Subgroups — C₃×#18006 #10636613

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3105	C₂	1	1	yes	maximal, characteristic
2	C₃×#2969	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1586	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃×#1495	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	#18006	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₃×M₁₆	—	2	1	no	—
18	C₃×#529	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
20	C₃×#484	C₄	1	1	yes	characteristic
21	C₃×#481	C₄	1	1	yes	characteristic
22	C₃×#478	C₄	1	1	yes	characteristic
23	C₃×#478	C₄	1	1	yes	characteristic
24	C₃×#463	C₄	1	1	yes	characteristic
25	C₃×#463	C₄	1	1	yes	characteristic
26	C₃×#450	C₄	1	1	yes	characteristic
27	C₃×#447	C₄	1	1	yes	characteristic
28	C₃×#446	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₄×SD₁₆	C₂²	1	1	yes	characteristic
31	C₃×C₄×SD₁₆	C₂²	1	1	yes	characteristic
32	C₃×C₄×D₈	C₂²	1	1	yes	characteristic
33	C₃×C₄×D₈	C₂²	1	1	yes	characteristic
34	C₃×#436	—	2	1	no	—
35	C₃×#436	C₂²	1	1	yes	characteristic
36	C₃×#436	C₂²	1	1	yes	characteristic
37	C₃×#436	C₂²	1	1	yes	characteristic, complemented
38	C₃×#435	—	2	1	no	—
39	C₃×#435	—	2	1	no	—
40	C₃×#435	—	2	1	no	—
41	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
42	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
43	C₃×C₈×D₄	C₂²	1	1	yes	characteristic, complemented
44	C₃×#433	—	2	1	no	—
45	C₃×#431	C₂²	1	1	yes	characteristic
46	C₃×#431	C₂²	1	1	yes	characteristic
47	C₃×#431	C₂²	1	1	yes	characteristic
48	C₃×#423	C₂²	1	1	yes	characteristic
49	C₃×#407	—	2	1	no	—
50	C₃×#405	—	2	1	no	—
51	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
52	C₃×#335	C₂²	1	1	yes	characteristic, complemented
53	C₃×#335	C₂²	1	1	yes	characteristic, complemented
54	C₃×#334	C₂²	1	1	yes	characteristic, complemented
55	C₃×#334	C₂²	1	1	yes	characteristic, complemented
56	C₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
58	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	C₃×#322	C₂²	1	1	yes	characteristic, complemented
65	C₃×#322	C₂²	1	1	yes	characteristic, complemented
66	C₃×#322	C₂²	1	1	yes	characteristic, complemented
67	C₃×C₈²	C₂²	1	1	yes	characteristic, complemented
68	#3105	C₂×C₃	1	1	yes	characteristic
69	#2969	C₂×C₃	1	1	yes	characteristic, complemented
70	D₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
71	#1594	C₂×C₃	1	1	yes	characteristic, complemented
72	#1590	C₂×C₃	1	1	yes	characteristic, complemented
73	#1588	C₂×C₃	1	1	yes	characteristic, complemented
74	#1586	C₂×C₃	1	1	yes	characteristic, complemented
75	#1582	C₂×C₃	1	1	yes	characteristic, complemented
76	#1581	C₂×C₃	1	1	yes	characteristic, complemented
77	SD₁₆⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
78	C₈×D₈	C₂×C₃	1	1	yes	characteristic, complemented
79	#1574	C₂×C₃	1	1	yes	characteristic, complemented
80	#1495	C₂×C₃	1	1	yes	characteristic, complemented
81	#1478	C₂×C₃	1	1	yes	characteristic, complemented
82	#1453	C₂×C₃	1	1	yes	characteristic, complemented
83	C₂²×C₃×D₄	C₂×C₄	1	1	yes	characteristic
84	C₂³×C₃×C₄	—	2	1	no	—
