Subgroups — #10592388⋊C₃ #13141455

Order 1728.

499 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3743⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#64081⋊C₃	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	#63642⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×(C₃⋊Dic₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#63630⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592388	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃×#4689	—	3	1	no	maximal
10	C₂×C₄×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
13	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
14	#3752⋊C₃	C₄	1	2	yes	—
15	#3750⋊C₃	C₄	1	2	yes	—
16	C₄×(C₃⋊Dic₃)⋊C₃	C₂²	1	1	yes	characteristic
17	C₂×(C₃⋊(C₃⋊C₈))⋊C₃	C₂²	1	1	yes	characteristic
18	C₂×(C₃⋊(C₃⋊C₈))⋊C₃	C₂²	1	1	yes	characteristic
19	#3743⋊C₃	C₄	1	4	yes	complemented
20	C₈×(C₃⋊D₃)⋊C₃	C₄	1	4	yes	complemented
21	C₂×#3743	C₂×C₃	1	1	yes	characteristic
22	C₂×C₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
23	#64081	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
25	C₂×C₃×D₃×C₈	—	3	1	no	—
26	C₃×#872	—	3	1	no	—
27	C₃²×C₄×C₈	—	2	1	no	—
28	C₃²×C₄×C₈	D₃	1	1	yes	characteristic, complemented
29	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	#63642	C₂×C₃	1	1	yes	characteristic, complemented
31	C₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
32	#63630	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#815	—	3	1	no	—
34	C₃×C₈×Dic₃	—	3	1	no	—
35	C₃×#803	—	3	1	no	—
36	C₂²×(C₃⋊D₃)⋊C₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
38	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
39	C₂×(C₃⋊Dic₃)⋊C₃	C₂×C₄	1	2	yes	—
40	C₂×(C₃⋊Dic₃)⋊C₃	C₂×C₄	1	1	yes	characteristic
41	C₄×(C₃⋊D₃)⋊C₃	C₂×C₄	1	2	yes	—
42	C₄×(C₃⋊D₃)⋊C₃	C₂×C₄	1	2	yes	—
43	C₈×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
44	C₈×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
45	(C₃⋊(C₃⋊C₈))⋊C₃	C₂×C₄	1	2	yes	—
46	(C₃⋊(C₃⋊C₈))⋊C₃	C₂×C₄	1	2	yes	—
47	C₃×#405	—	9	1	no	—
48	#4689	—	3	1	no	—
49	#4689	—	9	1	no	—
50	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄×D₃	—	3	1	no	—
52	C₂×C₃²×C₈	—	2	1	no	—
53	C₂×C₃²×C₈	—	2	1	no	—
54	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃²×C₄²	—	2	1	no	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	#3752	C₃×C₄	1	2	yes	—
62	#3750	C₃×C₄	1	2	yes	—
63	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
66	#3743	C₃×C₄	1	4	yes	complemented
67	C₈×C₃⋊D₃	C₃×C₄	1	4	yes	complemented
68	C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
69	C₃×Dic₃⋊C₄	—	3	2	no	—
70	C₃×C₄×Dic₃	—	3	1	no	—
71	C₂×C₃×C₃⋊C₈	—	3	1	no	—
72	C₂×C₃×C₃⋊C₈	—	3	1	no	—
73	C₃×ΓL(1,𝔽₂₅)	—	3	4	no	—
74	C₃×D₃×C₈	—	3	4	no	—
75	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
