
79	C₃²×C₄²	—	3	4	no	—
80	C₃²×C₄²	—	3	2	no	—
81	C₃²×C₄²	C₂×D₃	1	2	yes	—
82	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
83	C₃²×C₄²	C₂²×C₃	1	4	yes	—
84	C₃²×C₄²	C₂²×C₃	1	8	yes	—
85	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
86	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
87	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
88	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
89	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
90	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
91	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
92	C₃×C₄×Dic₃	C₂²×C₃	1	16	yes	—
93	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
94	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
95	C₃³×C₄	—	2	2	no	—
96	C₃³×C₄	—	2	2	no	—
97	C₃³×C₄	—	2	1	no	—
98	C₃³×C₄	C₂×Q₈	1	2	yes	—
99	C₃³×C₄	#41	1	2	yes	—
100	C₃²×Dic₃	—	2	4	no	—
101	C₃²×Dic₃	—	2	2	no	—
102	C₃²×Dic₃	C₂×Q₈	1	4	yes	—
103	C₃×C₄⋊Q₈	—	3	8	no	—
104	C₃×C₄⋊Q₈	—	3	4	no	—
105	C₃×#125	—	3	16	no	—
106	C₃×#125	—	3	8	no	—
107	C₃×C₄×Q₈	—	2	4	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	C₃×D₃	1	4	yes	—
112	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
113	#944	C₂×C₃²	1	2	yes	—
114	#889	C₂×C₃²	1	4	yes	—
115	#887	C₂×C₃²	1	2	yes	—
116	#870	C₂×C₃²	1	2	yes	—
117	#869	C₂×C₃²	1	1	yes	characteristic
118	C₄×Dic₆	C₂×C₃²	1	2	yes	—
119	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
120	C₃²×Q₈	—	2	16	no	—
121	C₃²×Q₈	—	2	2	no	—
122	C₃²×Q₈	—	2	8	no	—
123	C₃²×Q₈	—	3	4	no	—
124	C₃²×Q₈	—	6	4	no	—
125	C₂×C₃²×C₄	—	2	4	no	—
126	C₂×C₃²×C₄	—	2	8	no	—
127	C₂×C₃²×C₄	—	2	4	no	—
128	C₂×C₃²×C₄	—	2	8	no	—
129	C₂×C₃²×C₄	—	2	4	no	—
130	C₂×C₃²×C₄	—	3	2	no	—
131	C₂×C₃²×C₄	—	3	2	no	—
132	C₂×C₃²×C₄	—	3	4	no	—
133	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
134	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
136	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
139	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
140	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
141	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
142	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
143	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
144	C₂×C₃×Dic₃	C₂³×C₃	1	16	yes	—
145	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
146	C₃×Dic₆	—	2	16	no	—
147	C₃×Dic₆	C₃×Q₈	1	16	yes	complemented
148	#557	—	3	1	no	Sylow
149	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₃³	C₂×#41	1	1	yes	characteristic
151	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₃×Q₈	—	2	4	no	—
153	C₂×C₃×Q₈	—	3	4	no	—
154	C₂×C₃×Q₈	—	3	8	no	—
155	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
156	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
157	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
158	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
159	C₃×C₄⋊C₄	—	2	4	no	—
160	C₃×C₄⋊C₄	—	2	8	no	—
161	C₃×C₄⋊C₄	—	3	8	no	—
162	C₃×C₄⋊C₄	—	3	16	no	—
163	C₃×C₄⋊C₄	—	3	16	no	—
164	C₃×C₄⋊C₄	—	3	16	no	—
165	C₃×C₄⋊C₄	—	3	4	no	—
166	C₃×C₄⋊C₄	—	3	8	no	—
167	C₃×C₄⋊C₄	—	3	8	no	—
168	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
169	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
170	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
171	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
172	C₃×C₄²	—	2	8	no	—
173	C₃×C₄²	—	2	4	no	—
174	C₃×C₄²	—	3	16	no	—
175	C₃×C₄²	—	3	8	no	—
176	C₃×C₄²	C₂×C₃×D₃	1	8	yes	—
177	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
178	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
179	C₃×C₄²	C₂²×C₃²	1	2	yes	—
180	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
181	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
182	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
183	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
184	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	—
185	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	—
186	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	—
187	C₄×Dic₃	C₂²×C₃²	1	4	yes	—
188	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
189	C₂²×C₃²	—	2	4	no	—
190	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
191	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
192	C₃²×C₄	—	2	16	no	—
193	C₃²×C₄	—	2	2	no	—
194	C₃²×C₄	—	2	2	no	—
195	C₃²×C₄	—	2	1	no	—
196	C₃²×C₄	—	2	8	no	—
197	C₃²×C₄	—	2	8	no	—
198	C₃²×C₄	—	2	8	no	—
199	C₃²×C₄	—	2	8	no	—
200	C₃²×C₄	—	2	4	no	—
201	C₃²×C₄	—	3	4	no	—
202	C₃²×C₄	—	4	8	no	—
203	C₃²×C₄	—	4	4	no	—
204	C₃²×C₄	—	6	4	no	—
205	C₃²×C₄	—	6	2	no	—
206	C₃²×C₄	#235	1	2	yes	—
207	C₃²×C₄	D₃×Q₈	1	2	yes	—
