Subgroups — #10592533⋊C₃ #13142056

Order 1728.

507 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×(C₃⋊D₃)⋊C₃	C₂	1	2	yes	maximal, complemented
2	C₄×C₄⋊((C₃⋊D₃)⋊C₃)	C₂	1	1	yes	maximal, characteristic
3	H(3,𝔽₃)×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#63647⋊C₃	C₂	1	2	yes	maximal, complemented
5	C₄×(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592533	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#4834	—	3	1	no	maximal
8	C₂×C₄⋊((C₃⋊D₃)⋊C₃)	C₄	1	1	yes	characteristic
9	C₂×C₄×(C₃⋊D₃)⋊C₃	C₂²	1	2	yes	—
10	C₂×C₈×H(3,𝔽₃)	C₂²	1	2	yes	—
11	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	#3752⋊C₃	C₄	1	2	yes	—
13	C₄⋊((C₃⋊Dic₃)⋊C₃)	C₄	1	1	yes	characteristic
14	C₂×(C₃⋊(C₃⋊C₈))⋊C₃	C₂²	1	1	yes	characteristic
15	C₂×(C₃⋊(C₃⋊C₈))⋊C₃	C₂²	1	1	yes	characteristic
16	C₈×(C₃⋊D₃)⋊C₃	—	2	4	no	—
17	C₂×C₈×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
18	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×D₃×C₈	—	3	2	no	—
20	C₃×C₄×D₁₂	—	3	1	no	—
21	C₃²×C₄⋊C₈	—	2	1	no	—
22	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
23	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	#63647	C₂×C₃	1	2	yes	complemented
25	C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×(C₂×D₃)⋊C₈	—	3	2	no	—
27	C₃×C₄×C₃⋊C₈	—	3	1	no	—
28	C₂²×(C₃⋊D₃)⋊C₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
32	C₂×(C₃⋊Dic₃)⋊C₃	C₂×C₄	1	2	yes	—
33	C₄⋊((C₃⋊D₃)⋊C₃)	C₈	1	4	yes	complemented
34	C₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
35	C₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
36	C₈×H(3,𝔽₃)	—	2	2	no	—
37	(C₃⋊(C₃⋊C₈))⋊C₃	—	2	1	no	—
38	(C₃⋊(C₃⋊C₈))⋊C₃	D₄	1	2	yes	complemented
39	C₃×C₈×D₄	—	9	1	no	—
40	#4834	—	3	1	no	—
41	#4834	—	9	1	no	—
42	C₂×C₄⋊(C₃⋊D₃)	C₃×C₄	1	1	yes	characteristic
43	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
44	C₂×C₃×D₁₂	—	3	1	no	—
45	C₂×C₃×C₄×D₃	—	3	2	no	—
46	C₂×C₃²×C₈	—	2	2	no	—
47	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
48	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
49	C₃²×C₄²	—	2	1	no	—
50	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	#3752	C₃×C₄	1	2	yes	—
53	C₄⋊(C₃⋊Dic₃)	C₃×C₄	1	1	yes	characteristic
54	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
56	C₈×C₃⋊D₃	—	2	4	no	—
57	C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
58	C₃×C₄⋊Dic₃	—	3	1	no	—
59	C₂×C₃×C₃⋊C₈	—	3	1	no	—
60	C₂×C₃×C₃⋊C₈	—	3	1	no	—
61	C₃×D₃×C₈	—	6	4	no	—
62	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
63	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
