Subgroups — C₃×#61854 #10680461

Order 768.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3550	C₂	1	1	yes	maximal, characteristic
2	C₃×#3502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3387	C₂	1	2	yes	maximal, complemented
4	C₃×#3386	C₂	1	4	yes	maximal, complemented
5	C₃×Q₈×Q₁₆	C₂	1	2	yes	maximal, complemented
6	C₃×#3377	C₂	1	4	yes	maximal, complemented
7	C₃×#3375	C₂	1	2	yes	maximal, complemented
8	C₃×#3365	C₂	1	2	yes	maximal, complemented
9	C₃×#3243	C₂	1	4	yes	maximal, complemented
10	C₃×#3237	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#3228	C₂	1	1	yes	maximal, characteristic
12	C₃×#3121	C₂	1	4	yes	maximal, complemented
13	C₃×#3112	C₂	1	1	yes	maximal, characteristic
14	C₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃×#2985	C₂	1	1	yes	maximal, characteristic, complemented
16	#61854	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃×#563	—	2	2	no	—
18	C₃×#559	C₂²	1	2	yes	—
19	C₃×Q₈²	—	2	1	no	—
20	C₃×Q₈²	C₂²	1	2	yes	—
21	C₃×Q₈²	C₂²	1	2	yes	—
22	C₃×#557	—	2	1	no	—
23	C₃×#557	—	2	2	no	—
24	C₃×#557	C₂²	1	4	yes	—
25	C₃×#557	C₂²	1	2	yes	—
26	C₃×#556	—	2	1	no	—
27	C₃×#556	C₂²	1	2	yes	—
28	C₃×D₄×Q₈	—	2	1	no	—
29	C₃×D₄×Q₈	C₂²	1	2	yes	—
30	C₃×#544	—	2	2	no	—
31	C₃×#544	C₂²	1	2	yes	—
32	C₃×#542	—	2	2	no	—
33	C₃×#542	C₂²	1	4	yes	—
34	C₃×#536	—	2	2	no	—
35	C₃×#533	—	2	2	no	—
36	C₃×#533	C₂²	1	1	yes	characteristic
37	C₃×#533	C₂²	1	1	yes	characteristic
38	C₃×#533	C₂²	1	1	yes	characteristic
39	C₃×#532	C₂²	1	1	yes	characteristic
40	C₃×#519	C₂²	1	2	yes	—
41	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
42	C₃×#501	C₂²	1	2	yes	—
43	C₃×#501	C₂²	1	4	yes	—
44	C₃×#500	C₂²	1	4	yes	—
45	C₃×#500	C₂²	1	1	yes	characteristic
46	C₃×#498	C₂²	1	1	yes	characteristic
47	C₃×#491	C₂²	1	2	yes	—
48	C₃×#489	C₂²	1	4	yes	—
49	C₃×#488	C₂²	1	4	yes	—
50	C₃×#486	C₂²	1	1	yes	characteristic
51	C₃×#486	C₂²	1	1	yes	characteristic
52	C₃×#485	C₂²	1	4	yes	—
53	C₃×#484	C₂²	1	2	yes	—
54	C₃×#482	C₂²	1	4	yes	—
55	C₃×#480	C₂²	1	2	yes	—
56	C₃×#479	C₂²	1	4	yes	—
57	C₃×#478	C₂²	1	4	yes	—
58	C₃×#478	C₂²	1	4	yes	—
59	C₃×#477	C₂²	1	2	yes	—
60	C₃×#477	C₂²	1	4	yes	—
61	C₃×#477	C₂²	1	2	yes	—
62	C₃×#477	C₂²	1	2	yes	—
63	C₃×#477	C₂²	1	4	yes	—
64	C₃×#475	C₂²	1	4	yes	—
65	C₃×#475	C₂²	1	2	yes	—
66	C₃×#474	C₂²	1	4	yes	—
67	C₃×#464	C₂²	1	4	yes	—
68	C₃×#464	C₂²	1	4	yes	—
69	C₃×#462	C₂²	1	2	yes	—
70	C₃×#460	C₂²	1	2	yes	—
71	C₃×#451	C₂²	1	2	yes	—
72	C₃×#448	C₂²	1	2	yes	—
73	C₃×#446	C₂²	1	4	yes	—
74	C₃×#446	C₂²	1	2	yes	—
75	C₃×C₈×Q₈	C₂²	1	2	yes	—
76	C₃×#441	C₂²	1	4	yes	—
77	C₃×#440	C₂²	1	4	yes	—
78	C₃×#440	C₂²	1	4	yes	—
79	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
80	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
81	C₃×C₄×Q₁₆	C₂²	1	4	yes	—
