Subgroups — C₃×C₃⋊D₃×#137 #13179527

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Q₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64127	C₂	1	2	yes	maximal, complemented
7	C₃×Q₁₆×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
8	C₃×#64124	C₂	1	2	yes	maximal, complemented
9	C₃×SD₁₆×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
10	C₃×#64118	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×M₁₆×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃⋊D₃×#137	C₃	1	1	yes	maximal, characteristic, direct product factor
13	C₃×D₃×#137	—	3	4	no	maximal
14	C₃³×#41	C₂²	1	1	yes	characteristic
15	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₃×#3832	C₂²	1	1	yes	characteristic
17	C₃×Q₈×C₃⋊D₃	—	2	1	no	—
18	C₃×Q₈×C₃⋊D₃	—	2	1	no	—
19	C₃×Q₈×C₃⋊D₃	C₂²	1	2	yes	—
20	C₃×Q₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
21	C₃×Q₈×C₃⋊D₃	C₂²	1	2	yes	complemented
22	C₃×#3830	—	2	1	no	—
23	C₃×#3830	C₂²	1	1	yes	characteristic
24	C₃×D₄×C₃⋊D₃	—	2	1	no	—
25	C₃×D₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
26	C₃×#3828	—	2	1	no	—
27	C₃×#3828	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
29	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
30	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
31	C₂×C₃×#634	—	2	1	no	—
32	C₂×C₃×#634	C₂²	1	1	yes	characteristic, complemented
33	C₃³×Q₁₆	C₂²	1	2	yes	complemented
34	C₃³×SD₁₆	C₂²	1	2	yes	complemented
35	C₃³×M₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₃×#3756	C₂²	1	2	yes	complemented
37	C₃×#3756	C₂²	1	2	yes	complemented
38	C₃×#3755	C₂²	1	2	yes	complemented
39	C₃×#3754	C₂²	1	2	yes	complemented
40	C₃×#3748	C₂²	1	1	yes	characteristic, complemented
41	C₃×#3746	C₂²	1	2	yes	complemented
42	C₃×#3744	C₂²	1	2	yes	complemented
43	C₃×#3743	C₂²	1	2	yes	complemented
44	C₃×C₈×C₃⋊D₃	C₂²	1	2	yes	—
45	#41×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
46	C₂×Q₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
47	C₃×D₃×#41	—	3	4	no	—
48	C₂×C₃×D₃×Q₈	—	3	4	no	—
49	C₃²×#137	—	2	4	no	—
50	C₃²×#137	D₃	1	4	yes	complemented
51	C₃²×#137	C₂×C₃	1	1	yes	characteristic, complemented
52	#64160	C₂×C₃	1	1	yes	characteristic, complemented
53	#64151	C₂×C₃	1	1	yes	characteristic, complemented
54	#64127	C₂×C₃	1	2	yes	complemented
55	Q₁₆×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
56	#64124	C₂×C₃	1	2	yes	complemented
57	SD₁₆×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
58	#64118	C₂×C₃	1	1	yes	characteristic, complemented
59	M₁₆×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
60	C₃×#951	—	3	4	no	—
61	C₃×#942	—	3	4	no	—
62	C₃×#918	—	3	8	no	—
63	C₃×D₃×Q₁₆	—	3	8	no	—
64	C₃×#915	—	3	8	no	—
65	C₃×D₃×SD₁₆	—	3	8	no	—
66	C₃×#909	—	3	4	no	—
67	C₃×D₃×M₁₆	—	3	4	no	—
68	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
