Subgroups — C₃²×#5690 #13179068

Order 1728.

1592 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#897	C₂	1	2	yes	maximal, complemented
3	C₃²×#896	C₂	1	2	yes	maximal, complemented
4	C₃²×#896	C₂	1	2	yes	maximal, complemented
5	C₃²×#893	C₂	1	1	yes	maximal, characteristic
6	C₃²×#893	C₂	1	1	yes	maximal, characteristic
7	C₃²×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃²×#890	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
11	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃²×#554	—	3	1	no	maximal
13	C₃×#5690	C₃	1	4	yes	maximal, direct product factor
14	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
16	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
17	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
18	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
19	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
20	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃³×C₄²	C₂²	1	1	yes	characteristic
25	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
30	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
31	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃²×C₄⋊Q₈	—	3	1	no	—
38	C₃²×#125	—	2	4	no	—
39	C₃²×#125	—	3	1	no	—
40	C₃²×#125	D₃	1	1	yes	characteristic, complemented
41	C₃²×#125	C₂×C₃	1	4	yes	complemented
42	C₃²×C₂²⋊Q₈	—	3	2	no	—
43	C₃²×C₂²⋊Q₈	—	3	2	no	—
44	C₃²×C₂²⋊Q₈	—	3	2	no	—
45	C₃²×C₄×Q₈	—	3	1	no	—
46	C₃²×C₄×D₄	—	3	1	no	—
47	C₃²×C₄×D₄	—	3	1	no	—
48	C₃²×C₄×D₄	—	3	1	no	—
49	C₂×C₃²×C₄⋊C₄	—	3	2	no	—
50	C₃×#897	C₂×C₃	1	8	yes	complemented
51	C₃×#896	C₂×C₃	1	8	yes	complemented
52	C₃×#896	C₂×C₃	1	8	yes	complemented
53	C₃×#893	C₂×C₃	1	4	yes	—
54	C₃×#893	C₂×C₃	1	4	yes	—
55	C₃×D₃×C₄⋊C₄	C₂×C₃	1	8	yes	complemented
56	C₃×#890	C₂×C₃	1	4	yes	complemented
57	C₃×#888	C₂×C₃	1	4	yes	complemented
58	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
59	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	complemented
60	C₂²×C₃²×D₃	C₂³	1	2	yes	—
61	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
62	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
63	C₂×C₃³×C₄	C₂³	1	2	yes	—
64	C₂×C₃³×C₄	C₂³	1	2	yes	—
65	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
67	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
68	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
69	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
71	C₃²×D₁₂	Q₈	1	4	yes	complemented
72	C₃²×C₄×D₃	—	2	4	no	—
73	C₃²×C₄×D₃	—	4	2	no	—
74	C₃²×C₄×D₃	—	4	2	no	—
75	C₃²×Dic₆	—	2	2	no	—
76	C₃²×Dic₆	—	4	2	no	—
77	C₃×#554	—	3	4	no	—
78	#5690	C₃²	1	1	yes	characteristic, direct product factor
79	C₂×C₃²×Q₈	—	3	1	no	—
80	C₂×C₃²×Q₈	—	3	2	no	—
81	C₂×C₃²×D₄	—	3	1	no	—
82	C₂²×C₃²×C₄	—	3	2	no	—
83	C₂²×C₃²×C₄	—	3	2	no	—
84	C₂²×C₃²×C₄	—	3	2	no	—
85	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
86	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
87	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
88	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
89	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
90	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
91	C₃²×C₄⋊C₄	—	2	4	no	—
92	C₃²×C₄⋊C₄	—	2	4	no	—
93	C₃²×C₄⋊C₄	—	2	8	no	—
94	C₃²×C₄⋊C₄	—	2	8	no	—
95	C₃²×C₄⋊C₄	—	3	1	no	—
96	C₃²×C₄⋊C₄	—	3	2	no	—
97	C₃²×C₄⋊C₄	—	3	2	no	—
98	C₃²×C₄⋊C₄	—	3	1	no	—
99	C₃²×C₄⋊C₄	—	3	1	no	—
100	C₃²×C₄⋊C₄	—	3	1	no	—
101	C₃²×C₄⋊C₄	—	3	2	no	—
102	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
103	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
104	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
105	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
