Subgroups — C₂×C₃×#3585 #10681695

Order 768.

11138 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3585	C₂	1	32	yes	maximal, direct product factor
4	C₂×C₃×#577	C₂	1	8	yes	maximal, complemented
5	C₂×C₃×#576	C₂	1	8	yes	maximal, complemented
6	C₂×C₃×#575	C₂	1	6	yes	maximal, complemented
7	C₂²×C₃×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
8	C₂×C₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#3585	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃×#585	—	2	8	no	—
11	C₃×#585	C₂²	1	16	yes	complemented
12	C₂×C₃×2₋¹⁺⁴	—	2	1	no	—
13	C₂×C₃×2₋¹⁺⁴	C₂²	1	4	yes	complemented
14	C₂×C₃×H(5,𝔽₂)	—	2	4	no	—
15	C₂×C₃×H(5,𝔽₂)	—	2	8	no	—
16	C₂×C₃×H(5,𝔽₂)	—	2	3	no	—
17	C₂×C₃×H(5,𝔽₂)	C₂²	1	8	yes	complemented
18	C₂×C₃×H(5,𝔽₂)	C₂²	1	16	yes	complemented
19	C₂×C₃×H(5,𝔽₂)	C₂²	1	4	yes	complemented
20	C₂²×C₃×#41	—	2	4	no	—
21	C₂²×C₃×#41	—	2	1	no	—
22	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
23	C₂²×C₃×#41	C₂²	1	3	yes	complemented
24	C₂²×C₃×#41	C₂²	1	6	yes	complemented
25	C₂³×C₃×D₄	—	2	3	no	—
26	C₂³×C₃×D₄	C₂²	1	3	yes	complemented
27	C₃×#577	C₂²	1	128	yes	complemented
28	C₃×#576	C₂²	1	128	yes	complemented
29	C₃×#575	C₂²	1	96	yes	complemented
30	C₂×C₃×#137	C₂²	1	4	yes	complemented
31	C₂×C₃×#137	C₂²	1	12	yes	complemented
32	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	12	yes	complemented
33	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	24	yes	complemented
34	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	12	yes	complemented
35	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	96	yes	complemented
36	C₂×C₃×#135	C₂²	1	24	yes	complemented
37	C₂²×C₃×SD₁₆	C₂²	1	12	yes	complemented
38	C₂²×C₃×D₈	C₂²	1	12	yes	complemented
39	C₃×#568	C₂²	1	16	yes	complemented
40	C₂×C₃×#131	C₂²	1	8	yes	complemented
41	C₂²×C₃×M₁₆	C₂²	1	6	yes	complemented
42	C₂×#585	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂²×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
44	#3585	C₂×C₃	1	32	yes	direct product factor
45	C₂×#577	C₂×C₃	1	8	yes	complemented
46	C₂×#576	C₂×C₃	1	8	yes	complemented
47	C₂×#575	C₂×C₃	1	6	yes	complemented
48	C₂²×AGL(1,ℤ/8)	C₂×C₃	1	6	yes	complemented
49	C₂×#568	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁵×C₃	—	2	3	no	—
51	C₃×2₋¹⁺⁴	—	2	16	no	—
52	C₃×2₋¹⁺⁴	—	2	16	no	—
53	C₃×2₋¹⁺⁴	C₂³	1	32	yes	complemented
54	C₃×H(5,𝔽₂)	—	2	32	no	—
55	C₃×H(5,𝔽₂)	—	2	64	no	—
56	C₃×H(5,𝔽₂)	—	2	16	no	—
57	C₃×H(5,𝔽₂)	—	2	48	no	—
58	C₃×H(5,𝔽₂)	C₂³	1	64	yes	complemented
59	C₃×H(5,𝔽₂)	C₂³	1	32	yes	complemented
60	C₂×C₃×#41	—	2	64	no	—
61	C₂×C₃×#41	—	2	8	no	—
62	C₂×C₃×#41	—	2	16	no	—
63	C₂×C₃×#41	—	2	4	no	—
64	C₂×C₃×#41	—	2	12	no	—
