Subgroups — C₂²×C₃⋊D₃×#67 #13181660

Order 1728.

9116 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6312	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃²×#67	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃⋊D₃×#67	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#6292	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6290	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₃⋊D₃	—	3	1	no	maximal
10	C₂²×D₃×#67	—	3	4	no	maximal
11	C₂⁴×C₃³	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₃×C₃⋊D₃	—	2	8	no	—
14	C₂³×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₃⋊D₃	C₂²	1	12	yes	complemented
16	C₂³×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₃²×D₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₃×C₃⋊D₃	—	2	8	no	—
19	C₂²×D₃×C₃⋊D₃	C₂²	1	12	yes	complemented
20	C₂×#6312	C₂²	1	12	yes	complemented
21	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃²×#67	C₂²	1	12	yes	complemented
23	C₂²×C₃²×Dic₃	C₂²	1	1	yes	characteristic, complemented
24	C₃⋊D₃×#67	C₂²	1	64	yes	direct product factor
25	C₂×#6292	C₂²	1	12	yes	complemented
26	C₂×#6290	C₂²	1	12	yes	complemented
27	C₂×Dic₃×C₃⋊D₃	C₂²	1	12	yes	complemented
28	C₂⁴×C₃⋊D₃	—	3	1	no	—
29	C₂⁴×C₃⋊D₃	D₃	1	1	yes	characteristic, complemented
30	C₂⁴×C₃×D₃	—	3	4	no	—
31	C₂³×D₃²	—	3	4	no	—
32	C₂²×C₃²×D₄	—	3	1	no	—
33	C₂²×#638	—	3	1	no	—
34	C₂²×#638	—	3	4	no	—
35	C₂²×#638	—	3	1	no	—
36	C₂²×#638	—	6	4	no	—
37	C₂×D₄×C₃⋊D₃	—	3	24	no	—
38	C₂²×C₄⋊(C₃⋊D₃)	—	3	1	no	—
39	C₂²×C₄×C₃⋊D₃	—	3	1	no	—
40	C₂²×C₃×#67	D₃	1	4	yes	complemented
41	C₂×D₃×#67	—	3	96	no	—
42	C₂²×#626	—	3	4	no	—
43	C₂²×#625	—	3	4	no	—
44	C₂²×D₃×Dic₃	—	3	4	no	—
45	C₂³×C₃³	—	2	4	no	—
46	C₂³×C₃³	C₂³	1	1	yes	characteristic
47	C₂³×C₃³	C₂³	1	6	yes	complemented
48	C₂²×C₃⋊(C₃⋊D₃)	—	2	4	no	—
49	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	6	yes	complemented
50	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
51	C₂²×C₃×C₃⋊D₃	—	2	48	no	—
52	C₂²×C₃×C₃⋊D₃	D₄	1	8	yes	complemented
53	C₂²×C₃×C₃⋊D₃	C₂³	1	6	yes	—
54	C₂²×C₃×C₃⋊D₃	C₂³	1	16	yes	complemented
55	C₂²×C₃×C₃⋊D₃	C₂³	1	6	yes	complemented
56	C₂²×C₃²×D₃	—	2	4	no	—
57	C₂²×C₃²×D₃	C₂³	1	6	yes	complemented
58	C₂×D₃×C₃⋊D₃	—	2	48	no	—
59	C₂×D₃×C₃⋊D₃	C₂³	1	16	yes	complemented
