Subgroups — C₃×#27791 #10646398

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#466	C₂	1	1	yes	maximal, characteristic
3	C₃×#2491	C₂	1	2	yes	maximal, complemented
4	C₃×#2033	C₂	1	2	yes	maximal, complemented
5	C₃×#2023	C₂	1	2	yes	maximal, complemented
6	C₃×#1970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1970	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1925	C₂	1	2	yes	maximal, complemented
9	C₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×#107	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#27791	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃×#124	—	2	2	no	—
17	C₂×C₃×#121	C₂²	1	2	yes	—
18	C₃×#500	—	2	4	no	—
19	C₃×#466	—	2	4	no	—
20	C₂×C₃×#107	C₂²	1	2	yes	—
21	C₂×C₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃×#102	C₂²	1	2	yes	—
25	C₂×C₃×#102	C₂²	1	2	yes	—
26	C₂×C₃×#102	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×#394	—	2	2	no	—
31	C₃×#393	—	2	2	no	—
32	C₃×#390	C₂²	1	2	yes	—
33	C₃×#390	C₂²	1	2	yes	—
34	C₃×#388	—	2	2	no	—
35	C₃×#380	C₂²	1	2	yes	—
36	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
37	C₃×C₄×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₃×#340	C₂²	1	2	yes	—
39	C₃×#340	C₂²	1	2	yes	—
40	C₃×#336	C₂²	1	1	yes	characteristic
41	C₃×#336	C₂²	1	1	yes	characteristic
42	C₂×#500	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂×#466	C₂×C₃	1	1	yes	characteristic
44	#2491	C₂×C₃	1	2	yes	complemented
45	#2033	C₂×C₃	1	2	yes	complemented
46	#2023	C₂×C₃	1	2	yes	complemented
47	#1970	C₂×C₃	1	1	yes	characteristic, complemented
48	#1970	C₂×C₃	1	1	yes	characteristic, complemented
49	#1925	C₂×C₃	1	2	yes	complemented
50	#1808	C₂×C₃	1	1	yes	characteristic, complemented
51	C₄×#107	C₂×C₃	1	1	yes	characteristic, complemented
52	C₈×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂²×C₃×Q₈	—	2	2	no	—
54	C₂²×C₃×D₄	—	2	2	no	—
55	C₂²×C₃×D₄	C₂³	1	2	yes	—
56	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃×D₈	—	2	2	no	—
58	C₂×C₃×D₈	—	2	2	no	—
59	C₂×C₃×D₈	—	2	2	no	—
60	C₂²×C₃×C₈	—	2	2	no	—
61	C₂²×C₃×C₈	—	2	2	no	—
62	C₂²×C₃×C₈	—	2	2	no	—
63	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄⋊Q₈	—	2	4	no	—
68	C₃×C₄⋊Q₈	—	4	2	no	—
69	C₃×#124	—	4	8	no	—
70	C₃×#121	—	2	4	no	—
71	C₃×#121	—	4	2	no	—
72	C₂×C₃×C₄⋊C₄	—	2	2	no	—
73	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
75	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
76	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
80	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
81	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
82	C₂×C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
83	C₂×C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
84	C₂×C₃×C₄²	—	2	2	no	—
85	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
86	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
87	C₃×#107	—	2	2	no	—
88	C₃×#107	—	2	2	no	—
89	C₃×#107	—	2	4	no	—
90	C₃×#107	D₄	1	4	yes	—
91	C₃×#102	—	2	4	no	—
92	C₃×#102	—	2	4	no	—
93	C₃×#102	—	4	2	no	—
94	C₃×C₂²⋊C₈	—	2	2	no	—
95	C₃×C₄×C₈	—	2	2	no	—
96	C₃×C₄×C₈	—	2	2	no	—
97	C₃×#95	—	2	2	no	—
98	C₃×#95	—	2	2	no	—
99	C₃×#95	D₄	1	1	yes	characteristic
100	C₃×#95	D₄	1	1	yes	characteristic
101	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
102	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
104	C₂×#124	—	2	2	no	—
