Subgroups — C₃²×#5822 #13179202

Order 1728.

2228 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#951	C₂	1	2	yes	maximal, complemented
4	C₃²×#950	C₂	1	4	yes	maximal, complemented
5	C₃²×#949	C₂	1	2	yes	maximal, complemented
6	C₃²×#942	C₂	1	2	yes	maximal, complemented
7	C₃²×#932	C₂	1	2	yes	maximal, complemented
8	C₂×C₃²×#208	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#575	—	3	1	no	maximal
10	C₃×#5822	C₃	1	4	yes	maximal, direct product factor
11	C₃³×#41	—	2	1	no	—
12	C₃³×#41	—	2	1	no	—
13	C₃³×#41	C₂²	1	2	yes	—
14	C₃³×#41	C₂²	1	2	yes	complemented
15	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃³×D₄	—	2	1	no	—
17	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃³×C₄	—	2	1	no	—
19	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃²×#67	—	2	1	no	—
21	C₃²×#235	—	2	1	no	—
22	C₃²×#235	—	2	1	no	—
23	C₃²×#235	C₂²	1	2	yes	—
24	C₃²×#235	C₂²	1	2	yes	complemented
25	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₃²×C₄×D₃	—	2	1	no	—
27	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic, complemented
28	C₃²×#216	C₂²	1	4	yes	complemented
29	C₃²×#215	C₂²	1	4	yes	complemented
30	C₃²×#214	C₂²	1	4	yes	complemented
31	C₃²×#213	C₂²	1	4	yes	complemented
32	C₃²×#208	C₂²	1	2	yes	complemented
33	C₃²×#208	C₂²	1	2	yes	complemented
34	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	complemented
35	C₂×C₃²×#41	—	2	4	no	—
36	C₂×C₃²×#41	—	3	1	no	—
37	C₂×C₃²×#41	D₃	1	1	yes	characteristic, complemented
38	C₂×C₃²×#41	C₂×C₃	1	4	yes	complemented
39	C₂×C₃×#235	C₂×C₃	1	4	yes	complemented
40	C₃²×#137	—	3	2	no	—
41	C₃²×#137	—	3	2	no	—
42	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
43	C₃²×AGL(1,ℤ/8)	—	3	2	no	—
44	C₃²×#135	—	3	4	no	—
45	C₂×C₃²×M₁₆	—	3	1	no	—
46	C₃×#951	C₂×C₃	1	8	yes	complemented
47	C₃×#950	C₂×C₃	1	16	yes	complemented
48	C₃×#949	C₂×C₃	1	8	yes	complemented
49	C₃×#942	C₂×C₃	1	8	yes	complemented
50	C₃×#932	C₂×C₃	1	8	yes	complemented
51	C₂×C₃×#208	C₂×C₃	1	4	yes	complemented
52	C₂³×C₃³	—	2	1	no	—
53	C₂³×C₃³	D₄	1	1	yes	characteristic
54	C₂²×C₃²×D₃	—	2	1	no	—
55	C₃³×Q₈	—	2	1	no	—
56	C₃³×Q₈	C₂³	1	2	yes	—
57	C₃³×D₄	—	2	2	no	—
58	C₃³×D₄	—	2	2	no	—
59	C₃³×D₄	—	2	1	no	—
60	C₃³×D₄	C₂³	1	2	yes	—
61	C₂×C₃³×C₄	—	2	1	no	—
62	C₂×C₃³×C₄	—	2	2	no	—
63	C₂×C₃³×C₄	—	2	2	no	—
64	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
65	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
66	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
67	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
69	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
70	C₃²×#67	—	2	2	no	—
71	C₃²×#67	—	2	2	no	—
72	C₂×C₃²×Dic₃	—	2	1	no	—
73	C₃²×D₁₂	—	2	1	no	—
74	C₃²×D₁₂	C₂³	1	2	yes	—
