Subgroups — C₂×C₃×#3138 #10679220

Order 768.

2806 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	C₂×C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₃×#3138	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#490	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#486	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×#418	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#3138	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₃×D₄	—	2	2	no	—
10	C₂³×C₃×D₄	C₂²	1	2	yes	—
11	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
12	C₃×#533	—	2	4	no	—
13	C₃×#533	C₂²	1	8	yes	complemented
14	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
15	C₂×C₃×#127	—	2	2	no	—
16	C₂×C₃×#127	—	2	4	no	—
17	C₂×C₃×#127	C₂²	1	2	yes	—
18	C₂×C₃×#127	C₂²	1	2	yes	—
19	C₂×C₃×#127	C₂²	1	8	yes	—
20	C₂×C₃×#125	C₂²	1	2	yes	complemented
21	C₂×C₃×C₂²⋊Q₈	—	2	2	no	—
22	C₂×C₃×C₄×Q₈	—	2	2	no	—
23	C₂×C₃×#117	C₂²	1	2	yes	—
24	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#490	C₂²	1	32	yes	complemented
26	C₃×#486	C₂²	1	32	yes	complemented
27	C₃×#418	C₂²	1	32	yes	complemented
28	C₂×C₃×#102	C₂²	1	16	yes	complemented
29	C₃×C₄×M₁₆	C₂²	1	8	yes	complemented
30	C₂×C₃×#97	C₂²	1	2	yes	complemented
31	C₂×C₃×C₄×C₈	C₂²	1	2	yes	complemented
32	C₂×#533	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂²×#127	C₂×C₃	1	1	yes	characteristic
34	#3138	C₂×C₃	1	16	yes	direct product factor
35	C₂×#490	C₂×C₃	1	4	yes	complemented
36	C₂×#486	C₂×C₃	1	4	yes	complemented
37	C₂×#418	C₂×C₃	1	4	yes	complemented
38	C₂×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁵×C₃	—	2	2	no	—
40	C₂²×C₃×Q₈	—	2	2	no	—
41	C₂²×C₃×D₄	—	2	16	no	—
42	C₂²×C₃×D₄	—	2	4	no	—
43	C₂²×C₃×D₄	—	2	4	no	—
44	C₂²×C₃×D₄	—	2	4	no	—
45	C₂²×C₃×D₄	—	2	2	no	—
46	C₂²×C₃×D₄	—	2	16	no	—
47	C₂²×C₃×D₄	—	4	16	no	—
48	C₂²×C₃×D₄	C₂³	1	4	yes	—
49	C₂²×C₃×D₄	C₂³	1	8	yes	—
50	C₂³×C₃×C₄	D₄	1	2	yes	—
51	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂×C₃×M₁₆	—	2	8	no	—
53	C₂×C₃×M₁₆	C₂³	1	8	yes	—
54	C₂²×C₃×C₈	C₂³	1	4	yes	—
55	C₃×C₄⋊Q₈	—	2	4	no	—
56	C₃×C₄⋊Q₈	C₂³	1	8	yes	complemented
57	C₃×#127	—	2	16	no	—
58	C₃×#127	—	2	4	no	—
59	C₃×#127	—	2	4	no	—
60	C₃×#127	C₂³	1	8	yes	—
61	C₃×#127	C₂³	1	8	yes	—
62	C₃×#125	—	2	4	no	—
63	C₃×#125	C₂³	1	8	yes	complemented
64	C₃×C₂²⋊Q₈	—	2	16	no	—
65	C₃×C₄×Q₈	—	2	16	no	—
66	C₃×#117	—	2	4	no	—
67	C₃×#117	C₂³	1	8	yes	—
68	C₂×C₃×C₄⋊C₄	—	2	4	no	—
69	C₂×C₃×C₄⋊C₄	—	2	2	no	—
70	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
71	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₃×C₄²	—	2	2	no	—
73	C₂×C₃×C₄²	D₄	1	2	yes	—
74	C₂×C₃×C₄²	D₄	1	4	yes	—
75	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₄²	C₂³	1	4	yes	—
78	C₃×#102	C₂³	1	64	yes	complemented
79	C₃×#97	C₂³	1	8	yes	complemented
80	C₃×C₄×C₈	C₂³	1	8	yes	complemented
81	C₂³×D₄	—	2	2	no	—
82	C₂³×D₄	C₂²×C₃	1	2	yes	—
83	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
84	#533	—	2	4	no	—
85	#533	C₂²×C₃	1	8	yes	complemented