60	#6312	C₂³	1	16	yes	complemented
61	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	6	yes	complemented
62	C₃²×#67	C₂³	1	16	yes	complemented
63	C₂×C₃²×Dic₃	C₂³	1	6	yes	complemented
64	#6292	C₂³	1	16	yes	complemented
65	#6290	C₂³	1	16	yes	complemented
66	Dic₃×C₃⋊D₃	C₂³	1	16	yes	complemented
67	C₂³×#67	—	9	1	no	—
68	C₂²×D₃×D₄	—	9	4	no	—
69	C₂⁴×C₃²	—	2	4	no	—
70	C₂⁴×C₃²	—	3	1	no	—
71	C₂⁴×C₃²	C₂×D₃	1	4	yes	complemented
72	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic, complemented
73	C₂³×C₃⋊D₃	—	2	8	no	—
74	C₂³×C₃⋊D₃	—	3	1	no	—
75	C₂³×C₃⋊D₃	—	3	4	no	—
76	C₂³×C₃⋊D₃	—	3	12	no	—
77	C₂³×C₃⋊D₃	—	6	8	no	—
78	C₂³×C₃⋊D₃	—	6	4	no	—
79	C₂³×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
80	C₂³×C₃⋊D₃	C₂×D₃	1	12	yes	complemented
81	C₂³×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic, complemented
82	C₂³×C₃×D₃	—	3	4	no	—
83	C₂³×C₃×D₃	—	3	4	no	—
84	C₂³×C₃×D₃	—	3	48	no	—
85	C₂³×C₃×D₃	—	6	32	no	—
86	C₂³×C₃×D₃	C₂×D₃	1	4	yes	complemented
87	C₂²×D₃²	—	3	48	no	—
88	C₂²×D₃²	—	6	32	no	—
89	C₂×C₃²×D₄	—	3	12	no	—
90	C₂²×C₃²×C₄	—	3	1	no	—
91	C₂×#638	—	3	12	no	—
92	C₂×#638	—	3	48	no	—
93	C₂×#638	—	3	12	no	—
94	C₂×#638	—	6	48	no	—
95	C₂²×C₃⋊Dic₃	—	3	1	no	—
96	C₂²×C₃⋊Dic₃	—	3	4	no	—
97	C₂²×C₃⋊Dic₃	—	6	4	no	—
98	D₄×C₃⋊D₃	—	3	64	no	—
99	C₂×C₄⋊(C₃⋊D₃)	—	3	12	no	—
100	C₂×C₄×C₃⋊D₃	—	3	12	no	—
101	C₂×C₃×#67	C₂×D₃	1	48	yes	complemented
102	C₂²×C₃×Dic₃	C₂×D₃	1	4	yes	complemented
103	D₃×#67	—	3	256	no	—
104	C₂×#626	—	3	48	no	—
105	C₂×#625	—	3	48	no	—
106	C₂×D₃×Dic₃	—	3	48	no	—
107	C₂²×C₃³	—	2	12	no	—
108	C₂²×C₃³	C₂×D₄	1	4	yes	complemented
109	C₂²×C₃³	C₂⁴	1	3	yes	—
110	C₂²×C₃³	C₂⁴	1	4	yes	complemented
111	C₂×C₃⋊(C₃⋊D₃)	—	2	12	no	—
112	C₂×C₃⋊(C₃⋊D₃)	C₂⁴	1	4	yes	complemented
113	C₂×C₃×C₃⋊D₃	—	2	12	no	—
114	C₂×C₃×C₃⋊D₃	—	2	32	no	—
115	C₂×C₃×C₃⋊D₃	C₂×D₄	1	24	yes	complemented
116	C₂×C₃×C₃⋊D₃	C₂⁴	1	4	yes	—
117	C₂×C₃×C₃⋊D₃	C₂⁴	1	4	yes	complemented
118	C₂×C₃²×D₃	—	2	12	no	—
119	C₂×C₃²×D₃	C₂⁴	1	4	yes	complemented
120	D₃×C₃⋊D₃	—	2	32	no	—
121	C₃⋊(C₃⋊Dic₃)	C₂⁴	1	4	yes	complemented
122	C₃²×Dic₃	C₂⁴	1	4	yes	complemented
123	C₂⁵×C₃	—	9	1	no	—
124	C₂⁴×D₃	—	3	4	no	—
125	C₂⁴×D₃	—	9	4	no	—
126	C₂⁴×D₃	—	9	1	no	—
127	C₂²×C₃×D₄	—	3	4	no	—
128	C₂²×#67	—	9	4	no	—