69	C₂²×C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
70	C₃³×Q₈	—	2	1	no	—
71	C₃³×Q₈	C₂³	1	2	yes	—
72	C₃³×Q₈	C₂³	1	1	yes	characteristic
73	C₃³×D₄	—	2	1	no	—
74	C₃³×D₄	C₂³	1	1	yes	characteristic
75	C₂×C₃³×C₄	—	2	1	no	—
76	C₂×C₃³×C₄	—	2	1	no	—
77	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
78	C₃×#638	—	2	1	no	—
79	C₃×#638	—	2	1	no	—
80	C₃×#638	—	2	1	no	—
81	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
82	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
83	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
84	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
85	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
86	C₃×C₄×C₃⋊D₃	—	2	1	no	—
87	C₃×C₄×C₃⋊D₃	—	2	2	no	—
88	C₃×C₄×C₃⋊D₃	—	2	2	no	—
89	C₃×C₄×C₃⋊D₃	—	2	1	no	—
90	C₃×C₄×C₃⋊D₃	—	2	1	no	—
91	C₃×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
92	C₃×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
93	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
94	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
95	C₃×#634	—	2	1	no	—
96	C₃×#634	—	2	2	no	—
97	C₃×#634	—	2	2	no	—
98	C₃×#634	—	2	1	no	—
99	C₃×#634	C₂³	1	2	yes	—
100	C₃×#634	C₂³	1	1	yes	characteristic
101	C₃³×C₈	C₂³	1	2	yes	—
102	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
103	C₂×C₃×#137	—	9	1	no	—
104	D₃×#137	—	3	4	no	—
105	C₃²×#41	—	2	4	no	—
106	C₃²×#41	C₂×D₃	1	4	yes	—
107	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₃²×Q₈	—	2	4	no	—
109	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	complemented
110	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
111	#3832	C₂²×C₃	1	1	yes	characteristic
112	Q₈×C₃⋊D₃	—	2	1	no	—
113	Q₈×C₃⋊D₃	—	2	1	no	—
114	Q₈×C₃⋊D₃	C₂²×C₃	1	2	yes	—
115	Q₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
116	Q₈×C₃⋊D₃	C₂²×C₃	1	2	yes	complemented
117	#3830	—	2	1	no	—
118	#3830	C₂²×C₃	1	1	yes	characteristic
119	D₄×C₃⋊D₃	—	2	1	no	—
120	D₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
121	#3828	—	2	1	no	—
122	#3828	C₂²×C₃	1	1	yes	characteristic
123	C₂×C₄×C₃⋊D₃	—	2	1	no	—
124	C₂×C₄×C₃⋊D₃	—	2	1	no	—
125	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
126	C₂×#634	—	2	1	no	—
127	C₂×#634	C₂²×C₃	1	1	yes	characteristic, complemented
128	C₃×#239	—	3	4	no	—
129	C₃×D₃×Q₈	—	3	8	no	—
130	C₃×D₃×Q₈	—	3	8	no	—
131	C₃×D₃×Q₈	—	3	4	no	—
132	C₃×D₃×Q₈	—	6	4	no	—
133	C₃×D₃×Q₈	—	6	4	no	—
134	C₃×#237	—	3	4	no	—
135	C₃×#237	—	6	4	no	—
136	C₃×D₃×D₄	—	3	4	no	—
137	C₃×D₃×D₄	—	6	4	no	—
138	C₃×#235	—	3	4	no	—
139	C₃×#235	—	6	4	no	—
140	C₂×C₃×C₄×D₃	—	3	4	no	—
141	C₂×C₃×C₄×D₃	—	6	4	no	—
142	C₂×C₃×C₄×D₃	—	6	4	no	—
143	C₂×C₃×Dic₆	—	3	4	no	—
144	C₂×C₃×Dic₆	—	6	4	no	—
145	C₃²×Q₁₆	—	2	8	no	—
146	C₃²×Q₁₆	C₂×D₃	1	8	yes	complemented
147	C₃²×Q₁₆	C₂²×C₃	1	2	yes	complemented
148	C₃²×SD₁₆	—	2	8	no	—
149	C₃²×SD₁₆	C₂×D₃	1	8	yes	complemented
150	C₃²×SD₁₆	C₂²×C₃	1	2	yes	complemented
