Subgroups — C₃×#39369 #10657976

Order 768.

2154 conjugacy classes of subgroups.

The subgroup lattice has 690 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	2	yes	maximal, complemented
2	C₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2720	C₂	1	4	yes	maximal, complemented
4	C₃×#2677	C₂	1	4	yes	maximal, complemented
5	C₃×#2628	C₂	1	2	yes	maximal, complemented
6	C₃×#2552	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2551	C₂	1	2	yes	maximal, complemented
8	C₃×#2551	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2542	C₂	1	2	yes	maximal
10	C₃×#2506	C₂	1	2	yes	maximal, complemented
11	C₃×#2452	C₂	1	2	yes	maximal
12	C₃×#2441	C₂	1	2	yes	maximal, complemented
13	C₃×#2440	C₂	1	2	yes	maximal, complemented
14	C₃×#2354	C₂	1	1	yes	maximal, characteristic
15	C₃×D₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
16	C₃×C₄×#125	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃×#2293	C₂	1	1	yes	maximal, characteristic
18	#39369	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃×#544	—	2	16	no	—
20	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊Q₈	C₂²	1	4	yes	—
22	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₃×#125	C₂²	1	2	yes	—
27	C₂×C₃×#125	C₂²	1	4	yes	—
28	C₂×C₃×#125	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
30	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
31	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
32	C₂×C₃×C₄×D₄	C₂²	1	2	yes	—
33	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×#117	C₂²	1	2	yes	—
35	C₂×C₃×#117	C₂²	1	2	yes	—
36	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
37	C₃×#400	C₂²	1	4	yes	—
38	C₃×#399	C₂²	1	4	yes	—
39	C₃×#399	C₂²	1	4	yes	—
40	C₃×#398	C₂²	1	4	yes	—
41	C₃×#398	C₂²	1	4	yes	—
42	C₃×#396	C₂²	1	4	yes	—
43	C₃×#395	C₂²	1	4	yes	—
44	C₃×#395	C₂²	1	4	yes	—
45	C₃×#395	C₂²	1	4	yes	—
46	C₃×#393	C₂²	1	4	yes	—
47	C₃×#390	C₂²	1	1	yes	characteristic
48	C₃×#390	C₂²	1	2	yes	—
49	C₃×#390	C₂²	1	1	yes	characteristic
50	C₃×#389	C₂²	1	2	yes	—
51	C₃×#389	C₂²	1	2	yes	—
52	C₃×#389	C₂²	1	1	yes	characteristic
53	C₃×#389	C₂²	1	1	yes	characteristic
54	C₃×#389	C₂²	1	4	yes	—
55	C₃×#389	C₂²	1	2	yes	—
56	C₃×#389	C₂²	1	2	yes	—
57	C₃×#389	C₂²	1	1	yes	characteristic
58	C₃×#389	C₂²	1	2	yes	—
59	C₃×#389	C₂²	1	1	yes	characteristic
60	C₃×#388	C₂²	1	2	yes	—
61	C₃×#388	C₂²	1	4	yes	—
62	C₃×#388	C₂²	1	2	yes	—
63	C₃×#387	C₂²	1	2	yes	—
64	C₃×#387	C₂²	1	4	yes	—
65	C₃×#387	C₂²	1	2	yes	—
66	C₃×#385	C₂²	1	4	yes	—
67	C₃×#385	C₂²	1	4	yes	—
68	C₃×#384	C₂²	1	2	yes	—
69	C₃×#384	C₂²	1	2	yes	—
70	C₃×#384	C₂²	1	2	yes	—
71	C₃×#384	C₂²	1	2	yes	—
72	C₃×#384	C₂²	1	1	yes	characteristic
73	C₃×#384	C₂²	1	1	yes	characteristic
74	C₃×#382	C₂²	1	1	yes	characteristic
75	C₃×#380	C₂²	1	2	yes	—
76	C₃×#380	C₂²	1	2	yes	—
77	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
78	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
79	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
80	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
81	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
82	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
83	C₃×C₄×C₂²⋊C₄	C₂²	1	2	yes	—
84	C₃×C₄³	C₂²	1	1	yes	characteristic
