Subgroups — #13159530

Order 1728.

1180 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₄×#6289	C₂	1	1	yes	maximal, characteristic
3	C₂×#88785	C₂	1	1	yes	maximal, characteristic
4	C₃×#63642	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	#11722020	C₂	1	1	yes	maximal, characteristic, complemented
7	#11722019	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃⋊D₃×#97	—	3	1	no	maximal
9	#10590668	—	3	4	no	maximal
10	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
11	C₂×#6289	C₄	1	2	yes	—
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	C₈×C₃⋊(C₃⋊D₃)	C₄	1	4	yes	complemented
15	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	Dic₃×C₃⋊Dic₃	C₄	1	2	yes	—
20	#88785	C₄	1	4	yes	complemented
21	C₂×C₈×C₃⋊D₃	—	3	1	no	—
22	C₂×C₈×C₃⋊D₃	—	3	1	no	—
23	C₂×C₈×C₃⋊D₃	—	3	4	no	—
24	C₂×C₈×C₃⋊D₃	—	6	4	no	—
25	C₄²×C₃⋊D₃	—	3	1	no	—
26	C₄×#624	—	3	4	no	—
27	C₂×#3710	—	3	4	no	—
28	C₃²×#97	—	3	1	no	—
29	#63642	—	3	1	no	—
30	#63642	D₃	1	1	yes	characteristic, complemented
31	#63630	—	3	1	no	—
32	C₃×#815	—	3	4	no	—
33	C₃×#815	D₃	1	4	yes	complemented
34	#63554	—	3	4	no	—
35	#63554	—	3	4	no	—
36	C₂²×C₃⋊(C₃⋊D₃)	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
39	C₄×C₃⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
40	C₄×C₃⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
42	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
43	#6289	—	2	4	no	—
44	C₃³×C₈	C₂×C₄	1	2	yes	complemented
45	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	2	yes	complemented
46	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
47	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
48	D₃×#97	—	9	4	no	—
49	D₃×#97	—	9	1	no	—
50	C₂×C₄×C₃⋊D₃	—	3	1	no	—
51	C₂×C₄×C₃⋊D₃	—	3	4	no	—
52	C₂×C₄×C₃⋊D₃	—	3	2	no	—
53	C₂×C₄×C₃⋊D₃	—	6	4	no	—
54	C₂×#624	—	3	8	no	—
55	C₂×C₃²×C₈	—	2	4	no	—
56	C₂×C₃²×C₈	—	3	1	no	—
57	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
58	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
59	C₃²×C₄²	—	3	1	no	—
60	C₄×C₃⋊Dic₃	—	3	2	no	—
61	C₄×C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
63	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
64	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
65	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
66	C₈×C₃⋊D₃	—	3	4	no	—
67	C₈×C₃⋊D₃	—	3	16	no	—
68	C₈×C₃⋊D₃	—	3	4	no	—
69	C₈×C₃⋊D₃	—	6	16	no	—
70	C₃×C₄×Dic₃	—	3	4	no	—
71	C₃×C₄×Dic₃	C₂×D₃	1	4	yes	—
72	C₂×C₃×C₃⋊C₈	—	3	4	no	—
73	C₂×C₃×C₃⋊C₈	C₂×D₃	1	4	yes	—
74	Dic₃²	—	3	8	no	—
75	#3710	—	3	16	no	—