151	C₃²×M₁₆	—	2	4	no	—
152	C₃²×M₁₆	C₂×D₃	1	4	yes	complemented
153	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
154	#3756	C₂²×C₃	1	2	yes	complemented
155	#3756	C₂²×C₃	1	2	yes	complemented
156	#3755	C₂²×C₃	1	2	yes	complemented
157	#3754	C₂²×C₃	1	2	yes	complemented
158	#3748	C₂²×C₃	1	1	yes	characteristic, complemented
159	#3746	C₂²×C₃	1	2	yes	complemented
160	#3744	C₂²×C₃	1	2	yes	complemented
161	#3743	C₂²×C₃	1	2	yes	complemented
162	C₈×C₃⋊D₃	C₂²×C₃	1	2	yes	—
163	C₃×#216	—	3	8	no	—
164	C₃×#216	—	3	8	no	—
165	C₃×#215	—	3	8	no	—
166	C₃×#214	—	3	8	no	—
167	C₃×#208	—	3	4	no	—
168	C₃×Dic₁₂	—	3	8	no	—
169	C₃×#204	—	3	8	no	—
170	C₃×ΓL(1,𝔽₂₅)	—	3	8	no	—
171	C₃×D₃×C₈	—	3	8	no	—
172	C₂²×C₃³	—	2	1	no	—
173	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×C₃⋊D₃	—	2	2	no	—
175	C₂×C₃×C₃⋊D₃	—	2	1	no	—
176	C₂×C₃×C₃⋊D₃	—	4	2	no	—
177	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
178	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
179	C₃³×C₄	—	2	2	no	—
180	C₃³×C₄	—	2	1	no	—
181	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
182	C₃³×C₄	C₂⁴	1	1	yes	characteristic
183	C₃×C₃⋊Dic₃	—	2	2	no	—
184	C₃×C₃⋊Dic₃	—	2	1	no	—
185	C₃×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
186	C₃×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
187	C₂×C₃×#41	—	9	1	no	—
188	C₂²×C₃×Q₈	—	9	1	no	—
189	D₃×#41	—	3	4	no	—
190	C₂×D₃×Q₈	—	3	4	no	—
191	C₃×#137	—	2	4	no	—
192	C₃×#137	—	9	1	no	—
193	C₃×#137	—	9	1	no	—
194	C₃×#137	—	9	2	no	—
195	C₃×#137	—	9	2	no	—
196	C₃×#137	—	9	1	no	—
197	C₃×#137	C₃×D₃	1	4	yes	complemented
198	C₃×#137	C₃⋊D₃	1	1	yes	characteristic, direct product factor
199	C₂×C₃×Q₁₆	—	9	2	no	—
200	C₂×C₃×SD₁₆	—	9	2	no	—
201	C₂×C₃×M₁₆	—	9	1	no	—
202	#951	—	3	4	no	—
203	#942	—	3	4	no	—
204	#918	—	3	8	no	—
205	D₃×Q₁₆	—	3	8	no	—
206	#915	—	3	8	no	—
207	D₃×SD₁₆	—	3	8	no	—
208	#909	—	3	4	no	—
209	D₃×M₁₆	—	3	4	no	—
210	C₂²×C₃⋊D₃	—	2	1	no	—
211	C₂²×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
212	C₂²×C₃×D₃	—	3	4	no	—
213	C₂²×C₃×D₃	—	6	4	no	—
214	C₃²×Q₈	—	2	1	no	—
215	C₃²×Q₈	—	2	8	no	—
216	C₃²×Q₈	—	2	4	no	—
217	C₃²×Q₈	—	2	4	no	—
218	C₃²×Q₈	—	4	4	no	—
219	C₃²×Q₈	C₂²×D₃	1	8	yes	—
220	C₃²×Q₈	C₂²×D₃	1	4	yes	—
221	C₃²×Q₈	C₂³×C₃	1	2	yes	—
222	C₃²×Q₈	C₂³×C₃	1	1	yes	characteristic
223	C₃²×D₄	—	2	1	no	—
224	C₃²×D₄	—	2	4	no	—
225	C₃²×D₄	—	2	4	no	—
226	C₃²×D₄	—	4	4	no	—
227	C₃²×D₄	C₂²×D₃	1	4	yes	—
228	C₃²×D₄	C₂³×C₃	1	1	yes	characteristic
229	C₂×C₃²×C₄	—	2	1	no	—
230	C₂×C₃²×C₄	—	2	1	no	—
231	C₂×C₃²×C₄	—	2	4	no	—
232	C₂×C₃²×C₄	—	2	4	no	—
233	C₂×C₃²×C₄	—	2	4	no	—
234	C₂×C₃²×C₄	—	4	4	no	—
235	C₂×C₃²×C₄	—	4	4	no	—
236	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
237	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
238	#638	—	2	1	no	—
239	#638	—	2	1	no	—
240	#638	—	2	1	no	—
241	C₂×C₃⋊Dic₃	—	2	1	no	—
242	C₂×C₃⋊Dic₃	—	2	1	no	—