75	C₃²×C₄×D₃	—	2	2	no	—
76	C₃²×C₄×D₃	—	2	2	no	—
77	C₃²×Dic₆	—	2	1	no	—
78	C₃²×Dic₆	C₂³	1	2	yes	—
79	C₃²×C₃⋊C₈	C₂³	1	4	yes	—
80	C₃×#575	—	3	4	no	—
81	#5822	C₃²	1	1	yes	characteristic, direct product factor
82	C₃²×#41	—	2	8	no	—
83	C₃²×#41	—	2	8	no	—
84	C₃²×#41	—	2	8	no	—
85	C₃²×#41	—	2	8	no	—
86	C₃²×#41	—	2	1	no	—
87	C₃²×#41	—	2	1	no	—
88	C₃²×#41	—	2	4	no	—
89	C₃²×#41	—	2	4	no	—
90	C₃²×#41	—	3	2	no	—
91	C₃²×#41	—	3	2	no	—
92	C₃²×#41	—	6	1	no	—
93	C₃²×#41	—	6	1	no	—
94	C₃²×#41	C₂×D₃	1	2	yes	—
95	C₃²×#41	C₂×D₃	1	2	yes	complemented
96	C₃²×#41	C₂²×C₃	1	8	yes	—
97	C₃²×#41	C₂²×C₃	1	8	yes	complemented
98	C₂×C₃²×Q₈	—	2	4	no	—
99	C₂×C₃²×Q₈	—	3	1	no	—
100	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
101	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	complemented
102	C₂×C₃²×D₄	—	2	8	no	—
103	C₂×C₃²×D₄	—	2	4	no	—
104	C₂×C₃²×D₄	—	2	1	no	—
105	C₂×C₃²×D₄	—	2	4	no	—
106	C₂×C₃²×D₄	—	3	1	no	—
107	C₂×C₃²×D₄	—	6	1	no	—
108	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic, complemented
109	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	complemented
110	C₂²×C₃²×C₄	—	2	8	no	—
111	C₂²×C₃²×C₄	—	2	4	no	—
112	C₂²×C₃²×C₄	—	2	1	no	—
113	C₂²×C₃²×C₄	—	2	4	no	—
114	C₂²×C₃²×C₄	—	6	1	no	—
115	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
116	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
117	C₂×C₃×#67	—	2	4	no	—
118	C₃×#235	—	2	4	no	—
119	C₃×#235	—	2	4	no	—
120	C₃×#235	C₂²×C₃	1	8	yes	—
121	C₃×#235	C₂²×C₃	1	8	yes	complemented
122	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	complemented
123	C₂×C₃×C₄×D₃	—	2	4	no	—
124	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	complemented
125	C₃²×Q₁₆	—	3	4	no	—
126	C₃²×SD₁₆	—	3	4	no	—
127	C₃²×SD₁₆	—	3	4	no	—
128	C₃²×D₈	—	3	4	no	—
129	C₃²×M₁₆	—	3	2	no	—
130	C₃²×M₁₆	—	3	2	no	—
131	C₂×C₃²×C₈	—	3	2	no	—
132	C₃×#216	C₂²×C₃	1	16	yes	complemented
133	C₃×#215	C₂²×C₃	1	16	yes	complemented
134	C₃×#214	C₂²×C₃	1	16	yes	complemented
135	C₃×#213	C₂²×C₃	1	16	yes	complemented
136	C₃×#208	C₂²×C₃	1	8	yes	complemented
137	C₃×#208	C₂²×C₃	1	8	yes	complemented
138	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	complemented
139	C₂²×C₃³	—	2	2	no	—
140	C₂²×C₃³	—	4	1	no	—
141	C₂²×C₃³	—	4	2	no	—
142	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃²×D₃	—	2	2	no	—
146	C₂×C₃²×D₃	—	4	2	no	—
147	C₃³×C₄	—	2	2	no	—
148	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
149	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
150	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
151	C₃³×C₄	C₂⁴	1	1	yes	characteristic
152	C₃²×Dic₃	—	2	2	no	—
153	C₂×C₃×#41	—	2	4	no	—
154	C₂×C₃×#41	—	3	4	no	—
155	C₂×C₃×#41	C₃×D₃	1	4	yes	complemented
