Subgroups — C₃²×#5804 #13179181

Order 1728.

3020 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#953	C₂	1	2	yes	maximal, complemented
4	C₂×C₃²×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#939	C₂	1	4	yes	maximal, complemented
6	C₃²×#937	C₂	1	2	yes	maximal, complemented
7	C₃²×#935	C₂	1	2	yes	maximal, complemented
8	C₃²×#929	C₂	1	2	yes	maximal, complemented
9	C₃²×#534	—	3	1	no	maximal
10	C₃×#5804	C₃	1	4	yes	maximal, direct product factor
11	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
12	C₂⁴×C₃³	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₃²×D₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃³×D₄	—	2	4	no	—
15	C₂×C₃³×D₄	C₂²	1	2	yes	—
16	C₂×C₃³×D₄	C₂²	1	2	yes	complemented
17	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃²×#67	—	2	4	no	—
19	C₂×C₃²×#67	C₂²	1	2	yes	—
20	C₂×C₃²×#67	C₂²	1	2	yes	complemented
21	C₂×C₃²×#67	C₂²	1	4	yes	complemented
22	C₂²×C₃²×Dic₃	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₃²×Dic₃	C₂²	1	2	yes	complemented
24	C₃²×#217	C₂²	1	2	yes	complemented
25	C₃²×#217	C₂²	1	2	yes	complemented
26	C₃²×#217	C₂²	1	4	yes	complemented
27	C₃²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
28	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	complemented
29	C₂²×C₃²×D₄	—	2	4	no	—
30	C₂²×C₃²×D₄	—	3	1	no	—
31	C₂²×C₃²×D₄	D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃²×D₄	C₂×C₃	1	4	yes	complemented
33	C₂²×C₃×#67	C₂×C₃	1	4	yes	complemented
34	C₃²×#123	—	3	2	no	—
35	C₃²×#123	—	3	2	no	—
36	C₃²×#121	—	3	4	no	—
37	C₃²×#120	—	3	2	no	—
38	C₃²×#120	—	3	2	no	—
39	C₂×C₃²×C₂²⋊C₄	—	3	1	no	—
40	C₃×#953	C₂×C₃	1	8	yes	complemented
41	C₂×C₃×#217	C₂×C₃	1	4	yes	complemented
42	C₃×#939	C₂×C₃	1	16	yes	complemented
43	C₃×#937	C₂×C₃	1	8	yes	complemented
44	C₃×#935	C₂×C₃	1	8	yes	complemented
45	C₃×#929	C₂×C₃	1	8	yes	complemented
46	C₂³×C₃³	—	2	2	no	—
47	C₂³×C₃³	—	4	2	no	—
48	C₂³×C₃³	—	4	2	no	—
49	C₂³×C₃³	D₄	1	4	yes	complemented
50	C₂³×C₃³	C₂³	1	2	yes	—
51	C₂³×C₃³	C₂³	1	1	yes	characteristic
52	C₂³×C₃³	C₂³	1	1	yes	characteristic
53	C₂³×C₃³	C₂³	1	1	yes	characteristic
54	C₂²×C₃²×D₃	—	2	2	no	—
55	C₂²×C₃²×D₃	—	4	2	no	—
56	C₂²×C₃²×D₃	C₂³	1	2	yes	—
57	C₃³×D₄	—	2	4	no	—
58	C₃³×D₄	—	2	4	no	—
59	C₂×C₃³×C₄	—	2	2	no	—
60	C₂×C₃³×C₄	C₂³	1	2	yes	—
61	C₃²×#67	—	2	4	no	—
62	C₃²×#67	—	2	4	no	—
63	C₃²×#67	—	4	4	no	—
64	C₂×C₃²×Dic₃	—	2	2	no	—
65	C₂×C₃²×Dic₃	—	4	2	no	—
