Subgroups — C₃×#35264 #10653871

Order 768.

2654 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#535	C₂	1	1	yes	maximal, characteristic
3	C₃×#2861	C₂	1	2	yes	maximal, complemented
4	C₃×#2858	C₂	1	2	yes	maximal, complemented
5	C₃×#2641	C₂	1	2	yes	maximal, complemented
6	C₃×#2640	C₂	1	2	yes	maximal, complemented
7	C₃×#2491	C₂	1	4	yes	maximal, complemented
8	C₃×#2490	C₂	1	4	yes	maximal, complemented
9	C₃×#2472	C₂	1	8	yes	maximal, complemented
10	C₃×#2311	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#2309	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#2297	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic, complemented
15	#35264	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
17	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
18	C₃×#544	—	2	8	no	—
19	C₃×#535	—	2	8	no	—
20	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
21	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₂×C₃×#127	C₂²	1	2	yes	—
23	C₂×C₃×#125	C₂²	1	2	yes	complemented
24	C₂×C₃×#125	C₂²	1	2	yes	complemented
25	C₂×C₃×#124	C₂²	1	2	yes	—
26	C₂×C₃×#124	C₂²	1	8	yes	complemented
27	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
28	C₂×C₃×#121	C₂²	1	2	yes	—
29	C₂×C₃×#121	C₂²	1	2	yes	—
30	C₂×C₃×#120	C₂²	1	4	yes	—
31	C₂×C₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃×#117	C₂²	1	1	yes	characteristic
33	C₂×C₃×#117	C₂²	1	1	yes	characteristic
34	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
36	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
37	C₃×#397	C₂²	1	8	yes	complemented
38	C₃×#394	C₂²	1	8	yes	complemented
39	C₃×#394	C₂²	1	8	yes	complemented
40	C₃×#393	C₂²	1	8	yes	complemented
41	C₃×#391	C₂²	1	1	yes	characteristic, complemented
42	C₃×#391	C₂²	1	2	yes	complemented
43	C₃×#391	C₂²	1	1	yes	characteristic, complemented
44	C₃×#390	C₂²	1	2	yes	—
45	C₃×#390	C₂²	1	2	yes	—
46	C₃×#390	C₂²	1	2	yes	—
47	C₃×#390	C₂²	1	4	yes	—
48	C₃×#390	C₂²	1	4	yes	—
49	C₃×#390	C₂²	1	2	yes	—
50	C₃×#390	C₂²	1	1	yes	characteristic, complemented
51	C₃×#390	C₂²	1	2	yes	complemented
52	C₃×#390	C₂²	1	8	yes	complemented
53	C₃×#390	C₂²	1	1	yes	characteristic, complemented
54	C₃×#388	C₂²	1	8	yes	complemented
55	C₃×#388	C₂²	1	8	yes	complemented
56	C₃×#386	C₂²	1	4	yes	—
57	C₃×#386	C₂²	1	4	yes	—
58	C₃×#380	C₂²	1	4	yes	complemented
59	C₃×#379	C₂²	1	4	yes	—
60	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	complemented
61	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	complemented
62	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	—
63	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	complemented
64	C₃×#376	C₂²	1	2	yes	complemented
65	C₃×C₄³	C₂²	1	2	yes	complemented
66	C₂×#544	C₂×C₃	1	1	yes	characteristic, complemented
67	C₂×#535	C₂×C₃	1	1	yes	characteristic
68	#2861	C₂×C₃	1	2	yes	complemented
69	#2858	C₂×C₃	1	2	yes	complemented
70	#2641	C₂×C₃	1	2	yes	complemented
71	#2640	C₂×C₃	1	2	yes	complemented
72	#2491	C₂×C₃	1	4	yes	complemented