86	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
87	C₂×#127	—	2	2	no	—
88	C₂×#127	—	2	4	no	—
89	C₂×#127	C₂²×C₃	1	2	yes	—
90	C₂×#127	C₂²×C₃	1	2	yes	—
91	C₂×#127	C₂²×C₃	1	8	yes	—
92	C₂×#125	C₂²×C₃	1	2	yes	complemented
93	C₂×C₂²⋊Q₈	—	2	2	no	—
94	C₂×C₄×Q₈	—	2	2	no	—
95	C₂×#117	C₂²×C₃	1	2	yes	—
96	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
97	#490	C₂²×C₃	1	32	yes	complemented
98	#486	C₂²×C₃	1	32	yes	complemented
99	#418	C₂²×C₃	1	32	yes	complemented
100	C₂×#102	C₂²×C₃	1	16	yes	complemented
101	C₄×M₁₆	C₂²×C₃	1	8	yes	complemented
102	C₂×#97	C₂²×C₃	1	2	yes	complemented
103	C₂×C₄×C₈	C₂²×C₃	1	2	yes	complemented
104	C₂⁴×C₃	—	2	8	no	—
105	C₂⁴×C₃	—	2	4	no	—
106	C₂⁴×C₃	—	4	16	no	—
107	C₂⁴×C₃	—	4	8	no	—
108	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₃×Q₈	—	2	8	no	—
110	C₂×C₃×Q₈	—	4	8	no	—
111	C₂×C₃×D₄	—	2	8	no	—
112	C₂×C₃×D₄	—	2	4	no	—
113	C₂×C₃×D₄	—	2	16	no	—
114	C₂×C₃×D₄	—	2	16	no	—
115	C₂×C₃×D₄	—	2	16	no	—
116	C₂×C₃×D₄	—	2	32	no	—
117	C₂×C₃×D₄	—	4	16	no	—
118	C₂×C₃×D₄	—	4	64	no	—
119	C₂×C₃×D₄	—	4	16	no	—
120	C₂×C₃×D₄	—	4	8	no	—
121	C₂×C₃×D₄	—	4	16	no	—
122	C₂×C₃×D₄	C₂⁴	1	8	yes	—
123	C₂²×C₃×C₄	—	2	8	no	—
124	C₂²×C₃×C₄	—	2	4	no	—
125	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
126	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
127	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
128	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
130	C₂²×C₃×C₄	#41	1	8	yes	—
131	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
132	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
133	C₃×M₁₆	—	2	16	no	—
134	C₂×C₃×C₈	—	2	8	no	—
135	C₂×C₃×C₈	C₂⁴	1	8	yes	—
136	C₃×C₄⋊C₄	—	2	16	no	—
137	C₃×C₄⋊C₄	—	2	8	no	—
138	C₃×C₄⋊C₄	—	2	4	no	—
139	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
140	C₃×C₂²⋊C₄	—	2	8	no	—
141	C₃×C₄²	—	2	8	no	—
142	C₃×C₄²	C₂×D₄	1	8	yes	—
143	C₃×C₄²	C₂×D₄	1	2	yes	—
144	C₃×C₄²	C₂×D₄	1	2	yes	—
145	C₃×C₄²	C₂⁴	1	2	yes	—
146	C₃×C₄²	C₂⁴	1	2	yes	—
147	C₂⁵	—	2	2	no	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	—	2	4	no	—
151	C₂²×D₄	—	2	4	no	—
152	C₂²×D₄	—	2	16	no	—
153	C₂²×D₄	—	2	4	no	—
154	C₂²×D₄	—	2	16	no	—
155	C₂²×D₄	—	4	16	no	—
156	C₂²×D₄	C₂³×C₃	1	4	yes	—
157	C₂²×D₄	C₂³×C₃	1	8	yes	—
158	C₂³×C₄	C₃×D₄	1	2	yes	—
159	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	8	no	—
161	C₂×M₁₆	C₂³×C₃	1	8	yes	—
162	C₂²×C₈	C₂³×C₃	1	4	yes	—
163	C₄⋊Q₈	—	2	4	no	—
164	C₄⋊Q₈	C₂³×C₃	1	8	yes	complemented
165	#127	—	2	16	no	—
166	#127	—	2	4	no	—
167	#127	—	2	4	no	—
168	#127	C₂³×C₃	1	8	yes	—
169	#127	C₂³×C₃	1	8	yes	—
170	#125	—	2	4	no	—
171	#125	C₂³×C₃	1	8	yes	complemented
172	C₂²⋊Q₈	—	2	16	no	—
173	C₄×Q₈	—	2	16	no	—
174	#117	—	2	4	no	—
175	#117	C₂³×C₃	1	8	yes	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₃×D₄	1	2	yes	—
182	C₂×C₄²	C₃×D₄	1	4	yes	—
183	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₃	1	4	yes	—
186	#102	C₂³×C₃	1	64	yes	complemented
187	#97	C₂³×C₃	1	8	yes	complemented
188	C₄×C₈	C₂³×C₃	1	8	yes	complemented
