Subgroups — C₃×#10592926 #13163268

Order 1728.

1572 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3755	C₂	1	2	yes	maximal, complemented
3	C₃×#64139	C₂	1	1	yes	maximal, characteristic
4	C₃×#63662	C₂	1	2	yes	maximal, complemented
5	C₃×#63657	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592926	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5227	—	3	4	no	maximal
8	C₂×C₃³×Q₈	—	2	4	no	—
9	C₂×C₃³×Q₈	—	2	1	no	—
10	C₂×C₃³×Q₈	C₂²	1	2	yes	—
11	C₂²×C₃³×C₄	—	2	1	no	—
12	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×#638	—	2	1	no	—
14	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
15	C₃×#3755	—	2	4	no	—
16	C₃×#3752	—	2	1	no	—
17	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₂²×C₃²×Q₈	—	2	4	no	—
20	C₂²×C₃²×Q₈	D₃	1	4	yes	complemented
21	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×#3755	C₂×C₃	1	2	yes	complemented
23	#64139	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#215	—	3	8	no	—
25	C₃×#930	—	3	4	no	—
26	#63662	C₂×C₃	1	2	yes	complemented
27	#63657	C₂×C₃	1	1	yes	characteristic, complemented
28	C₃×#835	—	3	8	no	—
29	C₃×#830	—	3	4	no	—
30	C₂³×C₃³	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
32	C₃³×Q₈	—	2	4	no	—
33	C₃³×Q₈	—	2	2	no	—
34	C₃³×Q₈	D₄	1	4	yes	complemented
35	C₂×C₃³×C₄	—	2	4	no	—
36	C₂×C₃³×C₄	—	2	2	no	—
37	C₂×C₃³×C₄	—	2	2	no	—
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
40	C₃×#638	—	4	2	no	—
41	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
42	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
43	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
44	C₃×#448	—	9	1	no	—
45	#5227	—	3	4	no	—
46	C₂×C₃²×Q₈	—	2	32	no	—
47	C₂×C₃²×Q₈	—	2	8	no	—
48	C₂×C₃²×Q₈	—	2	8	no	—
49	C₂×C₃²×Q₈	—	2	16	no	—
50	C₂×C₃²×Q₈	—	2	4	no	—
51	C₂×C₃²×Q₈	—	2	4	no	—
52	C₂×C₃²×Q₈	—	2	1	no	—
53	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
54	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
55	C₂²×C₃²×C₄	—	2	8	no	—
56	C₂²×C₃²×C₄	—	2	4	no	—
57	C₂²×C₃²×C₄	—	2	4	no	—
58	C₂²×C₃²×C₄	—	2	1	no	—
59	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
60	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×#638	—	2	1	no	—
62	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×#67	—	6	4	no	—
64	C₂×C₃×D₁₂	—	3	4	no	—
65	#3755	—	2	4	no	—
66	#3752	—	2	1	no	—
67	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
69	C₃×#215	—	6	16	no	—
70	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
71	C₃×C₄⋊Dic₃	—	3	4	no	—
72	C₂×C₃×C₃⋊C₈	—	3	8	no	—
73	C₂²×C₃³	—	2	1	no	—
74	C₂²×C₃³	—	2	1	no	—
75	C₂²×C₃³	SD₁₆	1	2	yes	complemented
76	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₃⋊D₃	—	4	1	no	—
78	C₂×C₃×C₃⋊D₃	—	4	1	no	—
79	C₂×C₃×C₃⋊D₃	—	4	1	no	—
80	C₃³×C₄	—	2	4	no	—
81	C₃³×C₄	—	2	1	no	—
82	C₃³×C₄	C₂×D₄	1	2	yes	—
83	C₃×C₃⋊Dic₃	—	4	1	no	—
84	C₂²×C₃×Q₈	—	2	4	no	—