65	C₂×C₃×#41	—	2	6	no	—
66	C₂×C₃×#41	—	2	6	no	—
67	C₂×C₃×#41	—	2	12	no	—
68	C₂×C₃×#41	—	2	6	no	—
69	C₂×C₃×#41	—	2	24	no	—
70	C₂×C₃×#41	D₄	1	4	yes	—
71	C₂×C₃×#41	D₄	1	8	yes	—
72	C₂×C₃×#41	C₂³	1	12	yes	—
73	C₂×C₃×#41	C₂³	1	4	yes	—
74	C₂×C₃×#41	C₂³	1	4	yes	—
75	C₂×C₃×#41	C₂³	1	8	yes	—
76	C₂×C₃×#41	C₂³	1	24	yes	complemented
77	C₂×C₃×#41	C₂³	1	12	yes	complemented
78	C₂×C₃×#41	C₂³	1	48	yes	complemented
79	C₂²×C₃×Q₈	—	2	4	no	—
80	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
81	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
82	C₂²×C₃×Q₈	C₂³	1	6	yes	complemented
83	C₂²×C₃×D₄	—	2	3	no	—
84	C₂²×C₃×D₄	—	2	6	no	—
85	C₂²×C₃×D₄	—	2	24	no	—
86	C₂²×C₃×D₄	—	2	48	no	—
87	C₂²×C₃×D₄	—	2	12	no	—
88	C₂²×C₃×D₄	—	2	3	no	—
89	C₂²×C₃×D₄	—	2	6	no	—
90	C₂²×C₃×D₄	—	2	12	no	—
91	C₂²×C₃×D₄	D₄	1	3	yes	—
92	C₂²×C₃×D₄	C₂³	1	12	yes	—
93	C₂²×C₃×D₄	C₂³	1	3	yes	—
94	C₂²×C₃×D₄	C₂³	1	24	yes	complemented
95	C₂²×C₃×D₄	C₂³	1	6	yes	complemented
96	C₂³×C₃×C₄	—	2	3	no	—
97	C₂³×C₃×C₄	—	2	6	no	—
98	C₂³×C₃×C₄	C₂³	1	3	yes	—
99	C₃×#137	C₂³	1	32	yes	complemented
100	C₃×#137	C₂³	1	96	yes	complemented
101	C₃×AGL(1,ℤ/8)	C₂³	1	96	yes	complemented
102	C₃×AGL(1,ℤ/8)	C₂³	1	192	yes	complemented
103	C₃×AGL(1,ℤ/8)	C₂³	1	96	yes	complemented
104	C₃×#135	C₂³	1	192	yes	complemented
105	C₂×C₃×Q₁₆	C₂³	1	8	yes	complemented
106	C₂×C₃×SD₁₆	C₂³	1	24	yes	complemented
107	C₂×C₃×SD₁₆	C₂³	1	8	yes	complemented
108	C₂×C₃×SD₁₆	C₂³	1	96	yes	complemented
109	C₂×C₃×D₈	C₂³	1	24	yes	complemented
110	C₂×C₃×D₈	C₂³	1	96	yes	complemented
111	C₃×#131	C₂³	1	64	yes	complemented
112	C₂×C₃×M₁₆	C₂³	1	4	yes	—
113	C₂×C₃×M₁₆	C₂³	1	12	yes	complemented
114	C₂×C₃×M₁₆	C₂³	1	48	yes	complemented
115	C₂²×C₃×C₈	C₂³	1	12	yes	complemented
116	#585	—	2	8	no	—
117	#585	C₂²×C₃	1	16	yes	complemented
118	C₂×2₋¹⁺⁴	—	2	1	no	—
119	C₂×2₋¹⁺⁴	C₂²×C₃	1	4	yes	complemented
120	C₂×H(5,𝔽₂)	—	2	4	no	—
121	C₂×H(5,𝔽₂)	—	2	8	no	—
122	C₂×H(5,𝔽₂)	—	2	3	no	—
123	C₂×H(5,𝔽₂)	C₂²×C₃	1	8	yes	complemented
124	C₂×H(5,𝔽₂)	C₂²×C₃	1	16	yes	complemented
125	C₂×H(5,𝔽₂)	C₂²×C₃	1	4	yes	complemented
126	C₂²×#41	—	2	4	no	—
127	C₂²×#41	—	2	1	no	—
128	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
129	C₂²×#41	C₂²×C₃	1	3	yes	complemented
130	C₂²×#41	C₂²×C₃	1	6	yes	complemented
131	C₂³×D₄	—	2	3	no	—
132	C₂³×D₄	C₂²×C₃	1	3	yes	complemented
133	#577	C₂²×C₃	1	128	yes	complemented
134	#576	C₂²×C₃	1	128	yes	complemented
135	#575	C₂²×C₃	1	96	yes	complemented
136	C₂×#137	C₂²×C₃	1	4	yes	complemented
137	C₂×#137	C₂²×C₃	1	12	yes	complemented
138	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	12	yes	complemented
139	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	24	yes	complemented