106	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
107	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
108	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
109	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
110	C₃²×C₂²⋊C₄	—	3	2	no	—
111	C₃²×C₂²⋊C₄	—	3	2	no	—
112	C₃²×C₂²⋊C₄	—	3	2	no	—
113	C₃²×C₄²	—	2	4	no	—
114	C₃²×C₄²	—	3	1	no	—
115	C₃²×C₄²	—	3	1	no	—
116	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
117	C₃²×C₄²	C₂²×C₃	1	4	yes	—
118	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
119	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
120	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
121	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
122	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
123	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
124	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
125	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
126	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
127	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
128	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
129	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
130	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
131	C₂×C₃²×D₃	—	2	2	no	—
132	C₂×C₃²×D₃	—	2	2	no	—
133	C₂×C₃²×D₃	C₂×Q₈	1	4	yes	—
134	C₃³×C₄	—	2	1	no	—
135	C₃³×C₄	—	2	2	no	—
136	C₃³×C₄	—	2	2	no	—
137	C₃³×C₄	—	2	1	no	—
138	C₃³×C₄	C₂×Q₈	1	2	yes	—
139	C₃²×Dic₃	—	2	1	no	—
140	C₃²×Dic₃	—	2	1	no	—
141	C₃²×Dic₃	—	2	2	no	—
142	C₃²×Dic₃	—	2	1	no	—
143	C₃²×Dic₃	#41	1	2	yes	—
144	C₃×C₄⋊Q₈	—	3	4	no	—
145	C₃×#125	—	2	4	no	—
146	C₃×#125	—	3	4	no	—
147	C₃×#125	C₃×D₃	1	4	yes	complemented
148	C₃×#125	C₂×C₃²	1	1	yes	characteristic, complemented
149	C₃×C₂²⋊Q₈	—	3	8	no	—
150	C₃×C₂²⋊Q₈	—	3	8	no	—
151	C₃×C₂²⋊Q₈	—	3	8	no	—
152	C₃×C₄×Q₈	—	3	4	no	—
153	C₃×C₄×D₄	—	3	4	no	—
154	C₃×C₄×D₄	—	3	4	no	—
155	C₃×C₄×D₄	—	3	4	no	—
156	C₂×C₃×C₄⋊C₄	—	3	8	no	—
157	#897	C₂×C₃²	1	2	yes	complemented
158	#896	C₂×C₃²	1	2	yes	complemented
159	#896	C₂×C₃²	1	2	yes	complemented
160	#893	C₂×C₃²	1	1	yes	characteristic
161	#893	C₂×C₃²	1	1	yes	characteristic
162	D₃×C₄⋊C₄	C₂×C₃²	1	2	yes	complemented
163	#890	C₂×C₃²	1	1	yes	characteristic, complemented
164	#888	C₂×C₃²	1	1	yes	characteristic, complemented
165	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
166	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic, complemented
167	C₂³×C₃²	—	3	2	no	—
168	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
169	C₃²×Q₈	—	6	2	no	—
170	C₃²×Q₈	—	12	2	no	—
171	C₃²×D₄	—	3	4	no	—
172	C₂×C₃²×C₄	—	2	4	no	—
173	C₂×C₃²×C₄	—	2	4	no	—
174	C₂×C₃²×C₄	—	2	4	no	—
175	C₂×C₃²×C₄	—	2	8	no	—
176	C₂×C₃²×C₄	—	2	8	no	—
177	C₂×C₃²×C₄	—	3	1	no	—
178	C₂×C₃²×C₄	—	3	1	no	—
179	C₂×C₃²×C₄	—	3	1	no	—
180	C₂×C₃²×C₄	—	3	1	no	—
181	C₂×C₃²×C₄	—	3	2	no	—
182	C₂×C₃²×C₄	—	6	4	no	—
183	C₂×C₃²×C₄	—	12	2	no	—
184	C₂×C₃²×C₄	—	12	2	no	—
185	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
187	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
188	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
189	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
190	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
191	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
192	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
193	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
194	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
195	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
196	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