76	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃⋊(C₃⋊D₃)	C₄²	1	2	yes	—
78	C₂×C₃⋊(C₃⋊D₃)	M₁₆	1	4	yes	—
79	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₃⋊(C₃⋊Dic₃)	C₄²	1	2	yes	—
82	C₃×C₃⋊Dic₃	—	2	2	no	—
83	C₃²×Dic₃	—	2	2	no	—
84	C₂×D₃×C₈	—	9	4	no	—
85	C₂×D₃×C₈	—	9	4	no	—
86	C₂×D₃×C₈	—	9	1	no	—
87	C₂×D₃×C₈	—	9	1	no	—
88	C₂×D₃×C₈	—	18	4	no	—
89	C₄²×D₃	—	9	4	no	—
90	C₄²×D₃	—	9	1	no	—
91	C₃×#97	—	3	4	no	—
92	C₃×#97	—	9	1	no	—
93	#815	—	3	4	no	—
94	#815	—	9	4	no	—
95	#815	—	9	1	no	—
96	#815	C₃⋊D₃	1	1	yes	characteristic, complemented
97	#803	—	9	4	no	—
98	#803	—	9	1	no	—
99	C₂²×C₃⋊D₃	—	3	1	no	—
100	C₂²×C₃⋊D₃	—	3	4	no	—
101	C₂²×C₃⋊D₃	—	6	4	no	—
102	C₂×C₃²×C₄	—	2	4	no	—
103	C₂×C₃²×C₄	—	3	2	no	—
104	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃⋊Dic₃	—	3	1	no	—
107	C₂×C₃⋊Dic₃	—	3	4	no	—
108	C₂×C₃⋊Dic₃	—	6	4	no	—
109	C₂×C₃⋊Dic₃	C₄×D₃	1	2	yes	—
110	C₄×C₃⋊D₃	—	3	2	no	—
111	C₄×C₃⋊D₃	—	3	2	no	—
112	C₄×C₃⋊D₃	—	3	8	no	—
113	C₄×C₃⋊D₃	—	3	8	no	—
114	C₄×C₃⋊D₃	—	6	8	no	—
115	C₄×C₃⋊D₃	—	6	8	no	—
116	C₄×C₃⋊D₃	—	6	4	no	—
117	C₂×C₃×Dic₃	—	3	8	no	—
118	C₂×C₃×Dic₃	C₄×D₃	1	8	yes	—
119	#624	—	6	16	no	—
120	C₃²×C₈	—	2	8	no	—
121	C₃²×C₈	—	3	2	no	—
122	C₃²×C₈	C₄×D₃	1	8	yes	complemented
123	C₃²×C₈	C₄×D₃	1	2	yes	complemented
124	C₃⋊(C₃⋊C₈)	—	3	2	no	—
125	C₃⋊(C₃⋊C₈)	—	3	8	no	—
126	C₃⋊(C₃⋊C₈)	—	6	8	no	—
127	C₃⋊(C₃⋊C₈)	C₄×D₃	1	2	yes	complemented
128	C₃×C₃⋊C₈	—	3	8	no	—
129	C₃×C₃⋊C₈	C₄×D₃	1	8	yes	complemented
130	C₂×#97	—	27	1	no	Sylow
131	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₃³	C₂×M₁₆	1	2	yes	—
133	C₃⋊(C₃⋊D₃)	#97	1	4	yes	complemented
134	C₂×C₄×D₃	—	9	4	no	—
135	C₂×C₄×D₃	—	9	1	no	—
136	C₂×C₄×D₃	—	9	8	no	—
137	C₂×C₄×D₃	—	9	2	no	—
138	C₂×C₄×D₃	—	18	4	no	—
139	C₂×C₃×C₈	—	2	4	no	—
140	C₂×C₃×C₈	—	3	4	no	—
141	C₂×C₃×C₈	—	9	1	no	—
142	C₂×C₃×C₈	D₃²	1	4	yes	—
143	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
144	C₃×C₄²	—	3	4	no	—
145	C₃×C₄²	—	9	1	no	—
146	C₄×Dic₃	—	3	4	no	—
147	C₄×Dic₃	—	9	8	no	—
148	C₄×Dic₃	—	9	2	no	—
149	C₄×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	3	4	no	—
151	C₂×C₃⋊C₈	—	9	4	no	—
152	C₂×C₃⋊C₈	—	9	1	no	—
153	C₂×C₃⋊C₈	—	18	4	no	—
154	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
155	D₃×C₈	—	9	16	no	—
156	D₃×C₈	—	9	16	no	—
157	D₃×C₈	—	9	4	no	—
158	D₃×C₈	—	9	4	no	—
159	D₃×C₈	—	18	16	no	—
160	C₂²×C₃²	—	2	4	no	—
161	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
162	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂×C₃⋊D₃	—	3	2	no	—
164	C₂×C₃⋊D₃	—	3	4	no	—
165	C₂×C₃⋊D₃	—	3	8	no	—