73	#2490	C₂×C₃	1	4	yes	complemented
74	#2472	C₂×C₃	1	8	yes	complemented
75	#2311	C₂×C₃	1	1	yes	characteristic, complemented
76	#2309	C₂×C₃	1	1	yes	characteristic, complemented
77	#2297	C₂×C₃	1	1	yes	characteristic, complemented
78	C₂×#390	C₂×C₃	1	1	yes	characteristic, complemented
79	C₄×#117	C₂×C₃	1	1	yes	characteristic, complemented
80	C₂⁵×C₃	C₂³	1	2	yes	—
81	C₂×C₃×#41	—	2	4	no	—
82	C₂×C₃×#41	—	2	4	no	—
83	C₂×C₃×#41	—	2	4	no	—
84	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
85	C₂²×C₃×Q₈	C₂³	1	4	yes	—
86	C₂²×C₃×D₄	—	2	4	no	—
87	C₂²×C₃×D₄	—	2	4	no	—
88	C₂²×C₃×D₄	—	2	4	no	—
89	C₂²×C₃×D₄	C₂³	1	2	yes	—
90	C₂²×C₃×D₄	C₂³	1	2	yes	—
91	C₂²×C₃×D₄	C₂³	1	2	yes	—
92	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
93	C₂²×C₃×D₄	C₂³	1	4	yes	—
94	C₂²×C₃×D₄	C₂³	1	4	yes	—
95	C₂³×C₃×C₄	C₂³	1	2	yes	—
96	C₂³×C₃×C₄	C₂³	1	2	yes	—
97	C₂³×C₃×C₄	C₂³	1	2	yes	—
98	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
99	C₃×C₄⋊Q₈	—	2	8	no	—
100	C₃×C₄⋊Q₈	—	2	8	no	—
101	C₃×#127	—	2	8	no	—
102	C₃×#125	—	2	8	no	—
103	C₃×#125	—	2	8	no	—
104	C₃×#124	—	2	8	no	—
105	C₃×#124	—	4	16	no	—
106	C₃×C₂²⋊Q₈	—	2	16	no	—
107	C₃×#121	—	2	8	no	—
108	C₃×#121	—	2	8	no	—
109	C₃×#120	—	2	16	no	—
110	C₃×#117	—	2	4	no	—
111	C₃×#117	—	2	4	no	—
112	C₃×#117	D₄	1	8	yes	—
113	C₂×C₃×C₄⋊C₄	—	2	4	no	—
114	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
115	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
116	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
117	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
118	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
119	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
120	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
121	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
122	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
123	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
124	C₂×C₃×C₂²⋊C₄	—	4	8	no	—
125	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
127	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
129	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
133	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
134	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
135	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
136	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
137	C₂×C₃×C₄²	—	2	4	no	—
138	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
139	C₂×C₃×C₄²	C₂³	1	2	yes	—
140	C₂×C₃×C₄²	C₂³	1	4	yes	—
141	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
142	C₂×C₃×C₄²	C₂³	1	2	yes	—
143	C₂×C₃×C₄²	C₂³	1	2	yes	—
144	C₂×C₃×C₄²	C₂³	1	4	yes	—
145	C₂×C₃×C₄²	C₂³	1	2	yes	—
146	C₃×#95	C₂³	1	4	yes	—
147	C₃×#95	C₂³	1	4	yes	—
148	C₃×#95	C₂³	1	4	yes	—
149	C₃×#95	C₂³	1	4	yes	—
150	C₃×#95	C₂³	1	8	yes	—
151	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
152	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
153	#544	—	2	8	no	—
154	#535	—	2	8	no	—
