Subgroups — C₂×C₃×#64121 #13179507

Order 1728.

1900 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×C₃×#3830	C₂	1	2	yes	maximal, complemented
2	C₂×C₃³×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3754	C₂	1	2	yes	maximal, complemented
4	C₃×#64121	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#3746	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64121	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#912	—	3	4	no	maximal
9	C₂×C₃³×D₄	C₂²	1	2	yes	—
10	C₂×C₃×#638	—	2	2	no	—
11	C₂²×C₃×C₃⋊Dic₃	—	2	2	no	—
12	C₃×#3830	—	2	4	no	—
13	C₃×#3830	C₂²	1	8	yes	complemented
14	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×#634	C₂²	1	2	yes	—
16	C₃³×D₈	C₂²	1	4	yes	complemented
17	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
18	C₃×#3754	C₂²	1	8	yes	complemented
19	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₃×#3746	C₂²	1	4	yes	complemented
21	C₃×C₈×C₃⋊D₃	C₂²	1	4	yes	complemented
22	C₂×#3830	C₂×C₃	1	2	yes	complemented
23	C₂×C₃×#237	—	3	8	no	—
24	C₂×C₃²×D₈	—	2	4	no	—
25	C₂×C₃²×D₈	D₃	1	4	yes	complemented
26	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#3754	C₂×C₃	1	2	yes	complemented
28	#64121	C₂×C₃	1	8	yes	direct product factor
29	C₂×#3746	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×C₃×#214	—	3	8	no	—
32	C₃×#912	—	3	32	no	—
33	C₂×C₃×Dic₁₂	—	3	4	no	—
34	C₂×C₃×D₃×C₈	—	3	4	no	—
35	C₂³×C₃³	—	2	2	no	—
36	C₂²×C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
37	C₃³×D₄	—	2	2	no	—
38	C₃³×D₄	C₂³	1	4	yes	—
39	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
40	C₃×#638	—	2	8	no	—
41	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
42	C₂×C₃×C₃⋊Dic₃	—	2	8	no	—
43	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
44	C₃×C₄×C₃⋊D₃	D₄	1	2	yes	—
45	C₃×C₄×C₃⋊D₃	C₂³	1	2	yes	—
46	C₃×#634	—	2	2	no	—
47	C₃×#634	C₂³	1	4	yes	—
48	C₃³×C₈	C₂³	1	2	yes	—
49	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
50	C₂×C₃×#135	—	9	1	no	—
51	C₂×#912	—	3	4	no	—
52	C₂×C₃²×D₄	—	2	8	no	—
53	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
54	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
55	C₂×#638	—	2	2	no	—
56	C₂²×C₃⋊Dic₃	—	2	2	no	—
57	#3830	—	2	4	no	—
58	#3830	C₂²×C₃	1	8	yes	complemented
59	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
60	C₂×#634	C₂²×C₃	1	2	yes	—
61	C₂×C₃×#67	—	6	8	no	—
62	C₂²×C₃×Dic₃	—	6	8	no	—
63	C₃×#237	—	3	32	no	—
64	C₃×#237	—	6	16	no	—
65	C₂×C₃×C₄×D₃	—	3	4	no	—
66	C₂×C₃×Dic₆	—	3	8	no	—
67	C₃²×D₈	—	2	16	no	—
68	C₃²×D₈	C₂×D₃	1	16	yes	complemented
69	C₃²×D₈	C₂²×C₃	1	4	yes	complemented
70	C₂×C₃²×C₈	—	2	4	no	—
71	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
72	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
73	#3754	C₂²×C₃	1	8	yes	complemented
74	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
75	#3746	C₂²×C₃	1	4	yes	complemented
76	C₈×C₃⋊D₃	C₂²×C₃	1	4	yes	complemented
77	C₃×#214	—	3	32	no	—
78	C₂×C₃×C₃⋊C₈	—	3	4	no	—
79	C₃×Dic₁₂	—	3	16	no	—
80	C₃×D₃×C₈	—	3	16	no	—
81	C₂²×C₃³	—	2	4	no	—
82	C₂²×C₃³	—	4	4	no	—