85	C₂×#544	C₂×C₃	1	2	yes	complemented
86	#2868	C₂×C₃	1	1	yes	characteristic, complemented
87	#2720	C₂×C₃	1	4	yes	complemented
88	#2677	C₂×C₃	1	4	yes	complemented
89	#2628	C₂×C₃	1	2	yes	complemented
90	#2552	C₂×C₃	1	1	yes	characteristic, complemented
91	#2551	C₂×C₃	1	2	yes	complemented
92	#2551	C₂×C₃	1	1	yes	characteristic, complemented
93	#2542	C₂×C₃	1	2	yes	—
94	#2506	C₂×C₃	1	2	yes	complemented
95	#2452	C₂×C₃	1	2	yes	—
96	#2441	C₂×C₃	1	2	yes	complemented
97	#2440	C₂×C₃	1	2	yes	complemented
98	#2354	C₂×C₃	1	1	yes	characteristic
99	D₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
100	C₄×#125	C₂×C₃	1	1	yes	characteristic, complemented
101	#2293	C₂×C₃	1	1	yes	characteristic
102	C₂²×C₃×Q₈	C₂³	1	4	yes	—
103	C₂²×C₃×Q₈	C₂³	1	4	yes	—
104	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
105	C₂³×C₃×C₄	C₂³	1	2	yes	—
106	C₂³×C₃×C₄	C₂³	1	4	yes	—
107	C₃×C₄⋊Q₈	—	2	16	no	—
108	C₃×C₄⋊Q₈	—	2	8	no	—
109	C₃×C₄⋊Q₈	—	2	8	no	—
110	C₃×C₄⋊Q₈	—	4	4	no	—
111	C₃×C₄⋊Q₈	—	4	4	no	—
112	C₃×C₄⋊Q₈	—	4	4	no	—
113	C₃×#125	—	2	16	no	—
114	C₃×#125	—	2	8	no	—
115	C₃×#125	D₄	1	8	yes	complemented
116	C₃×C₂²⋊Q₈	—	2	16	no	—
117	C₃×C₂²⋊Q₈	—	2	16	no	—
118	C₃×C₂²⋊Q₈	—	4	8	no	—
119	C₃×C₄×D₄	—	4	4	no	—
120	C₃×C₄×D₄	Q₈	1	8	yes	complemented
121	C₃×#117	—	2	8	no	—
122	C₃×#117	—	2	8	no	—
123	C₂×C₃×C₄⋊C₄	—	2	8	no	—
124	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
125	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
126	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
127	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
128	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
129	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
130	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
131	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
132	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
133	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
134	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
135	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
136	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
137	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
138	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
139	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
140	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
141	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
142	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
143	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
144	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
145	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
146	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
147	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
148	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
149	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
150	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
151	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
152	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
153	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
154	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