105	C₂×#121	C₂²×C₃	1	2	yes	—
106	#500	—	2	4	no	—
107	#466	—	2	4	no	—
108	C₂×#107	C₂²×C₃	1	1	yes	characteristic
109	C₂×#107	C₂²×C₃	1	1	yes	characteristic
110	C₂×#107	C₂²×C₃	1	2	yes	—
111	C₂×#107	C₂²×C₃	1	1	yes	characteristic
112	C₂×#102	C₂²×C₃	1	2	yes	—
113	C₂×#102	C₂²×C₃	1	2	yes	—
114	C₂×#102	C₂²×C₃	1	2	yes	—
115	C₂×C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
118	#394	—	2	2	no	—
119	#393	—	2	2	no	—
120	#390	C₂²×C₃	1	2	yes	—
121	#390	C₂²×C₃	1	2	yes	—
122	#388	—	2	2	no	—
123	#380	C₂²×C₃	1	2	yes	—
124	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
125	C₄×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	#340	C₂²×C₃	1	2	yes	—
127	#340	C₂²×C₃	1	2	yes	—
128	#336	C₂²×C₃	1	1	yes	characteristic
129	#336	C₂²×C₃	1	1	yes	characteristic
130	C₂⁴×C₃	—	2	2	no	—
131	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×Q₈	—	4	4	no	—
133	C₂×C₃×Q₈	—	4	4	no	—
134	C₂×C₃×Q₈	—	8	2	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	—	2	2	no	—
137	C₂×C₃×D₄	—	2	2	no	—
138	C₂×C₃×D₄	—	2	2	no	—
139	C₂×C₃×D₄	—	2	2	no	—
140	C₂×C₃×D₄	—	2	2	no	—
141	C₂×C₃×D₄	—	4	4	no	—
142	C₂×C₃×D₄	—	4	4	no	—
143	C₂×C₃×D₄	—	8	2	no	—
144	C₂²×C₃×C₄	—	2	2	no	—
145	C₂²×C₃×C₄	—	2	1	no	—
146	C₂²×C₃×C₄	—	2	1	no	—
147	C₂²×C₃×C₄	—	2	1	no	—
148	C₂²×C₃×C₄	—	2	1	no	—
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	—	2	2	no	—
152	C₂²×C₃×C₄	—	2	2	no	—
153	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
159	C₃×D₈	—	4	4	no	—
160	C₂×C₃×C₈	—	2	1	no	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₂×C₃×C₈	—	2	1	no	—
163	C₂×C₃×C₈	—	2	1	no	—
164	C₂×C₃×C₈	—	2	1	no	—
165	C₂×C₃×C₈	—	2	1	no	—
166	C₂×C₃×C₈	—	2	1	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₂×C₃×C₈	—	4	4	no	—
169	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
170	C₂×C₃×C₈	#41	1	2	yes	—
171	C₂×C₃×C₈	#41	1	2	yes	—
172	C₂×C₃×C₈	#41	1	2	yes	—
173	C₃×C₄⋊C₄	—	2	2	no	—
174	C₃×C₄⋊C₄	—	2	2	no	—
175	C₃×C₄⋊C₄	—	2	2	no	—
176	C₃×C₄⋊C₄	—	2	2	no	—
177	C₃×C₄⋊C₄	—	2	2	no	—
178	C₃×C₄⋊C₄	—	4	4	no	—
179	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
180	C₃×C₂²⋊C₄	—	2	2	no	—
181	C₃×C₂²⋊C₄	—	4	4	no	—
182	C₃×C₂²⋊C₄	—	4	4	no	—
183	C₃×C₂²⋊C₄	—	4	4	no	—
184	C₃×C₂²⋊C₄	—	4	4	no	—
185	C₃×C₂²⋊C₄	—	4	4	no	—
186	C₃×C₂²⋊C₄	—	8	2	no	—
187	C₃×C₂²⋊C₄	D₈	1	4	yes	complemented
188	C₃×C₄²	—	2	1	no	—
189	C₃×C₄²	—	2	1	no	—
190	C₃×C₄²	—	2	1	no	—
191	C₃×C₄²	—	2	1	no	—
192	C₃×C₄²	—	4	4	no	—
193	C₂²×Q₈	—	2	2	no	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	C₂³×C₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
197	C₂×D₈	—	2	2	no	—
198	C₂×D₈	—	2	2	no	—
199	C₂×D₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
207	C₄⋊Q₈	—	2	4	no	—
208	C₄⋊Q₈	—	4	2	no	—
209	#124	—	4	8	no	—
210	#121	—	2	4	no	—
211	#121	—	4	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	2	no	—
229	#107	—	2	4	no	—
230	#107	C₃×D₄	1	4	yes	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	#102	—	4	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	C₃×D₄	1	1	yes	characteristic
240	#95	C₃×D₄	1	1	yes	characteristic
241	C₂³×C₃	—	2	1	no	—
242	C₂³×C₃	—	2	1	no	—
243	C₂³×C₃	—	2	1	no	—
244	C₂³×C₃	—	2	1	no	—
245	C₂³×C₃	—	2	1	no	—
246	C₂³×C₃	—	2	1	no	—
247	C₂³×C₃	—	4	2	no	—
248	C₂³×C₃	—	4	2	no	—
249	C₂³×C₃	—	4	2	no	—
250	C₂³×C₃	—	4	2	no	—
251	C₂³×C₃	—	4	2	no	—
252	C₂³×C₃	—	4	2	no	—
253	C₂³×C₃	—	4	2	no	—
