Subgroups — C₂×C₃²×D₃×#41 #13181720

Order 1728.

5996 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×D₃×#41	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃²×#239	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃²×#237	C₂	1	3	yes	maximal, complemented
6	C₂×C₃²×D₃×D₄	C₂	1	3	yes	maximal, complemented
7	C₂×C₃²×#235	C₂	1	3	yes	maximal, complemented
8	C₂²×C₃²×C₄×D₃	C₂	1	3	yes	maximal, complemented
9	C₂²×C₃²×#41	—	3	1	no	maximal
10	C₂×C₃×D₃×#41	C₃	1	4	yes	maximal, direct product factor
11	C₂³×C₃²×D₃	C₂²	1	3	yes	—
12	C₃³×#41	C₂²	1	8	yes	complemented
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃³×D₄	C₂²	1	3	yes	complemented
15	C₂²×C₃³×C₄	C₂²	1	3	yes	complemented
16	C₂×C₃²×#67	C₂²	1	6	yes	complemented
17	C₂²×C₃²×Dic₃	C₂²	1	3	yes	complemented
18	C₃²×#239	C₂²	1	8	yes	complemented
19	C₃²×D₃×Q₈	C₂²	1	8	yes	complemented
20	C₃²×#237	C₂²	1	24	yes	complemented
21	C₃²×D₃×D₄	C₂²	1	24	yes	complemented
22	C₃²×#235	C₂²	1	24	yes	complemented
23	C₂×C₃²×D₁₂	C₂²	1	3	yes	complemented
24	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
25	C₂×C₃²×C₄×D₃	C₂²	1	3	yes	—
26	C₂×C₃²×C₄×D₃	C₂²	1	24	yes	complemented
27	C₂×C₃²×C₄×D₃	C₂²	1	3	yes	complemented
28	C₂×C₃²×C₄×D₃	C₂²	1	3	yes	complemented
29	C₂×C₃²×Dic₆	C₂²	1	3	yes	complemented
30	C₂×C₃²×#41	—	2	4	no	—
31	C₂×C₃²×#41	—	3	16	no	—
32	C₂×C₃²×#41	—	3	1	no	—
33	C₂×C₃²×#41	—	3	3	no	—
34	C₂×C₃²×#41	—	3	3	no	—
35	C₂×C₃²×#41	D₃	1	1	yes	characteristic, direct product factor
36	C₂×C₃²×#41	C₂×C₃	1	4	yes	complemented
37	C₂²×C₃²×Q₈	—	3	1	no	—
38	C₂²×C₃²×D₄	—	3	3	no	—
39	C₂³×C₃²×C₄	—	3	3	no	—
40	C₃×D₃×#41	C₂×C₃	1	64	yes	direct product factor
41	C₂×C₃×#239	C₂×C₃	1	4	yes	complemented
42	C₂×C₃×D₃×Q₈	C₂×C₃	1	4	yes	complemented
43	C₂×C₃×#237	C₂×C₃	1	12	yes	complemented
44	C₂×C₃×D₃×D₄	C₂×C₃	1	12	yes	complemented
45	C₂×C₃×#235	C₂×C₃	1	12	yes	complemented
46	C₂²×C₃×C₄×D₃	C₂×C₃	1	12	yes	complemented
47	C₂³×C₃³	C₂³	1	3	yes	—
48	C₂²×C₃²×D₃	—	2	12	no	—
49	C₂²×C₃²×D₃	C₂³	1	3	yes	—
50	C₂²×C₃²×D₃	C₂³	1	12	yes	—
51	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
52	C₃³×Q₈	C₂³	1	4	yes	complemented
53	C₃³×D₄	C₂³	1	12	yes	complemented
54	C₂×C₃³×C₄	C₂³	1	3	yes	—
55	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃³×C₄	C₂³	1	12	yes	complemented
57	C₃²×#67	C₂³	1	24	yes	complemented
58	C₂×C₃²×Dic₃	C₂³	1	3	yes	—
59	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₃²×Dic₃	C₂³	1	12	yes	complemented
61	C₃²×D₁₂	C₂³	1	12	yes	complemented
62	C₃²×C₄×D₃	C₂³	1	4	yes	—
63	C₃²×C₄×D₃	C₂³	1	12	yes	—