82	C₃×C₄×SD₁₆	C₂²	1	4	yes	—
83	C₃×#433	C₂²	1	4	yes	—
84	C₃×#432	C₂²	1	1	yes	characteristic
85	C₃×#431	C₂²	1	1	yes	characteristic
86	#3550	C₂×C₃	1	1	yes	characteristic
87	#3502	C₂×C₃	1	1	yes	characteristic, complemented
88	#3387	C₂×C₃	1	2	yes	complemented
89	#3386	C₂×C₃	1	4	yes	complemented
90	Q₈×Q₁₆	C₂×C₃	1	2	yes	complemented
91	#3377	C₂×C₃	1	4	yes	complemented
92	#3375	C₂×C₃	1	2	yes	complemented
93	#3365	C₂×C₃	1	2	yes	complemented
94	#3243	C₂×C₃	1	4	yes	complemented
95	#3237	C₂×C₃	1	1	yes	characteristic, complemented
96	#3228	C₂×C₃	1	1	yes	characteristic
97	#3121	C₂×C₃	1	4	yes	complemented
98	#3112	C₂×C₃	1	1	yes	characteristic
99	#3104	C₂×C₃	1	1	yes	characteristic, complemented
100	#2985	C₂×C₃	1	1	yes	characteristic, complemented
101	C₂×C₃×#41	—	2	1	no	—
102	C₂²×C₃×Q₈	—	2	2	no	—
103	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
104	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
105	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
106	C₂×C₃×SD₁₆	C₂³	1	4	yes	—
107	C₃×C₄⋊Q₈	—	2	4	no	—
108	C₃×C₄⋊Q₈	—	2	2	no	—
109	C₃×C₄⋊Q₈	—	2	2	no	—
110	C₃×C₄⋊Q₈	—	2	2	no	—
111	C₃×C₄⋊Q₈	—	2	4	no	—
112	C₃×C₄⋊Q₈	—	2	2	no	—
113	C₃×C₄⋊Q₈	—	2	4	no	—
114	C₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
115	C₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
116	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
117	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
118	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
119	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
120	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
121	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
122	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
123	C₃×#127	C₂³	1	1	yes	characteristic
124	C₃×#126	—	2	4	no	—
125	C₃×#126	—	2	4	no	—
126	C₃×#125	—	2	4	no	—
127	C₃×#125	—	2	4	no	—
128	C₃×#125	—	2	4	no	—
129	C₃×#125	—	2	2	no	—
130	C₃×#125	—	2	1	no	—
131	C₃×#125	D₄	1	2	yes	—
132	C₃×#125	C₂³	1	2	yes	—
133	C₃×#125	C₂³	1	4	yes	—
134	C₃×#124	—	2	2	no	—
135	C₃×#124	—	2	4	no	—
136	C₃×#124	—	2	1	no	—
137	C₃×#124	C₂³	1	4	yes	—
138	C₃×#123	—	2	4	no	—
139	C₃×#123	—	2	4	no	—
140	C₃×C₂²⋊Q₈	—	2	2	no	—
141	C₃×C₂²⋊Q₈	—	2	1	no	—
142	C₃×C₂²⋊Q₈	—	2	1	no	—
143	C₃×C₂²⋊Q₈	—	2	1	no	—
144	C₃×C₂²⋊Q₈	—	2	4	no	—
145	C₃×C₂²⋊Q₈	—	2	4	no	—
146	C₃×C₂²⋊Q₈	—	2	2	no	—
147	C₃×C₂²⋊Q₈	—	2	4	no	—
148	C₃×C₂²⋊Q₈	D₄	1	2	yes	—
149	C₃×C₂²⋊Q₈	D₄	1	2	yes	—
150	C₃×C₂²⋊Q₈	C₂³	1	2	yes	—
151	C₃×#121	—	2	1	no	—
152	C₃×#121	C₂³	1	2	yes	—
153	C₃×C₄×Q₈	—	2	2	no	—
154	C₃×C₄×Q₈	—	2	2	no	—
155	C₃×C₄×Q₈	—	2	2	no	—
156	C₃×C₄×Q₈	—	2	2	no	—
157	C₃×C₄×Q₈	—	2	2	no	—
158	C₃×C₄×Q₈	—	2	4	no	—
159	C₃×C₄×Q₈	—	2	1	no	—
160	C₃×C₄×Q₈	—	2	1	no	—
161	C₃×C₄×Q₈	—	2	1	no	—