83	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₃⋊D₃	—	2	2	no	—
85	C₂×C₃×C₃⋊D₃	C₂×D₄	1	2	yes	—
86	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
87	C₃³×C₄	C₂⁴	1	1	yes	characteristic
88	C₃×C₃⋊Dic₃	—	2	4	no	—
89	C₃×C₃⋊Dic₃	C₂×D₄	1	2	yes	—
90	C₂×C₃×#41	—	9	2	no	—
91	C₂×#237	—	3	8	no	—
92	C₃×#135	—	9	8	no	—
93	C₂×C₃×Q₁₆	—	9	1	no	—
94	C₂×C₃×SD₁₆	—	9	2	no	—
95	C₂×C₃×D₈	—	2	4	no	—
96	C₂×C₃×D₈	C₃×D₃	1	4	yes	complemented
97	C₂×C₃×D₈	C₃⋊D₃	1	1	yes	characteristic, complemented
98	C₂²×C₃×C₈	—	9	1	no	—
99	C₂×#214	—	3	8	no	—
100	#912	—	3	32	no	—
101	C₂×Dic₁₂	—	3	4	no	—
102	C₂×D₃×C₈	—	3	4	no	—
103	C₂³×C₃²	—	2	2	no	—
104	C₂³×C₃²	—	2	8	no	—
105	C₂³×C₃²	—	4	8	no	—
106	C₂²×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₃×D₃	—	3	4	no	—
108	C₃²×D₄	—	2	2	no	—
109	C₃²×D₄	—	2	16	no	—
110	C₃²×D₄	—	2	8	no	—
111	C₃²×D₄	—	4	8	no	—
112	C₃²×D₄	C₂²×D₃	1	16	yes	—
113	C₃²×D₄	C₂³×C₃	1	4	yes	—
114	C₂×C₃²×C₄	—	2	4	no	—
115	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
116	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
117	#638	—	2	8	no	—
118	C₂×C₃⋊Dic₃	—	2	2	no	—
119	C₂×C₃⋊Dic₃	—	2	8	no	—
120	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
121	C₄×C₃⋊D₃	C₃×D₄	1	2	yes	—
122	C₄×C₃⋊D₃	C₂³×C₃	1	2	yes	—
123	#634	—	2	2	no	—
124	#634	C₂³×C₃	1	4	yes	—
125	C₃×#67	—	6	32	no	—
126	C₂×C₃×Dic₃	—	3	4	no	—
127	C₂×C₃×Dic₃	—	6	32	no	—
128	C₂×C₃×Dic₃	—	6	8	no	—
129	C₃×C₄×D₃	—	3	8	no	—
130	C₃×C₄×D₃	—	3	8	no	—
131	C₃×Dic₆	—	3	16	no	—
132	C₃×Dic₆	—	6	8	no	—
133	C₃²×C₈	—	2	8	no	—
134	C₃²×C₈	C₂²×D₃	1	8	yes	—
135	C₃²×C₈	C₂³×C₃	1	2	yes	—
136	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	2	yes	—
137	C₃×C₃⋊C₈	—	3	8	no	—
138	C₂×#135	—	9	1	no	Sylow
139	C₂×C₃³	—	4	4	no	—
140	C₂×C₃³	#135	1	2	yes	complemented
141	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
142	C₃×C₃⋊D₃	—	2	2	no	—
143	C₃×#41	—	9	8	no	—
144	C₃×#41	—	18	4	no	—
145	C₂×C₃×Q₈	—	9	2	no	—
146	C₂×C₃×D₄	—	2	8	no	—
147	C₂×C₃×D₄	—	18	2	no	—
148	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
149	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
150	C₂²×C₃×C₄	—	9	1	no	—
151	C₂²×C₃×C₄	—	18	2	no	—
152	C₂×#67	—	6	8	no	—
153	C₂²×Dic₃	—	6	8	no	—
154	#237	—	3	32	no	—
155	#237	—	6	16	no	—
156	C₂×C₄×D₃	—	3	4	no	—
157	C₂×Dic₆	—	3	8	no	—
158	C₃×Q₁₆	—	9	4	no	—
159	C₃×SD₁₆	—	9	8	no	—
160	C₃×D₈	—	2	16	no	—
161	C₃×D₈	C₂×C₃×D₃	1	16	yes	complemented
162	C₃×D₈	C₂×C₃⋊D₃	1	4	yes	complemented
163	C₂×C₃×C₈	—	2	4	no	—
164	C₂×C₃×C₈	—	9	1	no	—
165	C₂×C₃×C₈	—	9	4	no	—
166	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
167	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
168	#214	—	3	32	no	—
169	C₂×C₃⋊C₈	—	3	4	no	—
170	Dic₁₂	—	3	16	no	—
171	D₃×C₈	—	3	16	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	—	2	4	no	—
174	C₂²×C₃²	—	2	16	no	—
175	C₂²×C₃²	—	4	16	no	—
176	C₂²×C₃²	—	4	4	no	—
177	C₂²×C₃²	—	4	16	no	—
178	C₂²×C₃²	—	8	16	no	—
179	C₂²×C₃²	D₃×D₄	1	4	yes	—
180	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₃⋊D₃	—	2	2	no	—
182	C₂×C₃⋊D₃	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₃×D₃	—	3	8	no	—
