Subgroups — C₃×#10613771 #14478544

Order 1920.

720 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	C₃×#66103	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#66100	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#510	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65483	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#3960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65208	C₂	1	2	yes	maximal, complemented
7	C₃×#65207	C₂	1	2	yes	maximal, complemented
8	C₃×#65203	C₂	1	2	yes	maximal, complemented
9	C₃×D₅×SD₃₂	C₂	1	2	yes	maximal, complemented
10	C₃×#65197	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#65196	C₂	1	1	yes	maximal, characteristic, complemented
12	#10613771	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×#4149	—	2	1	no	—
14	C₃×#4147	—	2	1	no	—
15	C₃×C₅×#135	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₃×#4095	—	2	1	no	—
18	C₃×#4088	—	2	1	no	—
19	C₃×#4065	C₂²	1	2	yes	—
20	C₃×#4065	C₂²	1	2	yes	complemented
21	C₃×#4064	—	2	2	no	—
22	C₃×D₅×Q₁₆	C₂²	1	2	yes	complemented
23	C₃×#4060	—	2	2	no	—
24	C₃×D₅×D₈	C₂²	1	2	yes	—
25	C₃×#4050	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
27	C₃×#4047	C₂²	1	1	yes	characteristic
28	C₃×C₅×Q₃₂	C₂²	1	2	yes	complemented
29	C₃×C₅×SD₃₂	C₂²	1	2	yes	complemented
30	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic, complemented
31	C₃×#3961	C₂²	1	1	yes	characteristic, complemented
32	C₃×#3960	C₂²	1	2	yes	complemented
33	C₃×#3960	C₂²	1	2	yes	complemented
34	C₃×#3960	C₂²	1	1	yes	characteristic, complemented
35	C₃×#3959	C₂²	1	1	yes	characteristic, complemented
36	C₃×#3958	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₃×#672	C₂²	1	1	yes	characteristic, complemented
38	C₃×#3932	C₂²	1	2	yes	complemented
39	C₃×D₈₀	C₂²	1	2	yes	complemented
40	C₃×#3930	C₂²	1	2	yes	complemented
41	C₃×D₅×C₁₆	C₂²	1	2	yes	complemented
42	C₃×#3416	—	5	1	no	maximal, Hall
43	#66103	C₂×C₃	1	1	yes	characteristic, complemented
44	#66100	C₂×C₃	1	1	yes	characteristic, complemented
45	C₅×#510	C₂×C₃	1	1	yes	characteristic, complemented
46	#65483	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×#3960	C₂×C₃	1	1	yes	characteristic, complemented
48	#65208	C₂×C₃	1	2	yes	complemented
49	#65207	C₂×C₃	1	2	yes	complemented
50	#65203	C₂×C₃	1	2	yes	complemented
51	D₅×SD₃₂	C₂×C₃	1	2	yes	complemented
52	#65197	C₂×C₃	1	1	yes	characteristic, complemented
53	#65196	C₂×C₃	1	1	yes	characteristic, complemented
54	C₃×C₅×#41	—	2	1	no	—
55	C₂×C₃×C₅×Q₈	—	2	1	no	—
56	C₃×#713	—	2	2	no	—
57	C₃×#713	—	2	2	no	—
58	C₃×#713	—	4	1	no	—
