Subgroups — C₂×C₃×#3211 #10679383

Order 768.

2294 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3211	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#495	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#488	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#464	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#463	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#447	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#3211	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Q₁₆	C₂²	1	2	yes	complemented
17	C₂²×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂²×C₃×SD₁₆	C₂²	1	2	yes	complemented
19	C₂²×C₃×D₈	C₂²	1	2	yes	—
20	C₃×#529	—	2	4	no	—
21	C₃×#529	C₂²	1	8	yes	complemented
22	C₂×C₃×#126	—	2	1	no	—
23	C₂×C₃×#125	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×#124	—	2	1	no	—
25	C₂×C₃×#124	—	2	1	no	—
26	C₂×C₃×#124	—	2	4	no	—
27	C₂×C₃×#124	C₂²	1	2	yes	—
28	C₂×C₃×#124	C₂²	1	2	yes	—
29	C₂×C₃×#124	C₂²	1	8	yes	complemented
30	C₂×C₃×#124	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₃×#123	—	2	1	no	—
32	C₂×C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₃×#121	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₃×C₄×D₄	—	2	1	no	—
36	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₃×#117	—	2	1	no	—
38	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
39	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
40	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
41	C₃×#495	C₂²	1	16	yes	complemented
42	C₃×#488	C₂²	1	16	yes	complemented
43	C₃×#464	C₂²	1	16	yes	complemented
44	C₃×#463	C₂²	1	16	yes	complemented
45	C₃×#451	C₂²	1	8	yes	complemented
46	C₃×#450	C₂²	1	8	yes	complemented
47	C₃×#448	C₂²	1	8	yes	complemented
48	C₃×#447	C₂²	1	8	yes	—
49	C₃×#431	C₂²	1	8	yes	complemented
50	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
51	C₂×C₃×#103	C₂²	1	2	yes	complemented
52	C₂×C₃×#103	C₂²	1	2	yes	complemented
53	C₂×C₃×#102	C₂²	1	2	yes	—
54	C₂×C₃×#102	C₂²	1	2	yes	—
55	C₂×C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂×C₃×C₄×C₈	C₂²	1	2	yes	complemented
58	C₂×#529	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂²×#124	C₂×C₃	1	1	yes	characteristic, complemented
60	#3211	C₂×C₃	1	16	yes	direct product factor
61	C₂×#495	C₂×C₃	1	2	yes	complemented
62	C₂×#488	C₂×C₃	1	2	yes	complemented
63	C₂×#464	C₂×C₃	1	2	yes	complemented
64	C₂×#463	C₂×C₃	1	2	yes	complemented
65	C₂×#451	C₂×C₃	1	1	yes	characteristic, complemented
66	C₂×#448	C₂×C₃	1	1	yes	characteristic, complemented
67	C₂×#450	C₂×C₃	1	1	yes	characteristic, complemented
68	C₂×#447	C₂×C₃	1	1	yes	characteristic
69	C₂×#431	C₂×C₃	1	1	yes	characteristic, complemented