66	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
67	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
68	C₃×#534	—	3	4	no	—
69	#5804	C₃²	1	1	yes	characteristic, direct product factor
70	C₂⁴×C₃²	—	2	4	no	—
71	C₂⁴×C₃²	—	2	4	no	—
72	C₂⁴×C₃²	—	3	1	no	—
73	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
74	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic, complemented
75	C₂⁴×C₃²	C₂²×C₃	1	4	yes	—
76	C₂⁴×C₃²	C₂²×C₃	1	4	yes	complemented
77	C₂³×C₃×D₃	C₂²×C₃	1	4	yes	complemented
78	C₂×C₃²×D₄	—	2	32	no	—
79	C₂×C₃²×D₄	—	2	8	no	—
80	C₂×C₃²×D₄	—	2	8	no	—
81	C₂×C₃²×D₄	—	2	4	no	—
82	C₂×C₃²×D₄	—	2	16	no	—
83	C₂×C₃²×D₄	—	3	2	no	—
84	C₂×C₃²×D₄	—	3	2	no	—
85	C₂×C₃²×D₄	—	3	4	no	—
86	C₂×C₃²×D₄	—	6	4	no	—
87	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
88	C₂×C₃²×D₄	C₂×D₃	1	2	yes	complemented
89	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	—
90	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	complemented
91	C₂²×C₃²×C₄	—	2	4	no	—
92	C₂²×C₃²×C₄	—	3	1	no	—
93	C₂²×C₃²×C₄	—	3	2	no	—
94	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic, complemented
95	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	complemented
96	C₂×C₃×#67	—	2	16	no	—
97	C₂×C₃×#67	C₂²×C₃	1	8	yes	—
98	C₂×C₃×#67	C₂²×C₃	1	8	yes	complemented
99	C₂×C₃×#67	C₂²×C₃	1	16	yes	complemented
100	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	complemented
101	C₂²×C₃×Dic₃	C₂²×C₃	1	8	yes	complemented
102	C₃²×C₄⋊C₄	—	3	4	no	—
103	C₃²×C₂²⋊C₄	—	3	2	no	—
104	C₃²×C₂²⋊C₄	—	3	2	no	—
105	C₃²×C₂²⋊C₄	—	3	4	no	—
106	C₃²×C₂²⋊C₄	—	3	4	no	—
107	C₃×#217	C₂²×C₃	1	8	yes	complemented
108	C₃×#217	C₂²×C₃	1	8	yes	complemented
109	C₃×#217	C₂²×C₃	1	16	yes	complemented
110	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	complemented
111	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	complemented
112	C₂²×C₃³	—	2	2	no	—
113	C₂²×C₃³	—	2	2	no	—
114	C₂²×C₃³	—	2	2	no	—
115	C₂²×C₃³	—	2	2	no	—
116	C₂²×C₃³	—	2	4	no	—
117	C₂²×C₃³	—	4	4	no	—
118	C₂²×C₃³	—	4	4	no	—
119	C₂²×C₃³	C₂×D₄	1	2	yes	—
120	C₂²×C₃³	C₂×D₄	1	2	yes	—
121	C₂²×C₃³	C₂×D₄	1	2	yes	—
122	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
123	C₂×C₃²×D₃	—	2	2	no	—
124	C₂×C₃²×D₃	—	2	4	no	—
125	C₂×C₃²×D₃	—	4	4	no	—
126	C₃³×C₄	—	2	2	no	—
127	C₃²×Dic₃	—	2	2	no	—
128	C₃²×Dic₃	—	2	4	no	—
129	C₂²×C₃×D₄	—	2	4	no	—
130	C₂²×C₃×D₄	—	3	4	no	—
131	C₂²×C₃×D₄	C₃×D₃	1	4	yes	complemented
132	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