155	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
156	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
157	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
158	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
159	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
160	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
161	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
162	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
163	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
164	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
165	C₂×C₃×C₄²	C₂³	1	2	yes	—
166	C₂×C₃×C₄²	C₂³	1	2	yes	—
167	C₂×C₃×C₄²	C₂³	1	2	yes	—
168	C₂×C₃×C₄²	C₂³	1	4	yes	—
169	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
170	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
171	C₃×#95	C₂³	1	2	yes	—
172	C₃×#95	C₂³	1	2	yes	—
173	C₃×#95	C₂³	1	4	yes	—
174	C₃×#95	C₂³	1	2	yes	—
175	C₃×#95	C₂³	1	2	yes	—
176	C₃×#95	C₂³	1	2	yes	—
177	C₃×#95	C₂³	1	2	yes	—
178	C₃×#95	C₂³	1	2	yes	—
179	C₃×#95	C₂³	1	2	yes	—
180	C₃×#95	C₂³	1	2	yes	—
181	C₃×#95	C₂³	1	4	yes	—
182	C₃×#95	C₂³	1	4	yes	—
183	C₃×#95	C₂³	1	4	yes	—
184	#544	—	2	16	no	—
185	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
186	C₂×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
187	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
188	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
189	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
190	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
191	C₂×#125	C₂²×C₃	1	2	yes	—
192	C₂×#125	C₂²×C₃	1	4	yes	—
193	C₂×#125	C₂²×C₃	1	1	yes	characteristic
194	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
195	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
196	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
197	C₂×C₄×D₄	C₂²×C₃	1	2	yes	—
198	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
199	C₂×#117	C₂²×C₃	1	2	yes	—
200	C₂×#117	C₂²×C₃	1	2	yes	—
201	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
202	#400	C₂²×C₃	1	4	yes	—
203	#399	C₂²×C₃	1	4	yes	—
204	#399	C₂²×C₃	1	4	yes	—
205	#398	C₂²×C₃	1	4	yes	—
206	#398	C₂²×C₃	1	4	yes	—
207	#396	C₂²×C₃	1	4	yes	—
208	#395	C₂²×C₃	1	4	yes	—
209	#395	C₂²×C₃	1	4	yes	—
210	#395	C₂²×C₃	1	4	yes	—
211	#393	C₂²×C₃	1	4	yes	—
212	#390	C₂²×C₃	1	2	yes	—
213	#390	C₂²×C₃	1	1	yes	characteristic
214	#390	C₂²×C₃	1	1	yes	characteristic
215	#389	C₂²×C₃	1	1	yes	characteristic
216	#389	C₂²×C₃	1	1	yes	characteristic
217	#389	C₂²×C₃	1	2	yes	—
218	#389	C₂²×C₃	1	2	yes	—
219	#389	C₂²×C₃	1	2	yes	—
220	#389	C₂²×C₃	1	4	yes	—
221	#389	C₂²×C₃	1	2	yes	—
222	#389	C₂²×C₃	1	2	yes	—
223	#389	C₂²×C₃	1	1	yes	characteristic
224	#389	C₂²×C₃	1	1	yes	characteristic
225	#388	C₂²×C₃	1	4	yes	—
226	#388	C₂²×C₃	1	2	yes	—
227	#388	C₂²×C₃	1	2	yes	—
228	#387	C₂²×C₃	1	2	yes	—
229	#387	C₂²×C₃	1	2	yes	—
230	#387	C₂²×C₃	1	4	yes	—
231	#385	C₂²×C₃	1	4	yes	—
232	#385	C₂²×C₃	1	4	yes	—
233	#384	C₂²×C₃	1	1	yes	characteristic
234	#384	C₂²×C₃	1	1	yes	characteristic
235	#384	C₂²×C₃	1	2	yes	—
236	#384	C₂²×C₃	1	2	yes	—
237	#384	C₂²×C₃	1	2	yes	—
238	#384	C₂²×C₃	1	2	yes	—
239	#382	C₂²×C₃	1	1	yes	characteristic
240	#380	C₂²×C₃	1	2	yes	—
241	#380	C₂²×C₃	1	2	yes	—