254	C₂³×C₃	C₂×D₈	1	2	yes	—
255	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
256	C₃×Q₈	—	8	8	no	—
257	C₃×D₄	—	4	2	no	—
258	C₃×D₄	—	4	2	no	—
259	C₃×D₄	—	4	2	no	—
260	C₃×D₄	—	4	2	no	—
261	C₃×D₄	—	8	8	no	—
262	C₂×C₃×C₄	—	2	1	no	—
263	C₂×C₃×C₄	—	2	1	no	—
264	C₂×C₃×C₄	—	2	1	no	—
265	C₂×C₃×C₄	—	2	1	no	—
266	C₂×C₃×C₄	—	2	1	no	—
267	C₂×C₃×C₄	—	2	1	no	—
268	C₂×C₃×C₄	—	2	1	no	—
269	C₂×C₃×C₄	—	2	1	no	—
270	C₂×C₃×C₄	—	2	4	no	—
271	C₂×C₃×C₄	—	2	1	no	—
272	C₂×C₃×C₄	—	2	1	no	—
273	C₂×C₃×C₄	—	2	1	no	—
274	C₂×C₃×C₄	—	2	1	no	—
275	C₂×C₃×C₄	—	4	2	no	—
276	C₂×C₃×C₄	—	4	2	no	—
277	C₂×C₃×C₄	—	4	1	no	—
278	C₂×C₃×C₄	—	4	1	no	—
279	C₂×C₃×C₄	—	4	1	no	—
280	C₂×C₃×C₄	—	4	1	no	—
281	C₂×C₃×C₄	—	4	2	no	—
282	C₂×C₃×C₄	—	4	2	no	—
283	C₂×C₃×C₄	—	4	2	no	—
284	C₂×C₃×C₄	—	4	2	no	—
285	C₂×C₃×C₄	—	4	2	no	—
286	C₂×C₃×C₄	—	4	2	no	—
287	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
288	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
289	C₂×C₃×C₄	#135	1	2	yes	—
290	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
292	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
293	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
294	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
295	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
296	C₃×C₈	—	2	2	no	—
297	C₃×C₈	—	4	1	no	—
298	C₃×C₈	—	4	1	no	—
299	C₃×C₈	#124	1	4	yes	complemented
300	C₂⁴	—	2	2	no	—
301	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
302	C₂×Q₈	—	4	4	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×Q₈	—	8	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	4	4	no	—
312	C₂×D₄	—	4	4	no	—
313	C₂×D₄	—	8	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
324	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
325	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
326	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
327	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
329	D₈	—	4	4	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	4	4	no	—
339	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
340	C₂×C₈	C₃×#41	1	2	yes	—
341	C₂×C₈	C₃×#41	1	2	yes	—
342	C₂×C₈	C₃×#41	1	2	yes	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	4	no	—
349	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	4	4	no	—
352	C₂²⋊C₄	—	4	4	no	—
353	C₂²⋊C₄	—	4	4	no	—
354	C₂²⋊C₄	—	4	4	no	—
355	C₂²⋊C₄	—	4	4	no	—
356	C₂²⋊C₄	—	8	2	no	—
357	C₂²⋊C₄	C₃×D₈	1	4	yes	complemented
358	C₄²	—	2	1	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	4	4	no	—
363	C₂²×C₃	—	2	2	no	—
364	C₂²×C₃	—	2	2	no	—
365	C₂²×C₃	—	2	2	no	—
366	C₂²×C₃	—	4	1	no	—
367	C₂²×C₃	—	4	1	no	—
368	C₂²×C₃	—	4	1	no	—
369	C₂²×C₃	—	4	1	no	—
370	C₂²×C₃	—	8	2	no	—
371	C₂²×C₃	—	8	2	no	—
372	C₂²×C₃	—	8	2	no	—
373	C₂²×C₃	—	8	2	no	—
374	C₂²×C₃	—	8	2	no	—
375	C₂²×C₃	—	8	2	no	—
376	C₂²×C₃	—	8	2	no	—
377	C₂²×C₃	#532	1	1	yes	characteristic
378	C₂²×C₃	D₄²	1	1	yes	characteristic
379	C₂²×C₃	#547	1	1	yes	characteristic
380	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
381	C₂²×C₃	C₂×#135	1	1	yes	characteristic
382	C₂²×C₃	#576	1	1	yes	characteristic
383	C₂²×C₃	#578	1	1	yes	characteristic
384	C₃×C₄	—	2	2	no	—
385	C₃×C₄	—	2	2	no	—
386	C₃×C₄	—	2	2	no	—
387	C₃×C₄	—	4	1	no	—
388	C₃×C₄	—	4	1	no	—
389	C₃×C₄	—	4	4	no	—
390	C₃×C₄	—	8	2	no	—
391	C₃×C₄	—	8	2	no	—
392	C₃×C₄	C₂×#124	1	1	yes	characteristic