59	C₃×Q₈×D₅	—	2	2	no	—
60	C₃×Q₈×D₅	—	4	1	no	—
61	C₃×D₄×D₅	—	2	2	no	—
62	C₃×D₄×D₅	—	4	2	no	—
63	C₃×#709	—	2	1	no	—
64	C₃×#709	—	4	2	no	—
65	C₃×#709	—	4	1	no	—
66	C₃×#709	D₄	1	1	yes	characteristic
67	C₂×C₃×D₂₀	—	2	1	no	—
68	C₂×C₃×D₂₀	—	4	1	no	—
69	C₃×C₅×Q₁₆	—	2	1	no	—
70	C₃×C₅×Q₁₆	C₂³	1	2	yes	—
71	C₃×C₅×Q₁₆	C₂³	1	1	yes	characteristic
72	C₃×C₅×SD₁₆	—	2	1	no	—
73	C₃×C₅×D₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₃×#689	—	2	2	no	—
76	C₃×#688	—	2	2	no	—
77	C₃×#688	—	2	2	no	—
78	C₃×#686	—	2	2	no	—
79	C₃×#681	D₄	1	1	yes	characteristic
80	C₃×Dic₂₀	C₂³	1	1	yes	characteristic
81	C₃×D₄₀	—	2	1	no	—
82	C₃×D₄₀	C₂³	1	1	yes	characteristic
83	C₃×D₄₀	C₂³	1	2	yes	—
84	C₃×#677	—	2	1	no	—
85	C₃×#676	D₄	1	2	yes	—
86	C₃×C₈×D₅	C₂³	1	2	yes	—
87	C₃×C₅×C₁₆	C₂³	1	2	yes	—
88	C₃×#672	C₂³	1	1	yes	characteristic
89	C₃×#672	C₂³	1	1	yes	characteristic
90	C₃×#578	—	5	1	no	—
91	C₃×#576	—	5	1	no	—
92	C₃×#510	—	5	2	no	—
93	C₃×#510	D₅	1	1	yes	characteristic, complemented
94	C₃×#509	—	5	2	no	—
95	C₃×#509	—	5	1	no	—
96	C₃×#508	—	5	1	no	—
97	C₃×#508	—	5	2	no	—
98	C₂×C₃×SD₃₂	—	5	1	no	—
99	C₂×C₃×SD₃₂	—	5	2	no	—
100	C₃×#504	—	5	1	no	—
101	#4149	—	2	1	no	—
102	#4147	—	2	1	no	—
103	C₅×#135	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
105	#4095	—	2	1	no	—
106	#4088	—	2	1	no	—
107	#4065	C₂²×C₃	1	2	yes	—
108	#4065	C₂²×C₃	1	2	yes	complemented
109	#4064	—	2	2	no	—
110	D₅×Q₁₆	C₂²×C₃	1	2	yes	complemented
111	#4060	—	2	2	no	—
112	D₅×D₈	C₂²×C₃	1	2	yes	—
113	#4050	C₂²×C₃	1	1	yes	characteristic, complemented
114	C₂×D₄₀	C₂²×C₃	1	1	yes	characteristic
115	#4047	C₂²×C₃	1	1	yes	characteristic
116	C₅×Q₃₂	C₂²×C₃	1	2	yes	complemented
117	C₅×SD₃₂	C₂²×C₃	1	2	yes	complemented
118	C₅×M₃₂	C₂²×C₃	1	1	yes	characteristic, complemented
119	#3961	C₂²×C₃	1	1	yes	characteristic, complemented
120	#3960	C₂²×C₃	1	2	yes	complemented
121	#3960	C₂²×C₃	1	2	yes	complemented
122	#3960	C₂²×C₃	1	1	yes	characteristic, complemented
123	#3959	C₂²×C₃	1	1	yes	characteristic, complemented
124	#3958	C₂²×C₃	1	1	yes	characteristic, complemented
125	C₂×#672	C₂²×C₃	1	1	yes	characteristic, complemented
126	#3932	C₂²×C₃	1	2	yes	complemented
127	D₈₀	C₂²×C₃	1	2	yes	complemented
128	#3930	C₂²×C₃	1	2	yes	complemented
129	D₅×C₁₆	C₂²×C₃	1	2	yes	complemented
130	#3416	—	5	1	no	Sylow
131	C₂²×C₃×D₅	—	4	1	no	—
132	C₂²×C₃×D₅	—	4	2	no	—
133	C₃×C₅×Q₈	—	2	1	no	—
134	C₃×C₅×Q₈	—	2	2	no	—
135	C₃×C₅×Q₈	—	4	1	no	—
136	C₃×C₅×D₄	—	2	1	no	—
137	C₃×C₅×D₄	—	4	1	no	—
138	C₂×C₃×C₄×C₅	—	4	1	no	—
139	C₂×C₃×C₄×C₅	—	4	1	no	—
140	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
