Subgroups — C₃×#34255 #10652862

Order 768.

2682 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	C₂×C₃×#518	C₂	1	1	yes	maximal, characteristic
2	C₃×#2357	C₂	1	4	yes	maximal, complemented
3	C₃×#2354	C₂	1	4	yes	maximal, complemented
4	C₃×#2342	C₂	1	8	yes	maximal, complemented
5	C₃×#2319	C₂	1	4	yes	maximal, complemented
6	C₃×#2317	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2309	C₂	1	1	yes	maximal, characteristic
8	C₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2293	C₂	1	4	yes	maximal, complemented
10	C₂×C₃×#390	C₂	1	1	yes	maximal, characteristic
11	C₃×#2274	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	#34255	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₃×#518	C₄	1	16	yes	complemented
16	C₂×C₃×C₄×D₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄×D₄	C₂²	1	4	yes	—
18	C₂×C₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃×#117	C₂²	1	1	yes	characteristic
20	C₂×C₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₃×#117	C₂²	1	8	yes	complemented
23	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₃×#390	C₂²	1	2	yes	—
28	C₃×#390	C₂²	1	2	yes	—
29	C₃×#390	C₂²	1	4	yes	—
30	C₃×#390	C₂²	1	4	yes	—
31	C₃×#390	C₂²	1	2	yes	—
32	C₃×#390	C₂²	1	2	yes	—
33	C₃×#389	C₂²	1	2	yes	complemented
34	C₃×#389	C₂²	1	4	yes	complemented
35	C₃×#389	C₂²	1	2	yes	complemented
36	C₃×#388	C₂²	1	8	yes	complemented
37	C₃×#388	C₂²	1	8	yes	complemented
38	C₃×#388	C₂²	1	8	yes	complemented
39	C₃×#388	C₂²	1	8	yes	complemented
40	C₃×#385	C₂²	1	4	yes	—
41	C₃×#385	C₂²	1	4	yes	—
42	C₃×#384	C₂²	1	2	yes	complemented
43	C₃×#382	C₂²	1	2	yes	complemented
44	C₃×#381	C₂²	1	8	yes	complemented
45	C₃×#380	C₂²	1	8	yes	complemented
46	C₃×#379	C₂²	1	4	yes	—
47	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	complemented
48	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	complemented
49	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	complemented
50	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	complemented
51	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	complemented
52	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	—
53	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	—
54	C₃×C₄×C₂²⋊C₄	C₂²	1	8	yes	complemented
55	C₃×#376	C₂²	1	2	yes	complemented
56	C₃×C₄³	C₂²	1	2	yes	complemented
57	C₂×#518	C₂×C₃	1	1	yes	characteristic
58	#2357	C₂×C₃	1	4	yes	complemented
59	#2354	C₂×C₃	1	4	yes	complemented
60	#2342	C₂×C₃	1	8	yes	complemented
61	#2319	C₂×C₃	1	4	yes	complemented
62	#2317	C₂×C₃	1	1	yes	characteristic, complemented
63	#2309	C₂×C₃	1	1	yes	characteristic
64	#2296	C₂×C₃	1	1	yes	characteristic, complemented
65	#2293	C₂×C₃	1	4	yes	complemented
66	C₂×#390	C₂×C₃	1	1	yes	characteristic
67	#2274	C₂×C₃	1	1	yes	characteristic, complemented
68	C₄×#117	C₂×C₃	1	1	yes	characteristic, complemented
69	C₂⁵×C₃	C₂³	1	2	yes	—
70	C₂²×C₃×D₄	—	2	4	no	—
71	C₂²×C₃×D₄	—	2	4	no	—
72	C₂²×C₃×D₄	C₂×C₄	1	8	yes	—
73	C₂²×C₃×D₄	C₂³	1	2	yes	—
74	C₂²×C₃×D₄	C₂³	1	2	yes	—
75	C₂³×C₃×C₄	C₂³	1	2	yes	—
76	C₂³×C₃×C₄	C₂³	1	2	yes	—
77	C₂³×C₃×C₄	C₂³	1	8	yes	—
78	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄×D₄	C₂×C₄	1	32	yes	complemented
80	C₃×C₄×D₄	C₂×C₄	1	32	yes	complemented