85	C₂×C₃×SD₁₆	C₂×C₄	1	1	yes	characteristic
86	C₂×C₃×SD₁₆	C₂×C₄	1	1	yes	characteristic
87	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic
88	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic
89	C₂×C₃×M₁₆	—	2	1	no	—
90	C₂×C₃×M₁₆	—	2	1	no	—
91	C₂×C₃×M₁₆	—	2	1	no	—
92	C₂×C₃×M₁₆	—	2	1	no	—
93	C₂×C₃×M₁₆	—	4	4	no	—
94	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
95	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
96	C₂²×C₃×C₈	—	2	1	no	—
97	C₂²×C₃×C₈	—	2	1	no	—
98	C₂²×C₃×C₈	—	2	1	no	—
99	C₃×#126	—	2	1	no	—
100	C₃×#125	C₂×C₄	1	1	yes	characteristic
101	C₃×#124	C₂×C₄	1	1	yes	characteristic
102	C₃×#123	—	2	1	no	—
103	C₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
104	C₃×#121	C₂×C₄	1	1	yes	characteristic
105	C₃×C₄×Q₈	C₂³	1	1	yes	characteristic
106	C₃×C₄×D₄	—	2	1	no	—
107	C₃×C₄×D₄	—	2	1	no	—
108	C₃×C₄×D₄	—	2	1	no	—
109	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
110	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
111	C₃×C₄×D₄	C₂³	1	1	yes	characteristic
112	C₃×C₄×D₄	C₂³	1	1	yes	characteristic
113	C₃×C₄×D₄	C₂³	1	1	yes	characteristic
114	C₃×#117	—	2	1	no	—
115	C₂×C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
116	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
117	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
118	C₃×#107	C₂×C₄	1	1	yes	characteristic
119	C₃×#107	C₂×C₄	1	1	yes	characteristic
120	C₃×#106	C₂×C₄	1	1	yes	characteristic
121	C₃×#106	C₂×C₄	1	1	yes	characteristic
122	C₃×C₄⋊C₈	—	2	1	no	—
123	C₃×C₄⋊C₈	—	2	1	no	—
124	C₃×C₄⋊C₈	—	2	1	no	—
125	C₃×C₄⋊C₈	—	2	1	no	—
126	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
127	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
128	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
129	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
130	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
131	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
132	C₃×#103	C₂×C₄	1	1	yes	characteristic
133	C₃×#103	C₂×C₄	1	1	yes	characteristic
134	C₃×#102	C₂×C₄	1	1	yes	characteristic
135	C₃×#102	C₂×C₄	1	1	yes	characteristic
136	C₃×#102	C₂×C₄	1	1	yes	characteristic
137	C₃×#102	C₂×C₄	1	1	yes	characteristic
138	C₃×#102	C₂×C₄	1	1	yes	characteristic
139	C₃×#102	C₂×C₄	1	1	yes	characteristic
140	C₃×C₂²⋊C₈	—	2	1	no	—
141	C₃×C₂²⋊C₈	—	2	1	no	—
142	C₃×C₂²⋊C₈	—	2	1	no	—
143	C₃×C₂²⋊C₈	—	2	1	no	—
144	C₃×C₂²⋊C₈	—	2	1	no	—
145	C₃×C₂²⋊C₈	—	2	1	no	—
146	C₃×C₂²⋊C₈	—	2	1	no	—
147	C₃×C₂²⋊C₈	—	2	1	no	—
148	C₃×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
149	C₃×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
150	C₃×#97	—	2	1	no	—
151	C₃×#97	D₄	1	1	yes	characteristic
152	C₃×#97	D₄	1	1	yes	characteristic
153	C₃×C₄×C₈	—	2	1	no	—
154	C₃×C₄×C₈	—	2	1	no	—
155	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
156	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
157	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
158	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
159	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
160	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