141	C₃×C₂²⋊D₅	—	4	1	no	—
142	C₃×C₂²⋊D₅	—	4	2	no	—
143	C₃×C₂²⋊D₅	D₈	1	2	yes	—
144	C₃×D₂₀	—	2	1	no	—
145	C₃×D₂₀	—	2	2	no	—
146	C₃×D₂₀	—	4	2	no	—
147	C₃×D₂₀	—	4	2	no	—
148	C₃×D₂₀	—	4	1	no	—
149	C₃×D₂₀	D₈	1	1	yes	characteristic
150	C₃×C₄×D₅	—	4	2	no	—
151	C₃×C₄×D₅	—	4	1	no	—
152	C₃×C₄×D₅	—	4	2	no	—
153	C₃×C₄×D₅	—	4	2	no	—
154	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
155	C₃×Dic₁₀	—	2	1	no	—
156	C₃×Dic₁₀	—	4	2	no	—
157	C₃×Dic₁₀	D₈	1	1	yes	characteristic
158	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
159	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
160	C₃×#168	C₂×D₄	1	2	yes	—
161	C₃×2₋¹⁺⁴	—	10	1	no	—
162	C₃×H(5,𝔽₂)	—	10	1	no	—
163	C₃×#137	—	10	2	no	—
164	C₃×#137	—	10	1	no	—
165	C₃×AGL(1,ℤ/8)	—	10	2	no	—
166	C₃×AGL(1,ℤ/8)	—	10	1	no	—
167	C₃×#135	—	5	2	no	—
168	C₃×#135	—	5	2	no	—
169	C₃×#135	—	5	1	no	—
170	C₃×#135	C₂×D₅	1	1	yes	characteristic
171	C₂×C₃×Q₁₆	—	5	2	no	—
172	C₂×C₃×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
173	C₂×C₃×D₈	—	5	2	no	—
174	C₂×C₃×D₈	—	5	1	no	—
175	C₃×#131	—	5	1	no	—
176	C₃×Q₃₂	—	5	1	no	—
177	C₃×Q₃₂	C₂×D₅	1	2	yes	complemented
178	C₃×SD₃₂	—	5	2	no	—
179	C₃×SD₃₂	—	5	2	no	—
180	C₃×SD₃₂	—	5	1	no	—
181	C₃×SD₃₂	—	5	1	no	—
182	C₃×SD₃₂	—	5	2	no	—
183	C₃×SD₃₂	C₂×D₅	1	2	yes	complemented
184	C₃×D₁₆	—	5	1	no	—
185	C₃×D₁₆	—	5	2	no	—
186	C₃×M₃₂	—	5	2	no	—
187	C₃×M₃₂	C₂×D₅	1	1	yes	characteristic, complemented
188	C₂×C₃×C₁₆	—	5	1	no	—
189	C₂×C₃×C₁₆	—	5	2	no	—
190	C₅×#41	—	2	1	no	—
191	C₂×C₅×Q₈	—	2	1	no	—
192	#713	—	2	2	no	—
193	#713	—	2	2	no	—
194	#713	—	4	1	no	—
195	Q₈×D₅	—	2	2	no	—
196	Q₈×D₅	—	4	1	no	—
197	D₄×D₅	—	2	2	no	—
198	D₄×D₅	—	4	2	no	—
199	#709	—	2	1	no	—
200	#709	—	4	2	no	—
201	#709	—	4	1	no	—
202	#709	C₃×D₄	1	1	yes	characteristic
203	C₂×D₂₀	—	2	1	no	—
204	C₂×D₂₀	—	4	1	no	—
205	C₅×Q₁₆	—	2	1	no	—
206	C₅×Q₁₆	C₂³×C₃	1	2	yes	—
207	C₅×Q₁₆	C₂³×C₃	1	1	yes	characteristic
208	C₅×SD₁₆	—	2	1	no	—
209	C₅×D₈	C₂³×C₃	1	1	yes	characteristic
210	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
211	#689	—	2	2	no	—
212	#688	—	2	2	no	—
213	#688	—	2	2	no	—
214	#686	—	2	2	no	—
215	#681	C₃×D₄	1	1	yes	characteristic
216	Dic₂₀	C₂³×C₃	1	1	yes	characteristic
217	D₄₀	—	2	1	no	—
218	D₄₀	C₂³×C₃	1	1	yes	characteristic
219	D₄₀	C₂³×C₃	1	2	yes	—
220	#677	—	2	1	no	—
221	#676	C₃×D₄	1	2	yes	—
222	C₈×D₅	C₂³×C₃	1	2	yes	—
223	C₅×C₁₆	C₂³×C₃	1	2	yes	—
224	#672	C₂³×C₃	1	1	yes	characteristic
225	#672	C₂³×C₃	1	1	yes	characteristic
226	#578	—	5	1	no	—
227	#576	—	5	1	no	—
228	#510	—	5	2	no	—
229	#510	C₃×D₅	1	1	yes	characteristic, complemented