133	C₂²×#67	C₂×C₃²	1	1	yes	characteristic, complemented
134	C₃×#123	—	3	8	no	—
135	C₃×#123	—	3	8	no	—
136	C₃×#121	—	3	16	no	—
137	C₃×#120	—	3	8	no	—
138	C₃×#120	—	3	8	no	—
139	C₂×C₃×C₂²⋊C₄	—	3	4	no	—
140	#953	C₂×C₃²	1	2	yes	complemented
141	C₂×#217	C₂×C₃²	1	1	yes	characteristic, complemented
142	#939	C₂×C₃²	1	4	yes	complemented
143	#937	C₂×C₃²	1	2	yes	complemented
144	#935	C₂×C₃²	1	2	yes	complemented
145	#929	C₂×C₃²	1	2	yes	complemented
146	C₂³×C₃²	—	2	16	no	—
147	C₂³×C₃²	—	2	2	no	—
148	C₂³×C₃²	—	2	8	no	—
149	C₂³×C₃²	—	2	16	no	—
150	C₂³×C₃²	—	2	4	no	—
151	C₂³×C₃²	—	2	8	no	—
152	C₂³×C₃²	—	2	4	no	—
153	C₂³×C₃²	—	2	4	no	—
154	C₂³×C₃²	—	3	2	no	—
155	C₂³×C₃²	—	4	2	no	—
156	C₂³×C₃²	—	4	2	no	—
157	C₂³×C₃²	—	4	8	no	—
158	C₂³×C₃²	—	4	8	no	—
159	C₂³×C₃²	—	4	16	no	—
160	C₂³×C₃²	—	4	16	no	—
161	C₂³×C₃²	—	6	2	no	—
162	C₂³×C₃²	—	12	2	no	—
163	C₂³×C₃²	#67	1	4	yes	complemented
164	C₂³×C₃²	C₃×D₄	1	16	yes	complemented
165	C₂³×C₃²	C₂²×D₃	1	2	yes	—
166	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
167	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
168	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
169	C₂³×C₃²	C₂³×C₃	1	8	yes	—
170	C₂³×C₃²	C₂³×C₃	1	4	yes	—
171	C₂³×C₃²	C₂³×C₃	1	4	yes	—
172	C₂³×C₃²	C₂³×C₃	1	4	yes	—
173	C₂²×C₃×D₃	—	2	8	no	—
174	C₂²×C₃×D₃	—	4	8	no	—
175	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
176	C₃²×D₄	—	2	32	no	—
177	C₃²×D₄	—	2	32	no	—
178	C₃²×D₄	—	2	4	no	—
179	C₃²×D₄	—	2	4	no	—
180	C₃²×D₄	—	2	16	no	—
181	C₃²×D₄	—	2	16	no	—
182	C₃²×D₄	—	6	4	no	—
183	C₃²×D₄	—	6	4	no	—
184	C₃²×D₄	—	12	4	no	—
185	C₂×C₃²×C₄	—	2	16	no	—
186	C₂×C₃²×C₄	—	2	2	no	—
187	C₂×C₃²×C₄	—	2	8	no	—
188	C₂×C₃²×C₄	—	2	8	no	—
189	C₂×C₃²×C₄	—	3	4	no	—
190	C₂×C₃²×C₄	—	3	2	no	—
191	C₂×C₃²×C₄	—	6	2	no	—
192	C₂×C₃²×C₄	—	12	2	no	—
193	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
194	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
195	C₃×#67	—	2	16	no	—
196	C₃×#67	—	2	16	no	—
197	C₃×#67	—	4	16	no	—
198	C₂×C₃×Dic₃	—	2	8	no	—
199	C₂×C₃×Dic₃	—	4	8	no	—
200	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
201	C₂×C₃×Dic₃	C₂³×C₃	1	16	yes	—
202	#534	—	3	1	no	Sylow
203	C₂×C₃³	—	2	2	no	—
204	C₂×C₃³	—	2	2	no	—
205	C₂×C₃³	—	2	2	no	—
206	C₂×C₃³	—	4	2	no	—
207	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₃³	H(5,𝔽₂)	1	2	yes	—
209	C₃²×D₃	—	4	2	no	—
210	C₂⁴×C₃	—	2	4	no	—
211	C₂⁴×C₃	—	2	4	no	—
212	C₂⁴×C₃	—	3	4	no	—