161	C₂²×M₁₆	—	2	1	no	—
162	#529	C₂²×C₃	1	1	yes	characteristic
163	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
164	#484	C₃×C₄	1	1	yes	characteristic
165	#481	C₃×C₄	1	1	yes	characteristic
166	#478	C₃×C₄	1	1	yes	characteristic
167	#478	C₃×C₄	1	1	yes	characteristic
168	#463	C₃×C₄	1	1	yes	characteristic
169	#463	C₃×C₄	1	1	yes	characteristic
170	#450	C₃×C₄	1	1	yes	characteristic
171	#447	C₃×C₄	1	1	yes	characteristic
172	#446	C₂²×C₃	1	1	yes	characteristic, complemented
173	C₈×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
174	C₄×SD₁₆	C₂²×C₃	1	1	yes	characteristic
175	C₄×SD₁₆	C₂²×C₃	1	1	yes	characteristic
176	C₄×D₈	C₂²×C₃	1	1	yes	characteristic
177	C₄×D₈	C₂²×C₃	1	1	yes	characteristic
178	#436	—	2	1	no	—
179	#436	C₂²×C₃	1	1	yes	characteristic
180	#436	C₂²×C₃	1	1	yes	characteristic
181	#436	C₂²×C₃	1	1	yes	characteristic, complemented
182	#435	—	2	1	no	—
183	#435	—	2	1	no	—
184	#435	—	2	1	no	—
185	C₈×D₄	C₂²×C₃	1	1	yes	characteristic
186	C₈×D₄	C₂²×C₃	1	1	yes	characteristic
187	C₈×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
188	#433	—	2	1	no	—
189	#431	C₂²×C₃	1	1	yes	characteristic
190	#431	C₂²×C₃	1	1	yes	characteristic
191	#431	C₂²×C₃	1	1	yes	characteristic
192	#423	C₂²×C₃	1	1	yes	characteristic
193	#407	—	2	1	no	—
194	#405	—	2	1	no	—
195	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
196	#335	C₂²×C₃	1	1	yes	characteristic, complemented
197	#335	C₂²×C₃	1	1	yes	characteristic, complemented
198	#334	C₂²×C₃	1	1	yes	characteristic, complemented
199	#334	C₂²×C₃	1	1	yes	characteristic, complemented
200	Q₈⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
201	Q₈⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
202	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
203	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
204	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
205	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
206	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
207	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
208	#322	C₂²×C₃	1	1	yes	characteristic, complemented
209	#322	C₂²×C₃	1	1	yes	characteristic, complemented
210	#322	C₂²×C₃	1	1	yes	characteristic, complemented
211	C₈²	C₂²×C₃	1	1	yes	characteristic, complemented
212	C₂⁴×C₃	—	2	1	no	—
213	C₂×C₃×Q₈	C₂²×C₄	1	1	yes	characteristic
214	C₂×C₃×D₄	—	2	2	no	—
215	C₂×C₃×D₄	—	2	1	no	—
216	C₂×C₃×D₄	—	2	1	no	—
217	C₂×C₃×D₄	—	4	1	no	—
218	C₂×C₃×D₄	C₂²×C₄	1	1	yes	characteristic
219	C₂×C₃×D₄	C₂²×C₄	1	1	yes	characteristic
220	C₂×C₃×D₄	C₂²×C₄	1	1	yes	characteristic
221	C₂²×C₃×C₄	—	2	1	no	—
222	C₂²×C₃×C₄	—	2	1	no	—
223	C₂²×C₃×C₄	—	2	1	no	—
224	C₂²×C₃×C₄	—	2	1	no	—
225	C₂²×C₃×C₄	—	2	1	no	—
226	C₂²×C₃×C₄	—	2	1	no	—
227	C₂²×C₃×C₄	—	4	2	no	—
228	C₂²×C₃×C₄	—	4	2	no	—
229	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
230	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
231	C₂²×C₃×C₄	#41	1	1	yes	characteristic
232	C₃×SD₁₆	—	2	2	no	—
233	C₃×SD₁₆	—	4	1	no	—
234	C₃×D₈	—	2	2	no	—
235	C₃×D₈	—	4	1	no	—
236	C₃×M₁₆	—	2	2	no	—
237	C₃×M₁₆	—	4	1	no	—