140	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	12	yes	complemented
141	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	96	yes	complemented
142	C₂×#135	C₂²×C₃	1	24	yes	complemented
143	C₂²×SD₁₆	C₂²×C₃	1	12	yes	complemented
144	C₂²×D₈	C₂²×C₃	1	12	yes	complemented
145	#568	C₂²×C₃	1	16	yes	complemented
146	C₂×#131	C₂²×C₃	1	8	yes	complemented
147	C₂²×M₁₆	C₂²×C₃	1	6	yes	complemented
148	C₂⁴×C₃	—	2	3	no	—
149	C₂⁴×C₃	—	2	12	no	—
150	C₂⁴×C₃	—	2	24	no	—
151	C₂⁴×C₃	—	2	6	no	—
152	C₂⁴×C₃	—	8	12	no	—
153	C₂⁴×C₃	C₂×D₄	1	3	yes	—
154	C₃×#41	—	2	64	no	—
155	C₃×#41	—	2	32	no	—
156	C₃×#41	—	2	96	no	—
157	C₃×#41	—	2	8	no	—
158	C₃×#41	—	2	48	no	—
159	C₃×#41	—	2	24	no	—
160	C₃×#41	—	2	48	no	—
161	C₃×#41	—	2	48	no	—
162	C₃×#41	—	2	24	no	—
163	C₃×#41	C₂×D₄	1	32	yes	—
164	C₃×#41	C₂×D₄	1	16	yes	—
165	C₃×#41	C₂⁴	1	48	yes	—
166	C₃×#41	C₂⁴	1	16	yes	—
167	C₃×#41	C₂⁴	1	16	yes	—
168	C₃×#41	C₂⁴	1	48	yes	complemented
169	C₂×C₃×Q₈	—	2	32	no	—
170	C₂×C₃×Q₈	—	2	6	no	—
171	C₂×C₃×Q₈	—	2	2	no	—
172	C₂×C₃×Q₈	—	2	1	no	—
173	C₂×C₃×Q₈	—	2	12	no	—
174	C₂×C₃×Q₈	C₂×D₄	1	4	yes	—
175	C₂×C₃×Q₈	C₂×D₄	1	8	yes	—
176	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
177	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
178	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
179	C₂×C₃×Q₈	C₂⁴	1	24	yes	complemented
180	C₂×C₃×D₄	—	2	12	no	—
181	C₂×C₃×D₄	—	2	24	no	—
182	C₂×C₃×D₄	—	2	4	no	—
183	C₂×C₃×D₄	—	2	12	no	—
184	C₂×C₃×D₄	—	2	48	no	—
185	C₂×C₃×D₄	—	2	192	no	—
186	C₂×C₃×D₄	—	2	96	no	—
187	C₂×C₃×D₄	—	2	6	no	—
188	C₂×C₃×D₄	—	2	3	no	—
189	C₂×C₃×D₄	—	2	12	no	—
190	C₂×C₃×D₄	—	2	24	no	—
191	C₂×C₃×D₄	—	2	24	no	—
192	C₂×C₃×D₄	—	2	6	no	—
193	C₂×C₃×D₄	—	2	48	no	—
194	C₂×C₃×D₄	—	2	12	no	—
195	C₂×C₃×D₄	C₂×D₄	1	12	yes	—
196	C₂×C₃×D₄	C₂×D₄	1	24	yes	—
197	C₂×C₃×D₄	C₂⁴	1	4	yes	—
198	C₂×C₃×D₄	C₂⁴	1	6	yes	—
199	C₂×C₃×D₄	C₂⁴	1	48	yes	—
200	C₂×C₃×D₄	C₂⁴	1	12	yes	—
201	C₂×C₃×D₄	C₂⁴	1	24	yes	complemented
202	C₂²×C₃×C₄	—	2	1	no	—
203	C₂²×C₃×C₄	—	2	6	no	—
204	C₂²×C₃×C₄	—	2	4	no	—
205	C₂²×C₃×C₄	—	2	12	no	—
206	C₂²×C₃×C₄	—	2	24	no	—
207	C₂²×C₃×C₄	—	2	12	no	—
208	C₂²×C₃×C₄	—	2	48	no	—
209	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
210	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
211	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
212	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
213	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
214	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
215	C₃×Q₁₆	C₂⁴	1	32	yes	complemented
216	C₃×SD₁₆	C₂⁴	1	96	yes	complemented
217	C₃×SD₁₆	C₂⁴	1	32	yes	complemented