129	C₂²×#67	—	9	4	no	—
130	C₂²×#67	—	9	1	no	—
131	C₂²×#67	—	9	1	no	—
132	C₂²×#67	—	9	24	no	—
133	C₂²×#67	—	9	1	no	—
134	C₂²×#67	—	18	4	no	—
135	C₂²×#67	C₃⋊D₃	1	1	yes	characteristic, direct product factor
136	C₂³×Dic₃	—	9	1	no	—
137	C₂×D₃×D₄	—	9	96	no	—
138	C₂²×D₁₂	—	9	4	no	—
139	C₂²×C₄×D₃	—	9	4	no	—
140	C₂³×C₃²	—	2	4	no	—
141	C₂³×C₃²	—	2	4	no	—
142	C₂³×C₃²	—	2	24	no	—
143	C₂³×C₃²	—	2	16	no	—
144	C₂³×C₃²	—	3	6	no	—
145	C₂³×C₃²	—	4	16	no	—
146	C₂³×C₃²	—	6	4	no	—
147	C₂³×C₃²	C₂²×D₃	1	4	yes	—
148	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
149	C₂³×C₃²	C₂²×D₃	1	24	yes	complemented
150	C₂³×C₃²	C₂²×D₃	1	6	yes	complemented
151	C₂²×C₃⋊D₃	—	2	4	no	—
152	C₂²×C₃⋊D₃	—	2	48	no	—
153	C₂²×C₃⋊D₃	—	3	6	no	—
154	C₂²×C₃⋊D₃	—	3	16	no	—
155	C₂²×C₃⋊D₃	—	3	24	no	—
156	C₂²×C₃⋊D₃	—	6	24	no	—
157	C₂²×C₃⋊D₃	—	6	4	no	—
158	C₂²×C₃⋊D₃	—	6	48	no	—
159	C₂²×C₃⋊D₃	—	6	16	no	—
160	C₂²×C₃⋊D₃	—	12	16	no	—
161	C₂²×C₃⋊D₃	#67	1	8	yes	direct product factor
162	C₂²×C₃⋊D₃	C₂²×D₃	1	6	yes	—
163	C₂²×C₃⋊D₃	C₂²×D₃	1	16	yes	complemented
164	C₂²×C₃⋊D₃	C₂²×D₃	1	6	yes	complemented
165	C₂²×C₃×D₃	—	2	16	no	—
166	C₂²×C₃×D₃	—	3	32	no	—
167	C₂²×C₃×D₃	—	3	24	no	—
168	C₂²×C₃×D₃	—	3	24	no	—
169	C₂²×C₃×D₃	—	3	64	no	—
170	C₂²×C₃×D₃	—	6	192	no	—
171	C₂²×C₃×D₃	—	6	16	no	—
172	C₂²×C₃×D₃	C₂²×D₃	1	24	yes	complemented
173	C₂×D₃²	—	3	64	no	—
174	C₂×D₃²	—	6	192	no	—
175	C₃²×D₄	—	3	16	no	—
176	C₂×C₃²×C₄	—	3	6	no	—
177	#638	—	3	16	no	—
178	#638	—	3	16	no	—
179	#638	—	3	64	no	—
180	#638	—	6	64	no	—
181	C₂×C₃⋊Dic₃	—	3	6	no	—
182	C₂×C₃⋊Dic₃	—	3	24	no	—
183	C₂×C₃⋊Dic₃	—	6	24	no	—
184	C₄⋊(C₃⋊D₃)	—	3	16	no	—
185	C₄×C₃⋊D₃	—	3	16	no	—
186	C₃×#67	C₂²×D₃	1	64	yes	complemented
187	C₂×C₃×Dic₃	C₂²×D₃	1	24	yes	complemented
188	#626	—	3	64	no	—
189	#625	—	3	64	no	—
190	D₃×Dic₃	—	3	64	no	—
191	C₂³×D₄	—	27	1	no	Sylow
192	C₂×C₃³	—	2	4	no	—
193	C₂×C₃³	C₂²×D₄	1	6	yes	complemented
194	C₂×C₃³	C₂⁵	1	1	yes	characteristic
195	C₃⋊(C₃⋊D₃)	—	2	4	no	—
196	C₃×C₃⋊D₃	—	2	4	no	—
197	C₃×C₃⋊D₃	C₂²×D₄	1	8	yes	complemented
198	C₃²×D₃	—	2	4	no	—
199	C₂⁴×C₃	—	2	4	no	—
200	C₂⁴×C₃	—	3	4	no	—
201	C₂⁴×C₃	—	9	1	no	—
202	C₂⁴×C₃	—	9	1	no	—
203	C₂⁴×C₃	—	9	12	no	—
204	C₂⁴×C₃	—	18	8	no	—
205	C₂⁴×C₃	D₃²	1	4	yes	complemented
