Subgroups — C₃×#60470 #10679077

Order 768.

2604 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3586	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3585	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#577	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#3069	C₂	1	6	yes	maximal, complemented
6	C₃×#3068	C₂	1	6	yes	maximal, complemented
7	C₃×#3067	C₂	1	6	yes	maximal, complemented
8	C₂×C₃×#471	C₂	1	6	yes	maximal, complemented
9	C₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2889	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#2888	C₂	1	1	yes	maximal, characteristic, complemented
12	#60470	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#585	C₂²	1	1	yes	characteristic
14	C₂×C₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
15	C₂×C₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
16	C₃×#578	—	2	4	no	—
17	C₃×#577	—	2	4	no	—
18	C₃×#577	—	2	4	no	—
19	C₃×#577	—	2	4	no	—
20	C₃×#577	—	2	4	no	—
21	C₃×#576	—	2	4	no	—
22	C₃×#575	C₂²	1	6	yes	—
23	C₃×#575	C₂²	1	6	yes	complemented
24	C₂×C₃×#137	C₂²	1	6	yes	complemented
25	C₂×C₃×#137	C₂²	1	3	yes	complemented
26	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	6	yes	—
27	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	3	yes	complemented
28	C₂×C₃×#135	C₂²	1	3	yes	complemented
29	C₂²×C₃×SD₁₆	C₂²	1	3	yes	complemented
30	C₃×#568	—	2	4	no	—
31	C₃×#568	C₂²	1	1	yes	characteristic
32	C₃×#568	C₂²	1	1	yes	characteristic
33	C₃×#568	C₂²	1	1	yes	characteristic
34	C₃×#568	C₂²	1	6	yes	complemented
35	C₂×C₃×#131	—	2	4	no	—
36	C₂×C₃×#131	C₂²	1	1	yes	characteristic
37	C₂×C₃×#131	C₂²	1	6	yes	complemented
38	C₃×#473	C₂²	1	12	yes	complemented
39	C₃×#472	C₂²	1	12	yes	complemented
40	C₃×#471	C₂²	1	12	yes	complemented
41	C₃×#471	C₂²	1	24	yes	complemented
42	C₃×#471	C₂²	1	4	yes	complemented
43	C₃×#430	C₂²	1	6	yes	complemented
44	C₂×C₃×#108	C₂²	1	6	yes	complemented
45	C₃×#413	C₂²	1	6	yes	complemented
46	C₃×#413	C₂²	1	6	yes	complemented
47	C₂×C₃×#101	C₂²	1	6	yes	complemented
48	C₂×C₃×#100	C₂²	1	6	yes	complemented
49	#3586	C₂×C₃	1	1	yes	characteristic, complemented
