Subgroups — C₃²×#5725 #13179103

Order 1728.

2132 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#239	C₂	1	2	yes	maximal, complemented
2	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#947	C₂	1	2	yes	maximal, complemented
4	C₃²×#895	C₂	1	4	yes	maximal, complemented
5	C₃²×#893	C₂	1	4	yes	maximal, complemented
6	C₃²×#874	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#532	—	3	1	no	maximal
9	C₃×#5725	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃³×Q₈	C₂²	1	2	yes	—
11	C₃²×#239	—	2	8	no	—
12	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
13	C₂×C₃²×D₁₂	C₂²	1	4	yes	—
14	C₂×C₃²×D₁₂	C₂²	1	4	yes	—
15	C₂×C₃²×C₄×D₃	C₂²	1	4	yes	—
16	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
17	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
18	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
19	C₃³×C₄²	C₂²	1	1	yes	characteristic
20	C₃²×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
21	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
22	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
23	C₂×C₃²×#41	—	3	2	no	—
24	C₂×C₃×#239	C₂×C₃	1	8	yes	complemented
25	C₃²×C₄⋊Q₈	—	2	4	no	—
26	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
27	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
28	C₃²×#127	—	3	1	no	—
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×#947	C₂×C₃	1	8	yes	complemented
34	C₃×#895	C₂×C₃	1	16	yes	complemented
35	C₃×#893	C₂×C₃	1	16	yes	complemented
36	C₃×#874	C₂×C₃	1	4	yes	complemented
37	C₃×C₄²×D₃	C₂×C₃	1	4	yes	complemented
38	C₂²×C₃²×D₃	C₂³	1	4	yes	—
39	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
40	C₃³×Q₈	—	2	4	no	—
41	C₂×C₃³×C₄	C₂³	1	4	yes	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃³×C₄	C₂³	1	2	yes	—
44	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
45	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
46	C₃²×D₁₂	—	2	8	no	—
47	C₃²×D₁₂	—	2	4	no	—
48	C₃²×D₁₂	—	2	8	no	—
49	C₃²×C₄×D₃	—	2	4	no	—
50	C₃²×C₄×D₃	—	2	8	no	—
51	C₃²×C₄×D₃	D₄	1	4	yes	complemented
52	C₃×#532	—	3	4	no	—
53	#5725	C₃²	1	1	yes	characteristic, direct product factor
54	C₃²×#41	—	6	8	no	—
55	C₂×C₃²×Q₈	—	2	8	no	—
56	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
57	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
58	C₂×C₃²×D₄	—	3	4	no	—
59	C₂×C₃²×D₄	—	3	2	no	—
60	C₂×C₃²×D₄	—	3	4	no	—
61	C₂²×C₃²×C₄	—	3	1	no	—
62	C₂²×C₃²×C₄	—	3	4	no	—
63	C₂²×C₃²×C₄	—	3	2	no	—
64	C₃×#239	—	2	32	no	—
65	C₂×C₃×D₁₂	C₂²×C₃	1	16	yes	—
66	C₂×C₃×D₁₂	C₂²×C₃	1	16	yes	—
67	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
68	C₂×C₃×C₄×D₃	C₂²×C₃	1	16	yes	—
69	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
70	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
71	C₃²×C₄⋊C₄	—	2	16	no	—
72	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
73	C₃²×C₄⋊C₄	C₂²×C₃	1	16	yes	—
74	C₃²×C₂²⋊C₄	—	3	8	no	—
75	C₃²×C₄²	—	2	4	no	—
76	C₃²×C₄²	—	3	2	no	—
77	C₃²×C₄²	—	3	1	no	—
78	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
79	C₃²×C₄²	C₂²×C₃	1	4	yes	—
80	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	32	yes	—