206	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
207	C₂³×D₃	—	3	4	no	—
208	C₂³×D₃	—	3	4	no	—
209	C₂³×D₃	—	3	48	no	—
210	C₂³×D₃	—	6	32	no	—
211	C₂³×D₃	—	9	48	no	—
212	C₂³×D₃	—	9	4	no	—
213	C₂³×D₃	—	9	1	no	—
214	C₂³×D₃	—	9	12	no	—
215	C₂³×D₃	—	18	4	no	—
216	C₂³×D₃	—	18	8	no	—
217	C₂³×D₃	—	18	32	no	—
218	C₂³×D₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
219	C₂×C₃×D₄	—	3	48	no	—
220	C₂²×C₃×C₄	—	3	4	no	—
221	C₂×#67	—	9	48	no	—
222	C₂×#67	—	9	48	no	—
223	C₂×#67	—	9	12	no	—
224	C₂×#67	—	9	12	no	—
225	C₂×#67	—	9	64	no	—
226	C₂×#67	—	9	12	no	—
227	C₂×#67	—	18	48	no	—
228	C₂×#67	C₂×C₃⋊D₃	1	12	yes	direct product factor
229	C₂²×Dic₃	—	9	4	no	—
230	C₂²×Dic₃	—	9	1	no	—
231	C₂²×Dic₃	—	9	12	no	—
232	C₂²×Dic₃	—	18	4	no	—
233	C₂²×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
234	D₃×D₄	—	9	256	no	—
235	C₂×D₁₂	—	9	48	no	—
236	C₂×C₄×D₃	—	9	48	no	—
237	C₂²×C₃²	—	2	12	no	—
238	C₂²×C₃²	—	2	12	no	—
239	C₂²×C₃²	—	2	16	no	—
240	C₂²×C₃²	—	2	16	no	—
241	C₂²×C₃²	—	2	48	no	—
242	C₂²×C₃²	—	3	4	no	—
243	C₂²×C₃²	—	4	48	no	—
244	C₂²×C₃²	—	6	12	no	—
245	C₂²×C₃²	D₃×D₄	1	16	yes	complemented
246	C₂²×C₃²	C₂×#67	1	4	yes	complemented
247	C₂²×C₃²	C₂³×D₃	1	12	yes	—
248	C₂²×C₃²	C₂³×D₃	1	3	yes	—
249	C₂²×C₃²	C₂³×D₃	1	16	yes	complemented
250	C₂²×C₃²	C₂³×D₃	1	4	yes	complemented
251	C₂×C₃⋊D₃	—	2	12	no	—
252	C₂×C₃⋊D₃	—	2	32	no	—
253	C₂×C₃⋊D₃	—	3	4	no	—
254	C₂×C₃⋊D₃	—	3	16	no	—
255	C₂×C₃⋊D₃	—	6	16	no	—
256	C₂×C₃⋊D₃	—	6	12	no	—
257	C₂×C₃⋊D₃	—	6	32	no	—
258	C₂×C₃⋊D₃	—	6	48	no	—
259	C₂×C₃⋊D₃	—	12	48	no	—
260	C₂×C₃⋊D₃	C₂×#67	1	24	yes	direct product factor
261	C₂×C₃⋊D₃	C₂³×D₃	1	4	yes	—
262	C₂×C₃⋊D₃	C₂³×D₃	1	4	yes	complemented
263	C₂×C₃×D₃	—	2	48	no	—
264	C₂×C₃×D₃	—	3	96	no	—
265	C₂×C₃×D₃	—	3	16	no	—
266	C₂×C₃×D₃	—	3	16	no	—
267	C₂×C₃×D₃	—	6	128	no	—
268	C₂×C₃×D₃	—	6	48	no	—
269	C₂×C₃×D₃	C₂³×D₃	1	16	yes	complemented
270	D₃²	—	6	128	no	—
271	C₃²×C₄	—	3	4	no	—
272	C₃⋊Dic₃	—	3	4	no	—
273	C₃⋊Dic₃	—	3	16	no	—
274	C₃⋊Dic₃	—	6	16	no	—
275	C₃×Dic₃	C₂³×D₃	1	16	yes	complemented
276	C₂⁵	—	9	1	no	—
277	C₂⁵	—	27	1	no	—
278	C₂²×D₄	—	3	1	no	—
279	C₂²×D₄	—	27	1	no	—
280	C₂²×D₄	—	27	24	no	—
281	C₂²×D₄	—	27	1	no	—
282	C₂²×D₄	—	27	1	no	—
283	C₂³×C₄	—	27	1	no	—
284	C₃³	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