162	C₃×C₄×Q₈	—	2	1	no	—
163	C₃×C₄×Q₈	—	2	1	no	—
164	C₃×C₄×Q₈	C₂³	1	2	yes	—
165	C₃×C₄×Q₈	C₂³	1	2	yes	—
166	C₃×C₄×Q₈	C₂³	1	4	yes	—
167	C₃×C₄×Q₈	C₂³	1	4	yes	—
168	C₃×C₄×Q₈	C₂³	1	4	yes	—
169	C₃×C₄×Q₈	C₂³	1	2	yes	—
170	C₃×C₄×Q₈	C₂³	1	2	yes	—
171	C₃×C₄×Q₈	C₂³	1	2	yes	—
172	C₃×C₄×D₄	—	2	2	no	—
173	C₃×C₄×D₄	—	2	1	no	—
174	C₃×C₄×D₄	C₂³	1	4	yes	—
175	C₃×C₄×D₄	C₂³	1	2	yes	—
176	C₃×#117	—	2	1	no	—
177	C₃×#117	—	2	2	no	—
178	C₃×#117	—	2	4	no	—
179	C₃×#117	C₂³	1	2	yes	—
180	C₃×#117	C₂³	1	2	yes	—
181	C₂×C₃×C₄⋊C₄	—	2	1	no	—
182	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
183	C₂×C₃×C₄²	—	2	1	no	—
184	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
185	C₃×#107	C₂³	1	2	yes	—
186	C₃×#107	C₂³	1	4	yes	—
187	C₃×#107	C₂³	1	2	yes	—
188	C₃×#107	C₂³	1	4	yes	—
189	C₃×#107	C₂³	1	4	yes	—
190	C₃×#106	C₂³	1	4	yes	—
191	C₃×#106	C₂³	1	4	yes	—
192	C₃×C₄⋊C₈	C₂³	1	2	yes	—
193	C₃×C₄⋊C₈	C₂³	1	2	yes	—
194	C₃×C₄⋊C₈	C₂³	1	4	yes	—
195	C₃×C₄⋊C₈	C₂³	1	4	yes	—
196	C₃×#103	C₂³	1	2	yes	—
197	C₃×#103	C₂³	1	2	yes	—
198	C₃×#103	C₂³	1	4	yes	—
199	C₃×#103	C₂³	1	2	yes	—
200	C₃×#103	C₂³	1	2	yes	—
201	C₃×#103	C₂³	1	4	yes	—
202	C₃×#103	C₂³	1	4	yes	—
203	C₃×#102	C₂³	1	4	yes	—
204	C₃×#102	C₂³	1	4	yes	—
205	C₃×#102	C₂³	1	4	yes	—
206	C₃×C₂²⋊C₈	C₂³	1	2	yes	—
207	C₃×C₂²⋊C₈	C₂³	1	2	yes	—
208	C₃×#97	C₂³	1	2	yes	—
209	C₃×#97	C₂³	1	4	yes	—
210	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
211	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
212	C₃×C₄×C₈	C₂³	1	4	yes	—
213	#563	—	2	2	no	—
214	#559	C₂²×C₃	1	2	yes	—
215	Q₈²	—	2	1	no	—
216	Q₈²	C₂²×C₃	1	2	yes	—
217	Q₈²	C₂²×C₃	1	2	yes	—
218	#557	—	2	1	no	—
219	#557	—	2	2	no	—
220	#557	C₂²×C₃	1	2	yes	—
221	#557	C₂²×C₃	1	4	yes	—
222	#556	—	2	1	no	—
223	#556	C₂²×C₃	1	2	yes	—
224	D₄×Q₈	—	2	1	no	—
225	D₄×Q₈	C₂²×C₃	1	2	yes	—
226	#544	—	2	2	no	—
227	#544	C₂²×C₃	1	2	yes	—
228	#542	—	2	2	no	—
229	#542	C₂²×C₃	1	4	yes	—
230	#536	—	2	2	no	—
231	#533	—	2	2	no	—
232	#533	C₂²×C₃	1	1	yes	characteristic
233	#533	C₂²×C₃	1	1	yes	characteristic
234	#533	C₂²×C₃	1	1	yes	characteristic
235	#532	C₂²×C₃	1	1	yes	characteristic
236	#519	C₂²×C₃	1	2	yes	—
237	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
238	#501	C₂²×C₃	1	2	yes	—
239	#501	C₂²×C₃	1	4	yes	—
240	#500	C₂²×C₃	1	1	yes	characteristic
241	#500	C₂²×C₃	1	4	yes	—
242	#498	C₂²×C₃	1	1	yes	characteristic
243	#491	C₂²×C₃	1	2	yes	—
244	#489	C₂²×C₃	1	4	yes	—
245	#488	C₂²×C₃	1	4	yes	—
246	#486	C₂²×C₃	1	1	yes	characteristic
247	#486	C₂²×C₃	1	1	yes	characteristic
248	#485	C₂²×C₃	1	4	yes	—
249	#484	C₂²×C₃	1	2	yes	—
250	#482	C₂²×C₃	1	4	yes	—
251	#480	C₂²×C₃	1	2	yes	—
252	#479	C₂²×C₃	1	4	yes	—
253	#478	C₂²×C₃	1	4	yes	—
254	#478	C₂²×C₃	1	4	yes	—
255	#477	C₂²×C₃	1	2	yes	—
256	#477	C₂²×C₃	1	2	yes	—