242	C₄×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
243	C₄×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
246	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
248	C₄×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
249	C₄³	C₂²×C₃	1	1	yes	characteristic
250	C₂⁴×C₃	C₂⁴	1	2	yes	—
251	C₂×C₃×Q₈	—	2	8	no	—
252	C₂×C₃×Q₈	—	2	8	no	—
253	C₂×C₃×Q₈	—	4	4	no	—
254	C₂×C₃×Q₈	—	4	4	no	—
255	C₂×C₃×Q₈	—	4	4	no	—
256	C₂×C₃×Q₈	—	4	4	no	—
257	C₂×C₃×D₄	—	2	2	no	—
258	C₂×C₃×D₄	—	2	2	no	—
259	C₂×C₃×D₄	C₂×Q₈	1	4	yes	—
260	C₂²×C₃×C₄	—	2	4	no	—
261	C₂²×C₃×C₄	—	2	4	no	—
262	C₂²×C₃×C₄	—	2	8	no	—
263	C₂²×C₃×C₄	—	4	4	no	—
264	C₂²×C₃×C₄	—	4	4	no	—
265	C₂²×C₃×C₄	C₂×Q₈	1	8	yes	—
266	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
267	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
268	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
269	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
270	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
271	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
272	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
273	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
274	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
275	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
276	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
277	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
278	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
279	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
280	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
281	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
282	C₃×C₄⋊C₄	—	2	8	no	—
283	C₃×C₄⋊C₄	—	2	4	no	—
284	C₃×C₄⋊C₄	—	2	4	no	—
285	C₃×C₄⋊C₄	—	2	4	no	—
286	C₃×C₄⋊C₄	—	2	4	no	—
287	C₃×C₄⋊C₄	—	2	8	no	—
288	C₃×C₄⋊C₄	—	2	8	no	—
289	C₃×C₄⋊C₄	—	2	8	no	—
290	C₃×C₄⋊C₄	—	2	8	no	—
291	C₃×C₄⋊C₄	—	2	4	no	—
292	C₃×C₄⋊C₄	—	2	4	no	—
293	C₃×C₄⋊C₄	—	2	8	no	—
294	C₃×C₄⋊C₄	—	2	4	no	—
295	C₃×C₄⋊C₄	—	2	2	no	—
296	C₃×C₄⋊C₄	—	2	4	no	—
297	C₃×C₄⋊C₄	—	2	2	no	—
298	C₃×C₄⋊C₄	—	2	2	no	—
299	C₃×C₄⋊C₄	—	2	2	no	—
300	C₃×C₄⋊C₄	—	4	4	no	—
301	C₃×C₄⋊C₄	—	4	4	no	—
302	C₃×C₄⋊C₄	—	4	2	no	—
303	C₃×C₄⋊C₄	—	4	2	no	—
304	C₃×C₄⋊C₄	—	4	4	no	—
305	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
306	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
307	C₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
308	C₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
309	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
310	C₃×C₂²⋊C₄	—	2	4	no	—
311	C₃×C₂²⋊C₄	—	2	8	no	—
312	C₃×C₂²⋊C₄	—	2	4	no	—
313	C₃×C₂²⋊C₄	—	2	4	no	—
314	C₃×C₂²⋊C₄	—	2	4	no	—
315	C₃×C₂²⋊C₄	—	4	4	no	—
316	C₃×C₂²⋊C₄	—	4	4	no	—
317	C₃×C₂²⋊C₄	C₂×Q₈	1	8	yes	—
318	C₃×C₄²	—	2	8	no	—
319	C₃×C₄²	—	2	4	no	—
320	C₃×C₄²	—	2	4	no	—
321	C₃×C₄²	—	2	2	no	—
322	C₃×C₄²	—	2	2	no	—
323	C₃×C₄²	—	2	2	no	—
324	C₃×C₄²	—	2	2	no	—
325	C₃×C₄²	—	2	2	no	—
326	C₃×C₄²	—	2	4	no	—
327	C₃×C₄²	C₂×D₄	1	4	yes	—
328	C₃×C₄²	C₂×Q₈	1	4	yes	—
329	C₂²×Q₈	C₂³×C₃	1	4	yes	—
330	C₂²×Q₈	C₂³×C₃	1	4	yes	—
331	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₃	1	2	yes	—
333	C₂³×C₄	C₂³×C₃	1	4	yes	—
334	C₄⋊Q₈	—	2	16	no	—
335	C₄⋊Q₈	—	2	8	no	—