184	C₂×C₃×D₃	—	6	8	no	—
185	C₃²×C₄	—	2	4	no	—
186	C₃²×C₄	—	2	4	no	—
187	C₃²×C₄	D₃×D₄	1	4	yes	—
188	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₃²×C₄	C₂³×D₃	1	4	yes	—
190	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
191	C₃⋊Dic₃	—	2	4	no	—
192	C₃⋊Dic₃	C₂×C₃×D₄	1	2	yes	—
193	C₃×Dic₃	—	3	8	no	—
194	C₃×Dic₃	—	6	16	no	—
195	C₂×#41	—	9	2	no	—
196	#135	—	9	8	no	—
197	C₂×Q₁₆	—	9	1	no	—
198	C₂×SD₁₆	—	9	2	no	—
199	C₂×D₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
200	C₂²×C₈	—	9	1	no	—
201	C₃³	C₂×#135	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	2	8	no	—
203	C₂³×C₃	—	2	2	no	—
204	C₂³×C₃	—	4	8	no	—
205	C₂³×C₃	—	9	1	no	—
206	C₂²×D₃	—	3	4	no	—
207	C₃×Q₈	—	9	4	no	—
208	C₃×Q₈	—	18	2	no	—
209	C₃×D₄	—	2	8	no	—
210	C₃×D₄	—	2	2	no	—
211	C₃×D₄	—	2	16	no	—
212	C₃×D₄	—	4	8	no	—
213	C₃×D₄	—	18	8	no	—
214	C₃×D₄	C₂²×C₃×D₃	1	16	yes	—
215	C₃×D₄	C₂²×C₃⋊D₃	1	4	yes	—
216	C₂×C₃×C₄	—	2	4	no	—
217	C₂×C₃×C₄	—	9	1	no	—
218	C₂×C₃×C₄	—	9	2	no	—
219	C₂×C₃×C₄	—	9	2	no	—
220	C₂×C₃×C₄	—	18	8	no	—
221	C₂×C₃×C₄	—	18	2	no	—
222	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
223	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
224	#67	—	6	32	no	—
225	C₂×Dic₃	—	3	4	no	—
226	C₂×Dic₃	—	6	32	no	—
227	C₂×Dic₃	—	6	8	no	—
228	C₄×D₃	—	3	8	no	—
229	C₄×D₃	—	3	8	no	—
230	Dic₆	—	3	16	no	—
231	Dic₆	—	6	8	no	—
232	C₃×C₈	—	2	8	no	—
233	C₃×C₈	—	9	2	no	—
234	C₃×C₈	C₂²×C₃×D₃	1	8	yes	—
235	C₃×C₈	C₂²×C₃⋊D₃	1	2	yes	—
236	C₃⋊C₈	—	3	8	no	—
237	C₂×C₃²	—	2	8	no	—
238	C₂×C₃²	—	2	4	no	—
239	C₂×C₃²	—	4	4	no	—
240	C₂×C₃²	—	4	16	no	—
241	C₂×C₃²	—	8	16	no	—
242	C₂×C₃²	#912	1	8	yes	complemented
243	C₂×C₃²	C₃×#135	1	2	yes	complemented
244	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
245	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
246	C₃⋊D₃	—	2	2	no	—
247	C₃×D₃	—	6	8	no	—
248	#41	—	9	8	no	—
249	#41	—	18	4	no	—
250	C₂×Q₈	—	9	2	no	—
251	C₂×D₄	—	18	2	no	—
252	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
253	C₂²×C₄	—	9	1	no	—
254	C₂²×C₄	—	18	2	no	—
255	Q₁₆	—	9	4	no	—
256	SD₁₆	—	9	8	no	—
257	D₈	C₂×C₃×C₃⋊D₃	1	4	yes	complemented
258	C₂×C₈	—	9	1	no	—
259	C₂×C₈	—	9	4	no	—
260	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
261	C₂²×C₃	—	2	16	no	—
262	C₂²×C₃	—	2	4	no	—
263	C₂²×C₃	—	2	4	no	—
264	C₂²×C₃	—	4	16	no	—
265	C₂²×C₃	—	4	16	no	—
266	C₂²×C₃	—	4	4	no	—
267	C₂²×C₃	—	8	16	no	—
268	C₂²×C₃	—	9	2	no	—
269	C₂²×C₃	—	18	2	no	—
270	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
271	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
272	C₂×D₃	—	3	8	no	—
273	C₂×D₃	—	6	8	no	—
274	C₃×C₄	—	2	4	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	9	2	no	—
277	C₃×C₄	—	18	4	no	—
278	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
279	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
280	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
281	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
282	Dic₃	—	3	8	no	—
283	Dic₃	—	6	16	no	—
284	C₃²	—	2	4	no	—
285	C₃²	C₂×#912	1	4	yes	complemented