50	#3585	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×#577	C₂×C₃	1	1	yes	characteristic, complemented
52	#3573	C₂×C₃	1	1	yes	characteristic, complemented
53	#3069	C₂×C₃	1	6	yes	complemented
54	#3068	C₂×C₃	1	6	yes	complemented
55	#3067	C₂×C₃	1	6	yes	complemented
56	C₂×#471	C₂×C₃	1	6	yes	complemented
57	#2916	C₂×C₃	1	1	yes	characteristic, complemented
58	#2889	C₂×C₃	1	1	yes	characteristic, complemented
59	#2888	C₂×C₃	1	1	yes	characteristic, complemented
60	C₃×2₋¹⁺⁴	—	2	1	no	—
61	C₃×2₋¹⁺⁴	—	2	4	no	—
62	C₃×2₋¹⁺⁴	—	2	4	no	—
63	C₃×2₋¹⁺⁴	D₄	1	4	yes	—
64	C₃×H(5,𝔽₂)	—	2	3	no	—
65	C₃×H(5,𝔽₂)	—	2	4	no	—
66	C₃×H(5,𝔽₂)	—	2	4	no	—
67	C₃×H(5,𝔽₂)	D₄	1	4	yes	—
68	C₂×C₃×#41	—	2	4	no	—
69	C₂×C₃×#41	—	2	1	no	—
70	C₂×C₃×#41	—	2	3	no	—
71	C₂×C₃×#41	C₂³	1	6	yes	—
72	C₂×C₃×#41	C₂³	1	3	yes	—
73	C₂×C₃×#41	C₂³	1	3	yes	—
74	C₂×C₃×#41	C₂³	1	1	yes	characteristic
75	C₂²×C₃×Q₈	—	2	1	no	—
76	C₂²×C₃×Q₈	C₂³	1	3	yes	—
77	C₂²×C₃×D₄	—	2	3	no	—
78	C₂²×C₃×D₄	C₂³	1	3	yes	—
79	C₃×#137	—	2	12	no	—
80	C₃×#137	—	2	6	no	—
81	C₃×#137	—	2	6	no	—
82	C₃×#137	—	2	12	no	—
83	C₃×#137	C₂³	1	12	yes	—
84	C₃×#137	C₂³	1	4	yes	—
85	C₃×AGL(1,ℤ/8)	—	2	12	no	—
86	C₃×AGL(1,ℤ/8)	—	2	6	no	—
87	C₃×AGL(1,ℤ/8)	—	2	6	no	—
88	C₃×AGL(1,ℤ/8)	—	2	12	no	—
89	C₃×AGL(1,ℤ/8)	C₂³	1	12	yes	—
90	C₃×AGL(1,ℤ/8)	C₂³	1	4	yes	—
91	C₃×#135	—	2	12	no	—
92	C₃×#135	—	2	12	no	—
93	C₃×#135	—	2	6	no	—
94	C₃×#135	—	2	6	no	—
95	C₂×C₃×Q₁₆	—	2	6	no	—
96	C₂×C₃×Q₁₆	C₂³	1	3	yes	—
97	C₂×C₃×SD₁₆	—	2	6	no	—
98	C₂×C₃×SD₁₆	—	2	6	no	—
99	C₂×C₃×SD₁₆	—	2	6	no	—
100	C₂×C₃×SD₁₆	—	2	6	no	—
101	C₂×C₃×SD₁₆	C₂³	1	3	yes	—
102	C₂×C₃×SD₁₆	C₂³	1	6	yes	—
103	C₂×C₃×SD₁₆	C₂³	1	1	yes	characteristic
104	C₂×C₃×D₈	—	2	6	no	—
105	C₂×C₃×D₈	C₂³	1	3	yes	—
106	C₃×#131	—	2	4	no	—
107	C₃×#131	—	2	4	no	—
108	C₃×#131	—	2	4	no	—
109	C₃×#131	—	2	12	no	—
110	C₃×#131	—	2	6	no	—
111	C₃×#131	—	2	6	no	—
112	C₃×#131	D₄	1	4	yes	—
113	C₃×#131	D₄	1	4	yes	—
114	C₃×#131	C₂³	1	4	yes	—
115	C₃×#131	C₂³	1	12	yes	—
116	C₂×C₃×M₁₆	—	2	12	no	—
117	C₂×C₃×M₁₆	—	2	6	no	—
118	C₂×C₃×M₁₆	C₂³	1	6	yes	—
119	C₂×C₃×M₁₆	C₂³	1	6	yes	—
120	C₂×C₃×M₁₆	C₂³	1	6	yes	—
121	C₂×C₃×M₁₆	C₂³	1	3	yes	—
122	C₂²×C₃×C₈	—	2	6	no	—
123	C₂²×C₃×C₈	C₂³	1	3	yes	—