197	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
198	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
199	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
200	C₃×D₁₂	C₃×Q₈	1	16	yes	complemented
201	C₃×C₄×D₃	—	2	16	no	—
202	C₃×C₄×D₃	—	4	8	no	—
203	C₃×C₄×D₃	—	4	8	no	—
204	C₃×Dic₆	—	2	8	no	—
205	C₃×Dic₆	—	4	8	no	—
206	#554	—	3	1	no	Sylow
207	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
208	C₂×C₃³	C₂×#41	1	1	yes	characteristic
209	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
210	C₃²×D₃	—	2	4	no	—
211	C₂×C₃×Q₈	—	3	4	no	—
212	C₂×C₃×Q₈	—	3	8	no	—
213	C₂×C₃×D₄	—	3	4	no	—
214	C₂²×C₃×C₄	—	3	8	no	—
215	C₂²×C₃×C₄	—	3	8	no	—
216	C₂²×C₃×C₄	—	3	8	no	—
217	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
218	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
219	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
220	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
221	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
222	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
223	C₃×C₄⋊C₄	—	2	8	no	—
224	C₃×C₄⋊C₄	—	2	4	no	—
225	C₃×C₄⋊C₄	—	2	4	no	—
226	C₃×C₄⋊C₄	—	2	8	no	—
227	C₃×C₄⋊C₄	—	3	4	no	—
228	C₃×C₄⋊C₄	—	3	8	no	—
229	C₃×C₄⋊C₄	—	3	4	no	—
230	C₃×C₄⋊C₄	—	3	8	no	—
231	C₃×C₄⋊C₄	—	3	8	no	—
232	C₃×C₄⋊C₄	—	3	4	no	—
233	C₃×C₄⋊C₄	—	3	4	no	—
234	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
235	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
236	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
237	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
238	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
239	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
240	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
241	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
242	C₃×C₂²⋊C₄	—	3	8	no	—
243	C₃×C₂²⋊C₄	—	3	8	no	—
244	C₃×C₂²⋊C₄	—	3	8	no	—
245	C₃×C₄²	—	2	4	no	—
246	C₃×C₄²	—	3	4	no	—
247	C₃×C₄²	—	3	4	no	—
248	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
249	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
250	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
251	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
252	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
253	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
254	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
255	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
256	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic
257	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic
258	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
259	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
260	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
261	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
262	C₂²×C₃²	—	2	4	no	—
263	C₂²×C₃²	—	3	4	no	—
264	C₂²×C₃²	—	6	2	no	—
265	C₂²×C₃²	—	6	2	no	—
266	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
267	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
268	C₂×C₃×D₃	—	2	8	no	—
269	C₂×C₃×D₃	—	2	8	no	—
270	C₂×C₃×D₃	C₂×C₃×Q₈	1	16	yes	—
271	C₃²×C₄	—	2	2	no	—
272	C₃²×C₄	—	2	1	no	—
273	C₃²×C₄	—	2	1	no	—
274	C₃²×C₄	—	2	2	no	—
275	C₃²×C₄	—	2	8	no	—
276	C₃²×C₄	—	2	4	no	—
277	C₃²×C₄	—	2	8	no	—
278	C₃²×C₄	—	2	8	no	—
279	C₃²×C₄	—	2	4	no	—
280	C₃²×C₄	—	3	2	no	—
281	C₃²×C₄	—	4	8	no	—
282	C₃²×C₄	—	4	4	no	—
283	C₃²×C₄	—	4	4	no	—
284	C₃²×C₄	—	4	8	no	—
285	C₃²×C₄	—	6	1	no	—
286	C₃²×C₄	—	6	1	no	—
287	C₃²×C₄	—	6	2	no	—
288	C₃²×C₄	—	6	1	no	—
289	C₃²×C₄	D₃×Q₈	1	2	yes	—
290	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	—
291	C₃×Dic₃	—	2	4	no	—