70	C₂⁵×C₃	—	2	1	no	—
71	C₂²×C₃×Q₈	—	2	2	no	—
72	C₂²×C₃×Q₈	—	2	1	no	—
73	C₂²×C₃×Q₈	—	4	4	no	—
74	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
75	C₂²×C₃×Q₈	C₂³	1	2	yes	—
76	C₂²×C₃×Q₈	C₂³	1	4	yes	—
77	C₂²×C₃×D₄	—	2	1	no	—
78	C₂²×C₃×D₄	—	2	2	no	—
79	C₂²×C₃×D₄	—	2	1	no	—
80	C₂²×C₃×D₄	—	4	4	no	—
81	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
82	C₂²×C₃×D₄	C₂³	1	2	yes	—
83	C₂²×C₃×D₄	C₂³	1	4	yes	—
84	C₂³×C₃×C₄	—	2	1	no	—
85	C₂³×C₃×C₄	—	2	1	no	—
86	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
87	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
88	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
89	C₂×C₃×Q₁₆	—	4	4	no	—
90	C₂×C₃×Q₁₆	C₂³	1	8	yes	complemented
91	C₂×C₃×SD₁₆	—	4	4	no	—
92	C₂×C₃×SD₁₆	—	4	4	no	—
93	C₂×C₃×SD₁₆	C₂³	1	8	yes	—
94	C₂×C₃×SD₁₆	C₂³	1	8	yes	complemented
95	C₂×C₃×D₈	—	4	4	no	—
96	C₂×C₃×D₈	C₂³	1	8	yes	—
97	C₂²×C₃×C₈	C₂³	1	2	yes	—
98	C₂²×C₃×C₈	C₂³	1	2	yes	—
99	C₃×#126	—	2	8	no	—
100	C₃×#125	—	2	2	no	—
101	C₃×#125	C₂³	1	4	yes	complemented
102	C₃×#124	—	2	4	no	—
103	C₃×#124	—	2	4	no	—
104	C₃×#124	—	2	8	no	—
105	C₃×#124	—	2	8	no	—
106	C₃×#124	—	2	2	no	—
107	C₃×#124	C₂³	1	8	yes	—
108	C₃×#124	C₂³	1	8	yes	—
109	C₃×#124	C₂³	1	4	yes	complemented
110	C₃×#123	—	2	8	no	—
111	C₃×C₂²⋊Q₈	—	2	2	no	—
112	C₃×C₂²⋊Q₈	C₂³	1	4	yes	—
113	C₃×#121	—	2	2	no	—
114	C₃×#121	C₂³	1	4	yes	—
115	C₃×C₄×Q₈	—	2	2	no	—
116	C₃×C₄×Q₈	C₂³	1	4	yes	complemented
117	C₃×C₄×D₄	—	2	8	no	—
118	C₃×C₄×D₄	—	2	2	no	—
119	C₃×C₄×D₄	C₂³	1	4	yes	complemented
120	C₃×#117	—	2	8	no	—
121	C₂×C₃×C₄⋊C₄	—	2	1	no	—
122	C₂×C₃×C₄⋊C₄	—	2	1	no	—
123	C₂×C₃×C₄⋊C₄	—	2	1	no	—
124	C₂×C₃×C₄⋊C₄	—	2	1	no	—
125	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
126	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
127	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
128	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
129	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
130	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
131	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
132	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
133	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
134	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
135	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
136	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
137	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₃×C₄²	—	2	1	no	—
139	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
140	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
141	C₂×C₃×C₄²	D₄	1	4	yes	—
142	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