230	#509	—	5	2	no	—
231	#509	—	5	1	no	—
232	#508	—	5	1	no	—
233	#508	—	5	2	no	—
234	C₂×SD₃₂	—	5	1	no	—
235	C₂×SD₃₂	—	5	2	no	—
236	#504	—	5	1	no	—
237	C₂²×C₃×C₅	—	4	1	no	—
238	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
239	C₂×C₃×D₅	—	4	1	no	—
240	C₂×C₃×D₅	—	4	2	no	—
241	C₂×C₃×D₅	—	16	2	no	—
242	C₂×C₃×D₅	—	16	1	no	—
243	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
244	C₃×C₄×C₅	—	4	2	no	—
245	C₃×C₄×C₅	—	4	1	no	—
246	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
247	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
248	C₃×Dic₅	—	4	1	no	—
249	C₃×Dic₅	C₂×D₈	1	2	yes	—
250	C₃×#41	—	2	1	no	—
251	C₃×#41	—	5	1	no	—
252	C₃×#41	—	10	2	no	—
253	C₃×#41	—	10	2	no	—
254	C₃×#41	—	10	1	no	—
255	C₃×#41	—	20	2	no	—
256	C₃×#41	—	20	1	no	—
257	C₃×#41	—	20	1	no	—
258	C₂×C₃×Q₈	—	2	1	no	—
259	C₂×C₃×Q₈	—	10	2	no	—
260	C₂×C₃×Q₈	—	20	1	no	—
261	C₂×C₃×D₄	—	10	2	no	—
262	C₂×C₃×D₄	—	10	1	no	—
263	C₂×C₃×D₄	—	20	1	no	—
264	C₂×C₃×D₄	—	20	2	no	—
265	C₃×Q₁₆	—	2	1	no	—
266	C₃×Q₁₆	—	5	1	no	—
267	C₃×Q₁₆	—	10	2	no	—
268	C₃×Q₁₆	C₂²×D₅	1	2	yes	—
269	C₃×Q₁₆	C₂²×D₅	1	1	yes	characteristic
270	C₃×SD₁₆	—	2	1	no	—
271	C₃×SD₁₆	—	10	2	no	—
272	C₃×SD₁₆	—	10	2	no	—
273	C₃×SD₁₆	—	10	1	no	—
274	C₃×D₈	—	5	1	no	—
275	C₃×D₈	—	5	2	no	—
276	C₃×D₈	—	10	2	no	—
277	C₃×D₈	—	10	1	no	—
278	C₃×D₈	C₂²×D₅	1	1	yes	characteristic
279	C₃×M₁₆	—	5	1	no	—
280	C₃×M₁₆	—	5	2	no	—
281	C₂×C₃×C₈	—	5	2	no	—
282	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
283	C₃×C₁₆	—	5	1	no	—
284	C₃×C₁₆	—	5	1	no	—
285	C₃×C₁₆	C₂²×D₅	1	2	yes	—
286	C₂²×D₅	—	4	1	no	—
287	C₂²×D₅	—	4	2	no	—
288	C₅×Q₈	—	2	1	no	—
289	C₅×Q₈	—	2	2	no	—
290	C₅×Q₈	—	4	1	no	—
291	C₅×D₄	—	2	1	no	—
292	C₅×D₄	—	4	1	no	—
293	C₂×C₄×C₅	—	4	1	no	—
294	C₂×C₄×C₅	—	4	1	no	—
295	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
296	C₂²⋊D₅	—	4	1	no	—
297	C₂²⋊D₅	—	4	2	no	—
298	C₂²⋊D₅	C₃×D₈	1	2	yes	—
299	D₂₀	—	2	1	no	—
300	D₂₀	—	2	2	no	—
301	D₂₀	—	4	2	no	—
302	D₂₀	—	4	2	no	—
303	D₂₀	—	4	1	no	—
304	D₂₀	C₃×D₈	1	1	yes	characteristic
305	C₄×D₅	—	4	2	no	—
306	C₄×D₅	—	4	1	no	—
307	C₄×D₅	—	4	2	no	—
308	C₄×D₅	—	4	2	no	—
309	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
310	Dic₁₀	—	2	1	no	—
311	Dic₁₀	—	4	2	no	—
312	Dic₁₀	C₃×D₈	1	1	yes	characteristic
313	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
314	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
315	#168	C₂×C₃×D₄	1	2	yes	—
316	2₋¹⁺⁴	—	10	1	no	—
317	H(5,𝔽₂)	—	10	1	no	—
318	#137	—	10	2	no	—
319	#137	—	10	1	no	—
320	AGL(1,ℤ/8)	—	10	2	no	—
321	AGL(1,ℤ/8)	—	10	1	no	—
322	#135	—	5	2	no	—
323	#135	—	5	2	no	—