238	C₃×M₁₆	—	4	2	no	—
239	C₃×M₁₆	—	4	2	no	—
240	C₃×M₁₆	—	4	2	no	—
241	C₃×M₁₆	—	4	2	no	—
242	C₂×C₃×C₈	—	2	1	no	—
243	C₂×C₃×C₈	—	2	1	no	—
244	C₂×C₃×C₈	—	2	1	no	—
245	C₂×C₃×C₈	—	2	1	no	—
246	C₂×C₃×C₈	—	2	1	no	—
247	C₂×C₃×C₈	—	2	1	no	—
248	C₂×C₃×C₈	—	4	2	no	—
249	C₂×C₃×C₈	—	4	2	no	—
250	C₂×C₃×C₈	—	4	2	no	—
251	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
252	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
253	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
254	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
255	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
256	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
257	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
258	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
259	C₃×C₄⋊C₄	—	2	1	no	—
260	C₃×C₄⋊C₄	—	2	1	no	—
261	C₃×C₄⋊C₄	—	2	1	no	—
262	C₃×C₄⋊C₄	—	2	1	no	—
263	C₃×C₄⋊C₄	—	2	1	no	—
264	C₃×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
265	C₃×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
266	C₃×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
267	C₃×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
268	C₃×C₂²⋊C₄	—	2	1	no	—
269	C₃×C₂²⋊C₄	—	2	1	no	—
270	C₃×C₂²⋊C₄	—	2	1	no	—
271	C₃×C₂²⋊C₄	—	2	1	no	—
272	C₃×C₂²⋊C₄	—	2	1	no	—
273	C₃×C₂²⋊C₄	—	2	1	no	—
274	C₃×C₂²⋊C₄	—	2	1	no	—
275	C₃×C₂²⋊C₄	—	2	1	no	—
276	C₃×C₂²⋊C₄	—	2	1	no	—
277	C₃×C₄²	—	2	1	no	—
278	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
279	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
280	C₃×C₄²	#41	1	1	yes	characteristic
281	C₃×C₄²	C₂⁴	1	1	yes	characteristic
282	C₂²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	C₂×SD₁₆	C₂×C₃×C₄	1	1	yes	characteristic
285	C₂×SD₁₆	C₂×C₃×C₄	1	1	yes	characteristic
286	C₂×D₈	C₂×C₃×C₄	1	1	yes	characteristic
287	C₂×D₈	C₂×C₃×C₄	1	1	yes	characteristic
288	C₂×M₁₆	—	2	1	no	—
289	C₂×M₁₆	—	2	1	no	—
290	C₂×M₁₆	—	2	1	no	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	—	4	4	no	—
293	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
294	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
295	C₂²×C₈	—	2	1	no	—
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	—	2	1	no	—
298	#126	—	2	1	no	—
299	#125	C₂×C₃×C₄	1	1	yes	characteristic
300	#124	C₂×C₃×C₄	1	1	yes	characteristic
301	#123	—	2	1	no	—
302	C₂²⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
303	#121	C₂×C₃×C₄	1	1	yes	characteristic
304	C₄×Q₈	C₂³×C₃	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	C₂×C₃×C₄	1	1	yes	characteristic
309	C₄×D₄	C₂×C₃×C₄	1	1	yes	characteristic
310	C₄×D₄	C₂³×C₃	1	1	yes	characteristic
311	C₄×D₄	C₂³×C₃	1	1	yes	characteristic
312	C₄×D₄	C₂³×C₃	1	1	yes	characteristic
313	#117	—	2	1	no	—
314	C₂×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	—	2	1	no	—
316	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
317	#107	C₂×C₃×C₄	1	1	yes	characteristic
318	#107	C₂×C₃×C₄	1	1	yes	characteristic
319	#106	C₂×C₃×C₄	1	1	yes	characteristic
320	#106	C₂×C₃×C₄	1	1	yes	characteristic
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	—	2	1	no	—