155	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
156	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
157	C₂×#127	C₂²×C₃	1	2	yes	—
158	C₂×#125	C₂²×C₃	1	2	yes	complemented
159	C₂×#125	C₂²×C₃	1	2	yes	complemented
160	C₂×#124	C₂²×C₃	1	2	yes	—
161	C₂×#124	C₂²×C₃	1	8	yes	complemented
162	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
163	C₂×#121	C₂²×C₃	1	2	yes	—
164	C₂×#121	C₂²×C₃	1	2	yes	—
165	C₂×#120	C₂²×C₃	1	4	yes	—
166	C₂×#117	C₂²×C₃	1	1	yes	characteristic
167	C₂×#117	C₂²×C₃	1	1	yes	characteristic
168	C₂×#117	C₂²×C₃	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
170	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
171	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
172	#397	C₂²×C₃	1	8	yes	complemented
173	#394	C₂²×C₃	1	8	yes	complemented
174	#394	C₂²×C₃	1	8	yes	complemented
175	#393	C₂²×C₃	1	8	yes	complemented
176	#391	C₂²×C₃	1	2	yes	complemented
177	#391	C₂²×C₃	1	1	yes	characteristic, complemented
178	#391	C₂²×C₃	1	1	yes	characteristic, complemented
179	#390	C₂²×C₃	1	2	yes	—
180	#390	C₂²×C₃	1	2	yes	—
181	#390	C₂²×C₃	1	4	yes	—
182	#390	C₂²×C₃	1	2	yes	—
183	#390	C₂²×C₃	1	2	yes	—
184	#390	C₂²×C₃	1	4	yes	—
185	#390	C₂²×C₃	1	2	yes	complemented
186	#390	C₂²×C₃	1	1	yes	characteristic, complemented
187	#390	C₂²×C₃	1	8	yes	complemented
188	#390	C₂²×C₃	1	1	yes	characteristic, complemented
189	#388	C₂²×C₃	1	8	yes	complemented
190	#388	C₂²×C₃	1	8	yes	complemented
191	#386	C₂²×C₃	1	4	yes	—
192	#386	C₂²×C₃	1	4	yes	—
193	#380	C₂²×C₃	1	4	yes	complemented
194	#379	C₂²×C₃	1	4	yes	—
195	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
196	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
197	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
198	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
199	#376	C₂²×C₃	1	2	yes	complemented
200	C₄³	C₂²×C₃	1	2	yes	complemented
201	C₂⁴×C₃	—	2	4	no	—
202	C₂⁴×C₃	—	2	4	no	—
203	C₂⁴×C₃	—	2	4	no	—
204	C₂⁴×C₃	—	4	4	no	—
205	C₂⁴×C₃	—	4	4	no	—
206	C₂⁴×C₃	C₂⁴	1	4	yes	—
207	C₂⁴×C₃	C₂⁴	1	2	yes	—
208	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
209	C₃×#41	—	4	16	no	—
210	C₂×C₃×Q₈	—	2	2	no	—
211	C₂×C₃×Q₈	—	2	8	no	—
212	C₂×C₃×Q₈	—	2	2	no	—
213	C₂×C₃×Q₈	—	2	2	no	—
214	C₂×C₃×Q₈	—	4	8	no	—
215	C₂×C₃×D₄	—	2	4	no	—
216	C₂×C₃×D₄	—	2	4	no	—
217	C₂×C₃×D₄	—	2	4	no	—
218	C₂×C₃×D₄	—	2	4	no	—
219	C₂×C₃×D₄	—	2	4	no	—
220	C₂×C₃×D₄	—	2	2	no	—
221	C₂×C₃×D₄	—	2	8	no	—
222	C₂×C₃×D₄	—	2	8	no	—
223	C₂×C₃×D₄	—	2	4	no	—
224	C₂×C₃×D₄	—	2	4	no	—
225	C₂×C₃×D₄	—	2	4	no	—
226	C₂×C₃×D₄	—	2	4	no	—
227	C₂×C₃×D₄	—	2	2	no	—
228	C₂×C₃×D₄	—	2	2	no	—
229	C₂×C₃×D₄	—	4	8	no	—
230	C₂×C₃×D₄	—	4	8	no	—
231	C₂×C₃×D₄	—	4	16	no	—
232	C₂²×C₃×C₄	—	2	4	no	—
233	C₂²×C₃×C₄	—	2	4	no	—
234	C₂²×C₃×C₄	—	2	4	no	—
235	C₂²×C₃×C₄	—	2	4	no	—
236	C₂²×C₃×C₄	—	2	4	no	—
237	C₂²×C₃×C₄	—	2	4	no	—
238	C₂²×C₃×C₄	—	4	2	no	—
239	C₂²×C₃×C₄	—	4	4	no	—
240	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
241	C₂²×C₃×C₄	#41	1	4	yes	—
242	C₂²×C₃×C₄	#41	1	4	yes	—