124	C₃×#108	C₂³	1	4	yes	—
125	C₃×#108	C₂³	1	12	yes	—
126	C₃×#101	C₂³	1	12	yes	—
127	C₃×#101	C₂³	1	4	yes	—
128	C₃×#100	C₂³	1	12	yes	—
129	C₃×#100	C₂³	1	4	yes	—
130	#585	C₂²×C₃	1	1	yes	characteristic
131	C₂×2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic
132	C₂×H(5,𝔽₂)	C₂²×C₃	1	1	yes	characteristic
133	#578	—	2	4	no	—
134	#577	—	2	4	no	—
135	#577	—	2	4	no	—
136	#577	—	2	4	no	—
137	#577	—	2	4	no	—
138	#576	—	2	4	no	—
139	#575	C₂²×C₃	1	6	yes	—
140	#575	C₂²×C₃	1	6	yes	complemented
141	C₂×#137	C₂²×C₃	1	6	yes	complemented
142	C₂×#137	C₂²×C₃	1	3	yes	complemented
143	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	6	yes	—
144	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	3	yes	complemented
145	C₂×#135	C₂²×C₃	1	3	yes	complemented
146	C₂²×SD₁₆	C₂²×C₃	1	3	yes	complemented
147	#568	—	2	4	no	—
148	#568	C₂²×C₃	1	1	yes	characteristic
149	#568	C₂²×C₃	1	1	yes	characteristic
150	#568	C₂²×C₃	1	1	yes	characteristic
151	#568	C₂²×C₃	1	6	yes	complemented
152	C₂×#131	—	2	4	no	—
153	C₂×#131	C₂²×C₃	1	1	yes	characteristic
154	C₂×#131	C₂²×C₃	1	6	yes	complemented
155	#473	C₂²×C₃	1	12	yes	complemented
156	#472	C₂²×C₃	1	12	yes	complemented
157	#471	C₂²×C₃	1	12	yes	complemented
158	#471	C₂²×C₃	1	24	yes	complemented
159	#471	C₂²×C₃	1	4	yes	complemented
160	#430	C₂²×C₃	1	6	yes	complemented
161	C₂×#108	C₂²×C₃	1	6	yes	complemented
162	#413	C₂²×C₃	1	6	yes	complemented
163	#413	C₂²×C₃	1	6	yes	complemented
164	C₂×#101	C₂²×C₃	1	6	yes	complemented
165	C₂×#100	C₂²×C₃	1	6	yes	complemented
166	C₂⁴×C₃	—	2	3	no	—
167	C₃×#41	—	2	4	no	—
168	C₃×#41	—	2	12	no	—
169	C₃×#41	—	2	6	no	—
170	C₃×#41	—	2	6	no	—
171	C₃×#41	—	2	6	no	—
172	C₃×#41	—	2	6	no	—
173	C₃×#41	—	2	6	no	—
174	C₃×#41	—	2	6	no	—
175	C₃×#41	—	4	4	no	—
176	C₃×#41	—	4	12	no	—
177	C₃×#41	C₂×D₄	1	12	yes	—
178	C₃×#41	C₂×D₄	1	4	yes	—
179	C₃×#41	C₂×D₄	1	4	yes	—
180	C₃×#41	C₂⁴	1	4	yes	—
181	C₂×C₃×Q₈	—	2	4	no	—
182	C₂×C₃×Q₈	—	2	3	no	—
183	C₂×C₃×Q₈	—	2	6	no	—
184	C₂×C₃×Q₈	—	2	6	no	—
185	C₂×C₃×Q₈	—	4	4	no	—
186	C₂×C₃×Q₈	C₂×D₄	1	6	yes	—
187	C₂×C₃×Q₈	#41	1	1	yes	characteristic
188	C₂×C₃×Q₈	C₂⁴	1	3	yes	—
189	C₂×C₃×Q₈	C₂⁴	1	1	yes	characteristic