166	C₂×C₃⋊D₃	—	3	16	no	—
167	C₂×C₃⋊D₃	—	6	8	no	—
168	C₂×C₃⋊D₃	—	6	16	no	—
169	C₃²×C₄	—	2	4	no	—
170	C₃²×C₄	—	2	4	no	—
171	C₃²×C₄	—	6	2	no	—
172	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
173	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
174	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₃⋊Dic₃	—	2	2	no	—
177	C₃⋊Dic₃	—	3	2	no	—
178	C₃⋊Dic₃	—	3	8	no	—
179	C₃⋊Dic₃	—	6	8	no	—
180	C₃×Dic₃	—	2	8	no	—
181	C₃×Dic₃	—	6	8	no	—
182	C₂²×C₈	—	27	1	no	—
183	C₂²×C₈	—	27	1	no	—
184	C₂×C₄²	—	27	1	no	—
185	#97	—	3	1	no	—
186	#97	—	9	1	no	—
187	#97	—	27	1	no	—
188	#97	—	27	1	no	—
189	C₃³	C₂×#97	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₃	—	9	4	no	—
191	C₂²×D₃	—	9	1	no	—
192	C₂²×D₃	—	18	4	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	3	8	no	—
195	C₂×C₃×C₄	—	9	2	no	—
196	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
197	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
198	C₂×Dic₃	—	3	8	no	—
199	C₂×Dic₃	—	9	4	no	—
200	C₂×Dic₃	—	9	1	no	—
201	C₂×Dic₃	—	18	4	no	—
202	C₂×Dic₃	C₄×C₃⋊D₃	1	2	yes	—
203	C₄×D₃	—	9	8	no	—
204	C₄×D₃	—	9	8	no	—
205	C₄×D₃	—	9	2	no	—
206	C₄×D₃	—	9	2	no	—
207	C₄×D₃	—	18	8	no	—
208	C₄×D₃	—	18	8	no	—
209	C₄×D₃	—	18	4	no	—
210	C₄×D₃	—	18	16	no	—
211	C₃×C₈	—	2	8	no	—
212	C₃×C₈	—	3	8	no	—
213	C₃×C₈	—	9	2	no	—
214	C₃×C₈	#624	1	8	yes	complemented
215	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	complemented
216	C₃⋊C₈	—	3	8	no	—
217	C₃⋊C₈	—	9	8	no	—
218	C₃⋊C₈	—	9	2	no	—
219	C₃⋊C₈	—	18	8	no	—
220	C₃⋊C₈	C₄×C₃⋊D₃	1	2	yes	complemented
221	C₂×C₃²	—	2	8	no	—
222	C₂×C₃²	—	2	4	no	—
223	C₂×C₃²	C₄²×D₃	1	4	yes	—
224	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
225	C₂×C₃²	D₃×M₁₆	1	8	yes	—
226	C₂×C₃²	D₃×M₁₆	1	2	yes	—
227	C₃⋊D₃	—	3	4	no	—
228	C₃⋊D₃	—	3	16	no	—
229	C₃⋊D₃	—	6	16	no	—
230	C₂²×C₄	—	27	2	no	—
231	C₂²×C₄	—	27	1	no	—
232	C₂×C₈	—	3	1	no	—
233	C₂×C₈	—	9	1	no	—
234	C₂×C₈	—	27	4	no	—
235	C₂×C₈	—	27	1	no	—
236	C₂×C₈	—	27	4	no	—
237	C₂×C₈	D₃×C₃⋊D₃	1	1	yes	characteristic
238	C₄²	—	3	1	no	—
239	C₄²	—	9	1	no	—
240	C₄²	—	27	2	no	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	C₄×D₃²	1	4	yes	—
243	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
244	C₂×D₃	—	9	16	no	—
245	C₂×D₃	—	9	2	no	—
246	C₂×D₃	—	9	4	no	—
247	C₂×D₃	—	9	8	no	—
248	C₂×D₃	—	18	8	no	—
249	C₂×D₃	—	18	16	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	6	8	no	—
253	C₃×C₄	—	18	2	no	—
254	C₃×C₄	C₄×D₃²	1	4	yes	—
255	C₃×C₄	C₂×#624	1	4	yes	—
256	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