218	C₃×D₈	C₂⁴	1	96	yes	complemented
219	C₃×M₁₆	C₂⁴	1	16	yes	—
220	C₃×M₁₆	C₂⁴	1	48	yes	complemented
221	C₂×C₃×C₈	C₂⁴	1	8	yes	—
222	C₂×C₃×C₈	C₂⁴	1	48	yes	complemented
223	C₂⁵	—	2	3	no	—
224	2₋¹⁺⁴	—	2	16	no	—
225	2₋¹⁺⁴	—	2	16	no	—
226	2₋¹⁺⁴	C₂³×C₃	1	32	yes	complemented
227	H(5,𝔽₂)	—	2	32	no	—
228	H(5,𝔽₂)	—	2	64	no	—
229	H(5,𝔽₂)	—	2	16	no	—
230	H(5,𝔽₂)	—	2	48	no	—
231	H(5,𝔽₂)	C₂³×C₃	1	64	yes	complemented
232	H(5,𝔽₂)	C₂³×C₃	1	32	yes	complemented
233	C₂×#41	—	2	64	no	—
234	C₂×#41	—	2	8	no	—
235	C₂×#41	—	2	16	no	—
236	C₂×#41	—	2	4	no	—
237	C₂×#41	—	2	12	no	—
238	C₂×#41	—	2	6	no	—
239	C₂×#41	—	2	6	no	—
240	C₂×#41	—	2	12	no	—
241	C₂×#41	—	2	6	no	—
242	C₂×#41	—	2	24	no	—
243	C₂×#41	C₃×D₄	1	4	yes	—
244	C₂×#41	C₃×D₄	1	8	yes	—
245	C₂×#41	C₂³×C₃	1	12	yes	—
246	C₂×#41	C₂³×C₃	1	4	yes	—
247	C₂×#41	C₂³×C₃	1	4	yes	—
248	C₂×#41	C₂³×C₃	1	8	yes	—
249	C₂×#41	C₂³×C₃	1	24	yes	complemented
250	C₂×#41	C₂³×C₃	1	12	yes	complemented
251	C₂×#41	C₂³×C₃	1	48	yes	complemented
252	C₂²×Q₈	—	2	4	no	—
253	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
254	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
255	C₂²×Q₈	C₂³×C₃	1	6	yes	complemented
256	C₂²×D₄	—	2	3	no	—
257	C₂²×D₄	—	2	6	no	—
258	C₂²×D₄	—	2	24	no	—
259	C₂²×D₄	—	2	48	no	—
260	C₂²×D₄	—	2	12	no	—
261	C₂²×D₄	—	2	3	no	—
262	C₂²×D₄	—	2	6	no	—
263	C₂²×D₄	—	2	12	no	—
264	C₂²×D₄	C₃×D₄	1	3	yes	—
265	C₂²×D₄	C₂³×C₃	1	12	yes	—
266	C₂²×D₄	C₂³×C₃	1	3	yes	—
267	C₂²×D₄	C₂³×C₃	1	24	yes	complemented
268	C₂²×D₄	C₂³×C₃	1	6	yes	complemented
269	C₂³×C₄	—	2	3	no	—
270	C₂³×C₄	—	2	6	no	—
271	C₂³×C₄	C₂³×C₃	1	3	yes	—
272	#137	C₂³×C₃	1	32	yes	complemented
273	#137	C₂³×C₃	1	96	yes	complemented
274	AGL(1,ℤ/8)	C₂³×C₃	1	96	yes	complemented
275	AGL(1,ℤ/8)	C₂³×C₃	1	192	yes	complemented
276	AGL(1,ℤ/8)	C₂³×C₃	1	96	yes	complemented
277	#135	C₂³×C₃	1	192	yes	complemented
278	C₂×Q₁₆	C₂³×C₃	1	8	yes	complemented
279	C₂×SD₁₆	C₂³×C₃	1	24	yes	complemented
280	C₂×SD₁₆	C₂³×C₃	1	8	yes	complemented
281	C₂×SD₁₆	C₂³×C₃	1	96	yes	complemented
282	C₂×D₈	C₂³×C₃	1	24	yes	complemented
283	C₂×D₈	C₂³×C₃	1	96	yes	complemented
284	#131	C₂³×C₃	1	64	yes	complemented
285	C₂×M₁₆	C₂³×C₃	1	4	yes	—
286	C₂×M₁₆	C₂³×C₃	1	12	yes	complemented
287	C₂×M₁₆	C₂³×C₃	1	48	yes	complemented
288	C₂²×C₈	C₂³×C₃	1	12	yes	complemented
289	C₂³×C₃	—	2	12	no	—
290	C₂³×C₃	—	2	4	no	—
291	C₂³×C₃	—	2	6	no	—
292	C₂³×C₃	—	2	24	no	—
293	C₂³×C₃	—	2	48	no	—
294	C₂³×C₃	—	4	6	no	—
295	C₂³×C₃	—	4	12	no	—
296	C₂³×C₃	—	8	12	no	—
297	C₂³×C₃	—	8	24	no	—
298	C₂³×C₃	—	8	48	no	—
299	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
300	C₂³×C₃	C₂²×D₄	1	6	yes	—
301	C₂³×C₃	C₂²×D₄	1	12	yes	—
302	C₃×Q₈	—	2	4	no	—