213	C₂⁴×C₃	C₂×C₃×D₃	1	4	yes	—
214	C₂⁴×C₃	C₂×C₃×D₃	1	4	yes	complemented
215	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic
216	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic, complemented
217	C₂³×D₃	C₂²×C₃²	1	1	yes	characteristic, complemented
218	C₂×C₃×D₄	—	2	32	no	—
219	C₂×C₃×D₄	—	2	16	no	—
220	C₂×C₃×D₄	—	2	4	no	—
221	C₂×C₃×D₄	—	2	8	no	—
222	C₂×C₃×D₄	—	2	8	no	—
223	C₂×C₃×D₄	—	3	8	no	—
224	C₂×C₃×D₄	—	3	8	no	—
225	C₂×C₃×D₄	—	3	16	no	—
226	C₂×C₃×D₄	—	6	16	no	—
227	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
228	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	complemented
229	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
230	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	complemented
231	C₂²×C₃×C₄	—	2	4	no	—
232	C₂²×C₃×C₄	—	3	4	no	—
233	C₂²×C₃×C₄	—	3	8	no	—
234	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	complemented
235	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
236	C₂×#67	—	2	4	no	—
237	C₂×#67	C₂²×C₃²	1	2	yes	—
238	C₂×#67	C₂²×C₃²	1	2	yes	complemented
239	C₂×#67	C₂²×C₃²	1	4	yes	complemented
240	C₂²×Dic₃	C₂²×C₃²	1	1	yes	characteristic, complemented
241	C₂²×Dic₃	C₂²×C₃²	1	2	yes	complemented
242	C₃×C₄⋊C₄	—	3	16	no	—
243	C₃×C₂²⋊C₄	—	3	8	no	—
244	C₃×C₂²⋊C₄	—	3	8	no	—
245	C₃×C₂²⋊C₄	—	3	16	no	—
246	C₃×C₂²⋊C₄	—	3	16	no	—
247	#217	C₂²×C₃²	1	2	yes	complemented
248	#217	C₂²×C₃²	1	2	yes	complemented
249	#217	C₂²×C₃²	1	4	yes	complemented
250	(C₂×D₃)⋊C₄	C₂²×C₃²	1	4	yes	complemented
251	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	complemented
252	C₂²×C₃²	—	2	16	no	—
253	C₂²×C₃²	—	2	2	no	—
254	C₂²×C₃²	—	2	2	no	—
255	C₂²×C₃²	—	2	2	no	—
256	C₂²×C₃²	—	2	8	no	—
257	C₂²×C₃²	—	2	16	no	—
258	C₂²×C₃²	—	2	4	no	—
259	C₂²×C₃²	—	2	2	no	—
260	C₂²×C₃²	—	2	8	no	—
261	C₂²×C₃²	—	2	4	no	—
262	C₂²×C₃²	—	2	8	no	—
263	C₂²×C₃²	—	2	8	no	—
264	C₂²×C₃²	—	2	8	no	—
265	C₂²×C₃²	—	2	8	no	—
266	C₂²×C₃²	—	2	8	no	—
267	C₂²×C₃²	—	2	16	no	—
268	C₂²×C₃²	—	4	8	no	—
269	C₂²×C₃²	—	4	8	no	—
270	C₂²×C₃²	—	4	16	no	—
271	C₂²×C₃²	—	4	4	no	—
272	C₂²×C₃²	—	4	16	no	—
273	C₂²×C₃²	—	4	32	no	—
274	C₂²×C₃²	—	4	32	no	—
275	C₂²×C₃²	—	4	4	no	—
276	C₂²×C₃²	—	4	16	no	—
277	C₂²×C₃²	—	6	4	no	—
278	C₂²×C₃²	—	6	2	no	—
279	C₂²×C₃²	—	12	4	no	—
280	C₂²×C₃²	C₂×#67	1	2	yes	—
281	C₂²×C₃²	C₂×#67	1	2	yes	—
282	C₂²×C₃²	C₂×#67	1	2	yes	—
283	C₂²×C₃²	C₂×C₃×D₄	1	8	yes	—
284	C₂²×C₃²	C₂×C₃×D₄	1	8	yes	—
285	C₂²×C₃²	C₂×C₃×D₄	1	8	yes	—