64	C₃²×C₄×D₃	C₂³	1	12	yes	complemented
65	C₃²×C₄×D₃	C₂³	1	12	yes	complemented
66	C₃²×Dic₆	C₂³	1	12	yes	complemented
67	C₂²×C₃×#41	—	3	4	no	—
68	C₂×D₃×#41	C₃²	1	1	yes	characteristic, direct product factor
69	C₂⁴×C₃²	—	3	3	no	—
70	C₂³×C₃×D₃	C₂²×C₃	1	12	yes	—
71	C₃²×#41	—	2	32	no	—
72	C₃²×#41	—	3	8	no	—
73	C₃²×#41	—	3	24	no	—
74	C₃²×#41	—	3	24	no	—
75	C₃²×#41	C₂×D₃	1	8	yes	direct product factor
76	C₃²×#41	C₂²×C₃	1	32	yes	complemented
77	C₂×C₃²×Q₈	—	2	4	no	—
78	C₂×C₃²×Q₈	—	3	8	no	—
79	C₂×C₃²×Q₈	—	3	3	no	—
80	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
81	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	complemented
82	C₂×C₃²×D₄	—	2	12	no	—
83	C₂×C₃²×D₄	—	3	6	no	—
84	C₂×C₃²×D₄	—	3	24	no	—
85	C₂×C₃²×D₄	—	3	3	no	—
86	C₂×C₃²×D₄	C₂×D₃	1	3	yes	complemented
87	C₂×C₃²×D₄	C₂²×C₃	1	12	yes	complemented
88	C₂²×C₃²×C₄	—	2	12	no	—
89	C₂²×C₃²×C₄	—	3	3	no	—
90	C₂²×C₃²×C₄	—	3	3	no	—
91	C₂²×C₃²×C₄	—	3	3	no	—
92	C₂²×C₃²×C₄	—	3	1	no	—
93	C₂²×C₃²×C₄	—	3	3	no	—
94	C₂²×C₃²×C₄	—	3	24	no	—
95	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	complemented
96	C₂²×C₃²×C₄	C₂²×C₃	1	12	yes	complemented
97	C₂×C₃×#67	C₂²×C₃	1	24	yes	complemented
98	C₂²×C₃×Dic₃	C₂²×C₃	1	12	yes	complemented
99	C₃×#239	C₂²×C₃	1	32	yes	complemented
100	C₃×D₃×Q₈	C₂²×C₃	1	32	yes	complemented
101	C₃×#237	C₂²×C₃	1	96	yes	complemented
102	C₃×D₃×D₄	C₂²×C₃	1	96	yes	complemented
103	C₃×#235	C₂²×C₃	1	96	yes	complemented
104	C₂×C₃×D₁₂	C₂²×C₃	1	12	yes	complemented
105	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
106	C₂×C₃×C₄×D₃	C₂²×C₃	1	12	yes	—
107	C₂×C₃×C₄×D₃	C₂²×C₃	1	96	yes	complemented
108	C₂×C₃×C₄×D₃	C₂²×C₃	1	12	yes	complemented
109	C₂×C₃×C₄×D₃	C₂²×C₃	1	12	yes	complemented
110	C₂×C₃×Dic₆	C₂²×C₃	1	12	yes	complemented
111	C₂²×C₃³	—	2	6	no	—
112	C₂²×C₃³	C₂⁴	1	6	yes	—
113	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
114	C₂×C₃²×D₃	—	2	6	no	—
115	C₂×C₃²×D₃	—	2	24	no	—
116	C₂×C₃²×D₃	#41	1	4	yes	direct product factor
117	C₂×C₃²×D₃	C₂⁴	1	6	yes	—
118	C₂×C₃²×D₃	C₂⁴	1	2	yes	—
119	C₃³×C₄	C₂⁴	1	6	yes	—
120	C₃³×C₄	C₂⁴	1	2	yes	—
121	C₃²×Dic₃	C₂⁴	1	6	yes	—
122	C₃²×Dic₃	C₂⁴	1	2	yes	—
123	C₂×C₃×#41	—	2	4	no	—
124	C₂×C₃×#41	—	3	64	no	—
125	C₂×C₃×#41	—	3	4	no	—
126	C₂×C₃×#41	—	3	12	no	—
127	C₂×C₃×#41	—	3	12	no	—
128	C₂×C₃×#41	C₃×D₃	1	4	yes	direct product factor
129	C₂×C₃×#41	C₂×C₃²	1	1	yes	characteristic, complemented
130	C₂²×C₃×Q₈	—	3	4	no	—
131	C₂²×C₃×D₄	—	3	12	no	—
132	C₂³×C₃×C₄	—	3	12	no	—
133	D₃×#41	C₂×C₃²	1	16	yes	direct product factor
