Subgroups — C₃×#18000 #10636607

Order 768.

672 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3101	C₂	1	1	yes	maximal, characteristic
2	C₃×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×#1592	C₂	1	2	yes	maximal, complemented
5	C₃×#1588	C₂	1	2	yes	maximal, complemented
6	C₃×#1585	C₂	1	2	yes	maximal, complemented
7	C₃×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
11	C₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#18000	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×M₁₆	—	2	1	no	—
14	C₃×#533	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
16	C₃×#483	C₄	1	2	yes	—
17	C₃×#476	C₄	1	1	yes	characteristic
18	C₃×#476	C₄	1	1	yes	characteristic
19	C₃×#461	C₄	1	1	yes	characteristic
20	C₃×#461	C₄	1	1	yes	characteristic
21	C₃×#450	C₄	1	2	yes	—
22	C₃×#446	—	2	2	no	—
23	C₃×#446	C₂²	1	2	yes	complemented
24	C₃×C₈×Q₈	C₂²	1	2	yes	complemented
25	C₃×C₄×SD₁₆	C₂²	1	2	yes	—
26	C₃×C₄×SD₁₆	C₂²	1	2	yes	—
27	C₃×#436	C₂²	1	2	yes	—
28	C₃×#435	—	2	2	no	—
29	C₃×C₈×D₄	C₂²	1	2	yes	—
30	C₃×#431	C₂²	1	1	yes	characteristic
31	C₃×#424	—	2	1	no	—
32	C₃×#423	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×#407	—	2	1	no	—
35	C₃×#405	—	2	1	no	—
36	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
37	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
38	C₃×#335	C₂²	1	2	yes	complemented
39	C₃×#335	C₂²	1	2	yes	complemented
40	C₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
41	C₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
42	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₃×#322	C₂²	1	2	yes	complemented
47	C₃×#322	C₂²	1	2	yes	complemented
48	#3101	C₂×C₃	1	1	yes	characteristic
49	#2962	C₂×C₃	1	1	yes	characteristic, complemented
50	D₄×M₁₆	C₂×C₃	1	1	yes	characteristic
51	#1592	C₂×C₃	1	2	yes	complemented
52	#1588	C₂×C₃	1	2	yes	complemented
53	#1585	C₂×C₃	1	2	yes	complemented
54	SD₁₆⋊C₈	C₂×C₃	1	2	yes	complemented
55	#1571	C₂×C₃	1	1	yes	characteristic, complemented
56	#1493	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×D₄⋊C₈	C₂×C₃	1	1	yes	characteristic
58	#1451	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂²×C₃×D₄	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₃×C₄	—	2	1	no	—
61	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	—
62	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	—
63	C₂×C₃×M₁₆	—	2	1	no	—
64	C₂×C₃×M₁₆	—	2	2	no	—
65	C₂×C₃×M₁₆	—	4	4	no	—
66	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
67	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
68	C₂²×C₃×C₈	—	2	1	no	—
69	C₂²×C₃×C₈	—	2	1	no	—
70	C₂²×C₃×C₈	—	2	2	no	—
71	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
72	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
73	C₃×#125	—	2	1	no	—
74	C₃×C₂²⋊Q₈	—	2	1	no	—
75	C₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
76	C₃×C₄×Q₈	—	2	1	no	—
77	C₃×C₄×Q₈	C₂³	1	2	yes	—
78	C₃×C₄×D₄	—	2	1	no	—
79	C₃×C₄×D₄	—	2	1	no	—
80	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