76	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
77	C₂×(C₃⋊D₃)⋊C₃	C₄²	1	2	yes	—
78	C₄×H(3,𝔽₃)	—	2	2	no	—
79	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
80	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
81	(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
82	(C₃⋊Dic₃)⋊C₃	C₄²	1	2	yes	—
83	C₂×C₃×M₁₆	—	9	1	no	—
84	C₂²×C₃×C₈	—	9	1	no	—
85	C₃×#117	—	9	1	no	—
86	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
87	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
88	C₂×D₃×C₈	—	3	1	no	—
89	C₂×D₃×C₈	—	9	1	no	—
90	#872	—	3	1	no	—
91	#872	—	9	1	no	—
92	C₃×#97	—	9	1	no	—
93	C₃×#97	—	9	1	no	—
94	C₃×C₄×C₈	—	3	1	no	—
95	C₃×C₄×C₈	—	3	1	no	—
96	C₃×C₄×C₈	—	6	1	no	—
97	C₃×C₄×C₈	—	9	1	no	—
98	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
99	#815	—	3	1	no	—
100	#815	—	9	1	no	—
101	C₈×Dic₃	—	3	1	no	—
102	C₈×Dic₃	—	9	1	no	—
103	#803	—	3	1	no	—
104	#803	—	9	1	no	—
105	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂²×C₃×D₃	—	3	1	no	—
107	C₂×C₃²×C₄	—	2	2	no	—
108	C₂×C₃²×C₄	—	2	1	no	—
109	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
110	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
116	C₄×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₃×Dic₃	—	3	1	no	—
118	C₂×C₃×Dic₃	—	3	2	no	—
119	C₃×C₄×D₃	—	3	2	no	—
120	C₃×C₄×D₃	—	3	2	no	—
121	C₃²×C₈	—	2	2	no	—
122	C₃²×C₈	—	2	2	no	—
123	C₃²×C₈	C₄×D₃	1	2	yes	—
124	C₃²×C₈	C₄×D₃	1	2	yes	—
125	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
126	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
127	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	2	yes	—
128	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	2	yes	—
129	C₃×C₃⋊C₈	—	3	2	no	—
130	C₃×C₃⋊C₈	—	3	2	no	—
131	#405	—	9	1	no	Sylow
132	C₂×H(3,𝔽₃)	C₂×C₄²	1	1	yes	characteristic
133	C₂×H(3,𝔽₃)	#131	1	2	yes	—
134	(C₃⋊D₃)⋊C₃	—	2	2	no	—
135	C₂²×C₃×C₄	—	9	1	no	—
136	C₂×C₄×D₃	—	3	1	no	—
137	C₂×C₄×D₃	—	9	1	no	—
138	C₃×M₁₆	—	9	4	no	—
139	C₂×C₃×C₈	—	3	1	no	—
140	C₂×C₃×C₈	—	3	1	no	—
141	C₂×C₃×C₈	—	3	1	no	—
142	C₂×C₃×C₈	—	3	1	no	—
143	C₂×C₃×C₈	—	6	1	no	—
144	C₂×C₃×C₈	—	6	1	no	—
145	C₂×C₃×C₈	—	9	4	no	—
146	C₂×C₃×C₈	—	9	1	no	—
147	C₂×C₃×C₈	—	9	1	no	—
148	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	9	2	no	—
151	C₃×C₂²⋊C₄	—	9	2	no	—
152	C₃×C₄²	—	3	1	no	—
153	C₃×C₄²	—	3	1	no	—
154	C₃×C₄²	—	6	1	no	—
155	C₃×C₄²	—	9	1	no	—
156	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
157	(C₂×D₃)⋊C₄	—	3	2	no	—
158	(C₂×D₃)⋊C₄	—	9	2	no	—
159	Dic₃⋊C₄	—	3	2	no	—
160	Dic₃⋊C₄	—	9	2	no	—
161	C₄×Dic₃	—	3	1	no	—
162	C₄×Dic₃	—	9	1	no	—
163	C₂×C₃⋊C₈	—	3	1	no	—
164	C₂×C₃⋊C₈	—	3	1	no	—
165	C₂×C₃⋊C₈	—	9	1	no	—
166	C₂×C₃⋊C₈	—	9	1	no	—
167	ΓL(1,𝔽₂₅)	—	3	4	no	—
168	ΓL(1,𝔽₂₅)	—	9	4	no	—
169	D₃×C₈	—	3	4	no	—