285	C₂³×C₃	—	2	16	no	—
286	C₂³×C₃	—	2	4	no	—
287	C₂³×C₃	—	2	4	no	—
288	C₂³×C₃	—	2	24	no	—
289	C₂³×C₃	—	3	24	no	—
290	C₂³×C₃	—	4	16	no	—
291	C₂³×C₃	—	6	16	no	—
292	C₂³×C₃	—	9	6	no	—
293	C₂³×C₃	—	9	8	no	—
294	C₂³×C₃	—	9	6	no	—
295	C₂³×C₃	—	9	16	no	—
296	C₂³×C₃	—	18	48	no	—
297	C₂³×C₃	—	18	4	no	—
298	C₂³×C₃	C₂×D₃²	1	4	yes	—
299	C₂³×C₃	C₂×D₃²	1	24	yes	complemented
300	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
301	C₂³×C₃	C₂²×C₃⋊D₃	1	6	yes	complemented
302	C₂²×D₃	—	2	4	no	—
303	C₂²×D₃	—	3	24	no	—
304	C₂²×D₃	—	3	32	no	—
305	C₂²×D₃	—	3	24	no	—
306	C₂²×D₃	—	3	64	no	—
307	C₂²×D₃	—	6	192	no	—
308	C₂²×D₃	—	6	16	no	—
309	C₂²×D₃	—	9	64	no	—
310	C₂²×D₃	—	9	24	no	—
311	C₂²×D₃	—	9	6	no	—
312	C₂²×D₃	—	9	16	no	—
313	C₂²×D₃	—	18	24	no	—
314	C₂²×D₃	—	18	48	no	—
315	C₂²×D₃	—	18	192	no	—
316	C₂²×D₃	—	18	16	no	—
317	C₂²×D₃	—	18	4	no	—
318	C₂²×D₃	—	36	16	no	—
319	C₂²×D₃	C₂²×C₃⋊D₃	1	6	yes	complemented
320	C₃×D₄	—	3	64	no	—
321	C₂×C₃×C₄	—	3	24	no	—
322	#67	—	9	64	no	—
323	#67	—	9	64	no	—
324	#67	—	9	16	no	—
325	#67	—	9	16	no	—
326	#67	—	9	16	no	—
327	#67	—	18	64	no	—
328	#67	C₂²×C₃⋊D₃	1	16	yes	direct product factor
329	C₂×Dic₃	—	9	24	no	—
330	C₂×Dic₃	—	9	6	no	—
331	C₂×Dic₃	—	9	16	no	—
332	C₂×Dic₃	—	18	24	no	—
333	C₂×Dic₃	C₂²×C₃⋊D₃	1	6	yes	complemented
334	D₁₂	—	9	64	no	—
335	C₄×D₃	—	9	64	no	—
336	C₂×C₃²	—	2	4	no	—
337	C₂×C₃²	—	2	24	no	—
338	C₂×C₃²	—	2	4	no	—
339	C₂×C₃²	—	2	16	no	—
340	C₂×C₃²	—	4	16	no	—
341	C₂×C₃²	—	6	4	no	—
342	C₂×C₃²	C₂×D₃×D₄	1	24	yes	complemented
343	C₂×C₃²	C₂²×#67	1	6	yes	complemented
344	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
345	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
346	C₃⋊D₃	—	2	4	no	—
347	C₃⋊D₃	—	6	4	no	—
348	C₃⋊D₃	—	6	16	no	—
349	C₃⋊D₃	—	12	16	no	—
350	C₃⋊D₃	C₂²×#67	1	8	yes	direct product factor
351	C₃×D₃	—	2	16	no	—
352	C₃×D₃	—	3	32	no	—
353	C₃×D₃	—	6	16	no	—
354	C₂⁴	—	3	1	no	—
355	C₂⁴	—	9	1	no	—
356	C₂⁴	—	9	1	no	—
357	C₂⁴	—	9	12	no	—
358	C₂⁴	—	18	8	no	—
359	C₂⁴	—	27	12	no	—
360	C₂⁴	—	27	1	no	—
361	C₂⁴	—	54	8	no	—
362	C₂⁴	D₃×C₃⋊D₃	1	1	yes	characteristic, complemented
363	C₂×D₄	—	3	12	no	—
364	C₂×D₄	—	27	12	no	—
365	C₂×D₄	—	27	64	no	—
366	C₂×D₄	—	27	12	no	—
367	C₂×D₄	—	27	12	no	—