189	C₂³×C₃	—	2	4	no	—
190	C₂³×C₃	—	2	8	no	—
191	C₂³×C₃	—	4	32	no	—
192	C₂³×C₃	—	4	16	no	—
193	C₂³×C₃	—	4	8	no	—
194	C₂³×C₃	—	4	16	no	—
195	C₂³×C₃	—	8	32	no	—
196	C₂³×C₃	#124	1	4	yes	—
197	C₂³×C₃	C₂²×D₄	1	2	yes	—
198	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
199	C₃×Q₈	—	4	16	no	—
200	C₃×D₄	—	2	16	no	—
201	C₃×D₄	—	4	32	no	—
202	C₃×D₄	—	4	16	no	—
203	C₃×D₄	—	4	8	no	—
204	C₃×D₄	—	4	32	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	16	no	—
207	C₂×C₃×C₄	—	2	8	no	—
208	C₂×C₃×C₄	—	4	8	no	—
209	C₂×C₃×C₄	#124	1	8	yes	—
210	C₂×C₃×C₄	#124	1	4	yes	—
211	C₂×C₃×C₄	AGL(1,ℤ/8)	1	8	yes	—
212	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
213	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
214	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
216	C₂×C₃×C₄	C₂×#41	1	4	yes	—
217	C₂×C₃×C₄	C₂×#41	1	8	yes	—
218	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
219	C₃×C₈	—	2	8	no	—
220	C₂⁴	—	2	8	no	—
221	C₂⁴	—	2	4	no	—
222	C₂⁴	—	4	16	no	—
223	C₂⁴	—	4	8	no	—
224	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	16	no	—
230	C₂×D₄	—	2	16	no	—
231	C₂×D₄	—	2	16	no	—
232	C₂×D₄	—	2	32	no	—
233	C₂×D₄	—	4	16	no	—
234	C₂×D₄	—	4	64	no	—
235	C₂×D₄	—	4	16	no	—
236	C₂×D₄	—	4	8	no	—
237	C₂×D₄	—	4	16	no	—
238	C₂×D₄	C₂⁴×C₃	1	8	yes	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
242	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
243	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
245	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
246	C₂²×C₄	C₃×#41	1	8	yes	—
247	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
249	M₁₆	—	2	16	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	C₂⁴×C₃	1	8	yes	—
252	C₄⋊C₄	—	2	16	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₄²	—	2	8	no	—
258	C₄²	C₂×C₃×D₄	1	8	yes	—
259	C₄²	C₂×C₃×D₄	1	2	yes	—
260	C₄²	C₂×C₃×D₄	1	2	yes	—
261	C₄²	C₂⁴×C₃	1	2	yes	—
262	C₄²	C₂⁴×C₃	1	2	yes	—
263	C₂²×C₃	—	2	8	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	—	4	8	no	—
266	C₂²×C₃	—	4	16	no	—
267	C₂²×C₃	—	8	32	no	—
268	C₂²×C₃	—	8	16	no	—
269	C₂²×C₃	C₂×#124	1	4	yes	—
270	C₂²×C₃	C₂×#124	1	2	yes	—
271	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
272	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	4	8	no	—
275	C₃×C₄	C₂×#124	1	4	yes	—
276	C₃×C₄	C₂×#124	1	2	yes	—
277	C₃×C₄	C₂×AGL(1,ℤ/8)	1	8	yes	—
278	C₃×C₄	C₂²×#41	1	2	yes	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	8	no	—
281	C₂³	—	4	32	no	—
282	C₂³	—	4	16	no	—
283	C₂³	—	4	16	no	—
284	C₂³	—	4	8	no	—
285	C₂³	—	8	32	no	—
286	C₂³	C₃×#124	1	4	yes	—
287	C₂³	C₂²×C₃×D₄	1	2	yes	—
288	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
289	Q₈	—	4	16	no	—
290	D₄	—	2	16	no	—
291	D₄	—	4	32	no	—
292	D₄	—	4	32	no	—
293	D₄	—	4	16	no	—
294	D₄	—	4	8	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	16	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	C₃×#124	1	8	yes	—
300	C₂×C₄	C₃×#124	1	4	yes	—
301	C₂×C₄	C₃×AGL(1,ℤ/8)	1	8	yes	—
302	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