292	C₃×Dic₃	—	2	4	no	—
293	C₃×Dic₃	—	2	8	no	—
294	C₃×Dic₃	—	2	4	no	—
295	C₃×Dic₃	C₃×#41	1	8	yes	—
296	C₄⋊Q₈	—	3	1	no	—
297	#125	—	3	1	no	—
298	#125	C₃²×D₃	1	1	yes	characteristic, complemented
299	C₂²⋊Q₈	—	3	2	no	—
300	C₂²⋊Q₈	—	3	2	no	—
301	C₂²⋊Q₈	—	3	2	no	—
302	C₄×Q₈	—	3	1	no	—
303	C₄×D₄	—	3	1	no	—
304	C₄×D₄	—	3	1	no	—
305	C₄×D₄	—	3	1	no	—
306	C₂×C₄⋊C₄	—	3	2	no	—
307	C₃³	#554	1	1	yes	characteristic, complemented, Sylow
308	C₂³×C₃	—	3	8	no	—
309	C₂²×D₃	C₂³×C₃²	1	2	yes	—
310	C₃×Q₈	—	6	8	no	—
311	C₃×Q₈	—	12	8	no	—
312	C₃×D₄	—	3	16	no	—
313	C₂×C₃×C₄	—	2	4	no	—
314	C₂×C₃×C₄	—	2	4	no	—
315	C₂×C₃×C₄	—	2	4	no	—
316	C₂×C₃×C₄	—	2	8	no	—
317	C₂×C₃×C₄	—	2	8	no	—
318	C₂×C₃×C₄	—	3	4	no	—
319	C₂×C₃×C₄	—	3	4	no	—
320	C₂×C₃×C₄	—	3	4	no	—
321	C₂×C₃×C₄	—	3	4	no	—
322	C₂×C₃×C₄	—	3	8	no	—
323	C₂×C₃×C₄	—	6	16	no	—
324	C₂×C₃×C₄	—	12	8	no	—
325	C₂×C₃×C₄	—	12	8	no	—
326	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
327	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
328	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
329	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
330	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
331	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
332	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
333	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
334	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
335	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
336	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
337	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
338	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
339	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
340	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
341	D₁₂	C₃²×Q₈	1	4	yes	complemented
342	C₄×D₃	—	2	4	no	—
343	C₄×D₃	—	4	2	no	—
344	C₄×D₃	—	4	2	no	—
345	Dic₆	—	2	2	no	—
346	Dic₆	—	4	2	no	—
347	C₂×C₃²	—	2	4	no	—
348	C₂×C₃²	—	2	4	no	—
349	C₂×C₃²	—	2	4	no	—
350	C₂×C₃²	—	6	4	no	—
351	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
352	C₂×C₃²	D₃×#41	1	1	yes	characteristic
353	C₂×C₃²	#1009	1	1	yes	characteristic
354	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
355	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
356	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
357	C₃×D₃	—	2	16	no	—
358	C₂×Q₈	—	3	1	no	—
359	C₂×Q₈	—	3	2	no	—
360	C₂×D₄	—	3	1	no	—
361	C₂²×C₄	—	3	2	no	—
362	C₂²×C₄	—	3	2	no	—
363	C₂²×C₄	—	3	2	no	—
364	C₄⋊C₄	—	3	1	no	—
365	C₄⋊C₄	—	3	2	no	—
366	C₄⋊C₄	—	3	2	no	—
367	C₄⋊C₄	—	3	1	no	—
368	C₄⋊C₄	—	3	1	no	—
369	C₄⋊C₄	—	3	1	no	—
370	C₄⋊C₄	—	3	2	no	—
371	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
372	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
373	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
374	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
375	C₂²⋊C₄	—	3	2	no	—
376	C₂²⋊C₄	—	3	2	no	—
377	C₂²⋊C₄	—	3	2	no	—
378	C₄²	—	3	1	no	—
379	C₄²	—	3	1	no	—
380	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
381	C₂²×C₃	—	2	4	no	—
382	C₂²×C₃	—	3	16	no	—
383	C₂²×C₃	—	6	8	no	—
384	C₂²×C₃	—	6	8	no	—
385	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
386	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
387	C₂×D₃	—	2	2	no	—
388	C₂×D₃	—	2	2	no	—
389	C₂×D₃	C₂×C₃²×Q₈	1	4	yes	—
390	C₃×C₄	—	2	8	no	—
391	C₃×C₄	—	2	4	no	—
392	C₃×C₄	—	2	4	no	—
393	C₃×C₄	—	2	8	no	—
394	C₃×C₄	—	2	1	no	—
395	C₃×C₄	—	2	2	no	—
396	C₃×C₄	—	2	2	no	—
397	C₃×C₄	—	2	1	no	—
398	C₃×C₄	—	2	8	no	—
399	C₃×C₄	—	3	8	no	—
400	C₃×C₄	—	4	8	no	—
401	C₃×C₄	—	4	4	no	—
402	C₃×C₄	—	4	4	no	—
403	C₃×C₄	—	4	8	no	—