368	C₂²×C₄	—	3	1	no	—
369	C₂²×C₄	—	27	12	no	—
370	C₂²×C₄	—	27	1	no	—
371	C₂²×C₃	—	2	48	no	—
372	C₂²×C₃	—	2	12	no	—
373	C₂²×C₃	—	2	12	no	—
374	C₂²×C₃	—	2	16	no	—
375	C₂²×C₃	—	2	16	no	—
376	C₂²×C₃	—	3	16	no	—
377	C₂²×C₃	—	4	48	no	—
378	C₂²×C₃	—	6	48	no	—
379	C₂²×C₃	—	9	24	no	—
380	C₂²×C₃	—	9	4	no	—
381	C₂²×C₃	—	9	4	no	—
382	C₂²×C₃	—	18	32	no	—
383	C₂²×C₃	—	18	12	no	—
384	C₂²×C₃	D₃×#67	1	16	yes	complemented
385	C₂²×C₃	D₄×C₃⋊D₃	1	4	yes	complemented
386	C₂²×C₃	C₂²×D₃²	1	12	yes	—
387	C₂²×C₃	C₂²×D₃²	1	16	yes	complemented
388	C₂²×C₃	C₂³×C₃⋊D₃	1	3	yes	—
389	C₂²×C₃	C₂³×C₃⋊D₃	1	4	yes	complemented
390	C₂×D₃	—	2	12	no	—
391	C₂×D₃	—	3	96	no	—
392	C₂×D₃	—	3	16	no	—
393	C₂×D₃	—	3	16	no	—
394	C₂×D₃	—	6	128	no	—
395	C₂×D₃	—	6	48	no	—
396	C₂×D₃	—	9	16	no	—
397	C₂×D₃	—	9	4	no	—
398	C₂×D₃	—	18	16	no	—
399	C₂×D₃	—	18	32	no	—
400	C₂×D₃	—	18	128	no	—
401	C₂×D₃	—	18	48	no	—
402	C₂×D₃	—	18	12	no	—
403	C₂×D₃	—	36	48	no	—
404	C₂×D₃	C₂³×C₃⋊D₃	1	4	yes	complemented
405	C₃×C₄	—	3	16	no	—
406	Dic₃	—	9	16	no	—
407	Dic₃	—	9	4	no	—
408	Dic₃	—	18	16	no	—
409	Dic₃	C₂³×C₃⋊D₃	1	4	yes	complemented
410	C₃²	—	2	4	no	—
411	C₃²	C₂²×D₃×D₄	1	4	yes	complemented
412	C₃²	C₂³×#67	1	1	yes	characteristic, complemented
413	C₂³	—	2	4	no	—
414	C₂³	—	3	6	no	—
415	C₂³	—	6	4	no	—
416	C₂³	—	9	6	no	—
417	C₂³	—	9	8	no	—
418	C₂³	—	9	6	no	—
419	C₂³	—	9	16	no	—
420	C₂³	—	18	48	no	—
421	C₂³	—	18	4	no	—
422	C₂³	—	27	16	no	—
423	C₂³	—	27	6	no	—
424	C₂³	—	54	48	no	—
425	C₂³	—	54	4	no	—
426	C₂³	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
427	C₂³	C₂×D₃×C₃⋊D₃	1	6	yes	complemented
428	D₄	—	3	16	no	—
429	D₄	—	27	16	no	—
430	D₄	—	27	16	no	—
431	D₄	—	27	16	no	—
432	C₂×C₄	—	3	6	no	—
433	C₂×C₄	—	27	16	no	—
434	C₂×C₄	—	27	6	no	—
435	C₂×C₃	—	2	16	no	—
436	C₂×C₃	—	2	4	no	—
437	C₂×C₃	—	2	24	no	—
438	C₂×C₃	—	2	4	no	—
439	C₂×C₃	—	4	16	no	—
440	C₂×C₃	—	6	16	no	—
441	C₂×C₃	—	9	8	no	—
442	C₂×C₃	—	18	4	no	—
443	C₂×C₃	C₂×D₃×#67	1	24	yes	complemented
444	C₂×C₃	C₂×D₄×C₃⋊D₃	1	6	yes	complemented
445	C₂×C₃	C₂³×D₃²	1	4	yes	—
446	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
447	D₃	—	2	4	no	—
448	D₃	—	3	32	no	—
449	D₃	—	6	16	no	—
450	D₃	—	18	16	no	—
451	D₃	—	18	4	no	—
452	D₃	—	36	16	no	—
453	C₂²	—	2	12	no	—
454	C₂²	—	3	4	no	—