190	C₂×C₃×D₄	—	2	6	no	—
191	C₂×C₃×D₄	—	2	12	no	—
192	C₂×C₃×D₄	—	2	3	no	—
193	C₂×C₃×D₄	—	2	3	no	—
194	C₂×C₃×D₄	—	2	6	no	—
195	C₂×C₃×D₄	—	2	6	no	—
196	C₂×C₃×D₄	—	2	3	no	—
197	C₂×C₃×D₄	—	4	12	no	—
198	C₂×C₃×D₄	C₂×D₄	1	6	yes	—
199	C₂×C₃×D₄	#41	1	3	yes	—
200	C₂×C₃×D₄	C₂⁴	1	1	yes	characteristic
201	C₂×C₃×D₄	C₂⁴	1	3	yes	—
202	C₂²×C₃×C₄	—	2	6	no	—
203	C₂²×C₃×C₄	—	2	3	no	—
204	C₂²×C₃×C₄	—	2	3	no	—
205	C₂²×C₃×C₄	—	2	3	no	—
206	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
207	C₃×Q₁₆	—	2	12	no	—
208	C₃×Q₁₆	—	2	4	no	—
209	C₃×Q₁₆	—	2	6	no	—
210	C₃×SD₁₆	—	2	12	no	—
211	C₃×SD₁₆	—	2	4	no	—
212	C₃×SD₁₆	—	2	12	no	—
213	C₃×SD₁₆	—	2	4	no	—
214	C₃×SD₁₆	—	2	3	no	—
215	C₃×SD₁₆	—	2	3	no	—
216	C₃×SD₁₆	—	2	12	no	—
217	C₃×SD₁₆	—	2	1	no	—
218	C₃×SD₁₆	—	2	1	no	—
219	C₃×D₈	—	2	12	no	—
220	C₃×D₈	—	2	4	no	—
221	C₃×D₈	—	2	6	no	—
222	C₃×M₁₆	—	2	6	no	—
223	C₃×M₁₆	—	2	6	no	—
224	C₃×M₁₆	—	2	6	no	—
225	C₃×M₁₆	—	2	12	no	—
226	C₃×M₁₆	—	2	4	no	—
227	C₃×M₁₆	C₂×D₄	1	12	yes	—
228	C₃×M₁₆	C₂⁴	1	4	yes	—
229	C₃×M₁₆	C₂⁴	1	4	yes	—
230	C₃×M₁₆	C₂⁴	1	4	yes	—
231	C₂×C₃×C₈	—	2	6	no	—
232	C₂×C₃×C₈	—	2	6	no	—
233	C₂×C₃×C₈	—	2	6	no	—
234	C₂×C₃×C₈	—	2	4	no	—
235	C₂×C₃×C₈	—	2	12	no	—
236	C₂×C₃×C₈	C₂×D₄	1	6	yes	—
237	C₂×C₃×C₈	C₂×D₄	1	6	yes	—
238	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
239	C₂×C₃×C₈	C₂⁴	1	3	yes	—
240	2₋¹⁺⁴	—	2	1	no	—
241	2₋¹⁺⁴	—	2	4	no	—
242	2₋¹⁺⁴	—	2	4	no	—
243	2₋¹⁺⁴	C₃×D₄	1	4	yes	—
244	H(5,𝔽₂)	—	2	3	no	—
245	H(5,𝔽₂)	—	2	4	no	—
246	H(5,𝔽₂)	—	2	4	no	—
247	H(5,𝔽₂)	C₃×D₄	1	4	yes	—
248	C₂×#41	—	2	4	no	—
249	C₂×#41	—	2	1	no	—
250	C₂×#41	—	2	3	no	—
251	C₂×#41	C₂³×C₃	1	6	yes	—
252	C₂×#41	C₂³×C₃	1	3	yes	—
253	C₂×#41	C₂³×C₃	1	3	yes	—
254	C₂×#41	C₂³×C₃	1	1	yes	characteristic
255	C₂²×Q₈	—	2	1	no	—
256	C₂²×Q₈	C₂³×C₃	1	3	yes	—
257	C₂²×D₄	—	2	3	no	—
258	C₂²×D₄	C₂³×C₃	1	3	yes	—
259	#137	—	2	12	no	—
260	#137	—	2	6	no	—
261	#137	—	2	6	no	—
262	#137	—	2	12	no	—
263	#137	C₂³×C₃	1	12	yes	—
264	#137	C₂³×C₃	1	4	yes	—
265	AGL(1,ℤ/8)	—	2	12	no	—
266	AGL(1,ℤ/8)	—	2	6	no	—
267	AGL(1,ℤ/8)	—	2	6	no	—
268	AGL(1,ℤ/8)	—	2	12	no	—
269	AGL(1,ℤ/8)	C₂³×C₃	1	12	yes	—
270	AGL(1,ℤ/8)	C₂³×C₃	1	4	yes	—
271	#135	—	2	12	no	—