404	C₃×C₄	—	6	4	no	—
405	C₃×C₄	—	6	4	no	—
406	C₃×C₄	—	6	8	no	—
407	C₃×C₄	—	6	4	no	—
408	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
409	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	—
410	Dic₃	—	2	1	no	—
411	Dic₃	—	2	1	no	—
412	Dic₃	—	2	2	no	—
413	Dic₃	—	2	1	no	—
414	Dic₃	C₃²×#41	1	2	yes	—
415	C₃²	—	2	4	no	—
416	C₃²	#5690	1	1	yes	characteristic, direct product factor
417	C₃²	C₃×#554	1	4	yes	complemented
418	C₂³	—	3	2	no	—
419	Q₈	—	6	2	no	—
420	Q₈	—	12	2	no	—
421	D₄	—	3	4	no	—
422	C₂×C₄	—	3	1	no	—
423	C₂×C₄	—	3	1	no	—
424	C₂×C₄	—	3	1	no	—
425	C₂×C₄	—	3	1	no	—
426	C₂×C₄	—	3	2	no	—
427	C₂×C₄	—	6	4	no	—
428	C₂×C₄	—	12	2	no	—
429	C₂×C₄	—	12	2	no	—
430	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
431	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
432	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
433	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
434	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
435	C₂×C₃	—	2	4	no	—
436	C₂×C₃	—	2	4	no	—
437	C₂×C₃	—	2	4	no	—
438	C₂×C₃	—	6	16	no	—
439	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
440	C₂×C₃	C₃×D₃×#41	1	4	yes	—
441	C₂×C₃	C₃×#1009	1	4	yes	—
442	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
443	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
444	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
445	D₃	—	2	4	no	—
446	C₂²	—	3	4	no	—
447	C₂²	—	6	2	no	—
448	C₂²	—	6	2	no	—
449	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
450	C₄	—	2	2	no	—
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	2	no	—
454	C₄	—	3	2	no	—
455	C₄	—	6	1	no	—
456	C₄	—	6	1	no	—
457	C₄	—	6	2	no	—
458	C₄	—	6	1	no	—
459	C₄	C₃²×D₃×Q₈	1	2	yes	—
460	C₃	—	2	4	no	—
461	C₃	C₃×#5690	1	4	yes	direct product factor
462	C₃	C₃²×#554	1	1	yes	characteristic, complemented
463	C₂	—	6	4	no	—
464	C₂	C₂×C₃²×D₃×Q₈	1	1	yes	characteristic
465	C₂	C₃²×D₃×#41	1	1	yes	characteristic
466	C₂	C₃²×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#125	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×#125	D₃
hypocenter	C₃	C₃²×#554
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#125

Complements

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

N	H ≅ G/N	type
1. C₃³×#125	C₂	semidirect
2. C₃²×#897	C₂	semidirect
3. C₃²×#896	C₂	semidirect
4. C₃²×#896	C₂	semidirect
7. C₃²×D₃×C₄⋊C₄	C₂	semidirect
8. C₃²×#890	C₂	semidirect
9. C₃²×#888	C₂	semidirect
11. C₃²×C₄×Dic₆	C₂	semidirect
13. C₃×#5690	C₃	direct (N × H)
40. C₃²×#125	D₃	semidirect
41. C₃²×#125	C₂×C₃	semidirect
50. C₃×#897	C₂×C₃	semidirect
51. C₃×#896	C₂×C₃	semidirect
52. C₃×#896	C₂×C₃	semidirect
55. C₃×D₃×C₄⋊C₄	C₂×C₃	semidirect
56. C₃×#890	C₂×C₃	semidirect
57. C₃×#888	C₂×C₃	semidirect
59. C₃×C₄×Dic₆	C₂×C₃	semidirect
71. C₃²×D₁₂	Q₈	semidirect
78. #5690	C₃²	direct (N × H)
147. C₃×#125	C₃×D₃	semidirect
148. C₃×#125	C₂×C₃²	semidirect
157. #897	C₂×C₃²	semidirect
158. #896	C₂×C₃²	semidirect
159. #896	C₂×C₃²	semidirect
162. D₃×C₄⋊C₄	C₂×C₃²	semidirect
163. #890	C₂×C₃²	semidirect
164. #888	C₂×C₃²	semidirect
166. C₄×Dic₆	C₂×C₃²	semidirect
200. C₃×D₁₂	C₃×Q₈	semidirect
298. #125	C₃²×D₃	semidirect
307. C₃³	#554	semidirect
341. D₁₂	C₃²×Q₈	semidirect
416. C₃²	#5690	direct (N × H)
417. C₃²	C₃×#554	semidirect
461. C₃	C₃×#5690	direct (N × H)
462. C₃	C₃²×#554	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#125	2
C₃²×#897	2
C₃²×#896	2
C₃²×#893	2
C₃²×D₃×C₄⋊C₄	2
C₃²×#890	2
C₃²×#888	2
C₃²×C₄×D₁₂	2
C₃²×C₄×Dic₆	2
C₃²×#554	3
C₃×#5690	3

Subgroups — C32×#5690 #13179068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5690 #13179068