156	C₂×C₃×#41	C₂×C₃²	1	1	yes	characteristic, complemented
157	C₂×#235	C₂×C₃²	1	1	yes	characteristic, complemented
158	C₃×#137	—	3	8	no	—
159	C₃×#137	—	3	8	no	—
160	C₃×AGL(1,ℤ/8)	—	3	8	no	—
161	C₃×AGL(1,ℤ/8)	—	3	8	no	—
162	C₃×#135	—	3	16	no	—
163	C₂×C₃×M₁₆	—	3	4	no	—
164	#951	C₂×C₃²	1	2	yes	complemented
165	#950	C₂×C₃²	1	4	yes	complemented
166	#949	C₂×C₃²	1	2	yes	complemented
167	#942	C₂×C₃²	1	2	yes	complemented
168	#932	C₂×C₃²	1	2	yes	complemented
169	C₂×#208	C₂×C₃²	1	1	yes	characteristic, complemented
170	C₂³×C₃²	—	2	8	no	—
171	C₂³×C₃²	—	2	1	no	—
172	C₂³×C₃²	—	2	4	no	—
173	C₂³×C₃²	—	2	4	no	—
174	C₂³×C₃²	—	6	1	no	—
175	C₂³×C₃²	#67	1	1	yes	characteristic
176	C₂³×C₃²	C₃×D₄	1	4	yes	—
177	C₂²×C₃×D₃	—	2	4	no	—
178	C₃²×Q₈	—	2	8	no	—
179	C₃²×Q₈	—	2	1	no	—
180	C₃²×Q₈	—	2	4	no	—
181	C₃²×Q₈	—	2	8	no	—
182	C₃²×Q₈	—	3	2	no	—
183	C₃²×Q₈	—	6	1	no	—
184	C₃²×Q₈	C₂²×D₃	1	2	yes	—
185	C₃²×Q₈	C₂³×C₃	1	8	yes	—
186	C₃²×D₄	—	2	16	no	—
187	C₃²×D₄	—	2	16	no	—
188	C₃²×D₄	—	2	8	no	—
189	C₃²×D₄	—	2	2	no	—
190	C₃²×D₄	—	2	2	no	—
191	C₃²×D₄	—	2	8	no	—
192	C₃²×D₄	—	2	8	no	—
193	C₃²×D₄	—	2	1	no	—
194	C₃²×D₄	—	2	4	no	—
195	C₃²×D₄	—	2	8	no	—
196	C₃²×D₄	—	3	2	no	—
197	C₃²×D₄	—	6	2	no	—
198	C₃²×D₄	—	6	2	no	—
199	C₃²×D₄	—	6	1	no	—
200	C₃²×D₄	C₂²×D₃	1	2	yes	—
201	C₃²×D₄	C₂³×C₃	1	8	yes	—
202	C₂×C₃²×C₄	—	2	8	no	—
203	C₂×C₃²×C₄	—	2	16	no	—
204	C₂×C₃²×C₄	—	2	16	no	—
205	C₂×C₃²×C₄	—	2	1	no	—
206	C₂×C₃²×C₄	—	2	4	no	—
207	C₂×C₃²×C₄	—	2	2	no	—
208	C₂×C₃²×C₄	—	2	2	no	—
209	C₂×C₃²×C₄	—	2	8	no	—
210	C₂×C₃²×C₄	—	2	8	no	—
211	C₂×C₃²×C₄	—	2	4	no	—
212	C₂×C₃²×C₄	—	2	4	no	—
213	C₂×C₃²×C₄	—	2	4	no	—
214	C₂×C₃²×C₄	—	2	4	no	—
215	C₂×C₃²×C₄	—	2	4	no	—
216	C₂×C₃²×C₄	—	2	4	no	—
217	C₂×C₃²×C₄	—	6	1	no	—
218	C₂×C₃²×C₄	—	6	2	no	—
219	C₂×C₃²×C₄	—	6	2	no	—
220	C₂×C₃²×C₄	#67	1	1	yes	characteristic
221	C₂×C₃²×C₄	#67	1	1	yes	characteristic
222	C₂×C₃²×C₄	#67	1	1	yes	characteristic
223	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
224	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
225	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
226	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
227	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
228	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
229	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
230	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
231	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
232	C₃×#67	—	2	8	no	—
233	C₃×#67	—	2	8	no	—
234	C₂×C₃×Dic₃	—	2	4	no	—
235	C₃×D₁₂	—	2	4	no	—
236	C₃×D₁₂	C₂³×C₃	1	8	yes	—
237	C₃×C₄×D₃	—	2	8	no	—
238	C₃×C₄×D₃	—	2	8	no	—
239	C₃×Dic₆	—	2	4	no	—
240	C₃×Dic₆	C₂³×C₃	1	8	yes	—
241	C₃²×C₈	—	3	4	no	—