243	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
244	C₄⋊(C₃⋊D₃)	—	2	1	no	—
245	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	1	yes	characteristic
246	C₄×C₃⋊D₃	—	2	1	no	—
247	C₄×C₃⋊D₃	—	2	2	no	—
248	C₄×C₃⋊D₃	—	2	2	no	—
249	C₄×C₃⋊D₃	—	2	1	no	—
250	C₄×C₃⋊D₃	—	2	1	no	—
251	C₄×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
252	C₄×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
253	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
254	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
255	#634	—	2	1	no	—
256	#634	—	2	2	no	—
257	#634	—	2	2	no	—
258	#634	—	2	1	no	—
259	#634	C₂³×C₃	1	2	yes	—
260	#634	C₂³×C₃	1	1	yes	characteristic
261	C₃×#67	—	6	4	no	—
262	C₃×#67	—	6	4	no	—
263	C₃×#67	—	6	4	no	—
264	C₂×C₃×Dic₃	—	3	4	no	—
265	C₂×C₃×Dic₃	—	6	4	no	—
266	C₂×C₃×Dic₃	—	6	4	no	—
267	C₃×D₁₂	—	3	4	no	—
268	C₃×D₁₂	—	6	4	no	—
269	C₃×C₄×D₃	—	3	4	no	—
270	C₃×C₄×D₃	—	3	4	no	—
271	C₃×C₄×D₃	—	3	4	no	—
272	C₃×C₄×D₃	—	3	4	no	—
273	C₃×C₄×D₃	—	6	4	no	—
274	C₃×C₄×D₃	—	6	4	no	—
275	C₃×C₄×D₃	—	6	8	no	—
276	C₃×C₄×D₃	—	6	8	no	—
277	C₃×C₄×D₃	—	6	4	no	—
278	C₃×Dic₆	—	3	8	no	—
279	C₃×Dic₆	—	3	4	no	—
280	C₃×Dic₆	—	6	8	no	—
281	C₃×Dic₆	—	6	8	no	—
282	C₃×Dic₆	—	6	4	no	—
283	C₃×Dic₆	—	6	4	no	—
284	C₃²×C₈	—	2	8	no	—
285	C₃²×C₈	C₂²×D₃	1	8	yes	—
286	C₃²×C₈	C₂³×C₃	1	2	yes	—
287	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	2	yes	—
288	C₃×C₃⋊C₈	—	3	8	no	—
289	C₂×#137	—	9	1	no	Sylow
290	C₂×C₃³	—	2	1	no	—
291	C₂×C₃³	—	4	1	no	—
292	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
293	C₃×C₃⋊D₃	—	2	1	no	—
294	C₃×C₃⋊D₃	—	4	1	no	—
295	C₃×C₃⋊D₃	#137	1	2	yes	direct product factor
296	C₃×#41	—	2	4	no	—
297	C₃×#41	—	9	1	no	—
298	C₃×#41	—	9	1	no	—
299	C₃×#41	—	9	1	no	—
300	C₃×#41	—	18	1	no	—
301	C₃×#41	—	18	1	no	—
302	C₃×#41	C₂×C₃×D₃	1	4	yes	—
303	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
304	C₂×C₃×Q₈	—	2	4	no	—
305	C₂×C₃×Q₈	—	9	2	no	—
306	C₂×C₃×Q₈	—	9	1	no	—
307	C₂×C₃×Q₈	—	9	2	no	—
308	C₂×C₃×Q₈	—	9	1	no	—
309	C₂×C₃×Q₈	—	18	1	no	—
310	C₂×C₃×Q₈	—	18	1	no	—
311	C₂×C₃×Q₈	—	18	1	no	—
312	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	complemented
313	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
314	C₂×C₃×D₄	—	9	1	no	—
315	C₂×C₃×D₄	—	18	1	no	—
316	C₂²×C₃×C₄	—	9	1	no	—
317	C₂²×C₃×C₄	—	18	1	no	—
318	C₂²×C₃×C₄	—	18	1	no	—
319	#239	—	3	4	no	—
320	D₃×Q₈	—	3	8	no	—
321	D₃×Q₈	—	3	8	no	—
322	D₃×Q₈	—	3	4	no	—
323	D₃×Q₈	—	6	4	no	—
324	D₃×Q₈	—	6	4	no	—
325	#237	—	3	4	no	—
326	#237	—	6	4	no	—
327	D₃×D₄	—	3	4	no	—
328	D₃×D₄	—	6	4	no	—
329	#235	—	3	4	no	—
330	#235	—	6	4	no	—
331	C₂×C₄×D₃	—	3	4	no	—
332	C₂×C₄×D₃	—	6	4	no	—
333	C₂×C₄×D₃	—	6	4	no	—
334	C₂×Dic₆	—	3	4	no	—
335	C₂×Dic₆	—	6	4	no	—
336	C₃×Q₁₆	—	2	8	no	—
337	C₃×Q₁₆	—	9	2	no	—
338	C₃×Q₁₆	—	9	2	no	—
339	C₃×Q₁₆	—	9	2	no	—
340	C₃×Q₁₆	C₂×C₃×D₃	1	8	yes	complemented