81	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
82	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
83	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
84	C₂×C₃²×D₃	—	2	2	no	—
85	C₂×C₃²×D₃	—	2	4	no	—
86	C₂×C₃²×D₃	—	2	8	no	—
87	C₂×C₃²×D₃	C₂×D₄	1	2	yes	—
88	C₃³×C₄	—	2	4	no	—
89	C₃³×C₄	C₂×D₄	1	2	yes	—
90	C₃³×C₄	#41	1	4	yes	complemented
91	C₃²×Dic₃	—	2	2	no	—
92	C₃²×Dic₃	C₂×D₄	1	2	yes	—
93	C₂×C₃×#41	—	3	8	no	—
94	C₂×#239	C₂×C₃²	1	2	yes	complemented
95	C₃×C₄⋊Q₈	—	2	4	no	—
96	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
97	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
98	C₃×#127	—	3	4	no	—
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	#947	C₂×C₃²	1	2	yes	complemented
104	#895	C₂×C₃²	1	4	yes	complemented
105	#893	C₂×C₃²	1	4	yes	complemented
106	#874	C₂×C₃²	1	1	yes	characteristic, complemented
107	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic, complemented
108	C₂³×C₃²	—	3	1	no	—
109	C₂³×C₃²	—	3	4	no	—
110	C₂²×C₃×D₃	C₂³×C₃	1	16	yes	—
111	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
112	C₃²×Q₈	—	2	4	no	—
113	C₃²×Q₈	—	2	16	no	—
114	C₃²×Q₈	—	4	16	no	—
115	C₃²×D₄	—	6	8	no	—
116	C₃²×D₄	—	6	4	no	—
117	C₃²×D₄	—	6	8	no	—
118	C₂×C₃²×C₄	—	2	8	no	—
119	C₂×C₃²×C₄	—	2	4	no	—
120	C₂×C₃²×C₄	—	2	16	no	—
121	C₂×C₃²×C₄	—	3	1	no	—
122	C₂×C₃²×C₄	—	3	4	no	—
123	C₂×C₃²×C₄	—	3	2	no	—
124	C₂×C₃²×C₄	—	6	8	no	—
125	C₂×C₃²×C₄	—	6	4	no	—
126	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
127	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
129	C₂×C₃²×C₄	C₂³×C₃	1	16	yes	—
130	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
131	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
132	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
133	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
134	C₃×D₁₂	—	2	32	no	—
135	C₃×D₁₂	—	2	16	no	—
136	C₃×D₁₂	—	2	32	no	—
137	C₃×C₄×D₃	—	2	16	no	—
138	C₃×C₄×D₃	—	2	32	no	—
139	C₃×C₄×D₃	C₃×D₄	1	16	yes	complemented
140	#532	—	3	1	no	Sylow
141	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₃³	C₂×#41	1	2	yes	—
143	C₃²×D₃	—	2	2	no	—
144	C₃²×D₃	—	4	4	no	—
145	C₃×#41	—	6	32	no	—
146	C₂×C₃×Q₈	—	2	8	no	—
147	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
148	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
149	C₂×C₃×D₄	—	3	16	no	—
150	C₂×C₃×D₄	—	3	16	no	—
151	C₂×C₃×D₄	—	3	8	no	—
152	C₂²×C₃×C₄	—	3	4	no	—
153	C₂²×C₃×C₄	—	3	8	no	—
154	C₂²×C₃×C₄	—	3	16	no	—
155	#239	—	2	8	no	—
156	C₂×D₁₂	C₂²×C₃²	1	4	yes	—
157	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
158	C₂×D₁₂	C₂²×C₃²	1	4	yes	—
159	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
160	C₂×C₄×D₃	C₂²×C₃²	1	4	yes	—
161	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	2	16	no	—
163	C₃×C₄⋊C₄	C₂×C₃×D₃	1	16	yes	—
164	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
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	(C₂×D₃)⋊C₄	C₂²×C₃²	1	8	yes	—
172	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
173	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
174	C₂²×C₃²	—	2	4	no	—
175	C₂²×C₃²	—	3	2	no	—