81	C₃×#117	—	2	4	no	—
82	C₃×#117	—	2	4	no	—
83	C₃×#117	—	4	16	no	—
84	C₃×#117	C₂×C₄	1	8	yes	—
85	C₃×#117	C₂×C₄	1	8	yes	—
86	C₂×C₃×C₄⋊C₄	—	2	4	no	—
87	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
89	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
91	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
92	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
93	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
94	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
95	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
96	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
97	C₂×C₃×C₂²⋊C₄	—	4	8	no	—
98	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	32	yes	—
99	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
100	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
101	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
102	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
103	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
104	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
105	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
106	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
107	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
108	C₂×C₃×C₄²	—	2	4	no	—
109	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
110	C₂×C₃×C₄²	C₂³	1	2	yes	—
111	C₂×C₃×C₄²	C₂³	1	4	yes	—
112	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
113	C₂×C₃×C₄²	C₂³	1	2	yes	—
114	C₂×C₃×C₄²	C₂³	1	2	yes	—
115	C₂×C₃×C₄²	C₂³	1	4	yes	—
116	C₂×C₃×C₄²	C₂³	1	8	yes	—
117	C₂×C₃×C₄²	C₂³	1	2	yes	—
118	C₂×C₃×C₄²	C₂³	1	2	yes	—
119	C₃×#95	C₂³	1	4	yes	—
120	C₃×#95	C₂³	1	4	yes	—
121	C₃×#95	C₂³	1	8	yes	—
122	C₃×#95	C₂³	1	4	yes	—
123	C₃×#95	C₂³	1	8	yes	—
124	C₃×#95	C₂³	1	4	yes	—
125	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
126	#518	C₃×C₄	1	16	yes	complemented
127	C₂×C₄×D₄	C₂²×C₃	1	4	yes	—
128	C₂×C₄×D₄	C₂²×C₃	1	4	yes	—
129	C₂×#117	C₂²×C₃	1	1	yes	characteristic
130	C₂×#117	C₂²×C₃	1	1	yes	characteristic
131	C₂×#117	C₂²×C₃	1	1	yes	characteristic
132	C₂×#117	C₂²×C₃	1	1	yes	characteristic
133	C₂×#117	C₂²×C₃	1	8	yes	complemented
134	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
135	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
136	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
137	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
138	#390	C₂²×C₃	1	4	yes	—
139	#390	C₂²×C₃	1	2	yes	—
140	#390	C₂²×C₃	1	2	yes	—
141	#390	C₂²×C₃	1	2	yes	—
142	#390	C₂²×C₃	1	4	yes	—
143	#390	C₂²×C₃	1	2	yes	—
144	#389	C₂²×C₃	1	4	yes	complemented
145	#389	C₂²×C₃	1	2	yes	complemented
146	#389	C₂²×C₃	1	2	yes	complemented
147	#388	C₂²×C₃	1	8	yes	complemented
148	#388	C₂²×C₃	1	8	yes	complemented
149	#388	C₂²×C₃	1	8	yes	complemented
150	#388	C₂²×C₃	1	8	yes	complemented
151	#385	C₂²×C₃	1	4	yes	—
152	#385	C₂²×C₃	1	4	yes	—
153	#384	C₂²×C₃	1	2	yes	complemented
154	#382	C₂²×C₃	1	2	yes	complemented
155	#381	C₂²×C₃	1	8	yes	complemented
156	#380	C₂²×C₃	1	8	yes	complemented
157	#379	C₂²×C₃	1	4	yes	—
158	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
159	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