341	C₃×Q₁₆	C₂×C₃⋊D₃	1	2	yes	complemented
342	C₃×SD₁₆	—	2	8	no	—
343	C₃×SD₁₆	—	9	2	no	—
344	C₃×SD₁₆	—	9	2	no	—
345	C₃×SD₁₆	—	9	2	no	—
346	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
347	C₃×SD₁₆	C₂×C₃⋊D₃	1	2	yes	complemented
348	C₃×M₁₆	—	2	4	no	—
349	C₃×M₁₆	—	9	1	no	—
350	C₃×M₁₆	—	9	2	no	—
351	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	complemented
352	C₃×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
353	C₂×C₃×C₈	—	9	2	no	—
354	#216	—	3	8	no	—
355	#216	—	3	8	no	—
356	#215	—	3	8	no	—
357	#214	—	3	8	no	—
358	#208	—	3	4	no	—
359	Dic₁₂	—	3	8	no	—
360	#204	—	3	8	no	—
361	ΓL(1,𝔽₂₅)	—	3	8	no	—
362	D₃×C₈	—	3	8	no	—
363	C₂²×C₃²	—	2	1	no	—
364	C₂²×C₃²	—	2	4	no	—
365	C₂²×C₃²	—	2	4	no	—
366	C₂²×C₃²	—	4	4	no	—
367	C₂²×C₃²	D₃×D₄	1	4	yes	—
368	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
369	C₂×C₃⋊D₃	—	2	2	no	—
370	C₂×C₃⋊D₃	—	2	1	no	—
371	C₂×C₃⋊D₃	—	4	2	no	—
372	C₂×C₃⋊D₃	C₂×C₃×D₄	1	1	yes	characteristic
373	C₂×C₃⋊D₃	C₂×C₃×D₄	1	1	yes	characteristic
374	C₂×C₃×D₃	—	3	4	no	—
375	C₂×C₃×D₃	—	3	4	no	—
376	C₂×C₃×D₃	—	6	8	no	—
377	C₂×C₃×D₃	—	6	4	no	—
378	C₂×C₃×D₃	—	12	8	no	—
379	C₃²×C₄	—	2	2	no	—
380	C₃²×C₄	—	2	1	no	—
381	C₃²×C₄	—	2	4	no	—
382	C₃²×C₄	—	2	4	no	—
383	C₃²×C₄	—	2	8	no	—
384	C₃²×C₄	—	2	4	no	—
385	C₃²×C₄	—	4	4	no	—
386	C₃²×C₄	—	4	8	no	—
387	C₃²×C₄	D₃×D₄	1	4	yes	—
388	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
389	C₃²×C₄	C₂³×D₃	1	4	yes	—
390	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
391	C₃⋊Dic₃	—	2	2	no	—
392	C₃⋊Dic₃	—	2	1	no	—
393	C₃⋊Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
394	C₃⋊Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
395	C₃×Dic₃	—	3	4	no	—
396	C₃×Dic₃	—	3	4	no	—
397	C₃×Dic₃	—	6	8	no	—
398	C₃×Dic₃	—	6	4	no	—
399	C₂×#41	—	9	1	no	—
400	C₂²×Q₈	—	9	1	no	—
401	#137	—	9	1	no	—
402	#137	—	9	1	no	—
403	#137	—	9	2	no	—
404	#137	—	9	2	no	—
405	#137	—	9	1	no	—
406	#137	C₃×C₃⋊D₃	1	1	yes	characteristic, direct product factor
407	C₂×Q₁₆	—	9	2	no	—
408	C₂×SD₁₆	—	9	2	no	—
409	C₂×M₁₆	—	9	1	no	—
410	C₃³	C₂×#137	1	1	yes	characteristic, complemented, Sylow
411	C₂³×C₃	—	9	1	no	—
412	C₂³×C₃	—	18	1	no	—
413	C₂²×D₃	—	3	4	no	—
414	C₂²×D₃	—	6	4	no	—
415	C₃×Q₈	—	2	4	no	—
416	C₃×Q₈	—	2	1	no	—
417	C₃×Q₈	—	2	8	no	—
418	C₃×Q₈	—	2	4	no	—
419	C₃×Q₈	—	4	4	no	—
420	C₃×Q₈	—	9	2	no	—
421	C₃×Q₈	—	9	1	no	—
422	C₃×Q₈	—	18	2	no	—
423	C₃×Q₈	—	18	2	no	—
424	C₃×Q₈	—	18	1	no	—
425	C₃×Q₈	—	18	1	no	—
426	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
427	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
428	C₃×Q₈	C₂²×C₃⋊D₃	1	2	yes	—
429	C₃×Q₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
430	C₃×D₄	—	2	4	no	—
431	C₃×D₄	—	2	1	no	—
432	C₃×D₄	—	2	4	no	—
433	C₃×D₄	—	4	4	no	—
434	C₃×D₄	—	9	1	no	—
435	C₃×D₄	—	18	1	no	—
436	C₃×D₄	—	18	1	no	—
437	C₃×D₄	—	18	1	no	—
438	C₃×D₄	—	18	1	no	—