272	#135	—	2	12	no	—
273	#135	—	2	6	no	—
274	#135	—	2	6	no	—
275	C₂×Q₁₆	—	2	6	no	—
276	C₂×Q₁₆	C₂³×C₃	1	3	yes	—
277	C₂×SD₁₆	—	2	6	no	—
278	C₂×SD₁₆	—	2	6	no	—
279	C₂×SD₁₆	—	2	6	no	—
280	C₂×SD₁₆	—	2	6	no	—
281	C₂×SD₁₆	C₂³×C₃	1	3	yes	—
282	C₂×SD₁₆	C₂³×C₃	1	6	yes	—
283	C₂×SD₁₆	C₂³×C₃	1	1	yes	characteristic
284	C₂×D₈	—	2	6	no	—
285	C₂×D₈	C₂³×C₃	1	3	yes	—
286	#131	—	2	4	no	—
287	#131	—	2	4	no	—
288	#131	—	2	4	no	—
289	#131	—	2	12	no	—
290	#131	—	2	6	no	—
291	#131	—	2	6	no	—
292	#131	C₃×D₄	1	4	yes	—
293	#131	C₃×D₄	1	4	yes	—
294	#131	C₂³×C₃	1	4	yes	—
295	#131	C₂³×C₃	1	12	yes	—
296	C₂×M₁₆	—	2	12	no	—
297	C₂×M₁₆	—	2	6	no	—
298	C₂×M₁₆	C₂³×C₃	1	6	yes	—
299	C₂×M₁₆	C₂³×C₃	1	6	yes	—
300	C₂×M₁₆	C₂³×C₃	1	6	yes	—
301	C₂×M₁₆	C₂³×C₃	1	3	yes	—
302	C₂²×C₈	—	2	6	no	—
303	C₂²×C₈	C₂³×C₃	1	3	yes	—
304	#108	C₂³×C₃	1	4	yes	—
305	#108	C₂³×C₃	1	12	yes	—
306	#101	C₂³×C₃	1	12	yes	—
307	#101	C₂³×C₃	1	4	yes	—
308	#100	C₂³×C₃	1	12	yes	—
309	#100	C₂³×C₃	1	4	yes	—
310	C₂³×C₃	—	2	6	no	—
311	C₂³×C₃	—	2	1	no	—
312	C₂³×C₃	—	2	3	no	—
313	C₂³×C₃	—	4	6	no	—
314	C₂³×C₃	—	16	3	no	—
315	C₂³×C₃	C₂×#41	1	3	yes	—
316	C₃×Q₈	—	2	6	no	—
317	C₃×Q₈	—	2	3	no	—
318	C₃×Q₈	—	2	3	no	—
319	C₃×Q₈	—	2	1	no	—
320	C₃×Q₈	—	2	1	no	—
321	C₃×Q₈	—	4	6	no	—
322	C₃×Q₈	#121	1	4	yes	—
323	C₃×Q₈	C₂²×D₄	1	4	yes	—
324	C₃×D₄	—	2	12	no	—
325	C₃×D₄	—	2	4	no	—
326	C₃×D₄	—	2	6	no	—
327	C₃×D₄	—	2	1	no	—
328	C₃×D₄	—	2	3	no	—
329	C₃×D₄	—	2	1	no	—
330	C₃×D₄	—	2	3	no	—
331	C₃×D₄	—	4	6	no	—
332	C₃×D₄	—	4	6	no	—
333	C₃×D₄	—	4	6	no	—
334	C₃×D₄	#121	1	12	yes	—
335	C₃×D₄	C₂²×D₄	1	4	yes	—
336	C₂×C₃×C₄	—	2	6	no	—
337	C₂×C₃×C₄	—	2	12	no	—
338	C₂×C₃×C₄	—	2	4	no	—
339	C₂×C₃×C₄	—	2	1	no	—
340	C₂×C₃×C₄	—	2	3	no	—
341	C₂×C₃×C₄	—	4	6	no	—
342	C₂×C₃×C₄	—	4	6	no	—
343	C₂×C₃×C₄	—	4	6	no	—
344	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
345	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
346	C₂×C₃×C₄	C₂×#41	1	3	yes	—
347	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
348	C₃×C₈	—	2	4	no	—
349	C₃×C₈	—	2	4	no	—
350	C₃×C₈	—	2	4	no	—
351	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
352	C₃×C₈	C₂²×D₄	1	6	yes	—