455	C₂²	—	6	12	no	—
456	C₂²	—	9	24	no	—
457	C₂²	—	9	4	no	—
458	C₂²	—	9	4	no	—
459	C₂²	—	18	32	no	—
460	C₂²	—	18	12	no	—
461	C₂²	—	27	4	no	—
462	C₂²	—	54	32	no	—
463	C₂²	—	54	12	no	—
464	C₂²	C₃⋊D₃×#67	1	4	yes	direct product factor
465	C₂²	C₂²×D₃×C₃⋊D₃	1	3	yes	—
466	C₂²	C₂²×D₃×C₃⋊D₃	1	4	yes	complemented
467	C₄	—	3	4	no	—
468	C₄	—	27	4	no	—
469	C₃	—	2	4	no	—
470	C₃	C₂²×D₃×#67	1	4	yes	complemented
471	C₃	C₂²×D₄×C₃⋊D₃	1	1	yes	characteristic, complemented
472	C₂	—	2	4	no	—
473	C₂	—	6	4	no	—
474	C₂	—	9	8	no	—
475	C₂	—	18	4	no	—
476	C₂	—	54	4	no	—
477	C₂	C₂×C₃⋊D₃×#67	1	6	yes	direct product factor
478	C₂	C₂³×D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×C₃⋊D₃
commutator, cosocle	C₂×C₃³	C₂⁵
Frattini	C₂	C₂³×D₃×C₃⋊D₃
Fitting	C₂⁴×C₃³	C₂²
socle	C₂³×C₃³	C₂³
hypercenter	C₂⁴	D₃×C₃⋊D₃
hypocenter	C₃³	C₂³×D₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₃⋊D₃	C₂	semidirect
2. C₂³×D₃×C₃⋊D₃	C₂	semidirect
3. C₂²×#6312	C₂	semidirect
4. C₂²×C₃²×#67	C₂	semidirect
5. C₂×C₃⋊D₃×#67	C₂	direct (N × H)
6. C₂²×#6292	C₂	semidirect
7. C₂²×#6290	C₂	semidirect
8. C₂²×Dic₃×C₃⋊D₃	C₂	semidirect
11. C₂⁴×C₃³	C₂²	semidirect
12. C₂³×C₃⋊(C₃⋊D₃)	C₂²	semidirect
15. C₂³×C₃×C₃⋊D₃	C₂²	semidirect
16. C₂³×C₃×C₃⋊D₃	C₂²	semidirect
17. C₂³×C₃²×D₃	C₂²	semidirect
19. C₂²×D₃×C₃⋊D₃	C₂²	semidirect
20. C₂×#6312	C₂²	semidirect
21. C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
22. C₂×C₃²×#67	C₂²	semidirect
23. C₂²×C₃²×Dic₃	C₂²	semidirect
24. C₃⋊D₃×#67	C₂²	direct (N × H)
25. C₂×#6292	C₂²	semidirect
26. C₂×#6290	C₂²	semidirect
27. C₂×Dic₃×C₃⋊D₃	C₂²	semidirect
29. C₂⁴×C₃⋊D₃	D₃	semidirect
40. C₂²×C₃×#67	D₃	semidirect
47. C₂³×C₃³	C₂³	semidirect
49. C₂²×C₃⋊(C₃⋊D₃)	C₂³	semidirect
52. C₂²×C₃×C₃⋊D₃	D₄	semidirect
54. C₂²×C₃×C₃⋊D₃	C₂³	semidirect
55. C₂²×C₃×C₃⋊D₃	C₂³	semidirect
57. C₂²×C₃²×D₃	C₂³	semidirect
59. C₂×D₃×C₃⋊D₃	C₂³	semidirect
60. #6312	C₂³	semidirect
61. C₂×C₃⋊(C₃⋊Dic₃)	C₂³	semidirect
62. C₃²×#67	C₂³	semidirect
63. C₂×C₃²×Dic₃	C₂³	semidirect
64. #6292	C₂³	semidirect
65. #6290	C₂³	semidirect
66. Dic₃×C₃⋊D₃	C₂³	semidirect
71. C₂⁴×C₃²	C₂×D₃	semidirect
72. C₂⁴×C₃²	C₂×D₃	semidirect
80. C₂³×C₃⋊D₃	C₂×D₃	semidirect
81. C₂³×C₃⋊D₃	C₂×D₃	semidirect
86. C₂³×C₃×D₃	C₂×D₃	semidirect