160	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
161	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
162	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
163	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
164	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
165	C₄×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
166	#376	C₂²×C₃	1	2	yes	complemented
167	C₄³	C₂²×C₃	1	2	yes	complemented
168	C₂⁴×C₃	—	2	4	no	—
169	C₂⁴×C₃	—	2	4	no	—
170	C₂⁴×C₃	—	4	4	no	—
171	C₂⁴×C₃	—	4	4	no	—
172	C₂⁴×C₃	C₂²×C₄	1	8	yes	—
173	C₂⁴×C₃	C₂⁴	1	4	yes	—
174	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
175	C₂×C₃×D₄	—	2	32	no	—
176	C₂×C₃×D₄	—	2	4	no	—
177	C₂×C₃×D₄	—	2	4	no	—
178	C₂×C₃×D₄	—	2	4	no	—
179	C₂×C₃×D₄	—	2	4	no	—
180	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
181	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
182	C₂²×C₃×C₄	—	2	4	no	—
183	C₂²×C₃×C₄	—	2	4	no	—
184	C₂²×C₃×C₄	—	2	4	no	—
185	C₂²×C₃×C₄	—	4	8	no	—
186	C₂²×C₃×C₄	—	4	2	no	—
187	C₂²×C₃×C₄	—	4	8	no	—
188	C₂²×C₃×C₄	—	4	4	no	—
189	C₂²×C₃×C₄	C₂²×C₄	1	32	yes	—
190	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
191	C₂²×C₃×C₄	#41	1	4	yes	—
192	C₂²×C₃×C₄	#41	1	4	yes	—
193	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
194	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
195	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
196	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
197	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
198	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
199	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
200	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
201	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
202	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
203	C₃×C₄⋊C₄	—	2	4	no	—
204	C₃×C₄⋊C₄	—	2	4	no	—
205	C₃×C₄⋊C₄	—	2	4	no	—
206	C₃×C₄⋊C₄	—	2	2	no	—
207	C₃×C₄⋊C₄	—	2	2	no	—
208	C₃×C₄⋊C₄	—	2	8	no	—
209	C₃×C₄⋊C₄	—	4	4	no	—
210	C₃×C₄⋊C₄	—	4	8	no	—
211	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
212	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
213	C₃×C₂²⋊C₄	—	2	4	no	—
214	C₃×C₂²⋊C₄	—	2	4	no	—
215	C₃×C₂²⋊C₄	—	2	16	no	—
216	C₃×C₂²⋊C₄	—	4	8	no	—
217	C₃×C₂²⋊C₄	—	4	8	no	—
218	C₃×C₂²⋊C₄	—	4	8	no	—
219	C₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
220	C₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
221	C₃×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
222	C₃×C₄²	—	2	2	no	—
223	C₃×C₄²	—	2	4	no	—
224	C₃×C₄²	—	2	2	no	—
225	C₃×C₄²	—	2	4	no	—
226	C₃×C₄²	—	2	4	no	—
227	C₃×C₄²	—	2	8	no	—
228	C₃×C₄²	—	4	4	no	—
229	C₃×C₄²	—	4	8	no	—
230	C₃×C₄²	C₂²×C₄	1	8	yes	—
231	C₃×C₄²	C₂²×C₄	1	8	yes	—
232	C₂⁵	C₂³×C₃	1	2	yes	—
233	C₂²×D₄	—	2	4	no	—
234	C₂²×D₄	—	2	4	no	—
235	C₂²×D₄	C₂×C₃×C₄	1	8	yes	—
236	C₂²×D₄	C₂³×C₃	1	2	yes	—
237	C₂²×D₄	C₂³×C₃	1	2	yes	—
238	C₂³×C₄	C₂³×C₃	1	2	yes	—
239	C₂³×C₄	C₂³×C₃	1	2	yes	—
240	C₂³×C₄	C₂³×C₃	1	8	yes	—
241	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
242	C₄×D₄	C₂×C₃×C₄	1	32	yes	complemented
243	C₄×D₄	C₂×C₃×C₄	1	32	yes	complemented
244	#117	—	2	4	no	—
245	#117	—	2	4	no	—
246	#117	—	4	16	no	—
247	#117	C₂×C₃×C₄	1	8	yes	—