353	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
354	C₂⁴	—	2	3	no	—
355	#41	—	2	4	no	—
356	#41	—	2	12	no	—
357	#41	—	2	6	no	—
358	#41	—	2	6	no	—
359	#41	—	2	6	no	—
360	#41	—	2	6	no	—
361	#41	—	2	6	no	—
362	#41	—	2	6	no	—
363	#41	—	4	4	no	—
364	#41	—	4	12	no	—
365	#41	C₂×C₃×D₄	1	12	yes	—
366	#41	C₂×C₃×D₄	1	4	yes	—
367	#41	C₂×C₃×D₄	1	4	yes	—
368	#41	C₂⁴×C₃	1	4	yes	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	3	no	—
371	C₂×Q₈	—	2	6	no	—
372	C₂×Q₈	—	2	6	no	—
373	C₂×Q₈	—	4	4	no	—
374	C₂×Q₈	C₂×C₃×D₄	1	6	yes	—
375	C₂×Q₈	C₃×#41	1	1	yes	characteristic
376	C₂×Q₈	C₂⁴×C₃	1	3	yes	—
377	C₂×Q₈	C₂⁴×C₃	1	1	yes	characteristic
378	C₂×D₄	—	2	6	no	—
379	C₂×D₄	—	2	12	no	—
380	C₂×D₄	—	2	3	no	—
381	C₂×D₄	—	2	3	no	—
382	C₂×D₄	—	2	6	no	—
383	C₂×D₄	—	2	6	no	—
384	C₂×D₄	—	2	3	no	—
385	C₂×D₄	—	4	12	no	—
386	C₂×D₄	C₂×C₃×D₄	1	6	yes	—
387	C₂×D₄	C₃×#41	1	3	yes	—
388	C₂×D₄	C₂⁴×C₃	1	1	yes	characteristic
389	C₂×D₄	C₂⁴×C₃	1	3	yes	—
390	C₂²×C₄	—	2	6	no	—
391	C₂²×C₄	—	2	3	no	—
392	C₂²×C₄	—	2	3	no	—
393	C₂²×C₄	—	2	3	no	—
394	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
395	Q₁₆	—	2	12	no	—
396	Q₁₆	—	2	4	no	—
397	Q₁₆	—	2	6	no	—
398	SD₁₆	—	2	12	no	—
399	SD₁₆	—	2	4	no	—
400	SD₁₆	—	2	12	no	—
401	SD₁₆	—	2	4	no	—
402	SD₁₆	—	2	3	no	—
403	SD₁₆	—	2	3	no	—
404	SD₁₆	—	2	12	no	—
405	SD₁₆	—	2	1	no	—
406	SD₁₆	—	2	1	no	—
407	D₈	—	2	12	no	—
408	D₈	—	2	4	no	—
409	D₈	—	2	6	no	—
410	M₁₆	—	2	6	no	—
411	M₁₆	—	2	6	no	—
412	M₁₆	—	2	6	no	—
413	M₁₆	—	2	12	no	—
414	M₁₆	—	2	4	no	—
415	M₁₆	C₂×C₃×D₄	1	12	yes	—
416	M₁₆	C₂⁴×C₃	1	4	yes	—
417	M₁₆	C₂⁴×C₃	1	4	yes	—
418	M₁₆	C₂⁴×C₃	1	4	yes	—
419	C₂×C₈	—	2	6	no	—
420	C₂×C₈	—	2	6	no	—
421	C₂×C₈	—	2	6	no	—
422	C₂×C₈	—	2	4	no	—
423	C₂×C₈	—	2	12	no	—
424	C₂×C₈	C₂×C₃×D₄	1	6	yes	—
425	C₂×C₈	C₂×C₃×D₄	1	6	yes	—
426	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
427	C₂×C₈	C₂⁴×C₃	1	3	yes	—
428	C₂²×C₃	—	2	4	no	—
429	C₂²×C₃	—	4	3	no	—
430	C₂²×C₃	—	4	1	no	—
431	C₂²×C₃	—	4	3	no	—
432	C₂²×C₃	—	8	6	no	—
433	C₂²×C₃	—	8	1	no	—
434	C₂²×C₃	—	8	3	no	—
435	C₂²×C₃	—	16	6	no	—
436	C₂²×C₃	C₂×#121	1	6	yes	—
437	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
438	C₃×C₄	—	2	4	no	—
439	C₃×C₄	—	4	1	no	—