242	C₃×C₃⋊C₈	C₂³×C₃	1	16	yes	—
243	#575	—	3	1	no	Sylow
244	C₂×C₃³	—	2	1	no	—
245	C₂×C₃³	—	2	1	no	—
246	C₂×C₃³	—	2	1	no	—
247	C₂×C₃³	—	4	2	no	—
248	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
249	C₃²×D₃	—	4	2	no	—
250	C₃×#41	—	2	8	no	—
251	C₃×#41	—	2	8	no	—
252	C₃×#41	—	2	4	no	—
253	C₃×#41	—	2	4	no	—
254	C₃×#41	—	2	1	no	—
255	C₃×#41	—	2	1	no	—
256	C₃×#41	—	2	8	no	—
257	C₃×#41	—	2	8	no	—
258	C₃×#41	—	3	8	no	—
259	C₃×#41	—	3	8	no	—
260	C₃×#41	—	6	4	no	—
261	C₃×#41	—	6	4	no	—
262	C₃×#41	C₂×C₃×D₃	1	8	yes	—
263	C₃×#41	C₂×C₃×D₃	1	8	yes	complemented
264	C₃×#41	C₂²×C₃²	1	2	yes	—
265	C₃×#41	C₂²×C₃²	1	2	yes	complemented
266	C₂×C₃×Q₈	—	2	4	no	—
267	C₂×C₃×Q₈	—	3	4	no	—
268	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	complemented
269	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
270	C₂×C₃×D₄	—	2	8	no	—
271	C₂×C₃×D₄	—	2	4	no	—
272	C₂×C₃×D₄	—	2	1	no	—
273	C₂×C₃×D₄	—	2	4	no	—
274	C₂×C₃×D₄	—	3	4	no	—
275	C₂×C₃×D₄	—	6	4	no	—
276	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	complemented
277	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic, complemented
278	C₂²×C₃×C₄	—	2	8	no	—
279	C₂²×C₃×C₄	—	2	4	no	—
280	C₂²×C₃×C₄	—	2	1	no	—
281	C₂²×C₃×C₄	—	2	4	no	—
282	C₂²×C₃×C₄	—	6	4	no	—
283	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
284	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
285	C₂×#67	—	2	1	no	—
286	#235	—	2	1	no	—
287	#235	—	2	1	no	—
288	#235	C₂²×C₃²	1	2	yes	—
289	#235	C₂²×C₃²	1	2	yes	complemented
290	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic, complemented
291	C₂×C₄×D₃	—	2	1	no	—
292	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic, complemented
293	C₃×Q₁₆	—	3	16	no	—
294	C₃×SD₁₆	—	3	16	no	—
295	C₃×SD₁₆	—	3	16	no	—
296	C₃×D₈	—	3	16	no	—
297	C₃×M₁₆	—	3	8	no	—
298	C₃×M₁₆	—	3	8	no	—
299	C₂×C₃×C₈	—	3	8	no	—
300	#216	C₂²×C₃²	1	4	yes	complemented
301	#215	C₂²×C₃²	1	4	yes	complemented
302	#214	C₂²×C₃²	1	4	yes	complemented
303	#213	C₂²×C₃²	1	4	yes	complemented
304	#208	C₂²×C₃²	1	2	yes	complemented
305	#208	C₂²×C₃²	1	2	yes	complemented
306	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	complemented
307	C₂²×C₃²	—	2	16	no	—
308	C₂²×C₃²	—	2	2	no	—
309	C₂²×C₃²	—	2	8	no	—
310	C₂²×C₃²	—	2	4	no	—
311	C₂²×C₃²	—	2	4	no	—
312	C₂²×C₃²	—	2	4	no	—
313	C₂²×C₃²	—	4	1	no	—
314	C₂²×C₃²	—	4	4	no	—
315	C₂²×C₃²	—	4	8	no	—
316	C₂²×C₃²	—	4	16	no	—
317	C₂²×C₃²	—	4	8	no	—
318	C₂²×C₃²	—	4	2	no	—
319	C₂²×C₃²	—	6	2	no	—
320	C₂²×C₃²	—	12	2	no	—
321	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
322	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
323	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