143	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
144	C₂×C₃×C₄²	C₂³	1	4	yes	—
145	C₃×C₄⋊C₈	C₂³	1	8	yes	complemented
146	C₃×#103	C₂³	1	8	yes	complemented
147	C₃×#103	C₂³	1	8	yes	complemented
148	C₃×#102	C₂³	1	8	yes	—
149	C₃×#102	C₂³	1	8	yes	—
150	C₃×C₂²⋊C₈	C₂³	1	4	yes	—
151	C₃×C₂²⋊C₈	C₂³	1	4	yes	—
152	C₃×C₄×C₈	C₂³	1	8	yes	complemented
153	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
154	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
155	C₂²×Q₁₆	C₂²×C₃	1	2	yes	complemented
156	C₂²×SD₁₆	C₂²×C₃	1	2	yes	—
157	C₂²×SD₁₆	C₂²×C₃	1	2	yes	complemented
158	C₂²×D₈	C₂²×C₃	1	2	yes	—
159	#529	—	2	4	no	—
160	#529	C₂²×C₃	1	8	yes	complemented
161	C₂×#126	—	2	1	no	—
162	C₂×#125	C₂²×C₃	1	1	yes	characteristic, complemented
163	C₂×#124	—	2	1	no	—
164	C₂×#124	—	2	1	no	—
165	C₂×#124	—	2	4	no	—
166	C₂×#124	C₂²×C₃	1	2	yes	—
167	C₂×#124	C₂²×C₃	1	2	yes	—
168	C₂×#124	C₂²×C₃	1	8	yes	complemented
169	C₂×#124	C₂²×C₃	1	1	yes	characteristic, complemented
170	C₂×#123	—	2	1	no	—
171	C₂×C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
172	C₂×#121	C₂²×C₃	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
174	C₂×C₄×D₄	—	2	1	no	—
175	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
176	C₂×#117	—	2	1	no	—
177	C₂²×C₂²⋊C₄	—	2	1	no	—
178	C₂²×C₂²⋊C₄	—	2	1	no	—
179	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
180	#495	C₂²×C₃	1	16	yes	complemented
181	#488	C₂²×C₃	1	16	yes	complemented
182	#464	C₂²×C₃	1	16	yes	complemented
183	#463	C₂²×C₃	1	16	yes	complemented
184	#451	C₂²×C₃	1	8	yes	complemented
185	#450	C₂²×C₃	1	8	yes	complemented
186	#448	C₂²×C₃	1	8	yes	complemented
187	#447	C₂²×C₃	1	8	yes	—
188	#431	C₂²×C₃	1	8	yes	complemented
189	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
190	C₂×#103	C₂²×C₃	1	2	yes	complemented
191	C₂×#103	C₂²×C₃	1	2	yes	complemented
192	C₂×#102	C₂²×C₃	1	2	yes	—
193	C₂×#102	C₂²×C₃	1	2	yes	—
194	C₂×C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
195	C₂×C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₃	1	2	yes	complemented
197	C₂⁴×C₃	—	2	4	no	—
198	C₂⁴×C₃	—	2	2	no	—
199	C₂⁴×C₃	—	2	1	no	—
200	C₂⁴×C₃	—	4	4	no	—
201	C₂⁴×C₃	—	4	4	no	—
202	C₂⁴×C₃	—	4	4	no	—
203	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
204	C₂×C₃×Q₈	—	2	1	no	—
205	C₂×C₃×Q₈	—	2	4	no	—
206	C₂×C₃×Q₈	—	2	2	no	—
207	C₂×C₃×Q₈	—	2	4	no	—
208	C₂×C₃×Q₈	—	2	4	no	—
209	C₂×C₃×Q₈	—	2	4	no	—
210	C₂×C₃×Q₈	—	4	8	no	—
211	C₂×C₃×Q₈	—	4	4	no	—
212	C₂×C₃×Q₈	—	4	8	no	—
213	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
214	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
215	C₂×C₃×D₄	—	2	4	no	—
216	C₂×C₃×D₄	—	2	1	no	—
217	C₂×C₃×D₄	—	2	4	no	—