170	D₃×C₈	—	9	4	no	—
171	C₂²×C₃²	—	2	1	no	—
172	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₃⋊D₃	—	2	2	no	—
175	C₂×C₃⋊D₃	C₃×C₄²	1	2	yes	—
176	C₂×C₃×D₃	—	3	2	no	—
177	C₂×C₃×D₃	—	6	2	no	—
178	C₃²×C₄	—	2	4	no	—
179	C₃²×C₄	—	2	2	no	—
180	C₃²×C₄	—	2	2	no	—
181	C₃²×C₄	—	2	1	no	—
182	C₃²×C₄	—	2	1	no	—
183	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
184	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
186	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
187	C₃⋊Dic₃	—	2	2	no	—
188	C₃⋊Dic₃	C₃×C₄²	1	2	yes	—
189	C₃×Dic₃	—	3	2	no	—
190	C₃×Dic₃	—	6	2	no	—
191	C₂×M₁₆	—	9	1	no	—
192	C₂²×C₈	—	9	1	no	—
193	#117	—	9	1	no	—
194	#97	—	9	1	no	—
195	#97	—	9	1	no	—
196	C₄×C₈	—	9	1	no	—
197	C₄×C₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
198	H(3,𝔽₃)	#405	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	9	1	no	—
200	C₂²×D₃	—	3	1	no	—
201	C₂²×D₃	—	9	1	no	—
202	C₂×C₃×C₄	—	3	2	no	—
203	C₂×C₃×C₄	—	3	2	no	—
204	C₂×C₃×C₄	—	3	1	no	—
205	C₂×C₃×C₄	—	3	1	no	—
206	C₂×C₃×C₄	—	6	2	no	—
207	C₂×C₃×C₄	—	6	1	no	—
208	C₂×C₃×C₄	—	9	2	no	—
209	C₂×C₃×C₄	—	9	2	no	—
210	C₂×C₃×C₄	—	9	1	no	—
211	C₂×C₃×C₄	—	9	2	no	—
212	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
214	C₂×Dic₃	—	3	1	no	—
215	C₂×Dic₃	—	3	2	no	—
216	C₂×Dic₃	—	9	1	no	—
217	C₂×Dic₃	—	9	2	no	—
218	C₄×D₃	—	3	2	no	—
219	C₄×D₃	—	3	2	no	—
220	C₄×D₃	—	9	2	no	—
221	C₄×D₃	—	9	2	no	—
222	C₃×C₈	—	3	2	no	—
223	C₃×C₈	—	3	2	no	—
224	C₃×C₈	—	3	2	no	—
225	C₃×C₈	—	3	2	no	—
226	C₃×C₈	—	6	2	no	—
227	C₃×C₈	—	6	2	no	—
228	C₃×C₈	—	9	2	no	—
229	C₃×C₈	—	9	2	no	—
230	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
231	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
232	C₃⋊C₈	—	3	2	no	—
233	C₃⋊C₈	—	3	2	no	—
234	C₃⋊C₈	—	9	2	no	—
235	C₃⋊C₈	—	9	2	no	—
236	C₂×C₃²	—	2	2	no	—
237	C₂×C₃²	—	2	1	no	—
238	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
239	C₂×C₃²	#901	1	2	yes	—
240	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
241	C₂×C₃²	C₃×#131	1	2	yes	—
242	C₃⋊D₃	—	2	2	no	—
243	C₃×D₃	—	6	2	no	—
244	C₂²×C₄	—	9	1	no	—
245	M₁₆	—	9	4	no	—
246	C₂×C₈	—	9	1	no	—
247	C₂×C₈	—	9	1	no	—
248	C₂×C₈	—	9	4	no	—
249	C₂×C₈	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
250	C₂×C₈	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
251	C₄⋊C₄	—	9	2	no	—
252	C₂²⋊C₄	—	9	2	no	—
253	C₄²	—	9	1	no	—
254	C₄²	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
255	C₂²×C₃	—	3	1	no	—
256	C₂²×C₃	—	3	1	no	—
257	C₂²×C₃	—	6	1	no	—
258	C₂²×C₃	—	9	2	no	—
259	C₂²×C₃	—	18	2	no	—
260	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
261	C₂×D₃	—	3	2	no	—
262	C₂×D₃	—	6	2	no	—
263	C₂×D₃	—	9	2	no	—
264	C₂×D₃	—	18	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	3	1	no	—
267	C₃×C₄	—	3	1	no	—