439	C₃×D₄	C₂²×C₃×D₃	1	4	yes	—
440	C₃×D₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
441	C₂×C₃×C₄	—	2	4	no	—
442	C₂×C₃×C₄	—	2	1	no	—
443	C₂×C₃×C₄	—	2	4	no	—
444	C₂×C₃×C₄	—	2	1	no	—
445	C₂×C₃×C₄	—	2	4	no	—
446	C₂×C₃×C₄	—	4	4	no	—
447	C₂×C₃×C₄	—	4	4	no	—
448	C₂×C₃×C₄	—	9	1	no	—
449	C₂×C₃×C₄	—	9	1	no	—
450	C₂×C₃×C₄	—	9	1	no	—
451	C₂×C₃×C₄	—	9	1	no	—
452	C₂×C₃×C₄	—	9	1	no	—
453	C₂×C₃×C₄	—	18	1	no	—
454	C₂×C₃×C₄	—	18	1	no	—
455	C₂×C₃×C₄	—	18	2	no	—
456	C₂×C₃×C₄	—	18	2	no	—
457	C₂×C₃×C₄	—	18	1	no	—
458	C₂×C₃×C₄	—	18	1	no	—
459	C₂×C₃×C₄	—	18	1	no	—
460	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
461	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
462	#67	—	6	4	no	—
463	#67	—	6	4	no	—
464	#67	—	6	4	no	—
465	C₂×Dic₃	—	3	4	no	—
466	C₂×Dic₃	—	6	4	no	—
467	C₂×Dic₃	—	6	4	no	—
468	D₁₂	—	3	4	no	—
469	D₁₂	—	6	4	no	—
470	C₄×D₃	—	3	4	no	—
471	C₄×D₃	—	3	4	no	—
472	C₄×D₃	—	3	4	no	—
473	C₄×D₃	—	3	4	no	—
474	C₄×D₃	—	6	4	no	—
475	C₄×D₃	—	6	4	no	—
476	C₄×D₃	—	6	8	no	—
477	C₄×D₃	—	6	8	no	—
478	C₄×D₃	—	6	4	no	—
479	Dic₆	—	3	8	no	—
480	Dic₆	—	3	4	no	—
481	Dic₆	—	6	8	no	—
482	Dic₆	—	6	8	no	—
483	Dic₆	—	6	4	no	—
484	Dic₆	—	6	4	no	—
485	C₃×C₈	—	2	8	no	—
486	C₃×C₈	—	9	2	no	—
487	C₃×C₈	C₂²×C₃×D₃	1	8	yes	—
488	C₃×C₈	C₂²×C₃⋊D₃	1	2	yes	—
489	C₃⋊C₈	—	3	8	no	—
490	C₂×C₃²	—	2	1	no	—
491	C₂×C₃²	—	2	4	no	—
492	C₂×C₃²	—	2	4	no	—
493	C₂×C₃²	—	4	4	no	—
494	C₂×C₃²	—	4	1	no	—
495	C₂×C₃²	—	4	4	no	—
496	C₂×C₃²	—	8	4	no	—
497	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
498	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
499	C₃⋊D₃	—	2	1	no	—
500	C₃⋊D₃	—	4	1	no	—
501	C₃⋊D₃	C₃×#137	1	2	yes	direct product factor
502	C₃×D₃	—	3	8	no	—
503	C₃×D₃	—	6	4	no	—
504	C₃×D₃	—	12	4	no	—
505	#41	—	9	1	no	—
506	#41	—	9	1	no	—
507	#41	—	9	1	no	—
508	#41	—	18	1	no	—
509	#41	—	18	1	no	—
510	#41	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
511	C₂×Q₈	—	9	2	no	—
512	C₂×Q₈	—	9	1	no	—
513	C₂×Q₈	—	9	2	no	—
514	C₂×Q₈	—	9	1	no	—
515	C₂×Q₈	—	18	1	no	—
516	C₂×Q₈	—	18	1	no	—
517	C₂×Q₈	—	18	1	no	—
518	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
519	C₂×D₄	—	9	1	no	—
520	C₂×D₄	—	18	1	no	—
521	C₂²×C₄	—	9	1	no	—
522	C₂²×C₄	—	18	1	no	—
523	C₂²×C₄	—	18	1	no	—
524	Q₁₆	—	9	2	no	—
525	Q₁₆	—	9	2	no	—
526	Q₁₆	—	9	2	no	—
527	Q₁₆	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
528	SD₁₆	—	9	2	no	—
529	SD₁₆	—	9	2	no	—
530	SD₁₆	—	9	2	no	—
531	SD₁₆	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
532	M₁₆	—	9	1	no	—
533	M₁₆	—	9	2	no	—
534	M₁₆	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
535	C₂×C₈	—	9	2	no	—
536	C₂²×C₃	—	2	4	no	—
537	C₂²×C₃	—	2	1	no	—
538	C₂²×C₃	—	2	4	no	—
539	C₂²×C₃	—	4	4	no	—
540	C₂²×C₃	—	9	1	no	—
541	C₂²×C₃	—	9	1	no	—
542	C₂²×C₃	—	18	2	no	—
543	C₂²×C₃	—	18	1	no	—
544	C₂²×C₃	—	36	2	no	—
545	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