218	C₂×C₃×D₄	—	2	2	no	—
219	C₂×C₃×D₄	—	2	4	no	—
220	C₂×C₃×D₄	—	2	4	no	—
221	C₂×C₃×D₄	—	2	4	no	—
222	C₂×C₃×D₄	—	4	8	no	—
223	C₂×C₃×D₄	—	4	4	no	—
224	C₂×C₃×D₄	—	4	8	no	—
225	C₂×C₃×D₄	—	4	4	no	—
226	C₂×C₃×D₄	C₂⁴	1	4	yes	—
227	C₂×C₃×D₄	C₂⁴	1	2	yes	—
228	C₂²×C₃×C₄	—	2	4	no	—
229	C₂²×C₃×C₄	—	2	4	no	—
230	C₂²×C₃×C₄	—	2	4	no	—
231	C₂²×C₃×C₄	—	2	1	no	—
232	C₂²×C₃×C₄	—	2	1	no	—
233	C₂²×C₃×C₄	—	2	2	no	—
234	C₂²×C₃×C₄	—	2	1	no	—
235	C₂²×C₃×C₄	—	2	4	no	—
236	C₂²×C₃×C₄	—	4	2	no	—
237	C₂²×C₃×C₄	—	4	4	no	—
238	C₂²×C₃×C₄	—	4	4	no	—
239	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
240	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
241	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
242	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
243	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
244	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
245	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
246	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
247	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
248	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
249	C₃×Q₁₆	—	4	8	no	—
250	C₃×SD₁₆	—	4	8	no	—
251	C₃×SD₁₆	—	4	8	no	—
252	C₃×D₈	—	4	8	no	—
253	C₂×C₃×C₈	—	2	4	no	—
254	C₂×C₃×C₈	—	2	4	no	—
255	C₂×C₃×C₈	C₂⁴	1	4	yes	—
256	C₂×C₃×C₈	C₂⁴	1	4	yes	—
257	C₃×C₄⋊C₄	—	2	4	no	—
258	C₃×C₄⋊C₄	—	2	4	no	—
259	C₃×C₄⋊C₄	—	2	4	no	—
260	C₃×C₄⋊C₄	—	2	4	no	—
261	C₃×C₄⋊C₄	—	2	1	no	—
262	C₃×C₄⋊C₄	—	2	1	no	—
263	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
264	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
265	C₃×C₂²⋊C₄	—	2	4	no	—
266	C₃×C₂²⋊C₄	—	2	8	no	—
267	C₃×C₂²⋊C₄	—	2	8	no	—
268	C₃×C₂²⋊C₄	—	2	8	no	—
269	C₃×C₂²⋊C₄	—	2	8	no	—
270	C₃×C₂²⋊C₄	—	2	4	no	—
271	C₃×C₂²⋊C₄	—	2	4	no	—
272	C₃×C₂²⋊C₄	—	2	4	no	—
273	C₃×C₂²⋊C₄	—	2	4	no	—
274	C₃×C₄²	—	2	4	no	—
275	C₃×C₄²	C₂×D₄	1	4	yes	—
276	C₃×C₄²	C₂×D₄	1	2	yes	—
277	C₃×C₄²	C₂×D₄	1	2	yes	—
278	C₃×C₄²	C₂×D₄	1	4	yes	—
279	C₃×C₄²	C₂⁴	1	2	yes	—
280	C₃×C₄²	C₂⁴	1	2	yes	—
281	C₂⁵	—	2	1	no	—
282	C₂²×Q₈	—	2	1	no	—
283	C₂²×Q₈	—	2	2	no	—
284	C₂²×Q₈	—	4	4	no	—
285	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
286	C₂²×Q₈	C₂³×C₃	1	2	yes	—
287	C₂²×Q₈	C₂³×C₃	1	4	yes	—
288	C₂²×D₄	—	2	1	no	—
289	C₂²×D₄	—	2	1	no	—
290	C₂²×D₄	—	2	2	no	—
291	C₂²×D₄	—	4	4	no	—
292	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
293	C₂²×D₄	C₂³×C₃	1	2	yes	—
294	C₂²×D₄	C₂³×C₃	1	4	yes	—
295	C₂³×C₄	—	2	1	no	—
296	C₂³×C₄	—	2	1	no	—
297	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
298	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
299	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
300	C₂×Q₁₆	—	4	4	no	—