324	C₄⋊C₈	—	2	1	no	—
325	C₄⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
326	C₄⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
327	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
328	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
330	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
331	#103	C₂×C₃×C₄	1	1	yes	characteristic
332	#103	C₂×C₃×C₄	1	1	yes	characteristic
333	#102	C₂×C₃×C₄	1	1	yes	characteristic
334	#102	C₂×C₃×C₄	1	1	yes	characteristic
335	#102	C₂×C₃×C₄	1	1	yes	characteristic
336	#102	C₂×C₃×C₄	1	1	yes	characteristic
337	#102	C₂×C₃×C₄	1	1	yes	characteristic
338	#102	C₂×C₃×C₄	1	1	yes	characteristic
339	C₂²⋊C₈	—	2	1	no	—
340	C₂²⋊C₈	—	2	1	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₂²⋊C₈	—	2	1	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	C₂²⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
348	C₂²⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
349	#97	—	2	1	no	—
350	#97	C₃×D₄	1	1	yes	characteristic
351	#97	C₃×D₄	1	1	yes	characteristic
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
355	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
356	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
357	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
358	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
359	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
360	C₂³×C₃	—	2	1	no	—
361	C₂³×C₃	—	2	1	no	—
362	C₂³×C₃	—	2	1	no	—
363	C₂³×C₃	—	4	2	no	—
364	C₂³×C₃	—	4	2	no	—
365	C₂³×C₃	—	8	1	no	—
366	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
367	C₃×Q₈	—	2	1	no	—
368	C₃×Q₈	—	2	1	no	—
369	C₃×D₄	—	2	1	no	—
370	C₃×D₄	—	2	1	no	—
371	C₃×D₄	—	2	1	no	—
372	C₃×D₄	—	2	1	no	—
373	C₃×D₄	—	2	1	no	—
374	C₃×D₄	—	4	2	no	—
375	C₃×D₄	—	8	1	no	—
376	C₃×D₄	#131	1	2	yes	—
377	C₂×C₃×C₄	—	2	1	no	—
378	C₂×C₃×C₄	—	2	1	no	—
379	C₂×C₃×C₄	—	2	1	no	—
380	C₂×C₃×C₄	—	2	1	no	—
381	C₂×C₃×C₄	—	2	1	no	—
382	C₂×C₃×C₄	—	2	1	no	—
383	C₂×C₃×C₄	—	2	1	no	—
384	C₂×C₃×C₄	—	2	1	no	—
385	C₂×C₃×C₄	—	2	1	no	—
386	C₂×C₃×C₄	—	4	1	no	—
387	C₂×C₃×C₄	—	4	2	no	—
388	C₂×C₃×C₄	—	4	2	no	—
389	C₂×C₃×C₄	—	4	2	no	—
390	C₂×C₃×C₄	—	4	1	no	—
391	C₂×C₃×C₄	—	4	1	no	—
392	C₂×C₃×C₄	—	4	1	no	—
393	C₂×C₃×C₄	—	4	1	no	—
394	C₂×C₃×C₄	—	4	1	no	—
395	C₂×C₃×C₄	—	4	1	no	—
396	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
397	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
398	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
399	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
400	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
401	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
402	C₃×C₈	—	2	1	no	—
403	C₃×C₈	—	2	1	no	—
404	C₃×C₈	—	2	1	no	—
405	C₃×C₈	—	2	1	no	—
406	C₃×C₈	—	2	1	no	—
407	C₃×C₈	—	2	1	no	—
408	C₃×C₈	—	2	1	no	—
409	C₃×C₈	—	4	1	no	—
410	C₃×C₈	—	4	1	no	—
411	C₃×C₈	—	4	1	no	—
412	C₃×C₈	—	4	1	no	—
413	C₃×C₈	—	4	1	no	—
414	C₃×C₈	—	4	1	no	—
415	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
416	C₂⁴	—	2	1	no	—
417	C₂×Q₈	C₂²×C₃×C₄	1	1	yes	characteristic
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	4	1	no	—