257	#477	C₂²×C₃	1	2	yes	—
258	#477	C₂²×C₃	1	4	yes	—
259	#477	C₂²×C₃	1	4	yes	—
260	#475	C₂²×C₃	1	2	yes	—
261	#475	C₂²×C₃	1	4	yes	—
262	#474	C₂²×C₃	1	4	yes	—
263	#464	C₂²×C₃	1	4	yes	—
264	#464	C₂²×C₃	1	4	yes	—
265	#462	C₂²×C₃	1	2	yes	—
266	#460	C₂²×C₃	1	2	yes	—
267	#451	C₂²×C₃	1	2	yes	—
268	#448	C₂²×C₃	1	2	yes	—
269	#446	C₂²×C₃	1	2	yes	—
270	#446	C₂²×C₃	1	4	yes	—
271	C₈×Q₈	C₂²×C₃	1	2	yes	—
272	#441	C₂²×C₃	1	4	yes	—
273	#440	C₂²×C₃	1	4	yes	—
274	#440	C₂²×C₃	1	4	yes	—
275	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
276	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
277	C₄×Q₁₆	C₂²×C₃	1	4	yes	—
278	C₄×SD₁₆	C₂²×C₃	1	4	yes	—
279	#433	C₂²×C₃	1	4	yes	—
280	#432	C₂²×C₃	1	1	yes	characteristic
281	#431	C₂²×C₃	1	1	yes	characteristic
282	C₃×#41	—	8	2	no	—
283	C₂×C₃×Q₈	—	2	4	no	—
284	C₂×C₃×Q₈	—	2	1	no	—
285	C₂×C₃×Q₈	—	2	1	no	—
286	C₂×C₃×Q₈	—	2	1	no	—
287	C₂×C₃×Q₈	—	2	1	no	—
288	C₂×C₃×Q₈	—	2	4	no	—
289	C₂×C₃×Q₈	—	2	4	no	—
290	C₂×C₃×Q₈	—	2	1	no	—
291	C₂×C₃×Q₈	—	8	4	no	—
292	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
293	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
294	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
295	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
296	C₂×C₃×D₄	—	2	2	no	—
297	C₂×C₃×D₄	—	2	1	no	—
298	C₂×C₃×D₄	—	2	1	no	—
299	C₂×C₃×D₄	C₂⁴	1	2	yes	—
300	C₂²×C₃×C₄	—	2	1	no	—
301	C₂²×C₃×C₄	—	2	1	no	—
302	C₂²×C₃×C₄	—	2	2	no	—
303	C₂²×C₃×C₄	—	2	4	no	—
304	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
305	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
306	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
307	C₃×Q₁₆	—	2	4	no	—
308	C₃×Q₁₆	—	2	4	no	—
309	C₃×SD₁₆	—	2	8	no	—
310	C₂×C₃×C₈	C₂⁴	1	2	yes	—
311	C₂×C₃×C₈	C₂⁴	1	2	yes	—
312	C₂×C₃×C₈	C₂⁴	1	4	yes	—
313	C₃×C₄⋊C₄	—	2	2	no	—
314	C₃×C₄⋊C₄	—	2	2	no	—
315	C₃×C₄⋊C₄	—	2	1	no	—
316	C₃×C₄⋊C₄	—	2	1	no	—
317	C₃×C₄⋊C₄	—	2	4	no	—
318	C₃×C₄⋊C₄	—	2	4	no	—
319	C₃×C₄⋊C₄	—	2	1	no	—
320	C₃×C₄⋊C₄	—	2	1	no	—
321	C₃×C₄⋊C₄	—	2	1	no	—
322	C₃×C₄⋊C₄	—	2	2	no	—
323	C₃×C₄⋊C₄	—	2	1	no	—
324	C₃×C₄⋊C₄	—	2	2	no	—
325	C₃×C₄⋊C₄	—	2	4	no	—
326	C₃×C₄⋊C₄	—	2	1	no	—
327	C₃×C₄⋊C₄	—	2	4	no	—
328	C₃×C₄⋊C₄	—	2	4	no	—
329	C₃×C₄⋊C₄	—	2	2	no	—
330	C₃×C₄⋊C₄	—	2	4	no	—
331	C₃×C₄⋊C₄	—	2	4	no	—
332	C₃×C₄⋊C₄	—	2	2	no	—
333	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
334	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
335	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
336	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
337	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
338	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
339	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
340	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