393	C₃×C₄	#542	1	1	yes	characteristic
394	C₃×C₄	#548	1	1	yes	characteristic
395	C₃×C₄	#555	1	1	yes	characteristic
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	C₂×C₃×D₈	1	2	yes	—
410	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
411	Q₈	—	8	8	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	8	8	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
443	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
444	C₂×C₄	C₃×#135	1	2	yes	—
445	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
446	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
447	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
448	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
449	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
450	C₂×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	4	1	no	—
453	C₈	—	4	1	no	—
454	C₈	C₃×#124	1	4	yes	complemented
455	C₂×C₃	—	4	2	no	—
456	C₂×C₃	—	16	2	no	—
457	C₂×C₃	#2491	1	1	yes	characteristic
458	C₂×C₃	#3169	1	1	yes	characteristic
459	C₂×C₃	#3175	1	1	yes	characteristic
460	C₂×C₃	D₄×D₈	1	1	yes	characteristic
461	C₂×C₃	#3285	1	1	yes	characteristic
462	C₂×C₃	#3334	1	1	yes	characteristic
463	C₂×C₃	#3341	1	1	yes	characteristic
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	C₃×#532	1	1	yes	characteristic
479	C₂²	C₃×D₄²	1	1	yes	characteristic
480	C₂²	C₃×#547	1	1	yes	characteristic
481	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
482	C₂²	C₂×C₃×#135	1	1	yes	characteristic
483	C₂²	C₃×#576	1	1	yes	characteristic
484	C₂²	C₃×#578	1	1	yes	characteristic
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	4	no	—
490	C₄	—	4	1	no	—
491	C₄	—	8	2	no	—
492	C₄	—	8	2	no	—
493	C₄	C₂×C₃×#124	1	1	yes	characteristic
494	C₄	C₃×#542	1	1	yes	characteristic
495	C₄	C₃×#548	1	1	yes	characteristic
496	C₄	C₃×#555	1	1	yes	characteristic
497	C₃	#27791	1	1	yes	characteristic, direct product factor, Sylow
498	C₂	—	4	2	no	—
499	C₂	—	16	2	no	—
500	C₂	C₃×#2491	1	1	yes	characteristic
501	C₂	C₃×#3169	1	1	yes	characteristic
502	C₂	C₃×#3175	1	1	yes	characteristic
503	C₂	C₃×D₄×D₈	1	1	yes	characteristic
504	C₂	C₃×#3285	1	1	yes	characteristic
505	C₂	C₃×#3334	1	1	yes	characteristic
506	C₂	C₃×#3341	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×C₄×C₂²⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#500	C₂	semidirect
3. C₃×#2491	C₂	semidirect
4. C₃×#2033	C₂	semidirect
5. C₃×#2023	C₂	semidirect
6. C₃×#1970	C₂	semidirect
7. C₃×#1970	C₂	semidirect
8. C₃×#1925	C₂	semidirect
9. C₃×#1808	C₂	semidirect
10. C₃×C₄×#107	C₂	semidirect
11. C₃×C₈×C₂²⋊C₄	C₂	semidirect
12. #27791	C₃	direct (N × H)
42. C₂×#500	C₂×C₃	semidirect
44. #2491	C₂×C₃	semidirect
45. #2033	C₂×C₃	semidirect
46. #2023	C₂×C₃	semidirect
47. #1970	C₂×C₃	semidirect
48. #1970	C₂×C₃	semidirect
49. #1925	C₂×C₃	semidirect
50. #1808	C₂×C₃	semidirect
51. C₄×#107	C₂×C₃	semidirect
52. C₈×C₂²⋊C₄	C₂×C₃	semidirect
187. C₃×C₂²⋊C₄	D₈	semidirect
299. C₃×C₈	#124	semidirect
357. C₂²⋊C₄	C₃×D₈	semidirect
454. C₈	C₃×#124	semidirect
497. C₃	#27791	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×#500	2
C₂×C₃×#466	2
C₃×#2491	2
C₃×#2033	2
C₃×#2023	2
C₃×#1970	2
C₃×#1925	2
C₃×#1808	2
C₃×C₄×#107	2
C₃×C₈×C₂²⋊C₄	2
#27791	3

Subgroups — C3×#27791 #10646398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#27791 #10646398