422	C₂×D₄	C₂²×C₃×C₄	1	1	yes	characteristic
423	C₂×D₄	C₂²×C₃×C₄	1	1	yes	characteristic
424	C₂×D₄	C₂²×C₃×C₄	1	1	yes	characteristic
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
434	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
435	C₂²×C₄	C₃×#41	1	1	yes	characteristic
436	SD₁₆	—	2	2	no	—
437	SD₁₆	—	4	1	no	—
438	D₈	—	2	2	no	—
439	D₈	—	4	1	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	4	1	no	—
442	M₁₆	—	4	2	no	—
443	M₁₆	—	4	2	no	—
444	M₁₆	—	4	2	no	—
445	M₁₆	—	4	2	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	4	2	no	—
453	C₂×C₈	—	4	2	no	—
454	C₂×C₈	—	4	2	no	—
455	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
456	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
457	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
458	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
459	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
460	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
461	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
462	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	C₂²×C₃×C₄	1	1	yes	characteristic
469	C₄⋊C₄	C₂²×C₃×C₄	1	1	yes	characteristic
470	C₄⋊C₄	C₂²×C₃×C₄	1	1	yes	characteristic
471	C₄⋊C₄	C₂²×C₃×C₄	1	1	yes	characteristic
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
483	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
484	C₄²	C₃×#41	1	1	yes	characteristic
485	C₄²	C₂⁴×C₃	1	1	yes	characteristic
486	C₂²×C₃	—	2	1	no	—
487	C₂²×C₃	—	2	2	no	—
488	C₂²×C₃	—	2	1	no	—
489	C₂²×C₃	—	2	1	no	—
490	C₂²×C₃	—	4	1	no	—
491	C₂²×C₃	—	4	1	no	—
492	C₂²×C₃	—	4	1	no	—
493	C₂²×C₃	—	4	1	no	—
494	C₂²×C₃	—	4	1	no	—
495	C₂²×C₃	—	4	1	no	—
496	C₂²×C₃	—	4	1	no	—
497	C₂²×C₃	—	4	1	no	—
498	C₂²×C₃	—	8	4	no	—
499	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
500	C₃×C₄	—	2	1	no	—
501	C₃×C₄	—	2	1	no	—
502	C₃×C₄	—	2	1	no	—
503	C₃×C₄	—	2	1	no	—
504	C₃×C₄	—	4	1	no	—
505	C₃×C₄	—	4	1	no	—
506	C₃×C₄	—	4	1	no	—
507	C₃×C₄	—	4	1	no	—
508	C₃×C₄	—	4	1	no	—
509	C₃×C₄	C₂×#131	1	1	yes	characteristic
510	C₃×C₄	#568	1	1	yes	characteristic
511	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
512	C₃×C₄	#575	1	1	yes	characteristic
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	4	2	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	8	1	no	—
519	C₂³	C₃×C₄×D₄	1	1	yes	characteristic
520	Q₈	—	2	1	no	—
521	Q₈	—	2	1	no	—
522	D₄	—	2	1	no	—
523	D₄	—	2	1	no	—
524	D₄	—	2	1	no	—
525	D₄	—	2	1	no	—
526	D₄	—	2	1	no	—
527	D₄	—	4	2	no	—
528	D₄	—	8	1	no	—
529	D₄	C₃×#131	1	2	yes	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
550	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
551	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
552	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
553	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
554	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
555	C₈	—	2	1	no	—
556	C₈	—	2	1	no	—
557	C₈	—	2	1	no	—
558	C₈	—	2	1	no	—
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	2	1	no	—
562	C₈	—	4	1	no	—
563	C₈	—	4	1	no	—