546	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
547	C₂×D₃	—	3	4	no	—
548	C₂×D₃	—	3	4	no	—
549	C₂×D₃	—	6	8	no	—
550	C₂×D₃	—	6	4	no	—
551	C₂×D₃	—	12	8	no	—
552	C₃×C₄	—	2	8	no	—
553	C₃×C₄	—	2	4	no	—
554	C₃×C₄	—	2	1	no	—
555	C₃×C₄	—	2	2	no	—
556	C₃×C₄	—	2	4	no	—
557	C₃×C₄	—	2	4	no	—
558	C₃×C₄	—	4	4	no	—
559	C₃×C₄	—	4	8	no	—
560	C₃×C₄	—	9	1	no	—
561	C₃×C₄	—	9	1	no	—
562	C₃×C₄	—	18	1	no	—
563	C₃×C₄	—	18	2	no	—
564	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
565	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
566	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
567	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
568	Dic₃	—	3	4	no	—
569	Dic₃	—	3	4	no	—
570	Dic₃	—	6	8	no	—
571	Dic₃	—	6	4	no	—
572	C₃²	—	2	4	no	—
573	C₃²	D₃×#137	1	4	yes	complemented
574	C₃²	C₂×C₃×#137	1	1	yes	characteristic, complemented
575	C₂³	—	9	1	no	—
576	C₂³	—	18	1	no	—
577	Q₈	—	2	1	no	—
578	Q₈	—	9	2	no	—
579	Q₈	—	9	1	no	—
580	Q₈	—	18	2	no	—
581	Q₈	—	18	2	no	—
582	Q₈	—	18	1	no	—
583	Q₈	—	18	1	no	—
584	Q₈	C₂²×C₃×C₃⋊D₃	1	2	yes	—
585	Q₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
586	D₄	—	2	1	no	—
587	D₄	—	9	1	no	—
588	D₄	—	18	1	no	—
589	D₄	—	18	1	no	—
590	D₄	—	18	1	no	—
591	D₄	—	18	1	no	—
592	D₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	9	1	no	—
596	C₂×C₄	—	9	1	no	—
597	C₂×C₄	—	9	1	no	—
598	C₂×C₄	—	9	1	no	—
599	C₂×C₄	—	9	1	no	—
600	C₂×C₄	—	18	1	no	—
601	C₂×C₄	—	18	1	no	—
602	C₂×C₄	—	18	2	no	—
603	C₂×C₄	—	18	2	no	—
604	C₂×C₄	—	18	1	no	—
605	C₂×C₄	—	18	1	no	—
606	C₂×C₄	—	18	1	no	—
607	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
608	C₈	—	9	2	no	—
609	C₈	C₂²×C₃×C₃⋊D₃	1	2	yes	—
610	C₂×C₃	—	2	4	no	—
611	C₂×C₃	—	2	1	no	—
612	C₂×C₃	—	2	4	no	—
613	C₂×C₃	—	4	4	no	—
614	C₂×C₃	—	4	4	no	—
615	C₂×C₃	—	4	1	no	—
616	C₂×C₃	—	8	4	no	—
617	C₂×C₃	—	9	2	no	—
618	C₂×C₃	—	18	1	no	—
619	C₂×C₃	—	36	1	no	—
620	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
621	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
622	D₃	—	3	8	no	—
623	D₃	—	6	4	no	—
624	D₃	—	12	4	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	9	1	no	—
627	C₂²	—	9	1	no	—
628	C₂²	—	18	2	no	—
629	C₂²	—	18	1	no	—
630	C₂²	—	36	2	no	—
631	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
632	C₄	—	2	1	no	—
633	C₄	—	2	2	no	—
634	C₄	—	9	1	no	—
635	C₄	—	9	1	no	—
636	C₄	—	18	1	no	—
637	C₄	—	18	2	no	—
638	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
639	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
640	C₃	—	2	4	no	—
641	C₃	C₃×D₃×#137	1	4	yes	complemented
642	C₃	C₃⋊D₃×#137	1	1	yes	characteristic, direct product factor
643	C₂	—	2	1	no	—
644	C₂	—	4	1	no	—
645	C₂	—	9	2	no	—
646	C₂	—	18	1	no	—
647	C₂	—	36	1	no	—
648	C₂	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×#137	C₂
socle	C₂×C₃³	C₂²×D₄
prime radical	C₃×#41×C₃⋊D₃	C₂