336	C₄⋊Q₈	—	2	8	no	—
337	C₄⋊Q₈	—	4	4	no	—
338	C₄⋊Q₈	—	4	4	no	—
339	C₄⋊Q₈	—	4	4	no	—
340	#125	—	2	16	no	—
341	#125	—	2	8	no	—
342	#125	C₃×D₄	1	8	yes	complemented
343	C₂²⋊Q₈	—	2	16	no	—
344	C₂²⋊Q₈	—	2	16	no	—
345	C₂²⋊Q₈	—	4	8	no	—
346	C₄×D₄	—	4	4	no	—
347	C₄×D₄	C₃×Q₈	1	8	yes	complemented
348	#117	—	2	8	no	—
349	#117	—	2	8	no	—
350	C₂×C₄⋊C₄	—	2	8	no	—
351	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
360	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
363	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
365	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
387	C₂×C₄²	C₂³×C₃	1	2	yes	—
388	C₂×C₄²	C₂³×C₃	1	4	yes	—
389	C₂×C₄²	C₂³×C₃	1	2	yes	—
390	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₃	1	2	yes	—
396	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
398	#95	C₂³×C₃	1	2	yes	—
399	#95	C₂³×C₃	1	2	yes	—
400	#95	C₂³×C₃	1	4	yes	—
401	#95	C₂³×C₃	1	2	yes	—
402	#95	C₂³×C₃	1	2	yes	—
403	#95	C₂³×C₃	1	2	yes	—
404	#95	C₂³×C₃	1	2	yes	—
405	#95	C₂³×C₃	1	2	yes	—
406	#95	C₂³×C₃	1	2	yes	—
407	#95	C₂³×C₃	1	2	yes	—
408	#95	C₂³×C₃	1	4	yes	—
409	#95	C₂³×C₃	1	4	yes	—
410	#95	C₂³×C₃	1	4	yes	—
411	C₂³×C₃	—	2	2	no	—
412	C₂³×C₃	—	2	2	no	—
413	C₂³×C₃	—	2	2	no	—
414	C₂³×C₃	—	2	2	no	—
415	C₂³×C₃	—	2	2	no	—
416	C₂³×C₃	—	2	2	no	—
417	C₂³×C₃	C₂²×Q₈	1	4	yes	—
418	C₂³×C₃	C₂⁵	1	1	yes	characteristic
419	C₃×Q₈	—	8	8	no	—
420	C₃×Q₈	—	8	8	no	—
421	C₃×D₄	—	2	8	no	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	2	1	no	—
424	C₂×C₃×C₄	—	2	1	no	—
425	C₂×C₃×C₄	—	2	2	no	—
426	C₂×C₃×C₄	—	2	1	no	—
427	C₂×C₃×C₄	—	2	1	no	—
428	C₂×C₃×C₄	—	2	2	no	—
429	C₂×C₃×C₄	—	2	1	no	—
430	C₂×C₃×C₄	—	2	2	no	—
431	C₂×C₃×C₄	—	2	1	no	—
432	C₂×C₃×C₄	—	2	1	no	—
433	C₂×C₃×C₄	—	2	1	no	—
434	C₂×C₃×C₄	—	2	1	no	—
435	C₂×C₃×C₄	—	2	1	no	—
436	C₂×C₃×C₄	—	2	2	no	—
437	C₂×C₃×C₄	—	2	2	no	—
438	C₂×C₃×C₄	—	2	16	no	—
439	C₂×C₃×C₄	—	2	2	no	—
440	C₂×C₃×C₄	—	2	2	no	—
441	C₂×C₃×C₄	—	2	4	no	—
442	C₂×C₃×C₄	—	2	4	no	—
443	C₂×C₃×C₄	—	2	2	no	—
444	C₂×C₃×C₄	—	2	2	no	—
445	C₂×C₃×C₄	—	2	2	no	—
446	C₂×C₃×C₄	—	2	4	no	—
447	C₂×C₃×C₄	—	2	4	no	—
448	C₂×C₃×C₄	—	2	4	no	—
449	C₂×C₃×C₄	—	2	2	no	—
450	C₂×C₃×C₄	—	2	4	no	—
451	C₂×C₃×C₄	—	2	2	no	—
452	C₂×C₃×C₄	—	2	2	no	—
453	C₂×C₃×C₄	—	2	2	no	—
454	C₂×C₃×C₄	—	2	2	no	—
455	C₂×C₃×C₄	—	2	2	no	—
456	C₂×C₃×C₄	—	2	2	no	—
457	C₂×C₃×C₄	—	8	8	no	—
458	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
459	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
460	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
461	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
462	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
463	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
464	C₂×C₃×C₄	C₂²×Q₈	1	4	yes	—
465	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
466	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
467	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
468	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
469	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
470	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
471	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
472	C₂⁴	C₂⁴×C₃	1	2	yes	—
473	C₂×Q₈	—	2	8	no	—
474	C₂×Q₈	—	2	8	no	—