243	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
244	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
245	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
246	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
247	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
248	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
249	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
250	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
251	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
252	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
253	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
254	C₃×C₄⋊C₄	—	2	4	no	—
255	C₃×C₄⋊C₄	—	2	4	no	—
256	C₃×C₄⋊C₄	—	2	2	no	—
257	C₃×C₄⋊C₄	—	2	2	no	—
258	C₃×C₄⋊C₄	—	2	8	no	—
259	C₃×C₄⋊C₄	—	2	8	no	—
260	C₃×C₄⋊C₄	—	2	8	no	—
261	C₃×C₄⋊C₄	—	2	4	no	—
262	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
263	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
264	C₃×C₂²⋊C₄	—	2	4	no	—
265	C₃×C₂²⋊C₄	—	2	4	no	—
266	C₃×C₂²⋊C₄	—	2	8	no	—
267	C₃×C₂²⋊C₄	—	2	8	no	—
268	C₃×C₂²⋊C₄	—	2	16	no	—
269	C₃×C₂²⋊C₄	—	2	8	no	—
270	C₃×C₂²⋊C₄	—	4	8	no	—
271	C₃×C₂²⋊C₄	—	4	8	no	—
272	C₃×C₂²⋊C₄	—	4	4	no	—
273	C₃×C₂²⋊C₄	—	4	8	no	—
274	C₃×C₂²⋊C₄	—	4	8	no	—
275	C₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
276	C₃×C₄²	—	2	4	no	—
277	C₃×C₄²	—	2	2	no	—
278	C₃×C₄²	—	2	4	no	—
279	C₃×C₄²	—	2	2	no	—
280	C₃×C₄²	—	2	4	no	—
281	C₃×C₄²	—	2	8	no	—
282	C₃×C₄²	—	4	4	no	—
283	C₃×C₄²	C₂×D₄	1	8	yes	—
284	C₂⁵	C₂³×C₃	1	2	yes	—
285	C₂×#41	—	2	4	no	—
286	C₂×#41	—	2	4	no	—
287	C₂×#41	—	2	4	no	—
288	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
289	C₂²×Q₈	C₂³×C₃	1	4	yes	—
290	C₂²×D₄	—	2	4	no	—
291	C₂²×D₄	—	2	4	no	—
292	C₂²×D₄	—	2	4	no	—
293	C₂²×D₄	C₂³×C₃	1	2	yes	—
294	C₂²×D₄	C₂³×C₃	1	2	yes	—
295	C₂²×D₄	C₂³×C₃	1	2	yes	—
296	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
297	C₂²×D₄	C₂³×C₃	1	4	yes	—
298	C₂²×D₄	C₂³×C₃	1	4	yes	—
299	C₂³×C₄	C₂³×C₃	1	2	yes	—
300	C₂³×C₄	C₂³×C₃	1	2	yes	—
301	C₂³×C₄	C₂³×C₃	1	2	yes	—
302	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
303	C₄⋊Q₈	—	2	8	no	—
304	C₄⋊Q₈	—	2	8	no	—
305	#127	—	2	8	no	—
306	#125	—	2	8	no	—
307	#125	—	2	8	no	—
308	#124	—	2	8	no	—
309	#124	—	4	16	no	—
310	C₂²⋊Q₈	—	2	16	no	—
311	#121	—	2	8	no	—
312	#121	—	2	8	no	—
313	#120	—	2	16	no	—
314	#117	—	2	4	no	—
315	#117	—	2	4	no	—
316	#117	C₃×D₄	1	8	yes	—
317	C₂×C₄⋊C₄	—	2	4	no	—
318	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
324	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
325	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	4	8	no	—
329	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
341	C₂×C₄²	—	2	4	no	—
342	C₂×C₄²	C₂³×C₃	1	2	yes	—
343	C₂×C₄²	C₂³×C₃	1	2	yes	—
344	C₂×C₄²	C₂³×C₃	1	4	yes	—
345	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₃	1	2	yes	—
348	C₂×C₄²	C₂³×C₃	1	4	yes	—
349	C₂×C₄²	C₂³×C₃	1	2	yes	—
350	#95	C₂³×C₃	1	4	yes	—
351	#95	C₂³×C₃	1	4	yes	—
352	#95	C₂³×C₃	1	4	yes	—