303	C₃×Q₈	—	2	24	no	—
304	C₃×Q₈	—	2	4	no	—
305	C₃×Q₈	—	2	2	no	—
306	C₃×Q₈	C₂²×D₄	1	16	yes	—
307	C₃×Q₈	C₂⁵	1	8	yes	—
308	C₃×Q₈	C₂⁵	1	4	yes	—
309	C₃×D₄	—	2	48	no	—
310	C₃×D₄	—	2	16	no	—
311	C₃×D₄	—	2	48	no	—
312	C₃×D₄	—	2	12	no	—
313	C₃×D₄	—	2	48	no	—
314	C₃×D₄	—	2	6	no	—
315	C₃×D₄	—	2	4	no	—
316	C₃×D₄	—	2	24	no	—
317	C₃×D₄	C₂²×D₄	1	48	yes	—
318	C₃×D₄	C₂⁵	1	8	yes	—
319	C₃×D₄	C₂⁵	1	12	yes	—
320	C₂×C₃×C₄	—	2	4	no	—
321	C₂×C₃×C₄	—	2	24	no	—
322	C₂×C₃×C₄	—	2	2	no	—
323	C₂×C₃×C₄	—	2	4	no	—
324	C₂×C₃×C₄	—	2	16	no	—
325	C₂×C₃×C₄	—	2	48	no	—
326	C₂×C₃×C₄	—	2	48	no	—
327	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
328	C₂×C₃×C₄	C₂²×D₄	1	24	yes	—
329	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
330	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
331	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
332	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
333	C₂×C₃×C₄	C₂⁵	1	12	yes	—
334	C₂×C₃×C₄	C₂⁵	1	2	yes	—
335	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
336	C₃×C₈	C₂⁵	1	16	yes	—
337	C₂⁴	—	2	3	no	—
338	C₂⁴	—	2	6	no	—
339	C₂⁴	—	2	12	no	—
340	C₂⁴	—	2	24	no	—
341	C₂⁴	—	8	12	no	—
342	C₂⁴	C₂×C₃×D₄	1	3	yes	—
343	#41	—	2	64	no	—
344	#41	—	2	32	no	—
345	#41	—	2	96	no	—
346	#41	—	2	8	no	—
347	#41	—	2	48	no	—
348	#41	—	2	24	no	—
349	#41	—	2	48	no	—
350	#41	—	2	48	no	—
351	#41	—	2	24	no	—
352	#41	C₂×C₃×D₄	1	32	yes	—
353	#41	C₂×C₃×D₄	1	16	yes	—
354	#41	C₂⁴×C₃	1	48	yes	—
355	#41	C₂⁴×C₃	1	16	yes	—
356	#41	C₂⁴×C₃	1	16	yes	—
357	#41	C₂⁴×C₃	1	48	yes	complemented
358	C₂×Q₈	—	2	32	no	—
359	C₂×Q₈	—	2	6	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	12	no	—
363	C₂×Q₈	C₂×C₃×D₄	1	4	yes	—
364	C₂×Q₈	C₂×C₃×D₄	1	8	yes	—
365	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
366	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
367	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
368	C₂×Q₈	C₂⁴×C₃	1	24	yes	complemented
369	C₂×D₄	—	2	12	no	—
370	C₂×D₄	—	2	24	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	12	no	—
373	C₂×D₄	—	2	48	no	—
374	C₂×D₄	—	2	192	no	—
375	C₂×D₄	—	2	96	no	—
376	C₂×D₄	—	2	3	no	—
377	C₂×D₄	—	2	12	no	—
378	C₂×D₄	—	2	6	no	—
379	C₂×D₄	—	2	24	no	—
380	C₂×D₄	—	2	24	no	—
381	C₂×D₄	—	2	6	no	—
382	C₂×D₄	—	2	48	no	—
383	C₂×D₄	—	2	12	no	—
384	C₂×D₄	C₂×C₃×D₄	1	12	yes	—
385	C₂×D₄	C₂×C₃×D₄	1	24	yes	—
386	C₂×D₄	C₂⁴×C₃	1	4	yes	—
387	C₂×D₄	C₂⁴×C₃	1	6	yes	—
388	C₂×D₄	C₂⁴×C₃	1	48	yes	—
389	C₂×D₄	C₂⁴×C₃	1	12	yes	—
390	C₂×D₄	C₂⁴×C₃	1	24	yes	complemented
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	6	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	12	no	—
395	C₂²×C₄	—	2	24	no	—
396	C₂²×C₄	—	2	12	no	—