341	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
342	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
343	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
344	C₃×C₂²⋊C₄	—	2	2	no	—
345	C₃×C₂²⋊C₄	—	2	2	no	—
346	C₃×C₂²⋊C₄	—	2	4	no	—
347	C₃×C₂²⋊C₄	—	2	2	no	—
348	C₃×C₂²⋊C₄	—	2	4	no	—
349	C₃×C₂²⋊C₄	—	2	2	no	—
350	C₃×C₂²⋊C₄	—	2	4	no	—
351	C₃×C₂²⋊C₄	C₂×D₄	1	4	yes	—
352	C₃×C₄²	—	2	2	no	—
353	C₃×C₄²	—	2	2	no	—
354	C₃×C₄²	—	2	2	no	—
355	C₃×C₄²	—	2	4	no	—
356	C₃×C₄²	—	2	4	no	—
357	C₃×C₄²	—	2	4	no	—
358	C₃×C₄²	—	2	2	no	—
359	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
360	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
361	C₃×C₄²	C₂⁴	1	1	yes	characteristic
362	C₃×C₄²	C₂⁴	1	4	yes	—
363	C₃×C₄²	C₂⁴	1	1	yes	characteristic
364	C₂×#41	—	2	1	no	—
365	C₂²×Q₈	—	2	2	no	—
366	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
367	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
368	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
369	C₂×SD₁₆	C₂³×C₃	1	4	yes	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	—	2	4	no	—
372	C₄⋊Q₈	—	2	2	no	—
373	C₄⋊Q₈	—	2	4	no	—
374	C₄⋊Q₈	—	2	2	no	—
375	C₄⋊Q₈	—	2	4	no	—
376	C₄⋊Q₈	—	2	2	no	—
377	C₄⋊Q₈	C₃×D₄	1	1	yes	characteristic
378	C₄⋊Q₈	C₃×D₄	1	1	yes	characteristic
379	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
380	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
381	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
382	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
383	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
384	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
385	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
386	#127	C₂³×C₃	1	1	yes	characteristic
387	#126	—	2	4	no	—
388	#126	—	2	4	no	—
389	#125	—	2	4	no	—
390	#125	—	2	4	no	—
391	#125	—	2	4	no	—
392	#125	—	2	2	no	—
393	#125	—	2	1	no	—
394	#125	C₃×D₄	1	2	yes	—
395	#125	C₂³×C₃	1	2	yes	—
396	#125	C₂³×C₃	1	4	yes	—
397	#124	—	2	2	no	—
398	#124	—	2	4	no	—
399	#124	—	2	1	no	—
400	#124	C₂³×C₃	1	4	yes	—
401	#123	—	2	4	no	—
402	#123	—	2	4	no	—
403	C₂²⋊Q₈	—	2	2	no	—
404	C₂²⋊Q₈	—	2	1	no	—
405	C₂²⋊Q₈	—	2	1	no	—
406	C₂²⋊Q₈	—	2	1	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	4	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₂²⋊Q₈	C₃×D₄	1	2	yes	—
412	C₂²⋊Q₈	C₃×D₄	1	2	yes	—
413	C₂²⋊Q₈	C₂³×C₃	1	2	yes	—
414	#121	—	2	1	no	—
415	#121	C₂³×C₃	1	2	yes	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	2	no	—
418	C₄×Q₈	—	2	2	no	—
419	C₄×Q₈	—	2	2	no	—
420	C₄×Q₈	—	2	4	no	—
421	C₄×Q₈	—	2	2	no	—
422	C₄×Q₈	—	2	1	no	—
423	C₄×Q₈	—	2	1	no	—
424	C₄×Q₈	—	2	1	no	—
425	C₄×Q₈	—	2	1	no	—
426	C₄×Q₈	—	2	1	no	—
427	C₄×Q₈	C₂³×C₃	1	2	yes	—
428	C₄×Q₈	C₂³×C₃	1	2	yes	—