301	C₂×Q₁₆	C₂³×C₃	1	8	yes	complemented
302	C₂×SD₁₆	—	4	4	no	—
303	C₂×SD₁₆	—	4	4	no	—
304	C₂×SD₁₆	C₂³×C₃	1	8	yes	—
305	C₂×SD₁₆	C₂³×C₃	1	8	yes	complemented
306	C₂×D₈	—	4	4	no	—
307	C₂×D₈	C₂³×C₃	1	8	yes	—
308	C₂²×C₈	C₂³×C₃	1	2	yes	—
309	C₂²×C₈	C₂³×C₃	1	2	yes	—
310	#126	—	2	8	no	—
311	#125	—	2	2	no	—
312	#125	C₂³×C₃	1	4	yes	complemented
313	#124	—	2	4	no	—
314	#124	—	2	4	no	—
315	#124	—	2	8	no	—
316	#124	—	2	8	no	—
317	#124	—	2	2	no	—
318	#124	C₂³×C₃	1	8	yes	—
319	#124	C₂³×C₃	1	8	yes	—
320	#124	C₂³×C₃	1	4	yes	complemented
321	#123	—	2	8	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	C₂³×C₃	1	4	yes	—
324	#121	—	2	2	no	—
325	#121	C₂³×C₃	1	4	yes	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	C₂³×C₃	1	4	yes	complemented
328	C₄×D₄	—	2	8	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	C₂³×C₃	1	4	yes	complemented
331	#117	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	8	no	—
344	C₂×C₂²⋊C₄	—	2	8	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
351	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
352	C₂×C₄²	C₃×D₄	1	4	yes	—
353	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃	1	4	yes	—
356	C₄⋊C₈	C₂³×C₃	1	8	yes	complemented
357	#103	C₂³×C₃	1	8	yes	complemented
358	#103	C₂³×C₃	1	8	yes	complemented
359	#102	C₂³×C₃	1	8	yes	—
360	#102	C₂³×C₃	1	8	yes	—
361	C₂²⋊C₈	C₂³×C₃	1	4	yes	—
362	C₂²⋊C₈	C₂³×C₃	1	4	yes	—
363	C₄×C₈	C₂³×C₃	1	8	yes	complemented
364	C₂³×C₃	—	2	2	no	—
365	C₂³×C₃	—	2	2	no	—
366	C₂³×C₃	—	2	2	no	—
367	C₂³×C₃	—	2	4	no	—
368	C₂³×C₃	—	4	8	no	—
369	C₂³×C₃	—	4	8	no	—
370	C₂³×C₃	—	4	4	no	—
371	C₂³×C₃	—	4	2	no	—
372	C₂³×C₃	—	4	2	no	—
373	C₂³×C₃	—	4	8	no	—
374	C₂³×C₃	—	4	4	no	—
375	C₂³×C₃	—	4	2	no	—
376	C₂³×C₃	—	4	4	no	—
377	C₂³×C₃	—	8	16	no	—
378	C₂³×C₃	#135	1	4	yes	—
379	C₂³×C₃	C₂²×D₄	1	2	yes	—
380	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
381	C₃×Q₈	—	2	4	no	—
382	C₃×Q₈	—	2	4	no	—
383	C₃×Q₈	—	2	4	no	—
384	C₃×Q₈	—	4	8	no	—
385	C₃×Q₈	—	4	2	no	—
386	C₃×Q₈	—	4	4	no	—
387	C₃×D₄	—	2	4	no	—
388	C₃×D₄	—	2	4	no	—
389	C₃×D₄	—	2	4	no	—
390	C₃×D₄	—	4	8	no	—
391	C₃×D₄	—	4	8	no	—
392	C₃×D₄	—	4	2	no	—
393	C₃×D₄	—	4	4	no	—
394	C₂×C₃×C₄	—	2	2	no	—
395	C₂×C₃×C₄	—	2	8	no	—
396	C₂×C₃×C₄	—	2	2	no	—
397	C₂×C₃×C₄	—	2	2	no	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	2	2	no	—
400	C₂×C₃×C₄	—	2	4	no	—
401	C₂×C₃×C₄	—	2	4	no	—
402	C₂×C₃×C₄	—	2	2	no	—
403	C₂×C₃×C₄	—	2	2	no	—
404	C₂×C₃×C₄	—	2	4	no	—
405	C₂×C₃×C₄	—	2	2	no	—
406	C₂×C₃×C₄	—	4	4	no	—
407	C₂×C₃×C₄	—	4	8	no	—
408	C₂×C₃×C₄	—	4	8	no	—
409	C₂×C₃×C₄	—	4	4	no	—
410	C₂×C₃×C₄	—	4	2	no	—
411	C₂×C₃×C₄	—	4	2	no	—
412	C₂×C₃×C₄	—	4	2	no	—
413	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