353	#95	C₂³×C₃	1	4	yes	—
354	#95	C₂³×C₃	1	8	yes	—
355	C₂³×C₃	—	2	2	no	—
356	C₂³×C₃	—	2	2	no	—
357	C₂³×C₃	—	2	4	no	—
358	C₂³×C₃	—	2	2	no	—
359	C₂³×C₃	—	2	2	no	—
360	C₂³×C₃	—	2	2	no	—
361	C₂³×C₃	—	2	4	no	—
362	C₂³×C₃	—	2	4	no	—
363	C₂³×C₃	—	2	4	no	—
364	C₂³×C₃	—	2	8	no	—
365	C₂³×C₃	—	2	4	no	—
366	C₂³×C₃	—	2	8	no	—
367	C₂³×C₃	—	4	16	no	—
368	C₂³×C₃	—	4	8	no	—
369	C₂³×C₃	—	8	8	no	—
370	C₂³×C₃	—	8	4	no	—
371	C₂³×C₃	—	8	4	no	—
372	C₂³×C₃	C₂²×D₄	1	2	yes	—
373	C₂³×C₃	C₂×#41	1	2	yes	—
374	C₂³×C₃	C₂×#41	1	2	yes	—
375	C₂³×C₃	C₂⁵	1	1	yes	characteristic
376	C₃×Q₈	—	4	4	no	—
377	C₃×Q₈	—	8	8	no	—
378	C₃×D₄	—	4	8	no	—
379	C₃×D₄	—	4	8	no	—
380	C₃×D₄	—	4	8	no	—
381	C₃×D₄	—	4	4	no	—
382	C₃×D₄	—	8	8	no	—
383	C₃×D₄	—	8	8	no	—
384	C₂×C₃×C₄	—	2	2	no	—
385	C₂×C₃×C₄	—	2	4	no	—
386	C₂×C₃×C₄	—	2	2	no	—
387	C₂×C₃×C₄	—	2	4	no	—
388	C₂×C₃×C₄	—	2	2	no	—
389	C₂×C₃×C₄	—	2	2	no	—
390	C₂×C₃×C₄	—	2	2	no	—
391	C₂×C₃×C₄	—	2	4	no	—
392	C₂×C₃×C₄	—	2	2	no	—
393	C₂×C₃×C₄	—	2	4	no	—
394	C₂×C₃×C₄	—	2	2	no	—
395	C₂×C₃×C₄	—	2	2	no	—
396	C₂×C₃×C₄	—	2	4	no	—
397	C₂×C₃×C₄	—	2	8	no	—
398	C₂×C₃×C₄	—	2	4	no	—
399	C₂×C₃×C₄	—	2	2	no	—
400	C₂×C₃×C₄	—	2	4	no	—
401	C₂×C₃×C₄	—	2	4	no	—
402	C₂×C₃×C₄	—	2	2	no	—
403	C₂×C₃×C₄	—	4	8	no	—
404	C₂×C₃×C₄	—	4	8	no	—
405	C₂×C₃×C₄	—	8	4	no	—
406	C₂×C₃×C₄	#124	1	16	yes	complemented
407	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
408	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
409	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
410	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
411	C₂×C₃×C₄	C₂×#41	1	2	yes	—
412	C₂×C₃×C₄	C₂×#41	1	2	yes	—
413	C₂×C₃×C₄	C₂×#41	1	2	yes	—
414	C₂×C₃×C₄	C₂×#41	1	2	yes	—
415	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
416	C₂⁴	—	2	4	no	—
417	C₂⁴	—	2	4	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	4	4	no	—
420	C₂⁴	—	4	4	no	—
421	C₂⁴	C₂⁴×C₃	1	4	yes	—
422	C₂⁴	C₂⁴×C₃	1	2	yes	—
423	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
424	#41	—	4	16	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	8	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	4	8	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	8	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	—	4	8	no	—
446	C₂×D₄	—	4	16	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
456	C₂²×C₄	C₃×#41	1	4	yes	—
457	C₂²×C₄	C₃×#41	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
461	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
468	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	8	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
478	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	16	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	4	8	no	—
486	C₂²⋊C₄	—	4	8	no	—
487	C₂²⋊C₄	—	4	8	no	—
488	C₂²⋊C₄	—	4	8	no	—
489	C₂²⋊C₄	—	4	4	no	—
490	C₂²⋊C₄	C₂×C₃×D₄	1	8	yes	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	2	no	—
493	C₄²	—	2	4	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	2	8	no	—
497	C₄²	—	4	4	no	—