286	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
287	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
288	C₂×C₃×D₃	—	2	8	no	—
289	C₂×C₃×D₃	—	2	16	no	—
290	C₂×C₃×D₃	—	4	16	no	—
291	C₃²×C₄	—	2	16	no	—
292	C₃²×C₄	—	2	2	no	—
293	C₃²×C₄	—	2	8	no	—
294	C₃²×C₄	—	6	4	no	—
295	C₃²×C₄	—	6	2	no	—
296	C₃×Dic₃	—	2	8	no	—
297	C₃×Dic₃	—	2	16	no	—
298	C₂²×D₄	—	3	1	no	—
299	C₂²×D₄	C₃²×D₃	1	1	yes	characteristic, complemented
300	#123	—	3	2	no	—
301	#123	—	3	2	no	—
302	#121	—	3	4	no	—
303	#120	—	3	2	no	—
304	#120	—	3	2	no	—
305	C₂×C₂²⋊C₄	—	3	1	no	—
306	C₃³	#534	1	1	yes	characteristic, complemented, Sylow
307	C₂³×C₃	—	2	16	no	—
308	C₂³×C₃	—	2	8	no	—
309	C₂³×C₃	—	2	2	no	—
310	C₂³×C₃	—	2	16	no	—
311	C₂³×C₃	—	2	4	no	—
312	C₂³×C₃	—	2	8	no	—
313	C₂³×C₃	—	2	4	no	—
314	C₂³×C₃	—	2	4	no	—
315	C₂³×C₃	—	3	8	no	—
316	C₂³×C₃	—	4	8	no	—
317	C₂³×C₃	—	4	8	no	—
318	C₂³×C₃	—	4	2	no	—
319	C₂³×C₃	—	4	2	no	—
320	C₂³×C₃	—	4	16	no	—
321	C₂³×C₃	—	4	16	no	—
322	C₂³×C₃	—	6	8	no	—
323	C₂³×C₃	—	12	8	no	—
324	C₂³×C₃	C₃×#67	1	16	yes	complemented
325	C₂³×C₃	C₃²×D₄	1	4	yes	complemented
326	C₂³×C₃	C₂²×C₃×D₃	1	8	yes	—
327	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
328	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
329	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
330	C₂³×C₃	C₂³×C₃²	1	2	yes	—
331	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
332	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
333	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
334	C₂²×D₃	—	2	2	no	—
335	C₂²×D₃	—	4	2	no	—
336	C₂²×D₃	C₂³×C₃²	1	2	yes	—
337	C₃×D₄	—	2	32	no	—
338	C₃×D₄	—	2	32	no	—
339	C₃×D₄	—	2	16	no	—
340	C₃×D₄	—	2	16	no	—
341	C₃×D₄	—	2	4	no	—
342	C₃×D₄	—	2	4	no	—
343	C₃×D₄	—	6	16	no	—
344	C₃×D₄	—	6	16	no	—
345	C₃×D₄	—	12	16	no	—
346	C₂×C₃×C₄	—	2	16	no	—
347	C₂×C₃×C₄	—	2	8	no	—
348	C₂×C₃×C₄	—	2	2	no	—
349	C₂×C₃×C₄	—	2	8	no	—
350	C₂×C₃×C₄	—	3	16	no	—
351	C₂×C₃×C₄	—	3	8	no	—
352	C₂×C₃×C₄	—	6	8	no	—
353	C₂×C₃×C₄	—	12	8	no	—
354	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
355	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
356	#67	—	2	4	no	—
357	#67	—	2	4	no	—
358	#67	—	4	4	no	—
359	C₂×Dic₃	—	2	2	no	—
360	C₂×Dic₃	—	4	2	no	—
361	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
362	C₂×Dic₃	C₂³×C₃²	1	4	yes	—
363	C₂×C₃²	—	2	16	no	—
364	C₂×C₃²	—	2	2	no	—
365	C₂×C₃²	—	2	2	no	—
366	C₂×C₃²	—	2	2	no	—
367	C₂×C₃²	—	2	8	no	—