414	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
415	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
416	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
417	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
418	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
419	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
420	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
421	C₃×C₈	—	2	4	no	—
422	C₃×C₈	—	2	4	no	—
423	C₂⁴	—	2	1	no	—
424	C₂⁴	—	2	4	no	—
425	C₂⁴	—	2	2	no	—
426	C₂⁴	—	4	4	no	—
427	C₂⁴	—	4	4	no	—
428	C₂⁴	—	4	4	no	—
429	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	4	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	4	8	no	—
437	C₂×Q₈	—	4	4	no	—
438	C₂×Q₈	—	4	8	no	—
439	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
440	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	4	8	no	—
449	C₂×D₄	—	4	4	no	—
450	C₂×D₄	—	4	8	no	—
451	C₂×D₄	—	4	4	no	—
452	C₂×D₄	C₂⁴×C₃	1	4	yes	—
453	C₂×D₄	C₂⁴×C₃	1	2	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	4	2	no	—
463	C₂²×C₄	—	4	4	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
466	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
467	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
468	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
469	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
470	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
471	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
472	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
475	Q₁₆	—	4	8	no	—
476	SD₁₆	—	4	8	no	—
477	SD₁₆	—	4	8	no	—
478	D₈	—	4	8	no	—
479	C₂×C₈	—	2	4	no	—
480	C₂×C₈	—	2	4	no	—
481	C₂×C₈	C₂⁴×C₃	1	4	yes	—
482	C₂×C₈	C₂⁴×C₃	1	4	yes	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
490	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	8	no	—
493	C₂²⋊C₄	—	2	8	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	C₂×C₃×D₄	1	4	yes	—
502	C₄²	C₂×C₃×D₄	1	2	yes	—
503	C₄²	C₂×C₃×D₄	1	2	yes	—
504	C₄²	C₂×C₃×D₄	1	4	yes	—
505	C₄²	C₂⁴×C₃	1	2	yes	—
506	C₄²	C₂⁴×C₃	1	2	yes	—
507	C₂²×C₃	—	2	2	no	—
508	C₂²×C₃	—	2	8	no	—
509	C₂²×C₃	—	2	2	no	—
510	C₂²×C₃	—	4	4	no	—
511	C₂²×C₃	—	4	2	no	—
512	C₂²×C₃	—	4	2	no	—
513	C₂²×C₃	—	4	4	no	—
514	C₂²×C₃	—	4	2	no	—
515	C₂²×C₃	—	4	4	no	—
516	C₂²×C₃	—	8	16	no	—
517	C₂²×C₃	—	8	4	no	—
518	C₂²×C₃	—	8	8	no	—
519	C₂²×C₃	#535	1	2	yes	—
520	C₂²×C₃	C₂×#135	1	4	yes	—
521	C₂²×C₃	C₂×#135	1	2	yes	—
522	C₂²×C₃	#575	1	2	yes	—
523	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
524	C₃×C₄	—	2	2	no	—
525	C₃×C₄	—	2	2	no	—
526	C₃×C₄	—	2	2	no	—
527	C₃×C₄	—	2	2	no	—
528	C₃×C₄	—	2	2	no	—
529	C₃×C₄	—	4	4	no	—