324	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
325	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
326	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
327	C₂×C₃×D₃	—	2	8	no	—
328	C₂×C₃×D₃	—	4	8	no	—
329	C₃²×C₄	—	2	16	no	—
330	C₃²×C₄	—	2	2	no	—
331	C₃²×C₄	—	2	8	no	—
332	C₃²×C₄	—	2	4	no	—
333	C₃²×C₄	—	2	4	no	—
334	C₃²×C₄	—	2	4	no	—
335	C₃²×C₄	—	2	4	no	—
336	C₃²×C₄	—	6	2	no	—
337	C₃²×C₄	C₂×#67	1	1	yes	characteristic
338	C₃²×C₄	C₂×#67	1	1	yes	characteristic
339	C₃²×C₄	C₂×#67	1	1	yes	characteristic
340	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
341	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
342	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
343	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
344	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
345	C₃×Dic₃	—	2	8	no	—
346	C₂×#41	—	3	1	no	—
347	C₂×#41	C₃²×D₃	1	1	yes	characteristic, complemented
348	#137	—	3	2	no	—
349	#137	—	3	2	no	—
350	AGL(1,ℤ/8)	—	3	2	no	—
351	AGL(1,ℤ/8)	—	3	2	no	—
352	#135	—	3	4	no	—
353	C₂×M₁₆	—	3	1	no	—
354	C₃³	#575	1	1	yes	characteristic, complemented, Sylow
355	C₂³×C₃	—	2	8	no	—
356	C₂³×C₃	—	2	4	no	—
357	C₂³×C₃	—	2	1	no	—
358	C₂³×C₃	—	2	4	no	—
359	C₂³×C₃	—	6	4	no	—
360	C₂³×C₃	C₃×#67	1	4	yes	—
361	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
362	C₂²×D₃	—	2	1	no	—
363	C₃×Q₈	—	2	8	no	—
364	C₃×Q₈	—	2	4	no	—
365	C₃×Q₈	—	2	1	no	—
366	C₃×Q₈	—	2	8	no	—
367	C₃×Q₈	—	3	8	no	—
368	C₃×Q₈	—	6	4	no	—
369	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
370	C₃×Q₈	C₂³×C₃²	1	2	yes	—
371	C₃×D₄	—	2	16	no	—
372	C₃×D₄	—	2	16	no	—
373	C₃×D₄	—	2	8	no	—
374	C₃×D₄	—	2	8	no	—
375	C₃×D₄	—	2	8	no	—
376	C₃×D₄	—	2	2	no	—
377	C₃×D₄	—	2	2	no	—
378	C₃×D₄	—	2	4	no	—
379	C₃×D₄	—	2	1	no	—
380	C₃×D₄	—	2	8	no	—
381	C₃×D₄	—	3	8	no	—
382	C₃×D₄	—	6	8	no	—
383	C₃×D₄	—	6	8	no	—
384	C₃×D₄	—	6	4	no	—
385	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
386	C₃×D₄	C₂³×C₃²	1	2	yes	—
387	C₂×C₃×C₄	—	2	8	no	—
388	C₂×C₃×C₄	—	2	16	no	—
389	C₂×C₃×C₄	—	2	16	no	—
390	C₂×C₃×C₄	—	2	4	no	—
391	C₂×C₃×C₄	—	2	1	no	—
392	C₂×C₃×C₄	—	2	8	no	—
393	C₂×C₃×C₄	—	2	8	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	4	no	—
398	C₂×C₃×C₄	—	2	4	no	—
399	C₂×C₃×C₄	—	2	4	no	—
400	C₂×C₃×C₄	—	2	4	no	—
401	C₂×C₃×C₄	—	2	4	no	—
402	C₂×C₃×C₄	—	6	4	no	—
403	C₂×C₃×C₄	—	6	8	no	—
404	C₂×C₃×C₄	—	6	8	no	—
405	C₂×C₃×C₄	C₃×#67	1	4	yes	—
406	C₂×C₃×C₄	C₃×#67	1	4	yes	—
407	C₂×C₃×C₄	C₃×#67	1	4	yes	—
408	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
409	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
410	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
411	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