368	C₂×C₃²	—	2	8	no	—
369	C₂×C₃²	—	2	4	no	—
370	C₂×C₃²	—	2	8	no	—
371	C₂×C₃²	—	2	8	no	—
372	C₂×C₃²	—	4	8	no	—
373	C₂×C₃²	—	4	8	no	—
374	C₂×C₃²	—	4	16	no	—
375	C₂×C₃²	—	4	2	no	—
376	C₂×C₃²	—	4	8	no	—
377	C₂×C₃²	—	12	2	no	—
378	C₂×C₃²	#1004	1	2	yes	—
379	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
380	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
381	C₂×C₃²	C₃×H(5,𝔽₂)	1	8	yes	—
382	C₃×D₃	—	4	8	no	—
383	C₂⁴	—	3	1	no	—
384	C₂⁴	C₂×C₃²×D₃	1	1	yes	characteristic
385	C₂⁴	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	3	2	no	—
388	C₂×D₄	—	3	2	no	—
389	C₂×D₄	—	3	4	no	—
390	C₂×D₄	—	6	4	no	—
391	C₂×D₄	C₂×C₃²×D₃	1	2	yes	—
392	C₂×D₄	C₂×C₃²×D₃	1	2	yes	complemented
393	C₂²×C₄	—	3	1	no	—
394	C₂²×C₄	—	3	2	no	—
395	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
396	C₄⋊C₄	—	3	4	no	—
397	C₂²⋊C₄	—	3	2	no	—
398	C₂²⋊C₄	—	3	2	no	—
399	C₂²⋊C₄	—	3	4	no	—
400	C₂²⋊C₄	—	3	4	no	—
401	C₂²×C₃	—	2	16	no	—
402	C₂²×C₃	—	2	8	no	—
403	C₂²×C₃	—	2	8	no	—
404	C₂²×C₃	—	2	8	no	—
405	C₂²×C₃	—	2	2	no	—
406	C₂²×C₃	—	2	2	no	—
407	C₂²×C₃	—	2	2	no	—
408	C₂²×C₃	—	2	16	no	—
409	C₂²×C₃	—	2	16	no	—
410	C₂²×C₃	—	2	8	no	—
411	C₂²×C₃	—	2	2	no	—
412	C₂²×C₃	—	2	4	no	—
413	C₂²×C₃	—	2	4	no	—
414	C₂²×C₃	—	2	8	no	—
415	C₂²×C₃	—	2	8	no	—
416	C₂²×C₃	—	2	8	no	—
417	C₂²×C₃	—	4	8	no	—
418	C₂²×C₃	—	4	8	no	—
419	C₂²×C₃	—	4	16	no	—
420	C₂²×C₃	—	4	16	no	—
421	C₂²×C₃	—	4	4	no	—
422	C₂²×C₃	—	4	32	no	—
423	C₂²×C₃	—	4	32	no	—
424	C₂²×C₃	—	4	16	no	—
425	C₂²×C₃	—	4	4	no	—
426	C₂²×C₃	—	6	16	no	—
427	C₂²×C₃	—	6	8	no	—
428	C₂²×C₃	—	12	16	no	—
429	C₂²×C₃	C₂×C₃×#67	1	8	yes	—
430	C₂²×C₃	C₂×C₃×#67	1	8	yes	—
431	C₂²×C₃	C₂×C₃×#67	1	8	yes	—
432	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
433	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
434	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
435	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
436	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
437	C₂×D₃	—	2	2	no	—
438	C₂×D₃	—	2	4	no	—
439	C₂×D₃	—	4	4	no	—
440	C₃×C₄	—	2	16	no	—
441	C₃×C₄	—	2	8	no	—
442	C₃×C₄	—	2	2	no	—
443	C₃×C₄	—	6	16	no	—
444	C₃×C₄	—	6	8	no	—
445	Dic₃	—	2	2	no	—
446	Dic₃	—	2	4	no	—
447	C₃²	—	2	4	no	—
448	C₃²	#5804	1	1	yes	characteristic, direct product factor
449	C₃²	C₃×#534	1	4	yes	complemented
450	C₂³	—	2	2	no	—
451	C₂³	—	3	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	6	2	no	—
455	C₂³	—	12	2	no	—