324	#135	—	5	1	no	—
325	#135	C₂×C₃×D₅	1	1	yes	characteristic
326	C₂×Q₁₆	—	5	2	no	—
327	C₂×Q₁₆	C₂×C₃×D₅	1	1	yes	characteristic, complemented
328	C₂×D₈	—	5	2	no	—
329	C₂×D₈	—	5	1	no	—
330	#131	—	5	1	no	—
331	Q₃₂	—	5	1	no	—
332	Q₃₂	C₂×C₃×D₅	1	2	yes	complemented
333	SD₃₂	—	5	2	no	—
334	SD₃₂	—	5	2	no	—
335	SD₃₂	—	5	1	no	—
336	SD₃₂	—	5	1	no	—
337	SD₃₂	—	5	2	no	—
338	SD₃₂	C₂×C₃×D₅	1	2	yes	complemented
339	D₁₆	—	5	1	no	—
340	D₁₆	—	5	2	no	—
341	M₃₂	—	5	2	no	—
342	M₃₂	C₂×C₃×D₅	1	1	yes	characteristic, complemented
343	C₂×C₁₆	—	5	1	no	—
344	C₂×C₁₆	—	5	2	no	—
345	C₂×C₃×C₅	—	2	1	no	—
346	C₂×C₃×C₅	—	8	1	no	—
347	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
348	C₃×D₅	—	2	2	no	—
349	C₃×D₅	—	8	1	no	—
350	C₃×D₅	—	8	2	no	—
351	C₂³×C₃	—	20	2	no	—
352	C₂³×C₃	—	20	1	no	—
353	C₃×Q₈	—	2	1	no	—
354	C₃×Q₈	—	2	2	no	—
355	C₃×Q₈	—	4	1	no	—
356	C₃×Q₈	—	5	1	no	—
357	C₃×Q₈	—	10	1	no	—
358	C₃×Q₈	—	20	2	no	—
359	C₃×D₄	—	2	1	no	—
360	C₃×D₄	—	4	1	no	—
361	C₃×D₄	—	5	2	no	—
362	C₃×D₄	—	5	1	no	—
363	C₃×D₄	—	10	2	no	—
364	C₃×D₄	—	10	1	no	—
365	C₃×D₄	—	20	2	no	—
366	C₃×D₄	—	20	2	no	—
367	C₃×D₄	—	20	2	no	—
368	C₃×D₄	—	20	1	no	—
369	C₃×D₄	—	20	1	no	—
370	C₂×C₃×C₄	—	4	1	no	—
371	C₂×C₃×C₄	—	4	1	no	—
372	C₂×C₃×C₄	—	5	2	no	—
373	C₂×C₃×C₄	—	20	2	no	—
374	C₂×C₃×C₄	—	20	2	no	—
375	C₂×C₃×C₄	—	20	2	no	—
376	C₂×C₃×C₄	—	20	1	no	—
377	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
378	C₃×C₈	—	5	2	no	—
379	C₃×C₈	D₄×D₅	1	1	yes	characteristic
380	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
381	C₂²×C₅	—	4	1	no	—
382	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
383	C₂×D₅	—	4	1	no	—
384	C₂×D₅	—	4	2	no	—
385	C₂×D₅	—	16	2	no	—
386	C₂×D₅	—	16	1	no	—
387	C₂×D₅	C₂×C₃×D₈	1	2	yes	—
388	C₄×C₅	—	4	2	no	—
389	C₄×C₅	—	4	1	no	—
390	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
391	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
392	Dic₅	—	4	1	no	—
393	Dic₅	C₂×C₃×D₈	1	2	yes	—
394	#41	—	2	1	no	—
395	#41	—	5	1	no	—
396	#41	—	10	2	no	—
397	#41	—	10	2	no	—
398	#41	—	10	1	no	—
399	#41	—	20	2	no	—
400	#41	—	20	1	no	—
401	#41	—	20	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	10	2	no	—
404	C₂×Q₈	—	20	1	no	—
405	C₂×D₄	—	10	2	no	—
406	C₂×D₄	—	10	1	no	—
407	C₂×D₄	—	20	1	no	—
408	C₂×D₄	—	20	2	no	—
409	Q₁₆	—	2	1	no	—
410	Q₁₆	—	5	1	no	—
411	Q₁₆	—	10	2	no	—
412	Q₁₆	C₂²×C₃×D₅	1	2	yes	—
413	Q₁₆	C₂²×C₃×D₅	1	1	yes	characteristic
414	SD₁₆	—	2	1	no	—
415	SD₁₆	—	10	2	no	—
416	SD₁₆	—	10	2	no	—
417	SD₁₆	—	10	1	no	—
418	D₈	—	5	1	no	—
419	D₈	—	5	2	no	—
420	D₈	—	10	2	no	—
421	D₈	—	10	1	no	—