440	C₃×C₄	—	4	3	no	—
441	C₃×C₄	C₂×#121	1	6	yes	—
442	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
443	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
444	C₂³	—	2	6	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	3	no	—
447	C₂³	—	4	6	no	—
448	C₂³	—	16	3	no	—
449	C₂³	C₂×C₃×#41	1	3	yes	—
450	Q₈	—	2	6	no	—
451	Q₈	—	2	3	no	—
452	Q₈	—	2	3	no	—
453	Q₈	—	2	1	no	—
454	Q₈	—	2	1	no	—
455	Q₈	—	4	6	no	—
456	Q₈	C₃×#121	1	4	yes	—
457	Q₈	C₂²×C₃×D₄	1	4	yes	—
458	D₄	—	2	4	no	—
459	D₄	—	2	12	no	—
460	D₄	—	2	1	no	—
461	D₄	—	2	3	no	—
462	D₄	—	2	1	no	—
463	D₄	—	2	3	no	—
464	D₄	—	2	6	no	—
465	D₄	—	4	6	no	—
466	D₄	—	4	6	no	—
467	D₄	—	4	6	no	—
468	D₄	C₃×#121	1	12	yes	—
469	D₄	C₂²×C₃×D₄	1	4	yes	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	12	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	3	no	—
475	C₂×C₄	—	4	6	no	—
476	C₂×C₄	—	4	6	no	—
477	C₂×C₄	—	4	6	no	—
478	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
479	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
480	C₂×C₄	C₂×C₃×#41	1	3	yes	—
481	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
482	C₈	—	2	4	no	—
483	C₈	—	2	4	no	—
484	C₈	—	2	4	no	—
485	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
486	C₈	C₂²×C₃×D₄	1	6	yes	—
487	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
488	C₂×C₃	—	2	1	no	—
489	C₂×C₃	—	2	6	no	—
490	C₂×C₃	—	4	4	no	—
491	C₂×C₃	—	8	1	no	—
492	C₂×C₃	—	8	3	no	—
493	C₂×C₃	C₂²×#121	1	1	yes	characteristic
494	C₂²	—	2	4	no	—
495	C₂²	—	4	3	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	3	no	—
498	C₂²	—	8	6	no	—
499	C₂²	—	8	3	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	16	6	no	—
502	C₂²	C₂×C₃×#121	1	6	yes	—
503	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
504	C₄	—	2	4	no	—
505	C₄	—	4	1	no	—
506	C₄	—	4	3	no	—
507	C₄	C₂×C₃×#121	1	6	yes	—
508	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
509	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
510	C₃	#60470	1	1	yes	characteristic, direct product factor, Sylow
511	C₂	—	2	1	no	—
512	C₂	—	2	6	no	—
513	C₂	—	4	4	no	—
514	C₂	—	8	3	no	—
515	C₂	—	8	1	no	—