397	C₂²×C₄	—	2	48	no	—
398	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
399	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
400	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
401	C₂²×C₄	C₂⁴×C₃	1	6	yes	—
402	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
404	Q₁₆	C₂⁴×C₃	1	32	yes	complemented
405	SD₁₆	C₂⁴×C₃	1	96	yes	complemented
406	SD₁₆	C₂⁴×C₃	1	32	yes	complemented
407	D₈	C₂⁴×C₃	1	96	yes	complemented
408	M₁₆	C₂⁴×C₃	1	16	yes	—
409	M₁₆	C₂⁴×C₃	1	48	yes	complemented
410	C₂×C₈	C₂⁴×C₃	1	8	yes	—
411	C₂×C₈	C₂⁴×C₃	1	48	yes	complemented
412	C₂²×C₃	—	2	2	no	—
413	C₂²×C₃	—	2	12	no	—
414	C₂²×C₃	—	2	8	no	—
415	C₂²×C₃	—	2	12	no	—
416	C₂²×C₃	—	4	12	no	—
417	C₂²×C₃	—	4	24	no	—
418	C₂²×C₃	—	4	12	no	—
419	C₂²×C₃	—	4	8	no	—
420	C₂²×C₃	—	8	24	no	—
421	C₂²×C₃	—	8	48	no	—
422	C₂²×C₃	C₂³×D₄	1	2	yes	—
423	C₂²×C₃	C₂³×D₄	1	12	yes	—
424	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
425	C₃×C₄	—	2	4	no	—
426	C₃×C₄	—	2	8	no	—
427	C₃×C₄	C₂³×D₄	1	2	yes	—
428	C₃×C₄	C₂³×D₄	1	12	yes	—
429	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
430	C₃×C₄	C₂⁶	1	1	yes	characteristic
431	C₂³	—	2	4	no	—
432	C₂³	—	2	12	no	—
433	C₂³	—	2	6	no	—
434	C₂³	—	2	24	no	—
435	C₂³	—	2	48	no	—
436	C₂³	—	4	6	no	—
437	C₂³	—	4	12	no	—
438	C₂³	—	8	24	no	—
439	C₂³	—	8	12	no	—
440	C₂³	—	8	48	no	—
441	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
442	C₂³	C₂²×C₃×D₄	1	6	yes	—
443	C₂³	C₂²×C₃×D₄	1	12	yes	—
444	Q₈	—	2	4	no	—
445	Q₈	—	2	24	no	—
446	Q₈	—	2	4	no	—
447	Q₈	—	2	2	no	—
448	Q₈	C₂²×C₃×D₄	1	16	yes	—
449	Q₈	C₂⁵×C₃	1	8	yes	—
450	Q₈	C₂⁵×C₃	1	4	yes	—
451	D₄	—	2	48	no	—
452	D₄	—	2	16	no	—
453	D₄	—	2	48	no	—
454	D₄	—	2	12	no	—
455	D₄	—	2	48	no	—
456	D₄	—	2	6	no	—
457	D₄	—	2	4	no	—
458	D₄	—	2	24	no	—
459	D₄	C₂²×C₃×D₄	1	48	yes	—
460	D₄	C₂⁵×C₃	1	8	yes	—
461	D₄	C₂⁵×C₃	1	12	yes	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	24	no	—
466	C₂×C₄	—	2	16	no	—
467	C₂×C₄	—	2	48	no	—
468	C₂×C₄	—	2	48	no	—
469	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
470	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
471	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
472	C₂×C₄	C₂²×C₃×D₄	1	24	yes	—
473	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
474	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
475	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
476	C₂×C₄	C₂⁵×C₃	1	12	yes	—
477	C₂×C₄	C₂⁵×C₃	1	2	yes	—
478	C₈	C₂⁵×C₃	1	16	yes	—
479	C₂×C₃	—	2	2	no	—
480	C₂×C₃	—	2	12	no	—
481	C₂×C₃	—	4	12	no	—
482	C₂×C₃	—	4	8	no	—
483	C₂×C₃	#3585	1	2	yes	direct product factor
484	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
485	C₂²	—	2	2	no	—
486	C₂²	—	2	12	no	—