412	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
413	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
414	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
415	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
416	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
417	#67	—	2	2	no	—
418	#67	—	2	2	no	—
419	C₂×Dic₃	—	2	1	no	—
420	D₁₂	—	2	1	no	—
421	D₁₂	C₂³×C₃²	1	2	yes	—
422	C₄×D₃	—	2	2	no	—
423	C₄×D₃	—	2	2	no	—
424	Dic₆	—	2	1	no	—
425	Dic₆	C₂³×C₃²	1	2	yes	—
426	C₃×C₈	—	3	16	no	—
427	C₃⋊C₈	C₂³×C₃²	1	4	yes	—
428	C₂×C₃²	—	2	8	no	—
429	C₂×C₃²	—	2	4	no	—
430	C₂×C₃²	—	2	1	no	—
431	C₂×C₃²	—	2	1	no	—
432	C₂×C₃²	—	2	1	no	—
433	C₂×C₃²	—	2	4	no	—
434	C₂×C₃²	—	2	4	no	—
435	C₂×C₃²	—	2	4	no	—
436	C₂×C₃²	—	4	4	no	—
437	C₂×C₃²	—	4	4	no	—
438	C₂×C₃²	—	4	16	no	—
439	C₂×C₃²	—	4	8	no	—
440	C₂×C₃²	—	4	2	no	—
441	C₂×C₃²	—	12	2	no	—
442	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
443	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
444	C₃×D₃	—	4	8	no	—
445	#41	—	2	1	no	—
446	#41	—	2	1	no	—
447	#41	—	3	2	no	—
448	#41	—	3	2	no	—
449	#41	—	6	1	no	—
450	#41	—	6	1	no	—
451	#41	C₂×C₃²×D₃	1	2	yes	—
452	#41	C₂×C₃²×D₃	1	2	yes	complemented
453	C₂×Q₈	—	3	1	no	—
454	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	3	1	no	—
457	C₂×D₄	—	6	1	no	—
458	C₂×D₄	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	6	1	no	—
461	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
462	Q₁₆	—	3	4	no	—
463	SD₁₆	—	3	4	no	—
464	SD₁₆	—	3	4	no	—
465	D₈	—	3	4	no	—
466	M₁₆	—	3	2	no	—
467	M₁₆	—	3	2	no	—
468	C₂×C₈	—	3	2	no	—
469	C₂²×C₃	—	2	16	no	—
470	C₂²×C₃	—	2	8	no	—
471	C₂²×C₃	—	2	2	no	—
472	C₂²×C₃	—	2	4	no	—
473	C₂²×C₃	—	2	4	no	—
474	C₂²×C₃	—	2	4	no	—
475	C₂²×C₃	—	4	4	no	—
476	C₂²×C₃	—	4	1	no	—
477	C₂²×C₃	—	4	8	no	—
478	C₂²×C₃	—	4	16	no	—
479	C₂²×C₃	—	4	2	no	—
480	C₂²×C₃	—	4	8	no	—
481	C₂²×C₃	—	6	8	no	—
482	C₂²×C₃	—	12	8	no	—
483	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
484	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
485	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
486	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
487	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
488	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
489	C₂×D₃	—	2	2	no	—
490	C₂×D₃	—	4	2	no	—
491	C₃×C₄	—	2	16	no	—
492	C₃×C₄	—	2	8	no	—
493	C₃×C₄	—	2	2	no	—
494	C₃×C₄	—	2	4	no	—
495	C₃×C₄	—	2	4	no	—
496	C₃×C₄	—	2	4	no	—
497	C₃×C₄	—	2	4	no	—
498	C₃×C₄	—	6	8	no	—
499	C₃×C₄	C₂×C₃×#67	1	4	yes	—
500	C₃×C₄	C₂×C₃×#67	1	4	yes	—
501	C₃×C₄	C₂×C₃×#67	1	4	yes	—
502	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
503	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