85	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	complemented
86	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
87	C₂×C₃×SD₁₆	—	9	2	no	—
88	C₃×#121	—	9	1	no	—
89	C₂×#215	—	3	8	no	—
90	#930	—	3	4	no	—
91	C₃×#103	—	9	2	no	—
92	C₃×C₂²⋊C₈	—	9	1	no	—
93	#835	—	3	8	no	—
94	#830	—	3	4	no	—
95	C₂³×C₃²	—	2	4	no	—
96	C₂³×C₃²	#67	1	4	yes	—
97	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₃⋊D₃	—	2	1	no	—
99	C₂²×C₃×D₃	—	6	4	no	—
100	C₃²×Q₈	—	2	32	no	—
101	C₃²×Q₈	—	2	16	no	—
102	C₃²×Q₈	—	2	4	no	—
103	C₃²×Q₈	—	2	16	no	—
104	C₃²×Q₈	—	2	8	no	—
105	C₃²×Q₈	—	2	2	no	—
106	C₃²×Q₈	—	2	16	no	—
107	C₃²×Q₈	#67	1	16	yes	complemented
108	C₃²×Q₈	C₃×D₄	1	4	yes	complemented
109	C₂×C₃²×C₄	—	2	32	no	—
110	C₂×C₃²×C₄	—	2	16	no	—
111	C₂×C₃²×C₄	—	2	4	no	—
112	C₂×C₃²×C₄	—	2	16	no	—
113	C₂×C₃²×C₄	—	2	16	no	—
114	C₂×C₃²×C₄	—	2	8	no	—
115	C₂×C₃²×C₄	—	2	2	no	—
116	C₂×C₃²×C₄	—	2	8	no	—
117	C₂×C₃²×C₄	—	2	2	no	—
118	C₂×C₃²×C₄	—	2	4	no	—
119	C₂×C₃²×C₄	—	2	4	no	—
120	C₂×C₃²×C₄	#67	1	4	yes	—
121	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
122	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
123	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
124	#638	—	4	2	no	—
125	C₂×C₃⋊Dic₃	—	2	1	no	—
126	C₄⋊(C₃⋊D₃)	—	2	2	no	—
127	C₃×#67	—	12	8	no	—
128	C₂×C₃×Dic₃	—	6	4	no	—
129	C₃×D₁₂	—	6	8	no	—
130	C₃⋊(C₃⋊C₈)	—	2	2	no	—
131	C₃×C₃⋊C₈	—	6	8	no	—
132	#448	—	9	1	no	Sylow
133	C₂×C₃³	—	2	2	no	—
134	C₂×C₃³	#120	1	1	yes	characteristic
135	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₃³	#137	1	1	yes	characteristic
137	C₃×C₃⋊D₃	—	8	1	no	—
138	C₂×C₃×Q₈	—	2	32	no	—
139	C₂×C₃×Q₈	—	2	8	no	—
140	C₂×C₃×Q₈	—	2	16	no	—
141	C₂×C₃×Q₈	—	2	4	no	—
142	C₂×C₃×Q₈	—	2	4	no	—
143	C₂×C₃×Q₈	—	2	1	no	—
144	C₂×C₃×Q₈	—	2	8	no	—
145	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
146	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
147	C₂×C₃×D₄	—	9	1	no	—
148	C₂×C₃×D₄	—	18	1	no	—
149	C₂²×C₃×C₄	—	2	8	no	—
150	C₂²×C₃×C₄	—	2	4	no	—
151	C₂²×C₃×C₄	—	2	1	no	—
152	C₂²×C₃×C₄	—	2	4	no	—
153	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
154	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
155	C₂×#67	—	6	4	no	—
156	C₂×D₁₂	—	3	4	no	—
157	C₃×SD₁₆	—	18	4	no	—
158	C₂×C₃×C₈	—	9	2	no	—
159	C₃×C₄⋊C₄	—	9	1	no	—
160	C₃×C₂²⋊C₄	—	18	1	no	—
161	#215	—	6	16	no	—
162	(C₂×D₃)⋊C₄	—	6	4	no	—
163	C₄⋊Dic₃	—	3	4	no	—
164	C₂×C₃⋊C₈	—	3	8	no	—
165	C₂²×C₃²	—	2	8	no	—
166	C₂²×C₃²	—	2	8	no	—
167	C₂²×C₃²	—	2	4	no	—
168	C₂²×C₃²	—	2	4	no	—
169	C₂²×C₃²	—	2	1	no	—
170	C₂²×C₃²	—	2	1	no	—
171	C₂²×C₃²	—	2	8	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	#215	1	8	yes	complemented
174	C₂²×C₃²	C₃×SD₁₆	1	2	yes	complemented
175	C₂²×C₃²	C₂×#67	1	4	yes	—
176	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₃⋊D₃	—	4	1	no	—
178	C₂×C₃⋊D₃	—	4	1	no	—
179	C₂×C₃⋊D₃	—	4	1	no	—