498	C₄²	C₂×C₃×D₄	1	8	yes	—
499	C₂²×C₃	—	2	4	no	—
500	C₂²×C₃	—	2	4	no	—
501	C₂²×C₃	—	2	2	no	—
502	C₂²×C₃	—	2	2	no	—
503	C₂²×C₃	—	4	2	no	—
504	C₂²×C₃	—	4	4	no	—
505	C₂²×C₃	—	4	4	no	—
506	C₂²×C₃	—	4	8	no	—
507	C₂²×C₃	—	4	4	no	—
508	C₂²×C₃	—	4	8	no	—
509	C₂²×C₃	—	4	2	no	—
510	C₂²×C₃	—	4	4	no	—
511	C₂²×C₃	—	4	2	no	—
512	C₂²×C₃	—	4	4	no	—
513	C₂²×C₃	—	8	16	no	—
514	C₂²×C₃	C₂×#124	1	4	yes	—
515	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
516	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
517	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
518	C₂²×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
519	C₂²×C₃	#585	1	2	yes	—
520	C₃×C₄	—	2	4	no	—
521	C₃×C₄	—	2	4	no	—
522	C₃×C₄	—	2	4	no	—
523	C₃×C₄	—	4	2	no	—
524	C₃×C₄	—	4	4	no	—
525	C₃×C₄	—	4	2	no	—
526	C₃×C₄	—	4	4	no	—
527	C₃×C₄	—	4	2	no	—
528	C₃×C₄	—	4	2	no	—
529	C₃×C₄	C₂×#124	1	4	yes	—
530	C₃×C₄	C₂×#124	1	4	yes	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	8	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	2	8	no	—
543	C₂³	—	4	16	no	—
544	C₂³	—	4	8	no	—
545	C₂³	—	8	8	no	—
546	C₂³	—	8	4	no	—
547	C₂³	—	8	4	no	—
548	C₂³	C₂²×C₃×D₄	1	2	yes	—
549	C₂³	C₂×C₃×#41	1	2	yes	—
550	C₂³	C₂×C₃×#41	1	2	yes	—
551	C₂³	C₂⁵×C₃	1	1	yes	characteristic
552	Q₈	—	4	4	no	—
553	Q₈	—	8	8	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	8	no	—
556	D₄	—	4	8	no	—
557	D₄	—	4	4	no	—
558	D₄	—	8	8	no	—
559	D₄	—	8	8	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	8	4	no	—
582	C₂×C₄	C₃×#124	1	16	yes	complemented
583	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
584	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
585	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
586	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
587	C₂×C₄	C₂×C₃×#41	1	2	yes	—
588	C₂×C₄	C₂×C₃×#41	1	2	yes	—
589	C₂×C₄	C₂×C₃×#41	1	2	yes	—
590	C₂×C₄	C₂×C₃×#41	1	2	yes	—
591	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
592	C₂×C₃	—	2	4	no	—
593	C₂×C₃	—	8	2	no	—
594	C₂×C₃	—	8	4	no	—
595	C₂×C₃	C₂²×#124	1	1	yes	characteristic
596	C₂×C₃	#3460	1	2	yes	—
597	C₂×C₃	#3494	1	2	yes	—
598	C₂×C₃	#3498	1	1	yes	characteristic
599	C₂×C₃	#3500	1	1	yes	characteristic
600	C₂²	—	2	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	8	no	—
609	C₂²	—	4	4	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	4	4	no	—
614	C₂²	—	8	16	no	—
615	C₂²	C₂×C₃×#124	1	4	yes	—
616	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
617	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
618	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
619	C₂²	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
620	C₂²	C₃×#585	1	2	yes	—
621	C₄	—	2	4	no	—
622	C₄	—	2	4	no	—
623	C₄	—	2	4	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	4	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	4	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	C₂×C₃×#124	1	4	yes	—
631	C₄	C₂×C₃×#124	1	4	yes	—