81	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
82	C₃×C₄×D₄	C₂³	1	1	yes	characteristic
83	C₃×C₄×D₄	C₂³	1	1	yes	characteristic
84	C₃×#117	—	2	1	no	—
85	C₂×C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
87	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
88	C₃×#106	C₂×C₄	1	2	yes	—
89	C₃×#106	C₂×C₄	1	2	yes	—
90	C₃×C₄⋊C₈	—	2	2	no	—
91	C₃×C₄⋊C₈	—	2	2	no	—
92	C₃×C₄⋊C₈	—	2	1	no	—
93	C₃×C₄⋊C₈	—	2	1	no	—
94	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
98	C₃×C₄⋊C₈	C₂³	1	2	yes	—
99	C₃×#103	C₂×C₄	1	2	yes	—
100	C₃×#103	C₂×C₄	1	2	yes	—
101	C₃×#102	C₂×C₄	1	2	yes	—
102	C₃×#102	C₂×C₄	1	2	yes	—
103	C₃×C₂²⋊C₈	—	2	2	no	—
104	C₃×C₂²⋊C₈	—	2	2	no	—
105	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
106	C₃×#97	—	2	2	no	—
107	C₃×#97	D₄	1	2	yes	—
108	C₃×C₄×C₈	—	2	2	no	—
109	C₃×C₄×C₈	—	2	1	no	—
110	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
111	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
112	C₃×C₄×C₈	C₂³	1	2	yes	—
113	C₃×C₄×C₈	C₂³	1	2	yes	—
114	C₂²×M₁₆	—	2	1	no	—
115	#533	C₂²×C₃	1	1	yes	characteristic
116	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
117	#483	C₃×C₄	1	2	yes	—
118	#476	C₃×C₄	1	1	yes	characteristic
119	#476	C₃×C₄	1	1	yes	characteristic
120	#461	C₃×C₄	1	1	yes	characteristic
121	#461	C₃×C₄	1	1	yes	characteristic
122	#450	C₃×C₄	1	2	yes	—
123	#446	—	2	2	no	—
124	#446	C₂²×C₃	1	2	yes	complemented
125	C₈×Q₈	C₂²×C₃	1	2	yes	complemented
126	C₄×SD₁₆	C₂²×C₃	1	2	yes	—
127	C₄×SD₁₆	C₂²×C₃	1	2	yes	—
128	#436	C₂²×C₃	1	2	yes	—
129	#435	—	2	2	no	—
130	C₈×D₄	C₂²×C₃	1	2	yes	—
131	#431	C₂²×C₃	1	1	yes	characteristic
132	#424	—	2	1	no	—
133	#423	C₂²×C₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
135	#407	—	2	1	no	—
136	#405	—	2	1	no	—
137	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
139	#335	C₂²×C₃	1	2	yes	complemented
140	#335	C₂²×C₃	1	2	yes	complemented
141	Q₈⋊C₈	C₂²×C₃	1	2	yes	complemented
142	Q₈⋊C₈	C₂²×C₃	1	2	yes	complemented
143	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
144	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
145	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
146	D₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
147	#322	C₂²×C₃	1	2	yes	complemented
148	#322	C₂²×C₃	1	2	yes	complemented
149	C₂⁴×C₃	—	2	1	no	—
150	C₂×C₃×Q₈	—	2	1	no	—
151	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
152	C₂×C₃×D₄	—	2	2	no	—
153	C₂×C₃×D₄	—	2	1	no	—
154	C₂×C₃×D₄	—	4	1	no	—
155	C₂×C₃×D₄	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₃×D₄	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	—	2	1	no	—
158	C₂²×C₃×C₄	—	2	1	no	—
159	C₂²×C₃×C₄	—	2	1	no	—
160	C₂²×C₃×C₄	—	2	1	no	—
161	C₂²×C₃×C₄	—	2	1	no	—
162	C₂²×C₃×C₄	—	4	2	no	—
163	C₂²×C₃×C₄	—	4	2	no	—
164	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₃×C₄	#41	1	1	yes	characteristic
167	C₃×SD₁₆	—	2	4	no	—
168	C₃×SD₁₆	—	4	2	no	—
169	C₃×M₁₆	—	2	2	no	—
170	C₃×M₁₆	—	2	2	no	—