257	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
258	Dic₃	—	2	2	no	—
259	Dic₃	—	6	8	no	—
260	Dic₃	—	9	2	no	—
261	Dic₃	—	9	8	no	—
262	Dic₃	—	18	8	no	—
263	C₃²	—	2	4	no	—
264	C₃²	D₃×#97	1	4	yes	complemented
265	C₃²	D₃×#97	1	1	yes	characteristic, complemented
266	C₂³	—	27	1	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	9	2	no	—
269	C₂×C₄	—	27	1	no	—
270	C₂×C₄	—	27	2	no	—
271	C₂×C₄	—	27	2	no	—
272	C₂×C₄	—	54	4	no	—
273	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
274	C₈	—	3	2	no	—
275	C₈	—	9	2	no	—
276	C₈	—	27	2	no	—
277	C₈	#6289	1	2	yes	complemented
278	C₂×C₃	—	2	8	no	—
279	C₂×C₃	—	2	4	no	—
280	C₂×C₃	#63791	1	8	yes	—
281	C₂×C₃	C₄×#624	1	4	yes	—
282	C₂×C₃	C₄²×C₃⋊D₃	1	1	yes	characteristic
283	C₂×C₃	M₁₆×C₃⋊D₃	1	2	yes	—
284	D₃	—	9	4	no	—
285	D₃	—	9	16	no	—
286	D₃	—	18	16	no	—
287	C₂²	—	27	4	no	—
288	C₂²	—	27	2	no	—
289	C₂²	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
290	C₄	—	6	2	no	—
291	C₄	—	18	2	no	—
292	C₄	—	27	2	no	—
293	C₄	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
294	C₄	C₂×#6289	1	1	yes	characteristic
295	C₃	—	2	4	no	—
296	C₃	#10590668	1	4	yes	complemented
297	C₃	C₃⋊D₃×#97	1	1	yes	characteristic, complemented
298	C₂	—	27	4	no	—
299	C₂	#11723312	1	2	yes	—
300	C₂	C₄×#6289	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂×C₄²
Fitting	C₂×C₃³×C₈	C₂²
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂×C₈	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×#97
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#63642	C₂	semidirect
5. C₃²×#815	C₂	semidirect
6. #11722020	C₂	semidirect
7. #11722019	C₂	semidirect
14. C₈×C₃⋊(C₃⋊D₃)	C₄	semidirect
20. #88785	C₄	semidirect
30. #63642	D₃	semidirect
33. C₃×#815	D₃	semidirect
44. C₃³×C₈	C₂×C₄	semidirect
45. C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	semidirect
46. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
47. C₃²×C₃⋊C₈	C₂×C₄	semidirect
96. #815	C₃⋊D₃	semidirect
122. C₃²×C₈	C₄×D₃	semidirect
123. C₃²×C₈	C₄×D₃	semidirect
127. C₃⋊(C₃⋊C₈)	C₄×D₃	semidirect
129. C₃×C₃⋊C₈	C₄×D₃	semidirect
133. C₃⋊(C₃⋊D₃)	#97	semidirect
189. C₃³	C₂×#97	semidirect
214. C₃×C₈	#624	semidirect
215. C₃×C₈	C₄×C₃⋊D₃	semidirect
220. C₃⋊C₈	C₄×C₃⋊D₃	semidirect
264. C₃²	D₃×#97	semidirect
265. C₃²	D₃×#97	semidirect
277. C₈	#6289	semidirect
296. C₃	#10590668	semidirect
297. C₃	C₃⋊D₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×C₃⋊(C₃⋊D₃)	2
C₄×#6289	2
C₂×#88785	2
C₃×#63642	2
C₃²×#815	2
#11722020	2
#11722019	2
C₃⋊D₃×#97	3
#10590668	3