632	C₃	#35264	1	1	yes	characteristic, direct product factor, Sylow
633	C₂	—	2	4	no	—
634	C₂	—	8	4	no	—
635	C₂	—	8	2	no	—
636	C₂	C₂²×C₃×#124	1	1	yes	characteristic
637	C₂	C₃×#3460	1	2	yes	—
638	C₂	C₃×#3494	1	2	yes	—
639	C₂	C₃×#3498	1	1	yes	characteristic
640	C₂	C₃×#3500	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂×C₃×#535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#544	C₂	semidirect
3. C₃×#2861	C₂	semidirect
4. C₃×#2858	C₂	semidirect
5. C₃×#2641	C₂	semidirect
6. C₃×#2640	C₂	semidirect
7. C₃×#2491	C₂	semidirect
8. C₃×#2490	C₂	semidirect
9. C₃×#2472	C₂	semidirect
10. C₃×#2311	C₂	semidirect
11. C₃×#2309	C₂	semidirect
12. C₃×#2297	C₂	semidirect
13. C₂×C₃×#390	C₂	semidirect
14. C₃×C₄×#117	C₂	semidirect
15. #35264	C₃	direct (N × H)
20. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
23. C₂×C₃×#125	C₂²	semidirect
24. C₂×C₃×#125	C₂²	semidirect
26. C₂×C₃×#124	C₂²	semidirect
37. C₃×#397	C₂²	semidirect
38. C₃×#394	C₂²	semidirect
39. C₃×#394	C₂²	semidirect
40. C₃×#393	C₂²	semidirect
41. C₃×#391	C₂²	semidirect
42. C₃×#391	C₂²	semidirect
43. C₃×#391	C₂²	semidirect
50. C₃×#390	C₂²	semidirect
51. C₃×#390	C₂²	semidirect
52. C₃×#390	C₂²	semidirect
53. C₃×#390	C₂²	semidirect
54. C₃×#388	C₂²	semidirect
55. C₃×#388	C₂²	semidirect
58. C₃×#380	C₂²	semidirect
60. C₃×C₄×C₄⋊C₄	C₂²	semidirect
61. C₃×C₄×C₄⋊C₄	C₂²	semidirect
63. C₃×C₄×C₂²⋊C₄	C₂²	semidirect
64. C₃×#376	C₂²	semidirect
65. C₃×C₄³	C₂²	semidirect
66. C₂×#544	C₂×C₃	semidirect
68. #2861	C₂×C₃	semidirect
69. #2858	C₂×C₃	semidirect
70. #2641	C₂×C₃	semidirect
71. #2640	C₂×C₃	semidirect
72. #2491	C₂×C₃	semidirect
73. #2490	C₂×C₃	semidirect
74. #2472	C₂×C₃	semidirect
75. #2311	C₂×C₃	semidirect
76. #2309	C₂×C₃	semidirect
77. #2297	C₂×C₃	semidirect
78. C₂×#390	C₂×C₃	semidirect
79. C₄×#117	C₂×C₃	semidirect
155. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
156. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
158. C₂×#125	C₂²×C₃	semidirect
159. C₂×#125	C₂²×C₃	semidirect
161. C₂×#124	C₂²×C₃	semidirect
172. #397	C₂²×C₃	semidirect
173. #394	C₂²×C₃	semidirect
174. #394	C₂²×C₃	semidirect
175. #393	C₂²×C₃	semidirect
176. #391	C₂²×C₃	semidirect
177. #391	C₂²×C₃	semidirect
178. #391	C₂²×C₃	semidirect
185. #390	C₂²×C₃	semidirect
186. #390	C₂²×C₃	semidirect
187. #390	C₂²×C₃	semidirect
188. #390	C₂²×C₃	semidirect
189. #388	C₂²×C₃	semidirect
190. #388	C₂²×C₃	semidirect
193. #380	C₂²×C₃	semidirect
195. C₄×C₄⋊C₄	C₂²×C₃	semidirect
196. C₄×C₄⋊C₄	C₂²×C₃	semidirect
198. C₄×C₂²⋊C₄	C₂²×C₃	semidirect
199. #376	C₂²×C₃	semidirect
200. C₄³	C₂²×C₃	semidirect
406. C₂×C₃×C₄	#124	semidirect
582. C₂×C₄	C₃×#124	semidirect
632. C₃	#35264	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×#544	2
C₂×C₃×#535	2
C₃×#2861	2
C₃×#2858	2
C₃×#2641	2
C₃×#2640	2
C₃×#2491	2
C₃×#2490	2
C₃×#2472	2
C₃×#2311	2
C₃×#2309	2
C₃×#2297	2
C₂×C₃×#390	2
C₃×C₄×#117	2
#35264	3

Subgroups — C3×#35264 #10653871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#35264 #10653871