171	C₃×M₁₆	—	4	4	no	—
172	C₃×M₁₆	—	4	2	no	—
173	C₂×C₃×C₈	—	2	1	no	—
174	C₂×C₃×C₈	—	2	1	no	—
175	C₂×C₃×C₈	—	2	1	no	—
176	C₂×C₃×C₈	—	2	2	no	—
177	C₂×C₃×C₈	—	2	1	no	—
178	C₂×C₃×C₈	—	4	2	no	—
179	C₂×C₃×C₈	—	4	4	no	—
180	C₂×C₃×C₈	—	4	2	no	—
181	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
182	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
183	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
184	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
185	C₃×C₄⋊C₄	—	2	1	no	—
186	C₃×C₄⋊C₄	—	2	1	no	—
187	C₃×C₄⋊C₄	—	2	1	no	—
188	C₃×C₄⋊C₄	—	2	1	no	—
189	C₃×C₄⋊C₄	—	2	2	no	—
190	C₃×C₄⋊C₄	—	2	2	no	—
191	C₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
192	C₃×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
193	C₃×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
194	C₃×C₂²⋊C₄	—	2	2	no	—
195	C₃×C₂²⋊C₄	—	2	1	no	—
196	C₃×C₂²⋊C₄	—	2	1	no	—
197	C₃×C₂²⋊C₄	—	2	1	no	—
198	C₃×C₂²⋊C₄	—	2	1	no	—
199	C₃×C₄²	—	2	2	no	—
200	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
201	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
202	C₃×C₄²	#41	1	1	yes	characteristic
203	C₃×C₄²	C₂⁴	1	1	yes	characteristic
204	C₂²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
205	C₂³×C₄	—	2	1	no	—
206	C₂×SD₁₆	C₂×C₃×C₄	1	2	yes	—
207	C₂×SD₁₆	C₂×C₃×C₄	1	2	yes	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	4	4	no	—
211	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
212	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
217	C₄⋊Q₈	C₂×C₃×C₄	1	1	yes	characteristic
218	#125	—	2	1	no	—
219	C₂²⋊Q₈	—	2	1	no	—
220	C₂²⋊Q₈	C₂×C₃×C₄	1	2	yes	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	C₂³×C₃	1	2	yes	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	C₂×C₃×C₄	1	1	yes	characteristic
226	C₄×D₄	C₂×C₃×C₄	1	1	yes	characteristic
227	C₄×D₄	C₂³×C₃	1	1	yes	characteristic
228	C₄×D₄	C₂³×C₃	1	1	yes	characteristic
229	#117	—	2	1	no	—
230	C₂×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	—	2	1	no	—
232	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
233	#106	C₂×C₃×C₄	1	2	yes	—
234	#106	C₂×C₃×C₄	1	2	yes	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
240	C₄⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
241	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
242	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
243	C₄⋊C₈	C₂³×C₃	1	2	yes	—
244	#103	C₂×C₃×C₄	1	2	yes	—
245	#103	C₂×C₃×C₄	1	2	yes	—
246	#102	C₂×C₃×C₄	1	2	yes	—
247	#102	C₂×C₃×C₄	1	2	yes	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	C₂×C₃×C₄	1	2	yes	—
251	#97	—	2	2	no	—
252	#97	C₃×D₄	1	2	yes	—
253	C₄×C₈	—	2	2	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₃	1	2	yes	—
258	C₄×C₈	C₂³×C₃	1	2	yes	—
259	C₂³×C₃	—	2	1	no	—
260	C₂³×C₃	—	2	1	no	—
261	C₂³×C₃	—	4	2	no	—
262	C₂³×C₃	—	4	2	no	—
263	C₂³×C₃	—	8	1	no	—
264	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
265	C₃×Q₈	—	2	2	no	—
266	C₃×Q₈	—	2	2	no	—
267	C₃×Q₈	—	8	1	no	—
268	C₃×D₄	—	2	1	no	—