504	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
505	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
506	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
507	Dic₃	—	2	2	no	—
508	C₃²	—	2	4	no	—
509	C₃²	#5822	1	1	yes	characteristic, direct product factor
510	C₃²	C₃×#575	1	4	yes	complemented
511	C₂³	—	2	1	no	—
512	C₂³	—	6	1	no	—
513	C₂³	C₃²×#67	1	1	yes	characteristic
514	Q₈	—	2	1	no	—
515	Q₈	—	3	2	no	—
516	Q₈	—	6	1	no	—
517	Q₈	C₂²×C₃²×D₃	1	2	yes	—
518	D₄	—	2	2	no	—
519	D₄	—	2	2	no	—
520	D₄	—	2	1	no	—
521	D₄	—	3	2	no	—
522	D₄	—	6	2	no	—
523	D₄	—	6	2	no	—
524	D₄	—	6	1	no	—
525	D₄	C₂²×C₃²×D₃	1	2	yes	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	6	1	no	—
530	C₂×C₄	—	6	2	no	—
531	C₂×C₄	—	6	2	no	—
532	C₂×C₄	C₃²×#67	1	1	yes	characteristic
533	C₂×C₄	C₃²×#67	1	1	yes	characteristic
534	C₂×C₄	C₃²×#67	1	1	yes	characteristic
535	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
536	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
537	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
538	C₈	—	3	4	no	—
539	C₂×C₃	—	2	8	no	—
540	C₂×C₃	—	2	1	no	—
541	C₂×C₃	—	2	1	no	—
542	C₂×C₃	—	2	1	no	—
543	C₂×C₃	—	2	4	no	—
544	C₂×C₃	—	2	4	no	—
545	C₂×C₃	—	2	4	no	—
546	C₂×C₃	—	2	4	no	—
547	C₂×C₃	—	4	4	no	—
548	C₂×C₃	—	4	4	no	—
549	C₂×C₃	—	4	16	no	—
550	C₂×C₃	—	4	2	no	—
551	C₂×C₃	—	4	8	no	—
552	C₂×C₃	—	12	8	no	—
553	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
554	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
555	D₃	—	4	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	6	2	no	—
560	C₂²	—	12	2	no	—
561	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
562	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
563	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	6	2	no	—
566	C₄	C₂×C₃²×#67	1	1	yes	characteristic
567	C₄	C₂×C₃²×#67	1	1	yes	characteristic
568	C₄	C₂×C₃²×#67	1	1	yes	characteristic
569	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
570	C₃	—	2	4	no	—
571	C₃	C₃×#5822	1	4	yes	direct product factor
572	C₃	C₃²×#575	1	1	yes	characteristic, complemented
573	C₂	—	2	1	no	—
574	C₂	—	2	1	no	—
575	C₂	—	2	1	no	—
576	C₂	—	4	2	no	—
577	C₂	—	12	2	no	—
578	C₂	C₂²×C₃²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₂×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₂×C₃³×#41	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₂×C₃²×#41	D₃
hypocenter	C₃	C₃²×#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂×C₃²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₂×C₃²×#235	C₂	semidirect
3. C₃²×#951	C₂	semidirect
4. C₃²×#950	C₂	semidirect
5. C₃²×#949	C₂	semidirect
6. C₃²×#942	C₂	semidirect
7. C₃²×#932	C₂	semidirect
8. C₂×C₃²×#208	C₂	semidirect