456	C₂³	C₃²×#67	1	4	yes	complemented
457	C₂³	C₂²×C₃²×D₃	1	2	yes	—
458	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
459	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
460	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
461	D₄	—	2	4	no	—
462	D₄	—	2	4	no	—
463	D₄	—	6	4	no	—
464	D₄	—	6	4	no	—
465	D₄	—	12	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	3	4	no	—
468	C₂×C₄	—	3	2	no	—
469	C₂×C₄	—	6	2	no	—
470	C₂×C₄	—	12	2	no	—
471	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
472	C₂×C₃	—	2	16	no	—
473	C₂×C₃	—	2	8	no	—
474	C₂×C₃	—	2	8	no	—
475	C₂×C₃	—	2	8	no	—
476	C₂×C₃	—	2	2	no	—
477	C₂×C₃	—	2	2	no	—
478	C₂×C₃	—	2	2	no	—
479	C₂×C₃	—	2	8	no	—
480	C₂×C₃	—	2	4	no	—
481	C₂×C₃	—	4	8	no	—
482	C₂×C₃	—	4	8	no	—
483	C₂×C₃	—	4	16	no	—
484	C₂×C₃	—	4	8	no	—
485	C₂×C₃	—	4	2	no	—
486	C₂×C₃	—	12	8	no	—
487	C₂×C₃	C₃×#1004	1	8	yes	—
488	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
489	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
490	C₂×C₃	C₃²×H(5,𝔽₂)	1	2	yes	—
491	D₃	—	4	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	6	4	no	—
500	C₂²	—	6	2	no	—
501	C₂²	—	12	4	no	—
502	C₂²	C₂×C₃²×#67	1	2	yes	—
503	C₂²	C₂×C₃²×#67	1	2	yes	—
504	C₂²	C₂×C₃²×#67	1	2	yes	—
505	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	6	4	no	—
508	C₄	—	6	2	no	—
509	C₃	—	2	4	no	—
510	C₃	C₃×#5804	1	4	yes	direct product factor
511	C₃	C₃²×#534	1	1	yes	characteristic, complemented
512	C₂	—	2	2	no	—
513	C₂	—	2	2	no	—
514	C₂	—	2	2	no	—
515	C₂	—	4	2	no	—
516	C₂	—	12	2	no	—
517	C₂	C₃²×#1004	1	2	yes	—
518	C₂	C₂²×C₃²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₂²×C₃³×D₄	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₂²×C₃²×D₄	D₃
hypocenter	C₃	C₃²×#534
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×D₄	C₂	semidirect
2. C₂²×C₃²×#67	C₂	semidirect
3. C₃²×#953	C₂	semidirect
4. C₂×C₃²×#217	C₂	semidirect
5. C₃²×#939	C₂	semidirect
6. C₃²×#937	C₂	semidirect
7. C₃²×#935	C₂	semidirect
8. C₃²×#929	C₂	semidirect
10. C₃×#5804	C₃	direct (N × H)
12. C₂⁴×C₃³	C₂²	semidirect
13. C₂³×C₃²×D₃	C₂²	semidirect
16. C₂×C₃³×D₄	C₂²	semidirect
17. C₂²×C₃³×C₄	C₂²	semidirect
20. C₂×C₃²×#67	C₂²	semidirect
21. C₂×C₃²×#67	C₂²	semidirect
22. C₂²×C₃²×Dic₃	C₂²	semidirect
23. C₂²×C₃²×Dic₃	C₂²	semidirect
24. C₃²×#217	C₂²	semidirect
25. C₃²×#217	C₂²	semidirect
26. C₃²×#217	C₂²	semidirect
27. C₃²×(C₂×D₃)⋊C₄	C₂²	semidirect
28. C₃²×Dic₃⋊C₄	C₂²	semidirect