180	C₂×C₃×D₃	—	12	4	no	—
181	C₂×C₃×D₃	—	12	4	no	—
182	C₂×C₃×D₃	—	12	4	no	—
183	C₃²×C₄	—	2	32	no	—
184	C₃²×C₄	—	2	8	no	—
185	C₃²×C₄	—	2	16	no	—
186	C₃²×C₄	—	2	4	no	—
187	C₃²×C₄	—	2	4	no	—
188	C₃²×C₄	—	2	1	no	—
189	C₃²×C₄	—	2	8	no	—
190	C₃²×C₄	C₂×#67	1	8	yes	—
191	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
192	C₃⋊Dic₃	—	4	1	no	—
193	C₃×Dic₃	—	12	4	no	—
194	C₂²×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
195	C₂×SD₁₆	—	9	2	no	—
196	#121	—	9	1	no	—
197	#103	—	9	2	no	—
198	C₂²⋊C₈	—	9	1	no	—
199	C₃³	#448	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₃	—	2	4	no	—
201	C₂³×C₃	—	18	1	no	—
202	C₂³×C₃	C₃×#67	1	4	yes	—
203	C₂³×C₃	#638	1	1	yes	characteristic
204	C₂²×D₃	—	6	4	no	—
205	C₃×Q₈	—	2	32	no	—
206	C₃×Q₈	—	2	16	no	—
207	C₃×Q₈	—	2	4	no	—
208	C₃×Q₈	—	2	16	no	—
209	C₃×Q₈	—	2	8	no	—
210	C₃×Q₈	—	2	2	no	—
211	C₃×Q₈	—	2	16	no	—
212	C₃×Q₈	C₃×#67	1	16	yes	complemented
213	C₃×Q₈	#638	1	4	yes	complemented
214	C₃×D₄	—	18	2	no	—
215	C₃×D₄	—	36	2	no	—
216	C₂×C₃×C₄	—	2	32	no	—
217	C₂×C₃×C₄	—	2	16	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	16	no	—
220	C₂×C₃×C₄	—	2	16	no	—
221	C₂×C₃×C₄	—	2	8	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	8	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	—	18	1	no	—
228	C₂×C₃×C₄	C₃×#67	1	4	yes	—
229	C₂×C₃×C₄	#638	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
231	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
232	#67	—	12	8	no	—
233	C₂×Dic₃	—	6	4	no	—
234	D₁₂	—	6	8	no	—
235	C₃×C₈	—	18	2	no	—
236	C₃⋊C₈	—	6	8	no	—
237	C₂×C₃²	—	2	16	no	—
238	C₂×C₃²	—	2	8	no	—
239	C₂×C₃²	—	2	2	no	—
240	C₂×C₃²	—	2	4	no	—
241	C₂×C₃²	—	2	4	no	—
242	C₂×C₃²	—	2	4	no	—
243	C₂×C₃²	C₂×#215	1	4	yes	—
244	C₂×C₃²	#942	1	4	yes	—
245	C₂×C₃²	#953	1	4	yes	—
246	C₂×C₃²	C₃×#120	1	1	yes	characteristic
247	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
248	C₂×C₃²	C₃×#137	1	1	yes	characteristic
249	C₃⋊D₃	—	8	1	no	—
250	C₃×D₃	—	24	4	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
254	C₂×D₄	—	9	1	no	—
255	C₂×D₄	—	18	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
258	SD₁₆	—	18	4	no	—
259	C₂×C₈	—	9	2	no	—
260	C₄⋊C₄	—	9	1	no	—
261	C₂²⋊C₄	—	18	1	no	—
262	C₂²×C₃	—	2	8	no	—
263	C₂²×C₃	—	2	8	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	—	2	4	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	8	no	—
269	C₂²×C₃	—	2	4	no	—
270	C₂²×C₃	—	36	1	no	—
271	C₂²×C₃	—	36	1	no	—
272	C₂²×C₃	—	36	1	no	—
273	C₂²×C₃	C₃×#215	1	8	yes	complemented
274	C₂²×C₃	#3755	1	2	yes	complemented
275	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
276	C₂²×C₃	C₂×#638	1	1	yes	characteristic
277	C₂×D₃	—	12	4	no	—
278	C₂×D₃	—	12	4	no	—
279	C₂×D₃	—	12	4	no	—
280	C₃×C₄	—	2	32	no	—
281	C₃×C₄	—	2	8	no	—
282	C₃×C₄	—	2	16	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	8	no	—