134	C₂×#239	C₂×C₃²	1	1	yes	characteristic, complemented
135	C₂×D₃×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
136	C₂×#237	C₂×C₃²	1	3	yes	complemented
137	C₂×D₃×D₄	C₂×C₃²	1	3	yes	complemented
138	C₂×#235	C₂×C₃²	1	3	yes	complemented
139	C₂²×C₄×D₃	C₂×C₃²	1	3	yes	complemented
140	C₂³×C₃²	—	2	12	no	—
141	C₂³×C₃²	—	3	1	no	—
142	C₂³×C₃²	—	3	12	no	—
143	C₂³×C₃²	—	3	3	no	—
144	C₂³×C₃²	—	6	12	no	—
145	C₂³×C₃²	C₂²×D₃	1	3	yes	—
146	C₂³×C₃²	C₂³×C₃	1	12	yes	—
147	C₂²×C₃×D₃	—	2	48	no	—
148	C₂²×C₃×D₃	C₂³×C₃	1	12	yes	—
149	C₂²×C₃×D₃	C₂³×C₃	1	48	yes	—
150	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
151	C₃²×Q₈	—	2	16	no	—
152	C₃²×Q₈	—	3	12	no	—
153	C₃²×Q₈	C₂²×D₃	1	4	yes	complemented
154	C₃²×Q₈	C₂³×C₃	1	16	yes	complemented
155	C₃²×D₄	—	2	48	no	—
156	C₃²×D₄	—	3	24	no	—
157	C₃²×D₄	—	3	12	no	—
158	C₃²×D₄	C₂²×D₃	1	12	yes	complemented
159	C₃²×D₄	C₂³×C₃	1	48	yes	complemented
160	C₂×C₃²×C₄	—	2	4	no	—
161	C₂×C₃²×C₄	—	2	12	no	—
162	C₂×C₃²×C₄	—	2	48	no	—
163	C₂×C₃²×C₄	—	3	1	no	—
164	C₂×C₃²×C₄	—	3	3	no	—
165	C₂×C₃²×C₄	—	3	12	no	—
166	C₂×C₃²×C₄	—	3	4	no	—
167	C₂×C₃²×C₄	—	3	12	no	—
168	C₂×C₃²×C₄	—	3	12	no	—
169	C₂×C₃²×C₄	—	3	12	no	—
170	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
171	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
172	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	complemented
173	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
174	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
175	C₂×C₃²×C₄	C₂³×C₃	1	48	yes	complemented
176	C₃×#67	C₂³×C₃	1	96	yes	complemented
177	C₂×C₃×Dic₃	C₂³×C₃	1	12	yes	—
178	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
179	C₂×C₃×Dic₃	C₂³×C₃	1	48	yes	complemented
180	C₃×D₁₂	C₂³×C₃	1	48	yes	complemented
181	C₃×C₄×D₃	C₂³×C₃	1	16	yes	—
182	C₃×C₄×D₃	C₂³×C₃	1	48	yes	—
183	C₃×C₄×D₃	C₂³×C₃	1	48	yes	complemented
184	C₃×C₄×D₃	C₂³×C₃	1	48	yes	complemented
185	C₃×Dic₆	C₂³×C₃	1	48	yes	complemented
186	C₂²×#41	—	3	1	no	Sylow
187	C₂×C₃³	—	2	6	no	—
188	C₂×C₃³	C₂×#41	1	2	yes	complemented
189	C₂×C₃³	C₂⁵	1	1	yes	characteristic
190	C₃²×D₃	—	2	6	no	—
191	C₃²×D₃	C₂×#41	1	4	yes	direct product factor
192	C₂⁴×C₃	—	3	12	no	—
193	C₂³×D₃	C₂²×C₃²	1	3	yes	—
194	C₃×#41	—	2	32	no	—
195	C₃×#41	—	3	32	no	—
196	C₃×#41	—	3	96	no	—
197	C₃×#41	—	3	96	no	—
198	C₃×#41	C₂×C₃×D₃	1	32	yes	direct product factor
199	C₃×#41	C₂²×C₃²	1	8	yes	complemented
200	C₂×C₃×Q₈	—	2	4	no	—
201	C₂×C₃×Q₈	—	3	32	no	—
202	C₂×C₃×Q₈	—	3	12	no	—
203	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	complemented