429	C₄×Q₈	C₂³×C₃	1	4	yes	—
430	C₄×Q₈	C₂³×C₃	1	4	yes	—
431	C₄×Q₈	C₂³×C₃	1	2	yes	—
432	C₄×Q₈	C₂³×C₃	1	4	yes	—
433	C₄×Q₈	C₂³×C₃	1	2	yes	—
434	C₄×Q₈	C₂³×C₃	1	2	yes	—
435	C₄×D₄	—	2	2	no	—
436	C₄×D₄	—	2	1	no	—
437	C₄×D₄	C₂³×C₃	1	2	yes	—
438	C₄×D₄	C₂³×C₃	1	4	yes	—
439	#117	—	2	1	no	—
440	#117	—	2	2	no	—
441	#117	—	2	4	no	—
442	#117	C₂³×C₃	1	2	yes	—
443	#117	C₂³×C₃	1	2	yes	—
444	C₂×C₄⋊C₄	—	2	1	no	—
445	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
448	#107	C₂³×C₃	1	2	yes	—
449	#107	C₂³×C₃	1	4	yes	—
450	#107	C₂³×C₃	1	2	yes	—
451	#107	C₂³×C₃	1	4	yes	—
452	#107	C₂³×C₃	1	4	yes	—
453	#106	C₂³×C₃	1	4	yes	—
454	#106	C₂³×C₃	1	4	yes	—
455	C₄⋊C₈	C₂³×C₃	1	2	yes	—
456	C₄⋊C₈	C₂³×C₃	1	2	yes	—
457	C₄⋊C₈	C₂³×C₃	1	4	yes	—
458	C₄⋊C₈	C₂³×C₃	1	4	yes	—
459	#103	C₂³×C₃	1	2	yes	—
460	#103	C₂³×C₃	1	2	yes	—
461	#103	C₂³×C₃	1	4	yes	—
462	#103	C₂³×C₃	1	2	yes	—
463	#103	C₂³×C₃	1	2	yes	—
464	#103	C₂³×C₃	1	4	yes	—
465	#103	C₂³×C₃	1	4	yes	—
466	#102	C₂³×C₃	1	4	yes	—
467	#102	C₂³×C₃	1	4	yes	—
468	#102	C₂³×C₃	1	4	yes	—
469	C₂²⋊C₈	C₂³×C₃	1	2	yes	—
470	C₂²⋊C₈	C₂³×C₃	1	2	yes	—
471	#97	C₂³×C₃	1	2	yes	—
472	#97	C₂³×C₃	1	4	yes	—
473	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
474	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
475	C₄×C₈	C₂³×C₃	1	4	yes	—
476	C₂³×C₃	—	2	2	no	—
477	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
478	C₃×Q₈	—	2	4	no	—
479	C₃×Q₈	—	2	4	no	—
480	C₃×Q₈	—	2	2	no	—
481	C₃×Q₈	—	2	2	no	—
482	C₃×Q₈	—	2	2	no	—
483	C₃×Q₈	—	4	2	no	—
484	C₃×Q₈	—	4	2	no	—
485	C₃×Q₈	—	4	2	no	—
486	C₃×Q₈	—	8	1	no	—
487	C₃×Q₈	—	8	4	no	—
488	C₃×Q₈	—	8	4	no	—
489	C₃×Q₈	—	8	1	no	—
490	C₃×Q₈	2₋¹⁺⁴	1	4	yes	—
491	C₃×D₄	—	2	4	no	—
492	C₃×D₄	—	4	2	no	—
493	C₃×D₄	—	8	1	no	—
494	C₃×D₄	—	8	2	no	—
495	C₂×C₃×C₄	—	2	4	no	—
496	C₂×C₃×C₄	—	2	2	no	—
497	C₂×C₃×C₄	—	2	2	no	—
498	C₂×C₃×C₄	—	2	2	no	—
499	C₂×C₃×C₄	—	2	2	no	—
500	C₂×C₃×C₄	—	2	2	no	—
501	C₂×C₃×C₄	—	2	2	no	—
502	C₂×C₃×C₄	—	2	2	no	—
503	C₂×C₃×C₄	—	2	2	no	—
504	C₂×C₃×C₄	—	2	2	no	—
505	C₂×C₃×C₄	—	4	1	no	—
506	C₂×C₃×C₄	—	4	4	no	—
507	C₂×C₃×C₄	—	8	4	no	—
508	C₂×C₃×C₄	—	8	1	no	—
509	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
510	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
511	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
512	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
513	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
514	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
515	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
516	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
517	C₃×C₈	—	2	4	no	—
518	C₃×C₈	—	2	2	no	—
519	C₃×C₈	—	2	2	no	—
520	#41	—	8	2	no	—
521	C₂×Q₈	—	2	4	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	4	no	—