530	C₃×C₄	—	4	2	no	—
531	C₃×C₄	—	4	2	no	—
532	C₃×C₄	—	4	2	no	—
533	C₃×C₄	#535	1	2	yes	—
534	C₃×C₄	C₂×H(5,𝔽₂)	1	2	yes	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	4	8	no	—
540	C₂³	—	4	8	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	8	no	—
543	C₂³	—	4	4	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	4	no	—
546	C₂³	—	4	4	no	—
547	C₂³	—	4	2	no	—
548	C₂³	—	8	16	no	—
549	C₂³	C₃×#135	1	4	yes	—
550	C₂³	C₂²×C₃×D₄	1	2	yes	—
551	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
552	Q₈	—	2	4	no	—
553	Q₈	—	2	4	no	—
554	Q₈	—	2	4	no	—
555	Q₈	—	4	8	no	—
556	Q₈	—	4	4	no	—
557	Q₈	—	4	2	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	4	no	—
561	D₄	—	4	8	no	—
562	D₄	—	4	8	no	—
563	D₄	—	4	4	no	—
564	D₄	—	4	2	no	—
565	C₂×C₄	—	2	8	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	8	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
585	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
586	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
587	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
588	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
589	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
590	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
591	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
592	C₈	—	2	4	no	—
593	C₈	—	2	4	no	—
594	C₂×C₃	—	2	4	no	—
595	C₂×C₃	—	8	2	no	—
596	C₂×C₃	—	8	4	no	—
597	C₂×C₃	#3211	1	4	yes	direct product factor
598	C₂×C₃	C₂×#535	1	1	yes	characteristic
599	C₂×C₃	C₂²×#135	1	1	yes	characteristic
600	C₂×C₃	C₂×#575	1	1	yes	characteristic
601	C₂²	—	2	2	no	—
602	C₂²	—	2	8	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	4	no	—
610	C₂²	—	8	16	no	—
611	C₂²	—	8	8	no	—
612	C₂²	—	8	4	no	—
613	C₂²	C₃×#535	1	2	yes	—
614	C₂²	C₂×C₃×#135	1	4	yes	—
615	C₂²	C₂×C₃×#135	1	2	yes	—
616	C₂²	C₃×#575	1	2	yes	—
617	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	4	4	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	2	no	—
627	C₄	C₃×#535	1	2	yes	—
628	C₄	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
629	C₃	C₂×#3211	1	1	yes	characteristic, direct product factor, Sylow
630	C₂	—	2	4	no	—
631	C₂	—	8	4	no	—
632	C₂	—	8	2	no	—
633	C₂	C₃×#3211	1	4	yes	direct product factor
634	C₂	C₂×C₃×#535	1	1	yes	characteristic
635	C₂	C₂²×C₃×#135	1	1	yes	characteristic
636	C₂	C₂×C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂×C₃×#447	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×#529	C₂	semidirect
2. C₂²×C₃×#124	C₂	semidirect
3. C₃×#3211	C₂	direct (N × H)
4. C₂×C₃×#495	C₂	semidirect
5. C₂×C₃×#488	C₂	semidirect
6. C₂×C₃×#464	C₂	semidirect
7. C₂×C₃×#463	C₂	semidirect
8. C₂×C₃×#451	C₂	semidirect