248	#117	C₂×C₃×C₄	1	8	yes	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
251	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
252	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
253	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
256	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
257	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
258	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
259	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
260	C₂×C₂²⋊C₄	—	4	8	no	—
261	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	32	yes	—
262	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
263	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
264	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
265	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
266	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
267	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
268	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
270	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
271	C₂×C₄²	—	2	4	no	—
272	C₂×C₄²	C₂³×C₃	1	2	yes	—
273	C₂×C₄²	C₂³×C₃	1	2	yes	—
274	C₂×C₄²	C₂³×C₃	1	4	yes	—
275	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₃	1	2	yes	—
278	C₂×C₄²	C₂³×C₃	1	8	yes	—
279	C₂×C₄²	C₂³×C₃	1	4	yes	—
280	C₂×C₄²	C₂³×C₃	1	2	yes	—
281	C₂×C₄²	C₂³×C₃	1	2	yes	—
282	#95	C₂³×C₃	1	4	yes	—
283	#95	C₂³×C₃	1	4	yes	—
284	#95	C₂³×C₃	1	8	yes	—
285	#95	C₂³×C₃	1	4	yes	—
286	#95	C₂³×C₃	1	8	yes	—
287	#95	C₂³×C₃	1	4	yes	—
288	C₂³×C₃	—	2	2	no	—
289	C₂³×C₃	—	2	2	no	—
290	C₂³×C₃	—	2	4	no	—
291	C₂³×C₃	—	2	4	no	—
292	C₂³×C₃	—	2	8	no	—
293	C₂³×C₃	—	2	8	no	—
294	C₂³×C₃	—	2	16	no	—
295	C₂³×C₃	—	4	16	no	—
296	C₂³×C₃	—	8	8	no	—
297	C₂³×C₃	—	8	4	no	—
298	C₂³×C₃	—	8	4	no	—
299	C₂³×C₃	C₂³×C₄	1	8	yes	—
300	C₂³×C₃	C₂³×C₄	1	2	yes	—
301	C₂³×C₃	C₂×#41	1	2	yes	—
302	C₂³×C₃	C₂×#41	1	2	yes	—
303	C₂³×C₃	C₂⁵	1	1	yes	characteristic
304	C₃×D₄	—	2	16	no	—
305	C₃×D₄	—	2	16	no	—
306	C₂×C₃×C₄	—	2	2	no	—
307	C₂×C₃×C₄	—	2	4	no	—
308	C₂×C₃×C₄	—	2	4	no	—
309	C₂×C₃×C₄	—	2	2	no	—
310	C₂×C₃×C₄	—	2	4	no	—
311	C₂×C₃×C₄	—	2	4	no	—
312	C₂×C₃×C₄	—	2	2	no	—
313	C₂×C₃×C₄	—	2	4	no	—
314	C₂×C₃×C₄	—	2	2	no	—
315	C₂×C₃×C₄	—	2	2	no	—
316	C₂×C₃×C₄	—	2	4	no	—
317	C₂×C₃×C₄	—	2	4	no	—
318	C₂×C₃×C₄	—	2	4	no	—
319	C₂×C₃×C₄	—	2	8	no	—
320	C₂×C₃×C₄	—	2	2	no	—
321	C₂×C₃×C₄	—	2	4	no	—
322	C₂×C₃×C₄	—	2	4	no	—
323	C₂×C₃×C₄	—	2	2	no	—
324	C₂×C₃×C₄	—	4	32	no	—
325	C₂×C₃×C₄	—	4	8	no	—
326	C₂×C₃×C₄	—	8	4	no	—
327	C₂×C₃×C₄	#117	1	16	yes	complemented
328	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
329	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
330	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
331	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
332	C₂×C₃×C₄	C₂×#41	1	2	yes	—
333	C₂×C₃×C₄	C₂×#41	1	2	yes	—
334	C₂×C₃×C₄	C₂×#41	1	2	yes	—
335	C₂×C₃×C₄	C₂×#41	1	2	yes	—
336	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
337	C₂⁴	—	2	4	no	—
338	C₂⁴	—	2	4	no	—
339	C₂⁴	—	4	4	no	—
340	C₂⁴	—	4	4	no	—
341	C₂⁴	C₂²×C₃×C₄	1	8	yes	—
342	C₂⁴	C₂⁴×C₃	1	4	yes	—
343	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
344	C₂×D₄	—	2	32	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