287	C₃×C₄	—	36	1	no	—
288	C₃×C₄	C₂×C₃×#67	1	8	yes	—
289	C₃×C₄	C₂×#638	1	2	yes	—
290	Dic₃	—	12	4	no	—
291	C₃²	—	2	4	no	—
292	C₃²	#5227	1	4	yes	complemented
293	C₃²	C₃×#448	1	1	yes	characteristic, complemented
294	C₂³	—	18	1	no	—
295	C₂³	C₃×#638	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	2	2	no	—
298	Q₈	C₃×#638	1	4	yes	complemented
299	D₄	—	18	2	no	—
300	D₄	—	36	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	18	1	no	—
305	C₂×C₄	C₃×#638	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
307	C₈	—	18	2	no	—
308	C₂×C₃	—	2	16	no	—
309	C₂×C₃	—	2	8	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	2	4	no	—
312	C₂×C₃	—	2	4	no	—
313	C₂×C₃	—	2	4	no	—
314	C₂×C₃	—	72	1	no	—
315	C₂×C₃	C₂×C₃×#215	1	4	yes	—
316	C₂×C₃	C₃×#942	1	4	yes	—
317	C₂×C₃	C₃×#953	1	4	yes	—
318	C₂×C₃	C₂×#3755	1	1	yes	characteristic
319	C₂×C₃	#64151	1	1	yes	characteristic
320	C₂×C₃	#64162	1	1	yes	characteristic
321	D₃	—	24	4	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	36	1	no	—
325	C₂²	—	36	1	no	—
326	C₂²	—	36	1	no	—
327	C₂²	C₃×#3755	1	2	yes	complemented
328	C₂²	C₂×C₃×#638	1	1	yes	characteristic
329	C₄	—	2	4	no	—
330	C₄	—	2	1	no	—
331	C₄	—	36	1	no	—
332	C₄	C₂×C₃×#638	1	2	yes	—
333	C₃	—	2	4	no	—
334	C₃	C₃×#5227	1	4	yes	complemented
335	C₃	#10592926	1	1	yes	characteristic, direct product factor
336	C₂	—	2	2	no	—
337	C₂	—	72	1	no	—
338	C₂	C₂×C₃×#3755	1	1	yes	characteristic
339	C₂	C₃×#64151	1	1	yes	characteristic
340	C₂	C₃×#64162	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	C₂²×C₃³×Q₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃×#64139	C₂
hypercenter	C₂²×C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#448
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
2. C₂×C₃×#3755	C₂	semidirect
4. C₃×#63662	C₂	semidirect
5. C₃×#63657	C₂	semidirect
6. #10592926	C₃	direct (N × H)
20. C₂²×C₃²×Q₈	D₃	semidirect
21. C₂²×C₃²×Q₈	C₂×C₃	semidirect
22. C₂×#3755	C₂×C₃	semidirect
26. #63662	C₂×C₃	semidirect
27. #63657	C₂×C₃	semidirect
34. C₃³×Q₈	D₄	semidirect
75. C₂²×C₃³	SD₁₆	semidirect
85. C₂²×C₃×Q₈	C₃×D₃	semidirect
86. C₂²×C₃×Q₈	C₃⋊D₃	semidirect
107. C₃²×Q₈	#67	semidirect
108. C₃²×Q₈	C₃×D₄	semidirect
173. C₂²×C₃²	#215	semidirect
174. C₂²×C₃²	C₃×SD₁₆	semidirect
194. C₂²×Q₈	C₃×C₃⋊D₃	semidirect
199. C₃³	#448	semidirect
212. C₃×Q₈	C₃×#67	semidirect
213. C₃×Q₈	#638	semidirect
273. C₂²×C₃	C₃×#215	semidirect
274. C₂²×C₃	#3755	semidirect
292. C₃²	#5227	semidirect
293. C₃²	C₃×#448	semidirect
298. Q₈	C₃×#638	semidirect
327. C₂²	C₃×#3755	semidirect
334. C₃	C₃×#5227	semidirect
335. C₃	#10592926	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×Q₈	2
C₂×C₃×#3755	2
C₃×#64139	2
C₃×#63662	2
C₃×#63657	2
#10592926	3
C₃×#5227	3

Subgroups — C3×#10592926 #13163268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592926 #13163268