hypercenter	C₃×#137	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#137

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Q₈×C₃⋊D₃	C₂	semidirect
3. C₃³×#137	C₂	semidirect
4. C₃×#64160	C₂	semidirect
5. C₃×#64151	C₂	semidirect
6. C₃×#64127	C₂	semidirect
7. C₃×Q₁₆×C₃⋊D₃	C₂	semidirect
8. C₃×#64124	C₂	semidirect
9. C₃×SD₁₆×C₃⋊D₃	C₂	semidirect
10. C₃×#64118	C₂	semidirect
11. C₃×M₁₆×C₃⋊D₃	C₂	semidirect
12. C₃⋊D₃×#137	C₃	direct (N × H)
15. C₂×C₃³×Q₈	C₂²	semidirect
21. C₃×Q₈×C₃⋊D₃	C₂²	semidirect
32. C₂×C₃×#634	C₂²	semidirect
33. C₃³×Q₁₆	C₂²	semidirect
34. C₃³×SD₁₆	C₂²	semidirect
35. C₃³×M₁₆	C₂²	semidirect
36. C₃×#3756	C₂²	semidirect
37. C₃×#3756	C₂²	semidirect
38. C₃×#3755	C₂²	semidirect
39. C₃×#3754	C₂²	semidirect
40. C₃×#3748	C₂²	semidirect
41. C₃×#3746	C₂²	semidirect
42. C₃×#3744	C₂²	semidirect
43. C₃×#3743	C₂²	semidirect
46. C₂×Q₈×C₃⋊D₃	C₂×C₃	semidirect
50. C₃²×#137	D₃	semidirect
51. C₃²×#137	C₂×C₃	semidirect
52. #64160	C₂×C₃	semidirect
53. #64151	C₂×C₃	semidirect
54. #64127	C₂×C₃	semidirect
55. Q₁₆×C₃⋊D₃	C₂×C₃	semidirect
56. #64124	C₂×C₃	semidirect
57. SD₁₆×C₃⋊D₃	C₂×C₃	semidirect
58. #64118	C₂×C₃	semidirect
59. M₁₆×C₃⋊D₃	C₂×C₃	semidirect
109. C₂×C₃²×Q₈	C₂×D₃	semidirect
110. C₂×C₃²×Q₈	C₂²×C₃	semidirect
116. Q₈×C₃⋊D₃	C₂²×C₃	semidirect
127. C₂×#634	C₂²×C₃	semidirect
146. C₃²×Q₁₆	C₂×D₃	semidirect
147. C₃²×Q₁₆	C₂²×C₃	semidirect
149. C₃²×SD₁₆	C₂×D₃	semidirect
150. C₃²×SD₁₆	C₂²×C₃	semidirect
152. C₃²×M₁₆	C₂×D₃	semidirect
153. C₃²×M₁₆	C₂²×C₃	semidirect
154. #3756	C₂²×C₃	semidirect
155. #3756	C₂²×C₃	semidirect
156. #3755	C₂²×C₃	semidirect
157. #3754	C₂²×C₃	semidirect
158. #3748	C₂²×C₃	semidirect
159. #3746	C₂²×C₃	semidirect
160. #3744	C₂²×C₃	semidirect
161. #3743	C₂²×C₃	semidirect
197. C₃×#137	C₃×D₃	semidirect
198. C₃×#137	C₃⋊D₃	direct (N × H)
295. C₃×C₃⋊D₃	#137	direct (N × H)
312. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
313. C₂×C₃×Q₈	C₂×C₃⋊D₃	semidirect
340. C₃×Q₁₆	C₂×C₃×D₃	semidirect
341. C₃×Q₁₆	C₂×C₃⋊D₃	semidirect
346. C₃×SD₁₆	C₂×C₃×D₃	semidirect
347. C₃×SD₁₆	C₂×C₃⋊D₃	semidirect
351. C₃×M₁₆	C₂×C₃×D₃	semidirect
352. C₃×M₁₆	C₂×C₃⋊D₃	semidirect
406. #137	C₃×C₃⋊D₃	direct (N × H)
410. C₃³	C₂×#137	semidirect
501. C₃⋊D₃	C₃×#137	direct (N × H)
518. C₂×Q₈	C₂×C₃×C₃⋊D₃	semidirect
527. Q₁₆	C₂×C₃×C₃⋊D₃	semidirect
531. SD₁₆	C₂×C₃×C₃⋊D₃	semidirect
534. M₁₆	C₂×C₃×C₃⋊D₃	semidirect
573. C₃²	D₃×#137	semidirect
574. C₃²	C₂×C₃×#137	semidirect
641. C₃	C₃×D₃×#137	semidirect
642. C₃	C₃⋊D₃×#137	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×#41×C₃⋊D₃	2
C₂×C₃×Q₈×C₃⋊D₃	2
C₃³×#137	2
C₃×#64160	2
C₃×#64151	2
C₃×#64127	2
C₃×Q₁₆×C₃⋊D₃	2
C₃×#64124	2
C₃×SD₁₆×C₃⋊D₃	2
C₃×#64118	2
C₃×M₁₆×C₃⋊D₃	2
C₃⋊D₃×#137	3
C₃×D₃×#137	3

Subgroups — C3×C3⋊D3×#137 #13179527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊D₃×#137 #13179527