350	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	4	8	no	—
355	C₂²×C₄	—	4	2	no	—
356	C₂²×C₄	—	4	8	no	—
357	C₂²×C₄	—	4	4	no	—
358	C₂²×C₄	C₂²×C₃×C₄	1	32	yes	—
359	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
360	C₂²×C₄	C₃×#41	1	4	yes	—
361	C₂²×C₄	C₃×#41	1	4	yes	—
362	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
367	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
368	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
369	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
371	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	4	8	no	—
379	C₄⋊C₄	—	4	4	no	—
380	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
381	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
382	C₂²⋊C₄	—	2	4	no	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	2	16	no	—
385	C₂²⋊C₄	—	4	8	no	—
386	C₂²⋊C₄	—	4	8	no	—
387	C₂²⋊C₄	—	4	8	no	—
388	C₂²⋊C₄	C₂²×C₃×C₄	1	8	yes	—
389	C₂²⋊C₄	C₂²×C₃×C₄	1	8	yes	—
390	C₂²⋊C₄	C₂²×C₃×C₄	1	32	yes	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	4	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	4	no	—
396	C₄²	—	2	8	no	—
397	C₄²	—	4	8	no	—
398	C₄²	—	4	4	no	—
399	C₄²	C₂²×C₃×C₄	1	8	yes	—
400	C₄²	C₂²×C₃×C₄	1	8	yes	—
401	C₂²×C₃	—	2	4	no	—
402	C₂²×C₃	—	2	4	no	—
403	C₂²×C₃	—	2	2	no	—
404	C₂²×C₃	—	2	2	no	—
405	C₂²×C₃	—	2	8	no	—
406	C₂²×C₃	—	2	16	no	—
407	C₂²×C₃	—	4	8	no	—
408	C₂²×C₃	—	4	4	no	—
409	C₂²×C₃	—	4	4	no	—
410	C₂²×C₃	—	8	16	no	—
411	C₂²×C₃	C₂×#117	1	4	yes	—
412	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
413	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
414	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
415	C₂²×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
416	C₂²×C₃	#585	1	2	yes	—
417	C₃×C₄	—	2	8	no	—
418	C₃×C₄	—	2	8	no	—
419	C₃×C₄	—	2	4	no	—
420	C₃×C₄	—	4	4	no	—
421	C₃×C₄	—	4	2	no	—
422	C₃×C₄	—	4	2	no	—
423	C₃×C₄	—	4	4	no	—
424	C₃×C₄	—	4	2	no	—
425	C₃×C₄	C₂×#117	1	4	yes	—
426	C₃×C₄	C₂×#117	1	4	yes	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	8	no	—
432	C₂³	—	2	8	no	—
433	C₂³	—	2	16	no	—
434	C₂³	—	4	16	no	—
435	C₂³	—	8	8	no	—
436	C₂³	—	8	4	no	—
437	C₂³	—	8	4	no	—
438	C₂³	C₂³×C₃×C₄	1	8	yes	—
439	C₂³	C₂³×C₃×C₄	1	2	yes	—
440	C₂³	C₂×C₃×#41	1	2	yes	—
441	C₂³	C₂×C₃×#41	1	2	yes	—
442	C₂³	C₂⁵×C₃	1	1	yes	characteristic
443	D₄	—	2	16	no	—
444	D₄	—	2	16	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	4	32	no	—
464	C₂×C₄	—	4	8	no	—
465	C₂×C₄	—	8	4	no	—
466	C₂×C₄	C₃×#117	1	16	yes	complemented
467	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
468	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
469	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
470	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
471	C₂×C₄	C₂×C₃×#41	1	2	yes	—
472	C₂×C₄	C₂×C₃×#41	1	2	yes	—
473	C₂×C₄	C₂×C₃×#41	1	2	yes	—
474	C₂×C₄	C₂×C₃×#41	1	2	yes	—
475	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
476	C₂×C₃	—	2	4	no	—
477	C₂×C₃	—	4	8	no	—
478	C₂×C₃	C₂²×#117	1	1	yes	characteristic
479	C₂×C₃	C₄×H(5,𝔽₂)	1	2	yes	—
480	C₂×C₃	#3494	1	2	yes	—
481	C₂×C₃	#3508	1	1	yes	characteristic
482	C₂×C₃	#3510	1	1	yes	characteristic
483	C₂²	—	2	4	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	8	no	—
488	C₂²	—	2	16	no	—