31. C₂²×C₃²×D₄	D₃	semidirect
32. C₂²×C₃²×D₄	C₂×C₃	semidirect
33. C₂²×C₃×#67	C₂×C₃	semidirect
40. C₃×#953	C₂×C₃	semidirect
41. C₂×C₃×#217	C₂×C₃	semidirect
42. C₃×#939	C₂×C₃	semidirect
43. C₃×#937	C₂×C₃	semidirect
44. C₃×#935	C₂×C₃	semidirect
45. C₃×#929	C₂×C₃	semidirect
49. C₂³×C₃³	D₄	semidirect
69. #5804	C₃²	direct (N × H)
74. C₂⁴×C₃²	C₂×D₃	semidirect
76. C₂⁴×C₃²	C₂²×C₃	semidirect
77. C₂³×C₃×D₃	C₂²×C₃	semidirect
88. C₂×C₃²×D₄	C₂×D₃	semidirect
90. C₂×C₃²×D₄	C₂²×C₃	semidirect
94. C₂²×C₃²×C₄	C₂×D₃	semidirect
95. C₂²×C₃²×C₄	C₂²×C₃	semidirect
98. C₂×C₃×#67	C₂²×C₃	semidirect
99. C₂×C₃×#67	C₂²×C₃	semidirect
100. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
101. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
107. C₃×#217	C₂²×C₃	semidirect
108. C₃×#217	C₂²×C₃	semidirect
109. C₃×#217	C₂²×C₃	semidirect
110. C₃×(C₂×D₃)⋊C₄	C₂²×C₃	semidirect
111. C₃×Dic₃⋊C₄	C₂²×C₃	semidirect
131. C₂²×C₃×D₄	C₃×D₃	semidirect
132. C₂²×C₃×D₄	C₂×C₃²	semidirect
133. C₂²×#67	C₂×C₃²	semidirect
140. #953	C₂×C₃²	semidirect
141. C₂×#217	C₂×C₃²	semidirect
142. #939	C₂×C₃²	semidirect
143. #937	C₂×C₃²	semidirect
144. #935	C₂×C₃²	semidirect
145. #929	C₂×C₃²	semidirect
163. C₂³×C₃²	#67	semidirect
164. C₂³×C₃²	C₃×D₄	semidirect
214. C₂⁴×C₃	C₂×C₃×D₃	semidirect
216. C₂⁴×C₃	C₂²×C₃²	semidirect
217. C₂³×D₃	C₂²×C₃²	semidirect
228. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
230. C₂×C₃×D₄	C₂²×C₃²	semidirect
234. C₂²×C₃×C₄	C₂×C₃×D₃	semidirect
235. C₂²×C₃×C₄	C₂²×C₃²	semidirect
238. C₂×#67	C₂²×C₃²	semidirect
239. C₂×#67	C₂²×C₃²	semidirect
240. C₂²×Dic₃	C₂²×C₃²	semidirect
241. C₂²×Dic₃	C₂²×C₃²	semidirect
247. #217	C₂²×C₃²	semidirect
248. #217	C₂²×C₃²	semidirect
249. #217	C₂²×C₃²	semidirect
250. (C₂×D₃)⋊C₄	C₂²×C₃²	semidirect
251. Dic₃⋊C₄	C₂²×C₃²	semidirect
299. C₂²×D₄	C₃²×D₃	semidirect
306. C₃³	#534	semidirect
324. C₂³×C₃	C₃×#67	semidirect
325. C₂³×C₃	C₃²×D₄	semidirect
385. C₂⁴	C₂×C₃²×D₃	semidirect
392. C₂×D₄	C₂×C₃²×D₃	semidirect
395. C₂²×C₄	C₂×C₃²×D₃	semidirect
448. C₃²	#5804	direct (N × H)
449. C₃²	C₃×#534	semidirect
456. C₂³	C₃²×#67	semidirect
510. C₃	C₃×#5804	direct (N × H)
511. C₃	C₃²×#534	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃³×D₄	2
C₂²×C₃²×#67	2
C₃²×#953	2
C₂×C₃²×#217	2
C₃²×#939	2
C₃²×#937	2
C₃²×#935	2
C₃²×#929	2
C₃²×#534	3
C₃×#5804	3

Subgroups — C32×#5804 #13179181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5804 #13179181