9. C₂×C₃×#448	C₂	semidirect
10. C₂×C₃×#450	C₂	semidirect
12. C₂×C₃×#431	C₂	semidirect
13. C₂×#3211	C₃	direct (N × H)
16. C₂²×C₃×Q₁₆	C₂²	semidirect
18. C₂²×C₃×SD₁₆	C₂²	semidirect
21. C₃×#529	C₂²	semidirect
23. C₂×C₃×#125	C₂²	semidirect
29. C₂×C₃×#124	C₂²	semidirect
30. C₂×C₃×#124	C₂²	semidirect
34. C₂×C₃×C₄×Q₈	C₂²	semidirect
36. C₂×C₃×C₄×D₄	C₂²	semidirect
41. C₃×#495	C₂²	semidirect
42. C₃×#488	C₂²	semidirect
43. C₃×#464	C₂²	semidirect
44. C₃×#463	C₂²	semidirect
45. C₃×#451	C₂²	semidirect
46. C₃×#450	C₂²	semidirect
47. C₃×#448	C₂²	semidirect
49. C₃×#431	C₂²	semidirect
50. C₂×C₃×C₄⋊C₈	C₂²	semidirect
51. C₂×C₃×#103	C₂²	semidirect
52. C₂×C₃×#103	C₂²	semidirect
57. C₂×C₃×C₄×C₈	C₂²	semidirect
58. C₂×#529	C₂×C₃	semidirect
59. C₂²×#124	C₂×C₃	semidirect
60. #3211	C₂×C₃	direct (N × H)
61. C₂×#495	C₂×C₃	semidirect
62. C₂×#488	C₂×C₃	semidirect
63. C₂×#464	C₂×C₃	semidirect
64. C₂×#463	C₂×C₃	semidirect
65. C₂×#451	C₂×C₃	semidirect
66. C₂×#448	C₂×C₃	semidirect
67. C₂×#450	C₂×C₃	semidirect
69. C₂×#431	C₂×C₃	semidirect
90. C₂×C₃×Q₁₆	C₂³	semidirect
94. C₂×C₃×SD₁₆	C₂³	semidirect
101. C₃×#125	C₂³	semidirect
109. C₃×#124	C₂³	semidirect
116. C₃×C₄×Q₈	C₂³	semidirect
119. C₃×C₄×D₄	C₂³	semidirect
145. C₃×C₄⋊C₈	C₂³	semidirect
146. C₃×#103	C₂³	semidirect
147. C₃×#103	C₂³	semidirect
152. C₃×C₄×C₈	C₂³	semidirect
155. C₂²×Q₁₆	C₂²×C₃	semidirect
157. C₂²×SD₁₆	C₂²×C₃	semidirect
160. #529	C₂²×C₃	semidirect
162. C₂×#125	C₂²×C₃	semidirect
168. C₂×#124	C₂²×C₃	semidirect
169. C₂×#124	C₂²×C₃	semidirect
173. C₂×C₄×Q₈	C₂²×C₃	semidirect
175. C₂×C₄×D₄	C₂²×C₃	semidirect
180. #495	C₂²×C₃	semidirect
181. #488	C₂²×C₃	semidirect
182. #464	C₂²×C₃	semidirect
183. #463	C₂²×C₃	semidirect
184. #451	C₂²×C₃	semidirect
185. #450	C₂²×C₃	semidirect
186. #448	C₂²×C₃	semidirect
188. #431	C₂²×C₃	semidirect
189. C₂×C₄⋊C₈	C₂²×C₃	semidirect
190. C₂×#103	C₂²×C₃	semidirect
191. C₂×#103	C₂²×C₃	semidirect
196. C₂×C₄×C₈	C₂²×C₃	semidirect
301. C₂×Q₁₆	C₂³×C₃	semidirect
305. C₂×SD₁₆	C₂³×C₃	semidirect
312. #125	C₂³×C₃	semidirect
320. #124	C₂³×C₃	semidirect
327. C₄×Q₈	C₂³×C₃	semidirect
330. C₄×D₄	C₂³×C₃	semidirect
356. C₄⋊C₈	C₂³×C₃	semidirect
357. #103	C₂³×C₃	semidirect
358. #103	C₂³×C₃	semidirect
363. C₄×C₈	C₂³×C₃	semidirect
597. C₂×C₃	#3211	direct (N × H)
629. C₃	C₂×#3211	direct (N × H)
633. C₂	C₃×#3211	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×#529	2
C₂²×C₃×#124	2
C₃×#3211	2
C₂×C₃×#495	2
C₂×C₃×#488	2
C₂×C₃×#464	2
C₂×C₃×#463	2
C₂×C₃×#451	2
C₂×C₃×#448	2
C₂×C₃×#450	2
C₂×C₃×#447	2
C₂×C₃×#431	2
C₂×#3211	3

Subgroups — C2×C3×#3211 #10679383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3211 #10679383