176	C₂²×C₃²	—	6	2	no	—
177	C₂²×C₃²	—	6	4	no	—
178	C₂²×C₃²	—	6	8	no	—
179	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
180	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
181	C₂×C₃×D₃	—	2	8	no	—
182	C₂×C₃×D₃	—	2	16	no	—
183	C₂×C₃×D₃	—	2	32	no	—
184	C₂×C₃×D₃	C₂×C₃×D₄	1	8	yes	—
185	C₃²×C₄	—	2	4	no	—
186	C₃²×C₄	—	2	8	no	—
187	C₃²×C₄	—	2	16	no	—
188	C₃²×C₄	—	2	16	no	—
189	C₃²×C₄	—	3	2	no	—
190	C₃²×C₄	—	4	16	no	—
191	C₃²×C₄	—	6	2	no	—
192	C₃²×C₄	D₃×D₄	1	2	yes	—
193	C₃²×C₄	#239	1	4	yes	complemented
194	C₃²×C₄	C₂×C₃×D₄	1	8	yes	—
195	C₃²×C₄	C₃×#41	1	16	yes	complemented
196	C₃×Dic₃	—	2	8	no	—
197	C₃×Dic₃	C₂×C₃×D₄	1	8	yes	—
198	C₂×#41	—	3	2	no	—
199	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
200	#127	—	3	1	no	—
201	#124	—	3	2	no	—
202	#121	—	3	4	no	—
203	C₄×D₄	—	3	4	no	—
204	C₂×C₄²	—	3	1	no	—
205	C₃³	#532	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	3	4	no	—
207	C₂³×C₃	—	3	16	no	—
208	C₂²×D₃	C₂³×C₃²	1	4	yes	—
209	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
210	C₃×Q₈	—	2	16	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	4	16	no	—
213	C₃×D₄	—	6	32	no	—
214	C₃×D₄	—	6	16	no	—
215	C₃×D₄	—	6	32	no	—
216	C₂×C₃×C₄	—	2	8	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	16	no	—
219	C₂×C₃×C₄	—	3	4	no	—
220	C₂×C₃×C₄	—	3	16	no	—
221	C₂×C₃×C₄	—	3	8	no	—
222	C₂×C₃×C₄	—	6	32	no	—
223	C₂×C₃×C₄	—	6	16	no	—
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
225	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
226	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
227	C₂×C₃×C₄	C₂³×C₃²	1	4	yes	—
228	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
230	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
231	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
232	D₁₂	—	2	8	no	—
233	D₁₂	—	2	4	no	—
234	D₁₂	—	2	8	no	—
235	C₄×D₃	—	2	4	no	—
236	C₄×D₃	—	2	8	no	—
237	C₄×D₃	C₃²×D₄	1	4	yes	complemented
238	C₂×C₃²	—	2	4	no	—
239	C₂×C₃²	—	2	8	no	—
240	C₂×C₃²	—	6	2	no	—
241	C₂×C₃²	—	12	4	no	—
242	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
243	C₂×C₃²	C₂×#239	1	2	yes	—
244	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
245	C₂×C₃²	C₂×C₃×#41	1	8	yes	—
246	C₃×D₃	—	2	8	no	—
247	C₃×D₃	—	4	16	no	—
248	#41	—	6	8	no	—
249	C₂×Q₈	C₂×C₃²×D₃	1	2	yes	—
250	C₂×D₄	—	3	4	no	—
251	C₂×D₄	—	3	4	no	—
252	C₂×D₄	—	3	2	no	—
253	C₂²×C₄	—	3	1	no	—
254	C₂²×C₄	—	3	2	no	—
255	C₂²×C₄	—	3	4	no	—
256	C₄⋊C₄	C₂×C₃²×D₃	1	4	yes	—
257	C₂²⋊C₄	—	3	8	no	—
258	C₄²	—	3	2	no	—
259	C₄²	—	3	1	no	—
260	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
261	C₂²×C₃	—	2	4	no	—
262	C₂²×C₃	—	3	8	no	—
263	C₂²×C₃	—	6	8	no	—
264	C₂²×C₃	—	6	16	no	—
265	C₂²×C₃	—	6	32	no	—
266	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
267	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
268	C₂×D₃	—	2	2	no	—
269	C₂×D₃	—	2	4	no	—
270	C₂×D₃	—	2	8	no	—
271	C₂×D₃	C₂×C₃²×D₄	1	2	yes	—
272	C₃×C₄	—	2	16	no	—
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	2	8	no	—
275	C₃×C₄	—	2	16	no	—
276	C₃×C₄	—	3	8	no	—
277	C₃×C₄	—	4	16	no	—
278	C₃×C₄	—	6	8	no	—