208	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	—
209	C₃²×C₄	C₃×#41	1	8	yes	—
210	C₃×Dic₃	—	2	16	no	—
211	C₃×Dic₃	—	2	8	no	—
212	C₃×Dic₃	C₂×C₃×Q₈	1	16	yes	—
213	C₄⋊Q₈	—	3	2	no	—
214	C₄⋊Q₈	—	3	1	no	—
215	#125	—	3	4	no	—
216	#125	—	3	2	no	—
217	C₄×Q₈	—	3	2	no	—
218	C₄×Q₈	—	3	2	no	—
219	C₄×Q₈	—	3	1	no	—
220	C₄×Q₈	C₃²×D₃	1	1	yes	characteristic
221	C₃³	#557	1	1	yes	characteristic, complemented, Sylow
222	C₃×Q₈	—	2	16	no	—
223	C₃×Q₈	—	2	8	no	—
224	C₃×Q₈	—	2	2	no	—
225	C₃×Q₈	—	3	16	no	—
226	C₃×Q₈	—	6	16	no	—
227	C₂×C₃×C₄	—	2	4	no	—
228	C₂×C₃×C₄	—	2	8	no	—
229	C₂×C₃×C₄	—	2	4	no	—
230	C₂×C₃×C₄	—	2	8	no	—
231	C₂×C₃×C₄	—	2	4	no	—
232	C₂×C₃×C₄	—	3	8	no	—
233	C₂×C₃×C₄	—	3	8	no	—
234	C₂×C₃×C₄	—	3	16	no	—
235	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
236	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
237	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
238	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
239	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
240	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
241	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
242	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
245	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
246	C₂×Dic₃	C₂³×C₃²	1	4	yes	—
247	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
248	Dic₆	—	2	4	no	—
249	Dic₆	C₃²×Q₈	1	4	yes	complemented
250	C₂×C₃²	—	2	4	no	—
251	C₂×C₃²	—	2	4	no	—
252	C₂×C₃²	—	2	4	no	—
253	C₂×C₃²	C₂×#235	1	1	yes	characteristic
254	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
255	C₂×C₃²	#1010	1	1	yes	characteristic
256	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
257	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
258	C₂×C₃²	C₃×2₋¹⁺⁴	1	4	yes	—
259	C₂×Q₈	—	3	1	no	—
260	C₂×Q₈	—	3	2	no	—
261	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
262	C₄⋊C₄	—	3	2	no	—
263	C₄⋊C₄	—	3	4	no	—
264	C₄⋊C₄	—	3	1	no	—
265	C₄⋊C₄	—	3	4	no	—
266	C₄⋊C₄	—	3	2	no	—
267	C₄⋊C₄	—	3	4	no	—
268	C₄⋊C₄	—	3	2	no	—
269	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
271	C₄²	—	3	4	no	—
272	C₄²	—	3	2	no	—
273	C₄²	C₂×C₃²×D₃	1	2	yes	—
274	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
275	C₂²×C₃	—	2	4	no	—
276	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
277	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
278	C₃×C₄	—	2	16	no	—
279	C₃×C₄	—	2	8	no	—
280	C₃×C₄	—	2	8	no	—
281	C₃×C₄	—	2	4	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	8	no	—
286	C₃×C₄	—	2	8	no	—
287	C₃×C₄	—	3	16	no	—
288	C₃×C₄	—	4	8	no	—
289	C₃×C₄	—	4	4	no	—
290	C₃×C₄	—	6	16	no	—
291	C₃×C₄	—	6	8	no	—
292	C₃×C₄	C₃×#235	1	8	yes	—
293	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
294	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	—
295	C₃×C₄	C₃²×#41	1	2	yes	—
296	Dic₃	—	2	4	no	—
297	Dic₃	—	2	2	no	—
298	Dic₃	C₂×C₃²×Q₈	1	4	yes	—
299	C₃²	—	2	4	no	—
300	C₃²	#5567	1	1	yes	characteristic, direct product factor
301	C₃²	C₃×#557	1	4	yes	complemented
302	Q₈	—	2	2	no	—
303	Q₈	—	3	4	no	—
304	Q₈	—	6	4	no	—
305	C₂×C₄	—	3	2	no	—
306	C₂×C₄	—	3	2	no	—
307	C₂×C₄	—	3	4	no	—
308	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
309	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
311	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
313	C₂×C₃	—	2	4	no	—
314	C₂×C₃	—	2	4	no	—
315	C₂×C₃	—	2	4	no	—
316	C₂×C₃	C₂×C₃×#235	1	4	yes	—
317	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
318	C₂×C₃	C₃×#1010	1	4	yes	—
319	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
320	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
321	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
322	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	3	4	no	—
327	C₄	—	6	4	no	—
328	C₄	—	6	2	no	—
329	C₄	C₃²×#235	1	2	yes	—
330	C₄	C₃²×D₃×Q₈	1	2	yes	—
331	C₃	—	2	4	no	—
332	C₃	C₃×#5567	1	4	yes	direct product factor
333	C₃	C₃²×#557	1	1	yes	characteristic, complemented
334	C₂	C₂×C₃²×#235	1	1	yes	characteristic
335	C₂	C₂×C₃²×D₃×Q₈	1	1	yes	characteristic
336	C₂	C₃²×#1010	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×#5567	C₃	direct (N × H)
54. C₃²×Dic₆	Q₈	semidirect
56. #5567	C₃²	direct (N × H)
147. C₃×Dic₆	C₃×Q₈	semidirect
221. C₃³	#557	semidirect
249. Dic₆	C₃²×Q₈	semidirect
300. C₃²	#5567	direct (N × H)
301. C₃²	C₃×#557	semidirect
332. C₃	C₃×#5567	direct (N × H)
333. 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×#944	2
C₃²×#889	2
C₃²×#887	2
C₃²×#870	2
C₃²×#869	2
C₃²×C₄×Dic₆	2
C₃²×#557	3
C₃×#5567	3

Subgroups — C32×#5567 #13178944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5567 #13178944