516	C₂	C₂²×C₃×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3586	C₂	semidirect
2. C₃×#3585	C₂	semidirect
3. C₂×C₃×#577	C₂	semidirect
4. C₃×#3573	C₂	semidirect
5. C₃×#3069	C₂	semidirect
6. C₃×#3068	C₂	semidirect
7. C₃×#3067	C₂	semidirect
8. C₂×C₃×#471	C₂	semidirect
9. C₃×#2916	C₂	semidirect
10. C₃×#2889	C₂	semidirect
11. C₃×#2888	C₂	semidirect
12. #60470	C₃	direct (N × H)
23. C₃×#575	C₂²	semidirect
24. C₂×C₃×#137	C₂²	semidirect
25. C₂×C₃×#137	C₂²	semidirect
27. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
28. C₂×C₃×#135	C₂²	semidirect
29. C₂²×C₃×SD₁₆	C₂²	semidirect
34. C₃×#568	C₂²	semidirect
37. C₂×C₃×#131	C₂²	semidirect
38. C₃×#473	C₂²	semidirect
39. C₃×#472	C₂²	semidirect
40. C₃×#471	C₂²	semidirect
41. C₃×#471	C₂²	semidirect
42. C₃×#471	C₂²	semidirect
43. C₃×#430	C₂²	semidirect
44. C₂×C₃×#108	C₂²	semidirect
45. C₃×#413	C₂²	semidirect
46. C₃×#413	C₂²	semidirect
47. C₂×C₃×#101	C₂²	semidirect
48. C₂×C₃×#100	C₂²	semidirect
49. #3586	C₂×C₃	semidirect
50. #3585	C₂×C₃	semidirect
51. C₂×#577	C₂×C₃	semidirect
52. #3573	C₂×C₃	semidirect
53. #3069	C₂×C₃	semidirect
54. #3068	C₂×C₃	semidirect
55. #3067	C₂×C₃	semidirect
56. C₂×#471	C₂×C₃	semidirect
57. #2916	C₂×C₃	semidirect
58. #2889	C₂×C₃	semidirect
59. #2888	C₂×C₃	semidirect
140. #575	C₂²×C₃	semidirect
141. C₂×#137	C₂²×C₃	semidirect
142. C₂×#137	C₂²×C₃	semidirect
144. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
145. C₂×#135	C₂²×C₃	semidirect
146. C₂²×SD₁₆	C₂²×C₃	semidirect
151. #568	C₂²×C₃	semidirect
154. C₂×#131	C₂²×C₃	semidirect
155. #473	C₂²×C₃	semidirect
156. #472	C₂²×C₃	semidirect
157. #471	C₂²×C₃	semidirect
158. #471	C₂²×C₃	semidirect
159. #471	C₂²×C₃	semidirect
160. #430	C₂²×C₃	semidirect
161. C₂×#108	C₂²×C₃	semidirect
162. #413	C₂²×C₃	semidirect
163. #413	C₂²×C₃	semidirect
164. C₂×#101	C₂²×C₃	semidirect
165. C₂×#100	C₂²×C₃	semidirect
510. C₃	#60470	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×#3586	2
C₃×#3585	2
C₂×C₃×#577	2
C₃×#3573	2
C₃×#3069	2
C₃×#3068	2
C₃×#3067	2
C₂×C₃×#471	2
C₃×#2916	2
C₃×#2889	2
C₃×#2888	2
#60470	3

Subgroups — C3×#60470 #10679077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60470 #10679077