204	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
205	C₂×C₃×D₄	—	2	12	no	—
206	C₂×C₃×D₄	—	3	24	no	—
207	C₂×C₃×D₄	—	3	96	no	—
208	C₂×C₃×D₄	—	3	12	no	—
209	C₂×C₃×D₄	C₂×C₃×D₃	1	12	yes	complemented
210	C₂×C₃×D₄	C₂²×C₃²	1	3	yes	complemented
211	C₂²×C₃×C₄	—	2	12	no	—
212	C₂²×C₃×C₄	—	3	12	no	—
213	C₂²×C₃×C₄	—	3	4	no	—
214	C₂²×C₃×C₄	—	3	12	no	—
215	C₂²×C₃×C₄	—	3	12	no	—
216	C₂²×C₃×C₄	—	3	12	no	—
217	C₂²×C₃×C₄	—	3	96	no	—
218	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	complemented
219	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	complemented
220	C₂×#67	C₂²×C₃²	1	6	yes	complemented
221	C₂²×Dic₃	C₂²×C₃²	1	3	yes	complemented
222	#239	C₂²×C₃²	1	8	yes	complemented
223	D₃×Q₈	C₂²×C₃²	1	8	yes	complemented
224	#237	C₂²×C₃²	1	24	yes	complemented
225	D₃×D₄	C₂²×C₃²	1	24	yes	complemented
226	#235	C₂²×C₃²	1	24	yes	complemented
227	C₂×D₁₂	C₂²×C₃²	1	3	yes	complemented
228	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
229	C₂×C₄×D₃	C₂²×C₃²	1	3	yes	—
230	C₂×C₄×D₃	C₂²×C₃²	1	24	yes	complemented
231	C₂×C₄×D₃	C₂²×C₃²	1	3	yes	complemented
232	C₂×C₄×D₃	C₂²×C₃²	1	3	yes	complemented
233	C₂×Dic₆	C₂²×C₃²	1	3	yes	complemented
234	C₂²×C₃²	—	2	6	no	—
235	C₂²×C₃²	—	2	4	no	—
236	C₂²×C₃²	—	2	24	no	—
237	C₂²×C₃²	—	2	24	no	—
238	C₂²×C₃²	—	3	4	no	—
239	C₂²×C₃²	—	3	2	no	—
240	C₂²×C₃²	—	3	6	no	—
241	C₂²×C₃²	—	4	24	no	—
242	C₂²×C₃²	—	6	24	no	—
243	C₂²×C₃²	—	6	6	no	—
244	C₂²×C₃²	C₂³×D₃	1	6	yes	—
245	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
246	C₂²×C₃²	C₂⁴×C₃	1	24	yes	—
247	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
248	C₂×C₃×D₃	—	2	24	no	—
249	C₂×C₃×D₃	—	2	96	no	—
250	C₂×C₃×D₃	C₃×#41	1	16	yes	direct product factor
251	C₂×C₃×D₃	C₂⁴×C₃	1	24	yes	—
252	C₂×C₃×D₃	C₂⁴×C₃	1	8	yes	—
253	C₃²×C₄	—	2	8	no	—
254	C₃²×C₄	—	2	24	no	—
255	C₃²×C₄	—	3	2	no	—
256	C₃²×C₄	—	3	6	no	—
257	C₃²×C₄	C₂³×D₃	1	2	yes	—
258	C₃²×C₄	C₂³×D₃	1	6	yes	—
259	C₃²×C₄	C₂⁴×C₃	1	24	yes	—
260	C₃²×C₄	C₂⁴×C₃	1	8	yes	—
261	C₃×Dic₃	C₂⁴×C₃	1	24	yes	—
262	C₃×Dic₃	C₂⁴×C₃	1	8	yes	—
263	C₂×#41	—	3	16	no	—
264	C₂×#41	—	3	1	no	—
265	C₂×#41	—	3	3	no	—
266	C₂×#41	—	3	3	no	—
267	C₂×#41	C₃²×D₃	1	1	yes	characteristic, direct product factor
268	C₂²×Q₈	—	3	1	no	—
269	C₂²×D₄	—	3	3	no	—
270	C₂³×C₄	—	3	3	no	—
271	C₃³	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
272	C₂³×C₃	—	2	12	no	—
273	C₂³×C₃	—	3	4	no	—
274	C₂³×C₃	—	3	48	no	—
275	C₂³×C₃	—	3	12	no	—
276	C₂³×C₃	—	6	48	no	—
277	C₂³×C₃	C₂²×C₃×D₃	1	12	yes	—
278	C₂³×C₃	C₂³×C₃²	1	3	yes	—
279	C₂²×D₃	—	2	12	no	—