279	C₃×C₄	C₃×D₃×D₄	1	8	yes	—
280	C₃×C₄	C₃×#239	1	16	yes	complemented
281	C₃×C₄	C₂×C₃²×D₄	1	2	yes	—
282	C₃×C₄	C₃²×#41	1	4	yes	complemented
283	Dic₃	—	2	2	no	—
284	Dic₃	C₂×C₃²×D₄	1	2	yes	—
285	C₃²	—	2	4	no	—
286	C₃²	#5725	1	1	yes	characteristic, direct product factor
287	C₃²	C₃×#532	1	4	yes	complemented
288	C₂³	—	3	1	no	—
289	C₂³	—	3	4	no	—
290	Q₈	—	2	4	no	—
291	D₄	—	6	8	no	—
292	D₄	—	6	4	no	—
293	D₄	—	6	8	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	4	no	—
296	C₂×C₄	—	3	2	no	—
297	C₂×C₄	—	6	8	no	—
298	C₂×C₄	—	6	4	no	—
299	C₂×C₄	C₂²×C₃²×D₃	1	4	yes	—
300	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
302	C₂×C₃	—	2	4	no	—
303	C₂×C₃	—	2	8	no	—
304	C₂×C₃	—	6	8	no	—
305	C₂×C₃	—	12	16	no	—
306	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
307	C₂×C₃	C₂×C₃×#239	1	8	yes	—
308	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
309	C₂×C₃	C₂×C₃²×#41	1	2	yes	—
310	D₃	—	2	2	no	—
311	D₃	—	4	4	no	—
312	C₂²	—	3	2	no	—
313	C₂²	—	6	2	no	—
314	C₂²	—	6	4	no	—
315	C₂²	—	6	8	no	—
316	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	3	2	no	—
319	C₄	—	6	2	no	—
320	C₄	C₃²×D₃×D₄	1	2	yes	—
321	C₄	C₃²×#239	1	4	yes	complemented
322	C₃	—	2	4	no	—
323	C₃	C₃×#5725	1	4	yes	direct product factor
324	C₃	C₃²×#532	1	1	yes	characteristic, complemented
325	C₂	—	6	2	no	—
326	C₂	—	12	4	no	—
327	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic
328	C₂	C₂×C₃²×#239	1	2	yes	—

	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	C₂²×C₃³	C₂⁴
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#532
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
1. C₂×C₃²×#239	C₂	semidirect
2. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃²×#947	C₂	semidirect
4. C₃²×#895	C₂	semidirect
5. C₃²×#893	C₂	semidirect
6. C₃²×#874	C₂	semidirect
7. C₃²×C₄²×D₃	C₂	semidirect
9. C₃×#5725	C₃	direct (N × H)
24. C₂×C₃×#239	C₂×C₃	semidirect
26. C₃²×C₄⋊Q₈	D₃	semidirect
27. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
33. C₃×#947	C₂×C₃	semidirect
34. C₃×#895	C₂×C₃	semidirect
35. C₃×#893	C₂×C₃	semidirect
36. C₃×#874	C₂×C₃	semidirect
37. C₃×C₄²×D₃	C₂×C₃	semidirect
51. C₃²×C₄×D₃	D₄	semidirect
53. #5725	C₃²	direct (N × H)
90. C₃³×C₄	#41	semidirect
94. C₂×#239	C₂×C₃²	semidirect
96. C₃×C₄⋊Q₈	C₃×D₃	semidirect
97. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
103. #947	C₂×C₃²	semidirect
104. #895	C₂×C₃²	semidirect
105. #893	C₂×C₃²	semidirect
106. #874	C₂×C₃²	semidirect
107. C₄²×D₃	C₂×C₃²	semidirect
139. C₃×C₄×D₃	C₃×D₄	semidirect
193. C₃²×C₄	#239	semidirect
195. C₃²×C₄	C₃×#41	semidirect
199. C₄⋊Q₈	C₃²×D₃	semidirect
205. C₃³	#532	semidirect
237. C₄×D₃	C₃²×D₄	semidirect
280. C₃×C₄	C₃×#239	semidirect
282. C₃×C₄	C₃²×#41	semidirect
286. C₃²	#5725	direct (N × H)
287. C₃²	C₃×#532	semidirect
321. C₄	C₃²×#239	semidirect
323. C₃	C₃×#5725	direct (N × H)
324. 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×#239	2
C₃³×C₄⋊Q₈	2
C₃²×#947	2
C₃²×#895	2
C₃²×#893	2
C₃²×#874	2
C₃²×C₄²×D₃	2
C₃²×#532	3
C₃×#5725	3

Subgroups — C32×#5725 #13179103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5725 #13179103