269	C₃×D₄	—	2	1	no	—
270	C₃×D₄	—	2	1	no	—
271	C₃×D₄	—	4	2	no	—
272	C₃×D₄	#131	1	2	yes	—
273	C₂×C₃×C₄	—	2	1	no	—
274	C₂×C₃×C₄	—	2	1	no	—
275	C₂×C₃×C₄	—	2	2	no	—
276	C₂×C₃×C₄	—	2	2	no	—
277	C₂×C₃×C₄	—	2	1	no	—
278	C₂×C₃×C₄	—	2	1	no	—
279	C₂×C₃×C₄	—	4	1	no	—
280	C₂×C₃×C₄	—	4	2	no	—
281	C₂×C₃×C₄	—	4	1	no	—
282	C₂×C₃×C₄	—	4	1	no	—
283	C₂×C₃×C₄	—	4	1	no	—
284	C₂×C₃×C₄	—	4	1	no	—
285	C₂×C₃×C₄	—	4	2	no	—
286	C₂×C₃×C₄	—	4	2	no	—
287	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
288	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
289	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
290	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
291	C₂×C₃×C₄	#137	1	2	yes	—
292	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
293	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
295	C₃×C₈	—	2	2	no	—
296	C₃×C₈	—	2	2	no	—
297	C₃×C₈	—	2	2	no	—
298	C₃×C₈	—	2	2	no	—
299	C₃×C₈	—	2	2	no	—
300	C₃×C₈	—	4	1	no	—
301	C₃×C₈	—	4	1	no	—
302	C₃×C₈	—	4	1	no	—
303	C₃×C₈	—	4	2	no	—
304	C₂⁴	—	2	1	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	—	4	1	no	—
310	C₂×D₄	C₂²×C₃×C₄	1	1	yes	characteristic
311	C₂×D₄	C₂²×C₃×C₄	1	1	yes	characteristic
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
320	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
321	C₂²×C₄	C₃×#41	1	1	yes	characteristic
322	SD₁₆	—	2	4	no	—
323	SD₁₆	—	4	2	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	2	no	—
326	M₁₆	—	4	4	no	—
327	M₁₆	—	4	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	4	4	no	—
334	C₂×C₈	—	4	2	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
337	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
338	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
339	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	C₂²×C₃×C₄	1	2	yes	—
347	C₄⋊C₄	C₂²×C₃×C₄	1	1	yes	characteristic
348	C₄⋊C₄	C₂²×C₃×C₄	1	1	yes	characteristic
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	2	1	no	—
354	C₄²	—	2	2	no	—
355	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
356	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
357	C₄²	C₃×#41	1	1	yes	characteristic
358	C₄²	C₂⁴×C₃	1	1	yes	characteristic
359	C₂²×C₃	—	2	1	no	—
360	C₂²×C₃	—	2	1	no	—
361	C₂²×C₃	—	2	2	no	—
362	C₂²×C₃	—	4	1	no	—
363	C₂²×C₃	—	4	1	no	—
364	C₂²×C₃	—	4	1	no	—
365	C₂²×C₃	—	4	1	no	—
366	C₂²×C₃	—	4	1	no	—
367	C₂²×C₃	—	8	4	no	—
368	C₂²×C₃	#440	1	2	yes	—
369	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
370	C₃×C₄	—	2	1	no	—
371	C₃×C₄	—	2	1	no	—
372	C₃×C₄	—	2	1	no	—
373	C₃×C₄	—	4	2	no	—
374	C₃×C₄	—	4	1	no	—
375	C₃×C₄	—	4	1	no	—
376	C₃×C₄	—	4	2	no	—
377	C₃×C₄	#440	1	2	yes	—
378	C₃×C₄	C₂×#131	1	1	yes	characteristic
379	C₃×C₄	#568	1	1	yes	characteristic
380	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
381	C₃×C₄	C₂×#137	1	1	yes	characteristic
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	8	1	no	—
387	C₂³	C₃×C₄×D₄	1	1	yes	characteristic
388	Q₈	—	2	2	no	—