422	D₈	C₂²×C₃×D₅	1	1	yes	characteristic
423	M₁₆	—	5	1	no	—
424	M₁₆	—	5	2	no	—
425	C₂×C₈	—	5	2	no	—
426	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
427	C₁₆	—	5	1	no	—
428	C₁₆	—	5	1	no	—
429	C₁₆	C₂²×C₃×D₅	1	2	yes	—
430	C₃×C₅	#3416	1	1	yes	characteristic, complemented, Hall
431	C₂²×C₃	—	4	1	no	—
432	C₂²×C₃	—	5	2	no	—
433	C₂²×C₃	—	20	1	no	—
434	C₂²×C₃	—	20	2	no	—
435	C₂²×C₃	—	80	1	no	—
436	C₂²×C₃	—	80	2	no	—
437	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
438	C₃×C₄	—	4	1	no	—
439	C₃×C₄	—	4	2	no	—
440	C₃×C₄	—	5	2	no	—
441	C₃×C₄	—	20	1	no	—
442	C₃×C₄	D₅×D₈	1	1	yes	characteristic
443	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
444	C₂×C₅	—	2	1	no	—
445	C₂×C₅	—	8	1	no	—
446	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
447	D₅	—	2	2	no	—
448	D₅	—	8	2	no	—
449	D₅	—	8	1	no	—
450	C₂³	—	20	2	no	—
451	C₂³	—	20	1	no	—
452	Q₈	—	2	1	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	4	1	no	—
455	Q₈	—	5	1	no	—
456	Q₈	—	10	1	no	—
457	Q₈	—	20	2	no	—
458	D₄	—	2	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	5	2	no	—
461	D₄	—	5	1	no	—
462	D₄	—	10	2	no	—
463	D₄	—	10	1	no	—
464	D₄	—	20	2	no	—
465	D₄	—	20	2	no	—
466	D₄	—	20	2	no	—
467	D₄	—	20	1	no	—
468	D₄	—	20	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	5	2	no	—
472	C₂×C₄	—	20	2	no	—
473	C₂×C₄	—	20	2	no	—
474	C₂×C₄	—	20	2	no	—
475	C₂×C₄	—	20	1	no	—
476	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
477	C₈	—	5	2	no	—
478	C₈	C₃×D₄×D₅	1	1	yes	characteristic
479	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
480	C₂×C₃	—	2	1	no	—
481	C₂×C₃	—	8	1	no	—
482	C₂×C₃	—	10	2	no	—
483	C₂×C₃	—	40	2	no	—
484	C₂×C₃	—	40	1	no	—
485	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
486	C₅	C₃×#3416	1	1	yes	characteristic, complemented, Sylow
487	C₂²	—	4	1	no	—
488	C₂²	—	5	2	no	—
489	C₂²	—	20	1	no	—
490	C₂²	—	20	2	no	—
491	C₂²	—	80	1	no	—
492	C₂²	—	80	2	no	—
493	C₂²	C₃×D₅×D₈	1	1	yes	characteristic
494	C₄	—	4	1	no	—
495	C₄	—	4	2	no	—
496	C₄	—	5	2	no	—
497	C₄	—	20	1	no	—
498	C₄	C₃×D₅×D₈	1	1	yes	characteristic
499	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
500	C₃	#10613771	1	1	yes	characteristic, direct product factor, Sylow
501	C₂	—	2	1	no	—
502	C₂	—	8	1	no	—
503	C₂	—	10	2	no	—
504	C₂	—	40	2	no	—
505	C₂	—	40	1	no	—
506	C₂	C₂×C₃×D₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₅×D₈
commutator, cosocle	C₅×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₅
Fitting	C₃×C₅×#510	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₃×#510	D₅
hypocenter	C₅	C₃×#3416