487	C₂²	—	2	8	no	—
488	C₂²	—	2	12	no	—
489	C₂²	—	4	12	no	—
490	C₂²	—	4	24	no	—
491	C₂²	—	4	12	no	—
492	C₂²	—	4	8	no	—
493	C₂²	—	8	48	no	—
494	C₂²	—	8	24	no	—
495	C₂²	C₂³×C₃×D₄	1	2	yes	—
496	C₂²	C₂³×C₃×D₄	1	12	yes	—
497	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	8	no	—
500	C₄	C₂³×C₃×D₄	1	2	yes	—
501	C₄	C₂³×C₃×D₄	1	12	yes	—
502	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
503	C₄	C₂⁶×C₃	1	1	yes	characteristic
504	C₃	C₂×#3585	1	1	yes	characteristic, direct product factor, Sylow
505	C₂	—	2	2	no	—
506	C₂	—	2	12	no	—
507	C₂	—	4	12	no	—
508	C₂	—	4	8	no	—
509	C₂	C₃×#3585	1	2	yes	direct product factor
510	C₂	C₂⁴×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#585	C₂	semidirect
2. C₂²×C₃×H(5,𝔽₂)	C₂	semidirect
3. C₃×#3585	C₂	direct (N × H)
4. C₂×C₃×#577	C₂	semidirect
5. C₂×C₃×#576	C₂	semidirect
6. C₂×C₃×#575	C₂	semidirect
7. C₂²×C₃×AGL(1,ℤ/8)	C₂	semidirect
8. C₂×C₃×#568	C₂	semidirect
9. C₂×#3585	C₃	direct (N × H)
11. C₃×#585	C₂²	semidirect
13. C₂×C₃×2₋¹⁺⁴	C₂²	semidirect
17. C₂×C₃×H(5,𝔽₂)	C₂²	semidirect
18. C₂×C₃×H(5,𝔽₂)	C₂²	semidirect
19. C₂×C₃×H(5,𝔽₂)	C₂²	semidirect
23. C₂²×C₃×#41	C₂²	semidirect
24. C₂²×C₃×#41	C₂²	semidirect
26. C₂³×C₃×D₄	C₂²	semidirect
27. C₃×#577	C₂²	semidirect
28. C₃×#576	C₂²	semidirect
29. C₃×#575	C₂²	semidirect
30. C₂×C₃×#137	C₂²	semidirect
31. C₂×C₃×#137	C₂²	semidirect
32. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
33. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
34. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
35. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
36. C₂×C₃×#135	C₂²	semidirect
37. C₂²×C₃×SD₁₆	C₂²	semidirect
38. C₂²×C₃×D₈	C₂²	semidirect
39. C₃×#568	C₂²	semidirect
40. C₂×C₃×#131	C₂²	semidirect
41. C₂²×C₃×M₁₆	C₂²	semidirect
42. C₂×#585	C₂×C₃	semidirect
43. C₂²×H(5,𝔽₂)	C₂×C₃	semidirect
44. #3585	C₂×C₃	direct (N × H)
45. C₂×#577	C₂×C₃	semidirect
46. C₂×#576	C₂×C₃	semidirect
47. C₂×#575	C₂×C₃	semidirect
48. C₂²×AGL(1,ℤ/8)	C₂×C₃	semidirect
49. C₂×#568	C₂×C₃	semidirect
53. C₃×2₋¹⁺⁴	C₂³	semidirect
58. C₃×H(5,𝔽₂)	C₂³	semidirect
59. C₃×H(5,𝔽₂)	C₂³	semidirect
76. C₂×C₃×#41	C₂³	semidirect
77. C₂×C₃×#41	C₂³	semidirect
78. C₂×C₃×#41	C₂³	semidirect
82. C₂²×C₃×Q₈	C₂³	semidirect
94. C₂²×C₃×D₄	C₂³	semidirect
95. C₂²×C₃×D₄	C₂³	semidirect
99. C₃×#137	C₂³	semidirect
100. C₃×#137	C₂³	semidirect
101. C₃×AGL(1,ℤ/8)	C₂³	semidirect
102. C₃×AGL(1,ℤ/8)	C₂³	semidirect
103. C₃×AGL(1,ℤ/8)	C₂³	semidirect
104. C₃×#135	C₂³	semidirect
105. C₂×C₃×Q₁₆	C₂³	semidirect
106. C₂×C₃×SD₁₆	C₂³	semidirect
107. C₂×C₃×SD₁₆	C₂³	semidirect
108. C₂×C₃×SD₁₆	C₂³	semidirect
109. C₂×C₃×D₈	C₂³	semidirect