564	C₈	—	4	1	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
569	C₂×C₃	—	4	1	no	—
570	C₂×C₃	—	4	2	no	—
571	C₂×C₃	—	8	1	no	—
572	C₂×C₃	—	8	1	no	—
573	C₂×C₃	#2933	1	1	yes	characteristic
574	C₂×C₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
575	C₂×C₃	#2957	1	1	yes	characteristic
576	C₂²	—	2	1	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	8	4	no	—
589	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	C₂×C₃×#131	1	1	yes	characteristic
600	C₄	C₃×#568	1	1	yes	characteristic
601	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
602	C₄	C₃×#575	1	1	yes	characteristic
603	C₃	#18006	1	1	yes	characteristic, direct product factor, Sylow
604	C₂	—	4	1	no	—
605	C₂	—	4	2	no	—
606	C₂	—	8	1	no	—
607	C₂	—	8	1	no	—
608	C₂	C₃×#2933	1	1	yes	characteristic
609	C₂	C₃×C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
610	C₂	C₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₄²	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₃×#447	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2969	C₂	semidirect
3. C₃×D₄×M₁₆	C₂	semidirect
4. C₃×#1594	C₂	semidirect
5. C₃×#1590	C₂	semidirect
6. C₃×#1588	C₂	semidirect
7. C₃×#1586	C₂	semidirect
8. C₃×#1582	C₂	semidirect
9. C₃×#1581	C₂	semidirect
10. C₃×SD₁₆⋊C₈	C₂	semidirect
11. C₃×C₈×D₈	C₂	semidirect
12. C₃×#1574	C₂	semidirect
13. C₃×#1495	C₂	semidirect
14. C₃×#1478	C₂	semidirect
15. C₃×#1453	C₂	semidirect
16. #18006	C₃	direct (N × H)
28. C₃×#446	C₂²	semidirect
29. C₃×C₈×Q₈	C₂²	semidirect
37. C₃×#436	C₂²	semidirect
43. C₃×C₈×D₄	C₂²	semidirect
52. C₃×#335	C₂²	semidirect
53. C₃×#335	C₂²	semidirect
54. C₃×#334	C₂²	semidirect
55. C₃×#334	C₂²	semidirect
56. C₃×Q₈⋊C₈	C₂²	semidirect
57. C₃×Q₈⋊C₈	C₂²	semidirect
62. C₃×D₄⋊C₈	C₂²	semidirect
63. C₃×D₄⋊C₈	C₂²	semidirect
64. C₃×#322	C₂²	semidirect
65. C₃×#322	C₂²	semidirect
66. C₃×#322	C₂²	semidirect
67. C₃×C₈²	C₂²	semidirect
69. #2969	C₂×C₃	semidirect
70. D₄×M₁₆	C₂×C₃	semidirect
71. #1594	C₂×C₃	semidirect
72. #1590	C₂×C₃	semidirect
73. #1588	C₂×C₃	semidirect
74. #1586	C₂×C₃	semidirect
75. #1582	C₂×C₃	semidirect
76. #1581	C₂×C₃	semidirect
77. SD₁₆⋊C₈	C₂×C₃	semidirect
78. C₈×D₈	C₂×C₃	semidirect
79. #1574	C₂×C₃	semidirect
80. #1495	C₂×C₃	semidirect
81. #1478	C₂×C₃	semidirect
82. #1453	C₂×C₃	semidirect
172. #446	C₂²×C₃	semidirect
173. C₈×Q₈	C₂²×C₃	semidirect
181. #436	C₂²×C₃	semidirect
187. C₈×D₄	C₂²×C₃	semidirect
196. #335	C₂²×C₃	semidirect
197. #335	C₂²×C₃	semidirect
198. #334	C₂²×C₃	semidirect
199. #334	C₂²×C₃	semidirect
200. Q₈⋊C₈	C₂²×C₃	semidirect
201. Q₈⋊C₈	C₂²×C₃	semidirect
206. D₄⋊C₈	C₂²×C₃	semidirect
207. D₄⋊C₈	C₂²×C₃	semidirect
208. #322	C₂²×C₃	semidirect
209. #322	C₂²×C₃	semidirect
210. #322	C₂²×C₃	semidirect
211. C₈²	C₂²×C₃	semidirect
603. C₃	#18006	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3105	2
C₃×#2969	2
C₃×D₄×M₁₆	2
C₃×#1594	2
C₃×#1590	2
C₃×#1588	2
C₃×#1586	2
C₃×#1582	2
C₃×#1581	2
C₃×SD₁₆⋊C₈	2
C₃×C₈×D₈	2
C₃×#1574	2
C₃×#1495	2
C₃×#1478	2
C₃×#1453	2
#18006	3

Subgroups — C3×#18006 #10636613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#18006 #10636613