280	C₂²×D₃	C₂³×C₃²	1	3	yes	—
281	C₂²×D₃	C₂³×C₃²	1	12	yes	—
282	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
283	C₃×Q₈	—	2	16	no	—
284	C₃×Q₈	—	3	48	no	—
285	C₃×Q₈	C₂²×C₃×D₃	1	16	yes	complemented
286	C₃×Q₈	C₂³×C₃²	1	4	yes	complemented
287	C₃×D₄	—	2	48	no	—
288	C₃×D₄	—	3	96	no	—
289	C₃×D₄	—	3	48	no	—
290	C₃×D₄	C₂²×C₃×D₃	1	48	yes	complemented
291	C₃×D₄	C₂³×C₃²	1	12	yes	complemented
292	C₂×C₃×C₄	—	2	4	no	—
293	C₂×C₃×C₄	—	2	12	no	—
294	C₂×C₃×C₄	—	2	48	no	—
295	C₂×C₃×C₄	—	3	4	no	—
296	C₂×C₃×C₄	—	3	12	no	—
297	C₂×C₃×C₄	—	3	48	no	—
298	C₂×C₃×C₄	—	3	16	no	—
299	C₂×C₃×C₄	—	3	48	no	—
300	C₂×C₃×C₄	—	3	48	no	—
301	C₂×C₃×C₄	—	3	48	no	—
302	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
303	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
304	C₂×C₃×C₄	C₂²×C₃×D₃	1	48	yes	complemented
305	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
306	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
307	C₂×C₃×C₄	C₂³×C₃²	1	12	yes	complemented
308	#67	C₂³×C₃²	1	24	yes	complemented
309	C₂×Dic₃	C₂³×C₃²	1	3	yes	—
310	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
311	C₂×Dic₃	C₂³×C₃²	1	12	yes	complemented
312	D₁₂	C₂³×C₃²	1	12	yes	complemented
313	C₄×D₃	C₂³×C₃²	1	4	yes	—
314	C₄×D₃	C₂³×C₃²	1	12	yes	—
315	C₄×D₃	C₂³×C₃²	1	12	yes	complemented
316	C₄×D₃	C₂³×C₃²	1	12	yes	complemented
317	Dic₆	C₂³×C₃²	1	12	yes	complemented
318	C₂×C₃²	—	2	6	no	—
319	C₂×C₃²	—	2	8	no	—
320	C₂×C₃²	—	2	4	no	—
321	C₂×C₃²	—	2	24	no	—
322	C₂×C₃²	—	3	4	no	—
323	C₂×C₃²	—	4	24	no	—
324	C₂×C₃²	—	6	6	no	—
325	C₂×C₃²	D₃×#41	1	2	yes	direct product factor
326	C₂×C₃²	C₂×C₃×#41	1	8	yes	complemented
327	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
328	C₂×C₃²	C₂⁵×C₃	1	4	yes	—
329	C₃×D₃	—	2	24	no	—
330	C₃×D₃	C₂×C₃×#41	1	16	yes	direct product factor
331	C₂⁴	—	3	3	no	—
332	#41	—	3	8	no	—
333	#41	—	3	24	no	—
334	#41	—	3	24	no	—
335	#41	C₂×C₃²×D₃	1	8	yes	direct product factor
336	C₂×Q₈	—	3	8	no	—
337	C₂×Q₈	—	3	3	no	—
338	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
339	C₂×D₄	—	3	6	no	—
340	C₂×D₄	—	3	24	no	—
341	C₂×D₄	—	3	3	no	—
342	C₂×D₄	C₂×C₃²×D₃	1	3	yes	complemented
343	C₂²×C₄	—	3	3	no	—
344	C₂²×C₄	—	3	1	no	—
345	C₂²×C₄	—	3	3	no	—
346	C₂²×C₄	—	3	3	no	—
347	C₂²×C₄	—	3	3	no	—
348	C₂²×C₄	—	3	24	no	—
349	C₂²×C₄	C₂×C₃²×D₃	1	3	yes	complemented
350	C₂²×C₃	—	2	24	no	—
351	C₂²×C₃	—	2	6	no	—
352	C₂²×C₃	—	2	4	no	—
353	C₂²×C₃	—	2	24	no	—
354	C₂²×C₃	—	3	16	no	—
355	C₂²×C₃	—	3	8	no	—
356	C₂²×C₃	—	3	24	no	—
357	C₂²×C₃	—	4	24	no	—