303	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
304	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
305	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×#41	1	4	yes	—
307	C₂×C₄	C₂×C₃×#41	1	8	yes	—
308	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
309	C₈	—	2	8	no	—
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	—	8	8	no	—
312	C₂×C₃	#3138	1	4	yes	direct product factor
313	C₂×C₃	C₂²×#124	1	1	yes	characteristic
314	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
315	C₂²	—	2	8	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	4	8	no	—
318	C₂²	—	4	16	no	—
319	C₂²	—	8	32	no	—
320	C₂²	—	8	16	no	—
321	C₂²	C₂×C₃×#124	1	4	yes	—
322	C₂²	C₂×C₃×#124	1	2	yes	—
323	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	4	yes	—
324	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	4	8	no	—
327	C₄	C₂×C₃×#124	1	4	yes	—
328	C₄	C₂×C₃×#124	1	2	yes	—
329	C₄	C₂×C₃×AGL(1,ℤ/8)	1	8	yes	—
330	C₄	C₂²×C₃×#41	1	2	yes	—
331	C₃	C₂×#3138	1	1	yes	characteristic, direct product factor, Sylow
332	C₂	—	2	4	no	—
333	C₂	—	8	8	no	—
334	C₂	C₃×#3138	1	4	yes	direct product factor
335	C₂	C₂²×C₃×#124	1	1	yes	characteristic
336	C₂	C₂²×C₃×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂²×C₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#533	C₂	semidirect
3. C₃×#3138	C₂	direct (N × H)
4. C₂×C₃×#490	C₂	semidirect
5. C₂×C₃×#486	C₂	semidirect
6. C₂×C₃×#418	C₂	semidirect
7. C₂×C₃×C₄×M₁₆	C₂	semidirect
8. C₂×#3138	C₃	direct (N × H)
13. C₃×#533	C₂²	semidirect
14. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
20. C₂×C₃×#125	C₂²	semidirect
25. C₃×#490	C₂²	semidirect
26. C₃×#486	C₂²	semidirect
27. C₃×#418	C₂²	semidirect
28. C₂×C₃×#102	C₂²	semidirect
29. C₃×C₄×M₁₆	C₂²	semidirect
30. C₂×C₃×#97	C₂²	semidirect
31. C₂×C₃×C₄×C₈	C₂²	semidirect
32. C₂×#533	C₂×C₃	semidirect
34. #3138	C₂×C₃	direct (N × H)
35. C₂×#490	C₂×C₃	semidirect
36. C₂×#486	C₂×C₃	semidirect
37. C₂×#418	C₂×C₃	semidirect
38. C₂×C₄×M₁₆	C₂×C₃	semidirect
56. C₃×C₄⋊Q₈	C₂³	semidirect
63. C₃×#125	C₂³	semidirect
78. C₃×#102	C₂³	semidirect
79. C₃×#97	C₂³	semidirect
80. C₃×C₄×C₈	C₂³	semidirect
85. #533	C₂²×C₃	semidirect
86. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
92. C₂×#125	C₂²×C₃	semidirect
97. #490	C₂²×C₃	semidirect
98. #486	C₂²×C₃	semidirect
99. #418	C₂²×C₃	semidirect
100. C₂×#102	C₂²×C₃	semidirect
101. C₄×M₁₆	C₂²×C₃	semidirect
102. C₂×#97	C₂²×C₃	semidirect
103. C₂×C₄×C₈	C₂²×C₃	semidirect
164. C₄⋊Q₈	C₂³×C₃	semidirect
171. #125	C₂³×C₃	semidirect
186. #102	C₂³×C₃	semidirect
187. #97	C₂³×C₃	semidirect
188. C₄×C₈	C₂³×C₃	semidirect
312. C₂×C₃	#3138	direct (N × H)
331. C₃	C₂×#3138	direct (N × H)
334. C₂	C₃×#3138	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×#533	2
C₂²×C₃×#127	2
C₃×#3138	2
C₂×C₃×#490	2
C₂×C₃×#486	2
C₂×C₃×#418	2
C₂×C₃×C₄×M₁₆	2
C₂×#3138	3

Subgroups — C2×C3×#3138 #10679220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3138 #10679220