286	C₃²	C₂×C₃×#135	1	1	yes	characteristic, complemented
287	C₂³	—	2	2	no	—
288	C₂³	—	9	1	no	—
289	Q₈	—	9	4	no	—
290	Q₈	—	18	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	18	8	no	—
293	D₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	9	2	no	—
296	C₂×C₄	—	9	2	no	—
297	C₂×C₄	—	18	8	no	—
298	C₂×C₄	—	18	2	no	—
299	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
300	C₈	—	9	2	no	—
301	C₈	C₂²×C₃×C₃⋊D₃	1	2	yes	—
302	C₂×C₃	—	2	8	no	—
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	—	4	16	no	—
305	C₂×C₃	—	4	4	no	—
306	C₂×C₃	—	8	16	no	—
307	C₂×C₃	—	18	2	no	—
308	C₂×C₃	C₃×#912	1	8	yes	complemented
309	C₂×C₃	#64121	1	2	yes	direct product factor
310	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
311	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
312	D₃	—	6	8	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	9	2	no	—
316	C₂²	—	18	2	no	—
317	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
318	C₄	—	9	2	no	—
319	C₄	—	18	4	no	—
320	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
321	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
322	C₃	—	2	4	no	—
323	C₃	C₂×C₃×#912	1	4	yes	complemented
324	C₃	C₂×#64121	1	1	yes	characteristic, direct product factor
325	C₂	—	4	4	no	—
326	C₂	—	18	2	no	—
327	C₂	C₃×#64121	1	2	yes	direct product factor
328	C₂	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₂×C₃³×D₈	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₂×C₃×D₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3830	C₂	semidirect
2. C₂×C₃³×D₈	C₂	semidirect
3. C₂×C₃×#3754	C₂	semidirect
4. C₃×#64121	C₂	direct (N × H)
5. C₂×C₃×#3746	C₂	semidirect
6. C₂×C₃×C₈×C₃⋊D₃	C₂	semidirect
7. C₂×#64121	C₃	direct (N × H)
13. C₃×#3830	C₂²	semidirect
16. C₃³×D₈	C₂²	semidirect
18. C₃×#3754	C₂²	semidirect
20. C₃×#3746	C₂²	semidirect
21. C₃×C₈×C₃⋊D₃	C₂²	semidirect
22. C₂×#3830	C₂×C₃	semidirect
25. C₂×C₃²×D₈	D₃	semidirect
26. C₂×C₃²×D₈	C₂×C₃	semidirect
27. C₂×#3754	C₂×C₃	semidirect
28. #64121	C₂×C₃	direct (N × H)
29. C₂×#3746	C₂×C₃	semidirect
30. C₂×C₈×C₃⋊D₃	C₂×C₃	semidirect
58. #3830	C₂²×C₃	semidirect
68. C₃²×D₈	C₂×D₃	semidirect
69. C₃²×D₈	C₂²×C₃	semidirect
73. #3754	C₂²×C₃	semidirect
75. #3746	C₂²×C₃	semidirect
76. C₈×C₃⋊D₃	C₂²×C₃	semidirect
96. C₂×C₃×D₈	C₃×D₃	semidirect
97. C₂×C₃×D₈	C₃⋊D₃	semidirect
140. C₂×C₃³	#135	semidirect
161. C₃×D₈	C₂×C₃×D₃	semidirect
162. C₃×D₈	C₂×C₃⋊D₃	semidirect
199. C₂×D₈	C₃×C₃⋊D₃	semidirect
201. C₃³	C₂×#135	semidirect
242. C₂×C₃²	#912	semidirect
243. C₂×C₃²	C₃×#135	semidirect
257. D₈	C₂×C₃×C₃⋊D₃	semidirect
285. C₃²	C₂×#912	semidirect
286. C₃²	C₂×C₃×#135	semidirect
308. C₂×C₃	C₃×#912	semidirect
309. C₂×C₃	#64121	direct (N × H)
323. C₃	C₂×C₃×#912	semidirect
324. C₃	C₂×#64121	direct (N × H)
327. C₂	C₃×#64121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3830	2
C₂×C₃³×D₈	2
C₂×C₃×#3754	2
C₃×#64121	2
C₂×C₃×#3746	2
C₂×C₃×C₈×C₃⋊D₃	2
C₂×#64121	3
C₂×C₃×#912	3

Subgroups — C2×C3×#64121 #13179507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64121 #13179507