358	C₂²×C₃	—	6	96	no	—
359	C₂²×C₃	—	6	24	no	—
360	C₂²×C₃	C₂³×C₃×D₃	1	24	yes	—
361	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
362	C₂²×C₃	C₂⁴×C₃²	1	6	yes	—
363	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
364	C₂×D₃	—	2	6	no	—
365	C₂×D₃	—	2	24	no	—
366	C₂×D₃	C₃²×#41	1	4	yes	direct product factor
367	C₂×D₃	C₂⁴×C₃²	1	6	yes	—
368	C₂×D₃	C₂⁴×C₃²	1	2	yes	—
369	C₃×C₄	—	2	8	no	—
370	C₃×C₄	—	2	24	no	—
371	C₃×C₄	—	3	8	no	—
372	C₃×C₄	—	3	24	no	—
373	C₃×C₄	C₂³×C₃×D₃	1	8	yes	—
374	C₃×C₄	C₂³×C₃×D₃	1	24	yes	—
375	C₃×C₄	C₂⁴×C₃²	1	6	yes	—
376	C₃×C₄	C₂⁴×C₃²	1	2	yes	—
377	Dic₃	C₂⁴×C₃²	1	6	yes	—
378	Dic₃	C₂⁴×C₃²	1	2	yes	—
379	C₃²	—	2	4	no	—
380	C₃²	C₂×D₃×#41	1	1	yes	characteristic, direct product factor
381	C₃²	C₂²×C₃×#41	1	4	yes	complemented
382	C₂³	—	3	1	no	—
383	C₂³	—	3	12	no	—
384	C₂³	—	3	3	no	—
385	C₂³	—	6	12	no	—
386	C₂³	C₂²×C₃²×D₃	1	3	yes	—
387	Q₈	—	3	12	no	—
388	Q₈	C₂²×C₃²×D₃	1	4	yes	complemented
389	D₄	—	3	24	no	—
390	D₄	—	3	12	no	—
391	D₄	C₂²×C₃²×D₃	1	12	yes	complemented
392	C₂×C₄	—	3	1	no	—
393	C₂×C₄	—	3	3	no	—
394	C₂×C₄	—	3	12	no	—
395	C₂×C₄	—	3	4	no	—
396	C₂×C₄	—	3	12	no	—
397	C₂×C₄	—	3	12	no	—
398	C₂×C₄	—	3	12	no	—
399	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
400	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
401	C₂×C₄	C₂²×C₃²×D₃	1	12	yes	complemented
402	C₂×C₃	—	2	24	no	—
403	C₂×C₃	—	2	6	no	—
404	C₂×C₃	—	2	8	no	—
405	C₂×C₃	—	2	4	no	—
406	C₂×C₃	—	3	16	no	—
407	C₂×C₃	—	4	24	no	—
408	C₂×C₃	—	6	24	no	—
409	C₂×C₃	C₃×D₃×#41	1	8	yes	direct product factor
410	C₂×C₃	C₂×C₃²×#41	1	2	yes	complemented
411	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
412	C₂×C₃	C₂⁵×C₃²	1	1	yes	characteristic
413	D₃	—	2	6	no	—
414	D₃	C₂×C₃²×#41	1	4	yes	direct product factor
415	C₂²	—	2	6	no	—
416	C₂²	—	3	4	no	—
417	C₂²	—	3	2	no	—
418	C₂²	—	3	6	no	—
419	C₂²	—	6	24	no	—
420	C₂²	—	6	6	no	—
421	C₂²	C₂³×C₃²×D₃	1	6	yes	—
422	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
423	C₄	—	3	2	no	—
424	C₄	—	3	6	no	—
425	C₄	C₂³×C₃²×D₃	1	2	yes	—
426	C₄	C₂³×C₃²×D₃	1	6	yes	—
427	C₃	—	2	4	no	—
428	C₃	C₂×C₃×D₃×#41	1	4	yes	direct product factor
429	C₃	C₂²×C₃²×#41	1	1	yes	characteristic, complemented
430	C₂	—	2	6	no	—
431	C₂	—	3	4	no	—
432	C₂	—	6	6	no	—
433	C₂	C₃²×D₃×#41	1	2	yes	direct product factor
434	C₂	C₂⁴×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₃²
Frattini	C₂	C₂⁴×C₃²×D₃
Fitting	C₂×C₃³×#41	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₂×C₃²×#41	D₃
hypocenter	C₃	C₂²×C₃²×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₃²×D₃×#41	C₂	direct (N × H)