389	Q₈	—	2	2	no	—
390	Q₈	—	8	1	no	—
391	D₄	—	2	1	no	—
392	D₄	—	2	1	no	—
393	D₄	—	2	1	no	—
394	D₄	—	4	2	no	—
395	D₄	C₃×#131	1	2	yes	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
411	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
412	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
413	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
414	C₂×C₄	C₃×#137	1	2	yes	—
415	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
416	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
417	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	2	2	no	—
422	C₈	—	2	2	no	—
423	C₈	—	4	1	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	1	no	—
427	C₂×C₃	—	2	2	no	—
428	C₂×C₃	—	4	2	no	—
429	C₂×C₃	—	8	1	no	—
430	C₂×C₃	#2933	1	1	yes	characteristic
431	C₂×C₃	C₂×#440	1	1	yes	characteristic
432	C₂×C₃	#2956	1	1	yes	characteristic
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	4	1	no	—
437	C₂²	—	4	1	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	—	8	4	no	—
442	C₂²	C₃×#440	1	2	yes	—
443	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
444	C₄	—	2	1	no	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	2	no	—
451	C₄	C₃×#440	1	2	yes	—
452	C₄	C₂×C₃×#131	1	1	yes	characteristic
453	C₄	C₃×#568	1	1	yes	characteristic
454	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
455	C₄	C₂×C₃×#137	1	1	yes	characteristic
456	C₃	#18000	1	1	yes	characteristic, direct product factor, Sylow
457	C₂	—	2	2	no	—
458	C₂	—	4	2	no	—
459	C₂	—	8	1	no	—
460	C₂	C₃×#2933	1	1	yes	characteristic
461	C₂	C₂×C₃×#440	1	1	yes	characteristic
462	C₂	C₃×#2956	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	C₂²×C₃×D₄	C₂×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×#2962	C₂	semidirect
4. C₃×#1592	C₂	semidirect
5. C₃×#1588	C₂	semidirect
6. C₃×#1585	C₂	semidirect
7. C₃×SD₁₆⋊C₈	C₂	semidirect
8. C₃×#1571	C₂	semidirect
9. C₃×#1493	C₂	semidirect
11. C₃×#1451	C₂	semidirect
12. #18000	C₃	direct (N × H)
23. C₃×#446	C₂²	semidirect
24. C₃×C₈×Q₈	C₂²	semidirect
38. C₃×#335	C₂²	semidirect
39. C₃×#335	C₂²	semidirect
40. C₃×Q₈⋊C₈	C₂²	semidirect
41. C₃×Q₈⋊C₈	C₂²	semidirect
46. C₃×#322	C₂²	semidirect
47. C₃×#322	C₂²	semidirect
49. #2962	C₂×C₃	semidirect
51. #1592	C₂×C₃	semidirect
52. #1588	C₂×C₃	semidirect
53. #1585	C₂×C₃	semidirect
54. SD₁₆⋊C₈	C₂×C₃	semidirect
55. #1571	C₂×C₃	semidirect
56. #1493	C₂×C₃	semidirect
58. #1451	C₂×C₃	semidirect
124. #446	C₂²×C₃	semidirect
125. C₈×Q₈	C₂²×C₃	semidirect
139. #335	C₂²×C₃	semidirect
140. #335	C₂²×C₃	semidirect
141. Q₈⋊C₈	C₂²×C₃	semidirect
142. Q₈⋊C₈	C₂²×C₃	semidirect
147. #322	C₂²×C₃	semidirect
148. #322	C₂²×C₃	semidirect
456. C₃	#18000	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×#3101	2
C₃×#2962	2
C₃×D₄×M₁₆	2
C₃×#1592	2
C₃×#1588	2
C₃×#1585	2
C₃×SD₁₆⋊C₈	2
C₃×#1571	2
C₃×#1493	2
C₂×C₃×D₄⋊C₈	2
C₃×#1451	2
#18000	3

Subgroups — C3×#18000 #10636607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#18000 #10636607