527	C₂×Q₈	—	2	4	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	8	4	no	—
530	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
531	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
532	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
533	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
534	C₂×D₄	—	2	2	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	C₂⁴×C₃	1	2	yes	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
543	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
544	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
545	Q₁₆	—	2	4	no	—
546	Q₁₆	—	2	4	no	—
547	SD₁₆	—	2	8	no	—
548	C₂×C₈	C₂⁴×C₃	1	2	yes	—
549	C₂×C₈	C₂⁴×C₃	1	2	yes	—
550	C₂×C₈	C₂⁴×C₃	1	4	yes	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	4	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
572	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
573	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
574	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
575	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
576	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
577	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
578	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
579	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
580	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
581	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	2	no	—
586	C₂²⋊C₄	—	2	4	no	—
587	C₂²⋊C₄	—	2	2	no	—
588	C₂²⋊C₄	—	2	4	no	—
589	C₂²⋊C₄	C₂×C₃×D₄	1	4	yes	—
590	C₄²	—	2	2	no	—
591	C₄²	—	2	2	no	—
592	C₄²	—	2	2	no	—
593	C₄²	—	2	4	no	—
594	C₄²	—	2	4	no	—
595	C₄²	—	2	4	no	—
596	C₄²	—	2	2	no	—
597	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
598	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
599	C₄²	C₂⁴×C₃	1	1	yes	characteristic
600	C₄²	C₂⁴×C₃	1	4	yes	—
601	C₄²	C₂⁴×C₃	1	1	yes	characteristic
602	C₂²×C₃	—	2	1	no	—
603	C₂²×C₃	—	2	1	no	—
604	C₂²×C₃	—	2	1	no	—
605	C₂²×C₃	—	4	2	no	—
606	C₂²×C₃	—	4	2	no	—
607	C₂²×C₃	—	4	2	no	—
608	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
609	C₃×C₄	—	2	1	no	—
610	C₃×C₄	—	2	1	no	—
611	C₃×C₄	—	2	4	no	—
612	C₃×C₄	—	2	4	no	—
613	C₃×C₄	—	2	1	no	—
614	C₃×C₄	—	2	1	no	—
615	C₃×C₄	—	2	4	no	—
616	C₃×C₄	—	4	2	no	—
617	C₃×C₄	—	4	2	no	—
618	C₃×C₄	—	4	2	no	—
619	C₃×C₄	—	4	2	no	—
620	C₃×C₄	—	4	2	no	—
621	C₃×C₄	—	4	2	no	—
622	C₃×C₄	#578	1	2	yes	—
623	C₃×C₄	C₂×2₋¹⁺⁴	1	2	yes	—
624	C₂³	—	2	2	no	—
625	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
626	Q₈	—	2	4	no	—
627	Q₈	—	2	4	no	—
628	Q₈	—	2	2	no	—
629	Q₈	—	2	2	no	—
630	Q₈	—	2	2	no	—
631	Q₈	—	4	2	no	—
632	Q₈	—	4	2	no	—
633	Q₈	—	4	2	no	—
634	Q₈	—	8	4	no	—
635	Q₈	—	8	1	no	—
636	Q₈	—	8	1	no	—
637	Q₈	—	8	4	no	—
638	Q₈	C₃×2₋¹⁺⁴	1	4	yes	—