475	C₂×Q₈	—	4	4	no	—
476	C₂×Q₈	—	4	4	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	—	4	4	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	C₂×C₃×Q₈	1	4	yes	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	8	no	—
485	C₂²×C₄	—	4	4	no	—
486	C₂²×C₄	—	4	4	no	—
487	C₂²×C₄	C₂×C₃×Q₈	1	8	yes	—
488	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
496	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
504	C₄⋊C₄	—	2	8	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	8	no	—
512	C₄⋊C₄	—	2	8	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	8	no	—
516	C₄⋊C₄	—	2	4	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	4	4	no	—
523	C₄⋊C₄	—	4	4	no	—
524	C₄⋊C₄	—	4	2	no	—
525	C₄⋊C₄	—	4	2	no	—
526	C₄⋊C₄	—	4	4	no	—
527	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
528	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
529	C₄⋊C₄	C₂×C₃×D₄	1	8	yes	—
530	C₄⋊C₄	C₂×C₃×D₄	1	8	yes	—
531	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	8	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	4	4	no	—
538	C₂²⋊C₄	—	4	4	no	—
539	C₂²⋊C₄	C₂×C₃×Q₈	1	8	yes	—
540	C₄²	—	2	8	no	—
541	C₄²	—	2	4	no	—
542	C₄²	—	2	4	no	—
543	C₄²	—	2	2	no	—
544	C₄²	—	2	2	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	2	no	—
548	C₄²	—	2	4	no	—
549	C₄²	C₂×C₃×D₄	1	4	yes	—
550	C₄²	C₂×C₃×Q₈	1	4	yes	—
551	C₂²×C₃	—	2	4	no	—
552	C₂²×C₃	—	2	4	no	—
553	C₂²×C₃	—	2	4	no	—
554	C₂²×C₃	—	4	2	no	—
555	C₂²×C₃	—	4	2	no	—
556	C₂²×C₃	—	4	2	no	—
557	C₂²×C₃	—	4	2	no	—
558	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
559	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
560	C₂²×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
561	C₂²×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
562	C₂²×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
563	C₂²×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
564	C₂²×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
565	C₃×C₄	—	2	4	no	—
566	C₃×C₄	—	2	4	no	—
567	C₃×C₄	—	2	2	no	—
568	C₃×C₄	—	2	2	no	—
569	C₃×C₄	—	2	4	no	—
570	C₃×C₄	—	2	2	no	—
571	C₃×C₄	—	2	2	no	—
572	C₃×C₄	—	2	2	no	—
573	C₃×C₄	—	4	4	no	—
574	C₃×C₄	—	4	2	no	—
575	C₃×C₄	—	4	4	no	—
576	C₃×C₄	—	4	2	no	—
577	C₃×C₄	—	4	2	no	—
578	C₃×C₄	—	4	2	no	—
579	C₃×C₄	#544	1	4	yes	complemented
580	C₃×C₄	D₄×Q₈	1	4	yes	—
581	C₂³	—	2	2	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	2	no	—
587	C₂³	C₂²×C₃×Q₈	1	4	yes	—
588	C₂³	C₂⁵×C₃	1	1	yes	characteristic
589	Q₈	—	8	8	no	—
590	Q₈	—	8	8	no	—
591	D₄	—	2	8	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	16	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	4	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	8	8	no	—
628	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
629	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
630	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
631	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
632	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
633	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
634	C₂×C₄	C₂²×C₃×Q₈	1	4	yes	—
635	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
636	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