3. C₂×C₃²×#239	C₂	semidirect
4. C₂×C₃²×D₃×Q₈	C₂	semidirect
5. C₂×C₃²×#237	C₂	semidirect
6. C₂×C₃²×D₃×D₄	C₂	semidirect
7. C₂×C₃²×#235	C₂	semidirect
8. C₂²×C₃²×C₄×D₃	C₂	semidirect
10. C₂×C₃×D₃×#41	C₃	direct (N × H)
12. C₃³×#41	C₂²	semidirect
13. C₂×C₃³×Q₈	C₂²	semidirect
14. C₂×C₃³×D₄	C₂²	semidirect
15. C₂²×C₃³×C₄	C₂²	semidirect
16. C₂×C₃²×#67	C₂²	semidirect
17. C₂²×C₃²×Dic₃	C₂²	semidirect
18. C₃²×#239	C₂²	semidirect
19. C₃²×D₃×Q₈	C₂²	semidirect
20. C₃²×#237	C₂²	semidirect
21. C₃²×D₃×D₄	C₂²	semidirect
22. C₃²×#235	C₂²	semidirect
23. C₂×C₃²×D₁₂	C₂²	semidirect
26. C₂×C₃²×C₄×D₃	C₂²	semidirect
27. C₂×C₃²×C₄×D₃	C₂²	semidirect
28. C₂×C₃²×C₄×D₃	C₂²	semidirect
29. C₂×C₃²×Dic₆	C₂²	semidirect
35. C₂×C₃²×#41	D₃	direct (N × H)
36. C₂×C₃²×#41	C₂×C₃	semidirect
40. C₃×D₃×#41	C₂×C₃	direct (N × H)
41. C₂×C₃×#239	C₂×C₃	semidirect
42. C₂×C₃×D₃×Q₈	C₂×C₃	semidirect
43. C₂×C₃×#237	C₂×C₃	semidirect
44. C₂×C₃×D₃×D₄	C₂×C₃	semidirect
45. C₂×C₃×#235	C₂×C₃	semidirect
46. C₂²×C₃×C₄×D₃	C₂×C₃	semidirect
52. C₃³×Q₈	C₂³	semidirect
53. C₃³×D₄	C₂³	semidirect
56. C₂×C₃³×C₄	C₂³	semidirect
57. C₃²×#67	C₂³	semidirect
60. C₂×C₃²×Dic₃	C₂³	semidirect
61. C₃²×D₁₂	C₂³	semidirect
64. C₃²×C₄×D₃	C₂³	semidirect
65. C₃²×C₄×D₃	C₂³	semidirect
66. C₃²×Dic₆	C₂³	semidirect
68. C₂×D₃×#41	C₃²	direct (N × H)
75. C₃²×#41	C₂×D₃	direct (N × H)
76. C₃²×#41	C₂²×C₃	semidirect
80. C₂×C₃²×Q₈	C₂×D₃	semidirect
81. C₂×C₃²×Q₈	C₂²×C₃	semidirect
86. C₂×C₃²×D₄	C₂×D₃	semidirect
87. C₂×C₃²×D₄	C₂²×C₃	semidirect
95. C₂²×C₃²×C₄	C₂×D₃	semidirect
96. C₂²×C₃²×C₄	C₂²×C₃	semidirect
97. C₂×C₃×#67	C₂²×C₃	semidirect
98. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
99. C₃×#239	C₂²×C₃	semidirect
100. C₃×D₃×Q₈	C₂²×C₃	semidirect
101. C₃×#237	C₂²×C₃	semidirect
102. C₃×D₃×D₄	C₂²×C₃	semidirect
103. C₃×#235	C₂²×C₃	semidirect
104. C₂×C₃×D₁₂	C₂²×C₃	semidirect
107. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
108. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
109. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
110. C₂×C₃×Dic₆	C₂²×C₃	semidirect
116. C₂×C₃²×D₃	#41	direct (N × H)
128. C₂×C₃×#41	C₃×D₃	direct (N × H)
129. C₂×C₃×#41	C₂×C₃²	semidirect
133. D₃×#41	C₂×C₃²	direct (N × H)
134. C₂×#239	C₂×C₃²	semidirect
135. C₂×D₃×Q₈	C₂×C₃²	semidirect
136. C₂×#237	C₂×C₃²	semidirect
137. C₂×D₃×D₄	C₂×C₃²	semidirect
138. C₂×#235	C₂×C₃²	semidirect
139. C₂²×C₄×D₃	C₂×C₃²	semidirect
153. C₃²×Q₈	C₂²×D₃	semidirect
154. C₃²×Q₈	C₂³×C₃	semidirect