110. C₂×C₃×D₈	C₂³	semidirect
111. C₃×#131	C₂³	semidirect
113. C₂×C₃×M₁₆	C₂³	semidirect
114. C₂×C₃×M₁₆	C₂³	semidirect
115. C₂²×C₃×C₈	C₂³	semidirect
117. #585	C₂²×C₃	semidirect
119. C₂×2₋¹⁺⁴	C₂²×C₃	semidirect
123. C₂×H(5,𝔽₂)	C₂²×C₃	semidirect
124. C₂×H(5,𝔽₂)	C₂²×C₃	semidirect
125. C₂×H(5,𝔽₂)	C₂²×C₃	semidirect
129. C₂²×#41	C₂²×C₃	semidirect
130. C₂²×#41	C₂²×C₃	semidirect
132. C₂³×D₄	C₂²×C₃	semidirect
133. #577	C₂²×C₃	semidirect
134. #576	C₂²×C₃	semidirect
135. #575	C₂²×C₃	semidirect
136. C₂×#137	C₂²×C₃	semidirect
137. C₂×#137	C₂²×C₃	semidirect
138. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
139. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
140. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
141. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
142. C₂×#135	C₂²×C₃	semidirect
143. C₂²×SD₁₆	C₂²×C₃	semidirect
144. C₂²×D₈	C₂²×C₃	semidirect
145. #568	C₂²×C₃	semidirect
146. C₂×#131	C₂²×C₃	semidirect
147. C₂²×M₁₆	C₂²×C₃	semidirect
168. C₃×#41	C₂⁴	semidirect
179. C₂×C₃×Q₈	C₂⁴	semidirect
201. C₂×C₃×D₄	C₂⁴	semidirect
215. C₃×Q₁₆	C₂⁴	semidirect
216. C₃×SD₁₆	C₂⁴	semidirect
217. C₃×SD₁₆	C₂⁴	semidirect
218. C₃×D₈	C₂⁴	semidirect
220. C₃×M₁₆	C₂⁴	semidirect
222. C₂×C₃×C₈	C₂⁴	semidirect
226. 2₋¹⁺⁴	C₂³×C₃	semidirect
231. H(5,𝔽₂)	C₂³×C₃	semidirect
232. H(5,𝔽₂)	C₂³×C₃	semidirect
249. C₂×#41	C₂³×C₃	semidirect
250. C₂×#41	C₂³×C₃	semidirect
251. C₂×#41	C₂³×C₃	semidirect
255. C₂²×Q₈	C₂³×C₃	semidirect
267. C₂²×D₄	C₂³×C₃	semidirect
268. C₂²×D₄	C₂³×C₃	semidirect
272. #137	C₂³×C₃	semidirect
273. #137	C₂³×C₃	semidirect
274. AGL(1,ℤ/8)	C₂³×C₃	semidirect
275. AGL(1,ℤ/8)	C₂³×C₃	semidirect
276. AGL(1,ℤ/8)	C₂³×C₃	semidirect
277. #135	C₂³×C₃	semidirect
278. C₂×Q₁₆	C₂³×C₃	semidirect
279. C₂×SD₁₆	C₂³×C₃	semidirect
280. C₂×SD₁₆	C₂³×C₃	semidirect
281. C₂×SD₁₆	C₂³×C₃	semidirect
282. C₂×D₈	C₂³×C₃	semidirect
283. C₂×D₈	C₂³×C₃	semidirect
284. #131	C₂³×C₃	semidirect
286. C₂×M₁₆	C₂³×C₃	semidirect
287. C₂×M₁₆	C₂³×C₃	semidirect
288. C₂²×C₈	C₂³×C₃	semidirect
357. #41	C₂⁴×C₃	semidirect
368. C₂×Q₈	C₂⁴×C₃	semidirect
390. C₂×D₄	C₂⁴×C₃	semidirect
404. Q₁₆	C₂⁴×C₃	semidirect
405. SD₁₆	C₂⁴×C₃	semidirect
406. SD₁₆	C₂⁴×C₃	semidirect
407. D₈	C₂⁴×C₃	semidirect
409. M₁₆	C₂⁴×C₃	semidirect
411. C₂×C₈	C₂⁴×C₃	semidirect
483. C₂×C₃	#3585	direct (N × H)
504. C₃	C₂×#3585	direct (N × H)
509. C₂	C₃×#3585	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×#585	2
C₂²×C₃×H(5,𝔽₂)	2
C₃×#3585	2
C₂×C₃×#577	2
C₂×C₃×#576	2
C₂×C₃×#575	2
C₂²×C₃×AGL(1,ℤ/8)	2
C₂×C₃×#568	2
C₂×#3585	3

Subgroups — C2×C3×#3585 #10681695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3585 #10681695