101. C₂×C₃×#67	C₂×D₃	semidirect
102. C₂²×C₃×Dic₃	C₂×D₃	semidirect
108. C₂²×C₃³	C₂×D₄	semidirect
110. C₂²×C₃³	C₂⁴	semidirect
112. C₂×C₃⋊(C₃⋊D₃)	C₂⁴	semidirect
115. C₂×C₃×C₃⋊D₃	C₂×D₄	semidirect
117. C₂×C₃×C₃⋊D₃	C₂⁴	semidirect
119. C₂×C₃²×D₃	C₂⁴	semidirect
121. C₃⋊(C₃⋊Dic₃)	C₂⁴	semidirect
122. C₃²×Dic₃	C₂⁴	semidirect
135. C₂²×#67	C₃⋊D₃	direct (N × H)
149. C₂³×C₃²	C₂²×D₃	semidirect
150. C₂³×C₃²	C₂²×D₃	semidirect
161. C₂²×C₃⋊D₃	#67	direct (N × H)
163. C₂²×C₃⋊D₃	C₂²×D₃	semidirect
164. C₂²×C₃⋊D₃	C₂²×D₃	semidirect
172. C₂²×C₃×D₃	C₂²×D₃	semidirect
186. C₃×#67	C₂²×D₃	semidirect
187. C₂×C₃×Dic₃	C₂²×D₃	semidirect
193. C₂×C₃³	C₂²×D₄	semidirect
197. C₃×C₃⋊D₃	C₂²×D₄	semidirect
205. C₂⁴×C₃	D₃²	semidirect
206. C₂⁴×C₃	C₂×C₃⋊D₃	semidirect
218. C₂³×D₃	C₂×C₃⋊D₃	semidirect
228. C₂×#67	C₂×C₃⋊D₃	direct (N × H)
233. C₂²×Dic₃	C₂×C₃⋊D₃	semidirect
245. C₂²×C₃²	D₃×D₄	semidirect
246. C₂²×C₃²	C₂×#67	semidirect
249. C₂²×C₃²	C₂³×D₃	semidirect
250. C₂²×C₃²	C₂³×D₃	semidirect
260. C₂×C₃⋊D₃	C₂×#67	direct (N × H)
262. C₂×C₃⋊D₃	C₂³×D₃	semidirect
269. C₂×C₃×D₃	C₂³×D₃	semidirect
275. C₃×Dic₃	C₂³×D₃	semidirect
284. C₃³	C₂³×D₄	semidirect
299. C₂³×C₃	C₂×D₃²	semidirect
301. C₂³×C₃	C₂²×C₃⋊D₃	semidirect
319. C₂²×D₃	C₂²×C₃⋊D₃	semidirect
328. #67	C₂²×C₃⋊D₃	direct (N × H)
333. C₂×Dic₃	C₂²×C₃⋊D₃	semidirect
342. C₂×C₃²	C₂×D₃×D₄	semidirect
343. C₂×C₃²	C₂²×#67	semidirect
350. C₃⋊D₃	C₂²×#67	direct (N × H)
362. C₂⁴	D₃×C₃⋊D₃	semidirect
384. C₂²×C₃	D₃×#67	semidirect
385. C₂²×C₃	D₄×C₃⋊D₃	semidirect
387. C₂²×C₃	C₂²×D₃²	semidirect
389. C₂²×C₃	C₂³×C₃⋊D₃	semidirect
404. C₂×D₃	C₂³×C₃⋊D₃	semidirect
409. Dic₃	C₂³×C₃⋊D₃	semidirect
411. C₃²	C₂²×D₃×D₄	semidirect
412. C₃²	C₂³×#67	semidirect
427. C₂³	C₂×D₃×C₃⋊D₃	semidirect
443. C₂×C₃	C₂×D₃×#67	semidirect
444. C₂×C₃	C₂×D₄×C₃⋊D₃	semidirect
464. C₂²	C₃⋊D₃×#67	direct (N × H)
466. C₂²	C₂²×D₃×C₃⋊D₃	semidirect
470. C₃	C₂²×D₃×#67	semidirect
471. C₃	C₂²×D₄×C₃⋊D₃	semidirect
477. C₂	C₂×C₃⋊D₃×#67	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₃⋊D₃	2
C₂³×D₃×C₃⋊D₃	2
C₂²×#6312	2
C₂²×C₃²×#67	2
C₂×C₃⋊D₃×#67	2
C₂²×#6292	2
C₂²×#6290	2
C₂²×Dic₃×C₃⋊D₃	2
C₂²×D₄×C₃⋊D₃	3
C₂²×D₃×#67	3

Subgroups — C22×C3⋊D3×#67 #13181660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃⋊D₃×#67 #13181660