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#510

Complements

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

N	H ≅ G/N	type
1. C₃×#66103	C₂	semidirect
2. C₃×#66100	C₂	semidirect
3. C₃×C₅×#510	C₂	semidirect
4. C₃×#65483	C₂	semidirect
5. C₂×C₃×#3960	C₂	semidirect
6. C₃×#65208	C₂	semidirect
7. C₃×#65207	C₂	semidirect
8. C₃×#65203	C₂	semidirect
9. C₃×D₅×SD₃₂	C₂	semidirect
10. C₃×#65197	C₂	semidirect
11. C₃×#65196	C₂	semidirect
12. #10613771	C₃	direct (N × H)
16. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
20. C₃×#4065	C₂²	semidirect
22. C₃×D₅×Q₁₆	C₂²	semidirect
25. C₃×#4050	C₂²	semidirect
28. C₃×C₅×Q₃₂	C₂²	semidirect
29. C₃×C₅×SD₃₂	C₂²	semidirect
30. C₃×C₅×M₃₂	C₂²	semidirect
31. C₃×#3961	C₂²	semidirect
32. C₃×#3960	C₂²	semidirect
33. C₃×#3960	C₂²	semidirect
34. C₃×#3960	C₂²	semidirect
35. C₃×#3959	C₂²	semidirect
36. C₃×#3958	C₂²	semidirect
37. C₂×C₃×#672	C₂²	semidirect
38. C₃×#3932	C₂²	semidirect
39. C₃×D₈₀	C₂²	semidirect
40. C₃×#3930	C₂²	semidirect
41. C₃×D₅×C₁₆	C₂²	semidirect
43. #66103	C₂×C₃	semidirect
44. #66100	C₂×C₃	semidirect
45. C₅×#510	C₂×C₃	semidirect
46. #65483	C₂×C₃	semidirect
47. C₂×#3960	C₂×C₃	semidirect
48. #65208	C₂×C₃	semidirect
49. #65207	C₂×C₃	semidirect
50. #65203	C₂×C₃	semidirect
51. D₅×SD₃₂	C₂×C₃	semidirect
52. #65197	C₂×C₃	semidirect
53. #65196	C₂×C₃	semidirect
93. C₃×#510	D₅	semidirect
104. C₂×C₅×Q₁₆	C₂²×C₃	semidirect
108. #4065	C₂²×C₃	semidirect
110. D₅×Q₁₆	C₂²×C₃	semidirect
113. #4050	C₂²×C₃	semidirect
116. C₅×Q₃₂	C₂²×C₃	semidirect
117. C₅×SD₃₂	C₂²×C₃	semidirect
118. C₅×M₃₂	C₂²×C₃	semidirect
119. #3961	C₂²×C₃	semidirect
120. #3960	C₂²×C₃	semidirect
121. #3960	C₂²×C₃	semidirect
122. #3960	C₂²×C₃	semidirect
123. #3959	C₂²×C₃	semidirect
124. #3958	C₂²×C₃	semidirect
125. C₂×#672	C₂²×C₃	semidirect
126. #3932	C₂²×C₃	semidirect
127. D₈₀	C₂²×C₃	semidirect
128. #3930	C₂²×C₃	semidirect
129. D₅×C₁₆	C₂²×C₃	semidirect
172. C₂×C₃×Q₁₆	C₂×D₅	semidirect
177. C₃×Q₃₂	C₂×D₅	semidirect
183. C₃×SD₃₂	C₂×D₅	semidirect
187. C₃×M₃₂	C₂×D₅	semidirect
229. #510	C₃×D₅	semidirect
327. C₂×Q₁₆	C₂×C₃×D₅	semidirect
332. Q₃₂	C₂×C₃×D₅	semidirect
338. SD₃₂	C₂×C₃×D₅	semidirect
342. M₃₂	C₂×C₃×D₅	semidirect
430. C₃×C₅	#3416	semidirect
486. C₅	C₃×#3416	semidirect
500. C₃	#10613771	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×#66103	2
C₃×#66100	2
C₃×C₅×#510	2
C₃×#65483	2
C₂×C₃×#3960	2
C₃×#65208	2
C₃×#65207	2
C₃×#65203	2
C₃×D₅×SD₃₂	2
C₃×#65197	2
C₃×#65196	2
#10613771	3
C₃×#3416	5

Subgroups — C3×#10613771 #14478544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613771 #14478544