158. C₃²×D₄	C₂²×D₃	semidirect
159. C₃²×D₄	C₂³×C₃	semidirect
172. C₂×C₃²×C₄	C₂²×D₃	semidirect
175. C₂×C₃²×C₄	C₂³×C₃	semidirect
176. C₃×#67	C₂³×C₃	semidirect
179. C₂×C₃×Dic₃	C₂³×C₃	semidirect
180. C₃×D₁₂	C₂³×C₃	semidirect
183. C₃×C₄×D₃	C₂³×C₃	semidirect
184. C₃×C₄×D₃	C₂³×C₃	semidirect
185. C₃×Dic₆	C₂³×C₃	semidirect
188. C₂×C₃³	C₂×#41	semidirect
191. C₃²×D₃	C₂×#41	direct (N × H)
198. C₃×#41	C₂×C₃×D₃	direct (N × H)
199. C₃×#41	C₂²×C₃²	semidirect
203. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
204. C₂×C₃×Q₈	C₂²×C₃²	semidirect
209. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
210. C₂×C₃×D₄	C₂²×C₃²	semidirect
218. C₂²×C₃×C₄	C₂×C₃×D₃	semidirect
219. C₂²×C₃×C₄	C₂²×C₃²	semidirect
220. C₂×#67	C₂²×C₃²	semidirect
221. C₂²×Dic₃	C₂²×C₃²	semidirect
222. #239	C₂²×C₃²	semidirect
223. D₃×Q₈	C₂²×C₃²	semidirect
224. #237	C₂²×C₃²	semidirect
225. D₃×D₄	C₂²×C₃²	semidirect
226. #235	C₂²×C₃²	semidirect
227. C₂×D₁₂	C₂²×C₃²	semidirect
230. C₂×C₄×D₃	C₂²×C₃²	semidirect
231. C₂×C₄×D₃	C₂²×C₃²	semidirect
232. C₂×C₄×D₃	C₂²×C₃²	semidirect
233. C₂×Dic₆	C₂²×C₃²	semidirect
250. C₂×C₃×D₃	C₃×#41	direct (N × H)
267. C₂×#41	C₃²×D₃	direct (N × H)
271. C₃³	C₂²×#41	semidirect
285. C₃×Q₈	C₂²×C₃×D₃	semidirect
286. C₃×Q₈	C₂³×C₃²	semidirect
290. C₃×D₄	C₂²×C₃×D₃	semidirect
291. C₃×D₄	C₂³×C₃²	semidirect
304. C₂×C₃×C₄	C₂²×C₃×D₃	semidirect
307. C₂×C₃×C₄	C₂³×C₃²	semidirect
308. #67	C₂³×C₃²	semidirect
311. C₂×Dic₃	C₂³×C₃²	semidirect
312. D₁₂	C₂³×C₃²	semidirect
315. C₄×D₃	C₂³×C₃²	semidirect
316. C₄×D₃	C₂³×C₃²	semidirect
317. Dic₆	C₂³×C₃²	semidirect
325. C₂×C₃²	D₃×#41	direct (N × H)
326. C₂×C₃²	C₂×C₃×#41	semidirect
330. C₃×D₃	C₂×C₃×#41	direct (N × H)
335. #41	C₂×C₃²×D₃	direct (N × H)
338. C₂×Q₈	C₂×C₃²×D₃	semidirect
342. C₂×D₄	C₂×C₃²×D₃	semidirect
349. C₂²×C₄	C₂×C₃²×D₃	semidirect
366. C₂×D₃	C₃²×#41	direct (N × H)
380. C₃²	C₂×D₃×#41	direct (N × H)
381. C₃²	C₂²×C₃×#41	semidirect
388. Q₈	C₂²×C₃²×D₃	semidirect
391. D₄	C₂²×C₃²×D₃	semidirect
401. C₂×C₄	C₂²×C₃²×D₃	semidirect
409. C₂×C₃	C₃×D₃×#41	direct (N × H)
410. C₂×C₃	C₂×C₃²×#41	semidirect
414. D₃	C₂×C₃²×#41	direct (N × H)
428. C₃	C₂×C₃×D₃×#41	direct (N × H)
429. C₃	C₂²×C₃²×#41	semidirect
433. C₂	C₃²×D₃×#41	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×C₃³×#41	2
C₃²×D₃×#41	2
C₂×C₃²×#239	2
C₂×C₃²×D₃×Q₈	2
C₂×C₃²×#237	2
C₂×C₃²×D₃×D₄	2
C₂×C₃²×#235	2
C₂²×C₃²×C₄×D₃	2
C₂²×C₃²×#41	3
C₂×C₃×D₃×#41	3

Subgroups — C2×C32×D3×#41 #13181720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×D₃×#41 #13181720