268	C₃×C₄	—	3	1	no	—
269	C₃×C₄	—	3	1	no	—
270	C₃×C₄	—	6	2	no	—
271	C₃×C₄	—	6	2	no	—
272	C₃×C₄	—	6	4	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	—	6	1	no	—
275	C₃×C₄	—	9	2	no	—
276	C₃×C₄	—	18	2	no	—
277	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
278	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
279	Dic₃	—	3	2	no	—
280	Dic₃	—	6	2	no	—
281	Dic₃	—	9	2	no	—
282	Dic₃	—	18	2	no	—
283	C₃²	—	2	1	no	—
284	C₃²	#4689	1	1	yes	characteristic, complemented
285	C₃²	C₃×#405	1	1	yes	characteristic, complemented
286	C₂³	—	9	1	no	—
287	C₂×C₄	—	9	1	no	—
288	C₂×C₄	—	9	2	no	—
289	C₂×C₄	—	9	2	no	—
290	C₂×C₄	—	9	2	no	—
291	C₂×C₄	C₄×(C₃⋊D₃)⋊C₃	1	2	yes	—
292	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
293	C₈	—	9	2	no	—
294	C₈	—	9	2	no	—
295	C₈	C₄×(C₃⋊D₃)⋊C₃	1	2	yes	—
296	C₈	C₄×(C₃⋊D₃)⋊C₃	1	2	yes	—
297	C₂×C₃	—	3	2	no	—
298	C₂×C₃	—	3	2	no	—
299	C₂×C₃	—	3	1	no	—
300	C₂×C₃	—	3	1	no	—
301	C₂×C₃	—	6	2	no	—
302	C₂×C₃	—	6	1	no	—
303	C₂×C₃	—	18	2	no	—
304	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
305	C₂×C₃	C₃×#901	1	2	yes	—
306	D₃	—	6	2	no	—
307	D₃	—	18	2	no	—
308	C₂²	—	9	2	no	—
309	C₂²	—	18	2	no	—
310	C₂²	C₂×C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	9	2	no	—
313	C₄	—	18	2	no	—
314	C₄	C₂×C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
315	C₄	C₂×C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
316	C₃	—	3	1	no	—
317	C₃	—	3	1	no	—
318	C₃	—	6	1	no	—
319	C₃	C₃×#4689	1	1	yes	characteristic
320	C₂	—	18	2	no	—
321	C₂	C₄²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
322	C₂	#64110⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×(C₃⋊D₃)⋊C₃
commutator	C₂×C₃²	C₂×C₃×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×C₈×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₂×C₃×C₄×D₃
prime radical	C₂²×(C₃⋊D₃)⋊C₃	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×#405
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₈×H(3,𝔽₃)	C₂	semidirect
5. #63642⋊C₃	C₂	semidirect
6. C₈×(C₃⋊Dic₃)⋊C₃	C₂	semidirect
7. #63630⋊C₃	C₂	semidirect
8. #10592388	C₃	semidirect
19. #3743⋊C₃	C₄	semidirect
20. C₈×(C₃⋊D₃)⋊C₃	C₄	semidirect
28. C₃²×C₄×C₈	D₃	semidirect
29. C₃²×C₄×C₈	C₂×C₃	semidirect
30. #63642	C₂×C₃	semidirect
31. C₈×C₃⋊Dic₃	C₂×C₃	semidirect
32. #63630	C₂×C₃	semidirect
66. #3743	C₃×C₄	semidirect
67. C₈×C₃⋊D₃	C₃×C₄	semidirect
197. C₄×C₈	(C₃⋊D₃)⋊C₃	semidirect
198. H(3,𝔽₃)	#405	semidirect
284. C₃²	#4689	semidirect
285. C₃²	C₃×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3743⋊C₃	2
C₂×C₈×(C₃⋊D₃)⋊C₃	2
#64081⋊C₃	2
C₄×C₈×H(3,𝔽₃)	2
#63642⋊C₃	2
C₈×(C₃⋊Dic₃)⋊C₃	2
#63630⋊C₃	2
#10592388	3
C₃×#4689	3

Subgroups — #10592388⋊C3 #13141455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10592388⋊C₃ #13141455