639	D₄	—	2	4	no	—
640	D₄	—	4	2	no	—
641	D₄	—	8	1	no	—
642	D₄	—	8	2	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	4	no	—
655	C₂×C₄	—	8	4	no	—
656	C₂×C₄	—	8	1	no	—
657	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
658	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
659	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
660	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
661	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
662	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
663	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
664	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
665	C₈	—	2	4	no	—
666	C₈	—	2	2	no	—
667	C₈	—	2	2	no	—
668	C₂×C₃	—	4	1	no	—
669	C₂×C₃	—	8	2	no	—
670	C₂×C₃	#3486	1	1	yes	characteristic
671	C₂×C₃	C₂×#578	1	1	yes	characteristic
672	C₂×C₃	#3585	1	1	yes	characteristic
673	C₂²	—	2	1	no	—
674	C₂²	—	2	1	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	4	2	no	—
679	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
680	C₄	—	2	4	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	1	no	—
683	C₄	—	2	4	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	1	no	—
686	C₄	—	2	4	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	—	4	2	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	2	no	—
692	C₄	—	4	2	no	—
693	C₄	C₃×#578	1	2	yes	—
694	C₄	C₂×C₃×2₋¹⁺⁴	1	2	yes	—
695	C₃	#61854	1	1	yes	characteristic, direct product factor, Sylow
696	C₂	—	4	1	no	—
697	C₂	—	8	2	no	—
698	C₂	C₃×#3486	1	1	yes	characteristic
699	C₂	C₂×C₃×#578	1	1	yes	characteristic
700	C₂	C₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#533 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#3502	C₂	semidirect
3. C₃×#3387	C₂	semidirect
4. C₃×#3386	C₂	semidirect
5. C₃×Q₈×Q₁₆	C₂	semidirect
6. C₃×#3377	C₂	semidirect
7. C₃×#3375	C₂	semidirect
8. C₃×#3365	C₂	semidirect
9. C₃×#3243	C₂	semidirect
10. C₃×#3237	C₂	semidirect
12. C₃×#3121	C₂	semidirect
14. C₃×#3104	C₂	semidirect
15. C₃×#2985	C₂	semidirect
16. #61854	C₃	direct (N × H)
87. #3502	C₂×C₃	semidirect
88. #3387	C₂×C₃	semidirect
89. #3386	C₂×C₃	semidirect
90. Q₈×Q₁₆	C₂×C₃	semidirect
91. #3377	C₂×C₃	semidirect
92. #3375	C₂×C₃	semidirect
93. #3365	C₂×C₃	semidirect
94. #3243	C₂×C₃	semidirect
95. #3237	C₂×C₃	semidirect
97. #3121	C₂×C₃	semidirect
99. #3104	C₂×C₃	semidirect
100. #2985	C₂×C₃	semidirect
695. C₃	#61854	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×#3550	2
C₃×#3502	2
C₃×#3387	2
C₃×#3386	2
C₃×Q₈×Q₁₆	2
C₃×#3377	2
C₃×#3375	2
C₃×#3365	2
C₃×#3243	2
C₃×#3237	2
C₃×#3228	2
C₃×#3121	2
C₃×#3112	2
C₃×#3104	2
C₃×#2985	2
#61854	3

Subgroups — C3×#61854 #10680461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#61854 #10680461