10. C₃×#5822	C₃	direct (N × H)
14. C₃³×#41	C₂²	semidirect
15. C₂×C₃³×Q₈	C₂²	semidirect
17. C₂×C₃³×D₄	C₂²	semidirect
24. C₃²×#235	C₂²	semidirect
25. C₂×C₃²×D₁₂	C₂²	semidirect
27. C₂×C₃²×Dic₆	C₂²	semidirect
28. C₃²×#216	C₂²	semidirect
29. C₃²×#215	C₂²	semidirect
30. C₃²×#214	C₂²	semidirect
31. C₃²×#213	C₂²	semidirect
32. C₃²×#208	C₂²	semidirect
33. C₃²×#208	C₂²	semidirect
34. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
37. C₂×C₃²×#41	D₃	semidirect
38. C₂×C₃²×#41	C₂×C₃	semidirect
39. C₂×C₃×#235	C₂×C₃	semidirect
46. C₃×#951	C₂×C₃	semidirect
47. C₃×#950	C₂×C₃	semidirect
48. C₃×#949	C₂×C₃	semidirect
49. C₃×#942	C₂×C₃	semidirect
50. C₃×#932	C₂×C₃	semidirect
51. C₂×C₃×#208	C₂×C₃	semidirect
81. #5822	C₃²	direct (N × H)
95. C₃²×#41	C₂×D₃	semidirect
97. C₃²×#41	C₂²×C₃	semidirect
100. C₂×C₃²×Q₈	C₂×D₃	semidirect
101. C₂×C₃²×Q₈	C₂²×C₃	semidirect
108. C₂×C₃²×D₄	C₂×D₃	semidirect
109. C₂×C₃²×D₄	C₂²×C₃	semidirect
121. C₃×#235	C₂²×C₃	semidirect
122. C₂×C₃×D₁₂	C₂²×C₃	semidirect
124. C₂×C₃×Dic₆	C₂²×C₃	semidirect
132. C₃×#216	C₂²×C₃	semidirect
133. C₃×#215	C₂²×C₃	semidirect
134. C₃×#214	C₂²×C₃	semidirect
135. C₃×#213	C₂²×C₃	semidirect
136. C₃×#208	C₂²×C₃	semidirect
137. C₃×#208	C₂²×C₃	semidirect
138. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
155. C₂×C₃×#41	C₃×D₃	semidirect
156. C₂×C₃×#41	C₂×C₃²	semidirect
157. C₂×#235	C₂×C₃²	semidirect
164. #951	C₂×C₃²	semidirect
165. #950	C₂×C₃²	semidirect
166. #949	C₂×C₃²	semidirect
167. #942	C₂×C₃²	semidirect
168. #932	C₂×C₃²	semidirect
169. C₂×#208	C₂×C₃²	semidirect
263. C₃×#41	C₂×C₃×D₃	semidirect
265. C₃×#41	C₂²×C₃²	semidirect
268. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
269. C₂×C₃×Q₈	C₂²×C₃²	semidirect
276. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
277. C₂×C₃×D₄	C₂²×C₃²	semidirect
289. #235	C₂²×C₃²	semidirect
290. C₂×D₁₂	C₂²×C₃²	semidirect
292. C₂×Dic₆	C₂²×C₃²	semidirect
300. #216	C₂²×C₃²	semidirect
301. #215	C₂²×C₃²	semidirect
302. #214	C₂²×C₃²	semidirect
303. #213	C₂²×C₃²	semidirect
304. #208	C₂²×C₃²	semidirect
305. #208	C₂²×C₃²	semidirect
306. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
347. C₂×#41	C₃²×D₃	semidirect
354. C₃³	#575	semidirect
452. #41	C₂×C₃²×D₃	semidirect
454. C₂×Q₈	C₂×C₃²×D₃	semidirect
458. C₂×D₄	C₂×C₃²×D₃	semidirect
509. C₃²	#5822	direct (N × H)
510. C₃²	C₃×#575	semidirect
571. C₃	C₃×#5822	direct (N × H)
572. C₃	C₃²×#575	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃³×#41	2
C₂×C₃²×#235	2
C₃²×#951	2
C₃²×#950	2
C₃²×#949	2
C₃²×#942	2
C₃²×#932	2
C₂×C₃²×#208	2
C₃²×#575	3
C₃×#5822	3

Subgroups — C32×#5822 #13179202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5822 #13179202