637	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
638	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
639	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
640	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
641	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
642	C₂×C₃	—	4	4	no	—
643	C₂×C₃	C₂×#544	1	1	yes	characteristic
644	C₂×C₃	C₂×D₄×Q₈	1	1	yes	characteristic
645	C₂×C₃	#3498	1	1	yes	characteristic
646	C₂×C₃	#3499	1	1	yes	characteristic
647	C₂×C₃	#3503	1	1	yes	characteristic
648	C₂×C₃	#3544	1	1	yes	characteristic
649	C₂×C₃	#3552	1	1	yes	characteristic
650	C₂²	—	2	4	no	—
651	C₂²	—	2	4	no	—
652	C₂²	—	2	4	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	2	no	—
656	C₂²	—	4	2	no	—
657	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
658	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
659	C₂²	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
660	C₂²	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
661	C₂²	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
662	C₂²	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
663	C₂²	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
664	C₄	—	2	4	no	—
665	C₄	—	2	4	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	4	no	—
669	C₄	—	2	2	no	—
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	4	4	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	4	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	2	no	—
678	C₄	C₃×#544	1	4	yes	complemented
679	C₄	C₃×D₄×Q₈	1	4	yes	—
680	C₃	#39369	1	1	yes	characteristic, direct product factor, Sylow
681	C₂	—	4	4	no	—
682	C₂	C₂×C₃×#544	1	1	yes	characteristic
683	C₂	C₂×C₃×D₄×Q₈	1	1	yes	characteristic
684	C₂	C₃×#3498	1	1	yes	characteristic
685	C₂	C₃×#3499	1	1	yes	characteristic
686	C₂	C₃×#3503	1	1	yes	characteristic
687	C₂	C₃×#3544	1	1	yes	characteristic
688	C₂	C₃×#3552	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂²×C₃×D₄	C₂³
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
2. C₃×#2868	C₂	semidirect
3. C₃×#2720	C₂	semidirect
4. C₃×#2677	C₂	semidirect
5. C₃×#2628	C₂	semidirect
6. C₃×#2552	C₂	semidirect
7. C₃×#2551	C₂	semidirect
8. C₃×#2551	C₂	semidirect
10. C₃×#2506	C₂	semidirect
12. C₃×#2441	C₂	semidirect
13. C₃×#2440	C₂	semidirect
16. C₃×C₄×#125	C₂	semidirect
18. #39369	C₃	direct (N × H)
85. C₂×#544	C₂×C₃	semidirect
86. #2868	C₂×C₃	semidirect
87. #2720	C₂×C₃	semidirect
88. #2677	C₂×C₃	semidirect
89. #2628	C₂×C₃	semidirect
90. #2552	C₂×C₃	semidirect
91. #2551	C₂×C₃	semidirect
92. #2551	C₂×C₃	semidirect
94. #2506	C₂×C₃	semidirect
96. #2441	C₂×C₃	semidirect
97. #2440	C₂×C₃	semidirect
100. C₄×#125	C₂×C₃	semidirect
115. C₃×#125	D₄	semidirect
120. C₃×C₄×D₄	Q₈	semidirect
342. #125	C₃×D₄	semidirect
347. C₄×D₄	C₃×Q₈	semidirect
579. C₃×C₄	#544	semidirect
678. C₄	C₃×#544	semidirect
680. C₃	#39369	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
C₃×#2868	2
C₃×#2720	2
C₃×#2677	2
C₃×#2628	2
C₃×#2552	2
C₃×#2551	2
C₃×#2542	2
C₃×#2506	2
C₃×#2452	2
C₃×#2441	2
C₃×#2440	2
C₃×#2354	2
C₃×D₄×C₄⋊C₄	2
C₃×C₄×#125	2
C₃×#2293	2
#39369	3

Subgroups — C3×#39369 #10657976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#39369 #10657976