489	C₂²	—	4	8	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	8	16	no	—
493	C₂²	C₂×C₃×#117	1	4	yes	—
494	C₂²	C₂⁴×C₃×C₄	1	1	yes	characteristic
495	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
496	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
497	C₂²	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
498	C₂²	C₃×#585	1	2	yes	—
499	C₄	—	2	8	no	—
500	C₄	—	2	8	no	—
501	C₄	—	2	4	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	2	no	—
507	C₄	C₂×C₃×#117	1	4	yes	—
508	C₄	C₂×C₃×#117	1	4	yes	—
509	C₃	#34255	1	1	yes	characteristic, direct product factor, Sylow
510	C₂	—	2	4	no	—
511	C₂	—	4	8	no	—
512	C₂	C₂²×C₃×#117	1	1	yes	characteristic
513	C₂	C₃×C₄×H(5,𝔽₂)	1	2	yes	—
514	C₂	C₃×#3494	1	2	yes	—
515	C₂	C₃×#3508	1	1	yes	characteristic
516	C₂	C₃×#3510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁵
commutator	C₂²	C₂⁴×C₃×C₄
Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2357	C₂	semidirect
3. C₃×#2354	C₂	semidirect
4. C₃×#2342	C₂	semidirect
5. C₃×#2319	C₂	semidirect
6. C₃×#2317	C₂	semidirect
8. C₃×#2296	C₂	semidirect
9. C₃×#2293	C₂	semidirect
11. C₃×#2274	C₂	semidirect
12. C₃×C₄×#117	C₂	semidirect
13. #34255	C₃	direct (N × H)
15. C₃×#518	C₄	semidirect
22. C₂×C₃×#117	C₂²	semidirect
33. C₃×#389	C₂²	semidirect
34. C₃×#389	C₂²	semidirect
35. C₃×#389	C₂²	semidirect
36. C₃×#388	C₂²	semidirect
37. C₃×#388	C₂²	semidirect
38. C₃×#388	C₂²	semidirect
39. C₃×#388	C₂²	semidirect
42. C₃×#384	C₂²	semidirect
43. C₃×#382	C₂²	semidirect
44. C₃×#381	C₂²	semidirect
45. C₃×#380	C₂²	semidirect
47. C₃×C₄×C₄⋊C₄	C₂²	semidirect
48. C₃×C₄×C₄⋊C₄	C₂²	semidirect
49. C₃×C₄×C₄⋊C₄	C₂²	semidirect
50. C₃×C₄×C₄⋊C₄	C₂²	semidirect
51. C₃×C₄×C₄⋊C₄	C₂²	semidirect
54. C₃×C₄×C₂²⋊C₄	C₂²	semidirect
55. C₃×#376	C₂²	semidirect
56. C₃×C₄³	C₂²	semidirect
58. #2357	C₂×C₃	semidirect
59. #2354	C₂×C₃	semidirect
60. #2342	C₂×C₃	semidirect
61. #2319	C₂×C₃	semidirect
62. #2317	C₂×C₃	semidirect
64. #2296	C₂×C₃	semidirect
65. #2293	C₂×C₃	semidirect
67. #2274	C₂×C₃	semidirect
68. C₄×#117	C₂×C₃	semidirect
79. C₃×C₄×D₄	C₂×C₄	semidirect
80. C₃×C₄×D₄	C₂×C₄	semidirect
126. #518	C₃×C₄	semidirect
133. C₂×#117	C₂²×C₃	semidirect
144. #389	C₂²×C₃	semidirect
145. #389	C₂²×C₃	semidirect
146. #389	C₂²×C₃	semidirect
147. #388	C₂²×C₃	semidirect
148. #388	C₂²×C₃	semidirect
149. #388	C₂²×C₃	semidirect
150. #388	C₂²×C₃	semidirect
153. #384	C₂²×C₃	semidirect
154. #382	C₂²×C₃	semidirect
155. #381	C₂²×C₃	semidirect
156. #380	C₂²×C₃	semidirect
158. C₄×C₄⋊C₄	C₂²×C₃	semidirect
159. C₄×C₄⋊C₄	C₂²×C₃	semidirect
160. C₄×C₄⋊C₄	C₂²×C₃	semidirect
161. C₄×C₄⋊C₄	C₂²×C₃	semidirect
162. C₄×C₄⋊C₄	C₂²×C₃	semidirect
165. C₄×C₂²⋊C₄	C₂²×C₃	semidirect
166. #376	C₂²×C₃	semidirect
167. C₄³	C₂²×C₃	semidirect
242. C₄×D₄	C₂×C₃×C₄	semidirect
243. C₄×D₄	C₂×C₃×C₄	semidirect
327. C₂×C₃×C₄	#117	semidirect
466. C₂×C₄	C₃×#117	semidirect
509. C₃	#34255	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×#518	2
C₃×#2357	2
C₃×#2354	2
C₃×#2342	2
C₃×#2319	2
C₃×#2317	2
C₃×#2309	2
C₃×#2296	2
C₃×#2293	2
C₂×C₃×#390	2
C₃×#2274	2
C₃×C₄×#117	2
#34255	3

Subgroups — C3×#34255 #10652862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#34255 #10652862