64	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
65	C₂×(C₃⋊D₃)⋊C₃	C₂×C₈	1	4	yes	complemented
66	C₄×H(3,𝔽₃)	—	2	1	no	—
67	C₄×H(3,𝔽₃)	C₂×C₈	1	2	yes	complemented
68	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
69	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
70	(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
71	C₂²×C₃×C₈	—	9	2	no	—
72	C₃×C₄×D₄	—	9	1	no	—
73	C₂×D₃×C₈	—	3	2	no	—
74	C₂×D₃×C₈	—	9	2	no	—
75	C₄×D₁₂	—	3	1	no	—
76	C₄×D₁₂	—	9	1	no	—
77	C₃×C₄⋊C₈	—	3	1	no	—
78	C₃×C₄⋊C₈	—	3	1	no	—
79	C₃×C₄⋊C₈	—	6	1	no	—
80	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
81	C₃×C₂²⋊C₈	—	9	2	no	—
82	C₃×C₄×C₈	—	9	1	no	—
83	(C₂×D₃)⋊C₈	—	3	2	no	—
84	(C₂×D₃)⋊C₈	—	9	2	no	—
85	C₄×C₃⋊C₈	—	3	1	no	—
86	C₄×C₃⋊C₈	—	9	1	no	—
87	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
88	C₂²×C₃×D₃	—	3	2	no	—
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	1	no	—
92	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
99	C₄⋊(C₃⋊D₃)	C₃×C₈	1	4	yes	complemented
100	C₄×C₃⋊D₃	—	2	2	no	—
101	C₄×C₃⋊D₃	—	2	2	no	—
102	C₂×C₃×Dic₃	—	3	2	no	—
103	C₃×D₁₂	—	3	4	no	—
104	C₃×C₄×D₃	—	6	2	no	—
105	C₃×C₄×D₃	—	6	2	no	—
106	C₃²×C₈	—	2	2	no	—
107	C₃²×C₈	—	2	2	no	—
108	C₃²×C₈	—	4	2	no	—
109	C₃⋊(C₃⋊C₈)	—	2	1	no	—
110	C₃⋊(C₃⋊C₈)	C₃×D₄	1	2	yes	complemented
111	C₃×C₃⋊C₈	—	3	2	no	—
112	C₃×C₃⋊C₈	—	6	1	no	—
113	C₈×D₄	—	9	1	no	Sylow
114	C₂×H(3,𝔽₃)	C₄×D₄	1	1	yes	characteristic
115	C₂×H(3,𝔽₃)	C₂²×C₈	1	1	yes	characteristic
116	C₂×H(3,𝔽₃)	#131	1	1	yes	characteristic
117	(C₃⋊D₃)⋊C₃	—	2	4	no	—
118	C₂×C₃×D₄	—	9	1	no	—
119	C₂²×C₃×C₄	—	9	2	no	—
120	C₂×D₁₂	—	3	1	no	—
121	C₂×D₁₂	—	9	1	no	—
122	C₂×C₄×D₃	—	3	2	no	—
123	C₂×C₄×D₃	—	9	2	no	—
124	C₂×C₃×C₈	—	3	2	no	—
125	C₂×C₃×C₈	—	3	2	no	—
126	C₂×C₃×C₈	—	6	2	no	—
127	C₂×C₃×C₈	—	9	1	no	—
128	C₂×C₃×C₈	—	9	1	no	—
129	C₂×C₃×C₈	—	18	4	no	—
130	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
131	C₃×C₄⋊C₄	—	9	1	no	—
132	C₃×C₂²⋊C₄	—	9	2	no	—
133	C₃×C₄²	—	3	1	no	—
134	C₃×C₄²	—	3	1	no	—
135	C₃×C₄²	—	6	1	no	—
136	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
137	(C₂×D₃)⋊C₄	—	3	2	no	—
138	(C₂×D₃)⋊C₄	—	9	2	no	—
139	C₄⋊Dic₃	—	3	1	no	—
140	C₄⋊Dic₃	—	9	1	no	—
141	C₂×C₃⋊C₈	—	3	1	no	—
142	C₂×C₃⋊C₈	—	3	1	no	—
143	C₂×C₃⋊C₈	—	9	1	no	—
144	C₂×C₃⋊C₈	—	9	1	no	—
145	D₃×C₈	—	6	4	no	—
146	D₃×C₈	—	18	4	no	—
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₂×C₃⋊D₃	—	2	2	no	—
151	C₂×C₃⋊D₃	—	2	2	no	—
152	C₂×C₃⋊D₃	C₂×C₃×C₈	1	4	yes	complemented
153	C₂×C₃×D₃	—	3	4	no	—
154	C₂×C₃×D₃	—	6	2	no	—
155	C₂×C₃×D₃	—	6	2	no	—
156	C₃²×C₄	—	2	1	no	—
157	C₃²×C₄	—	2	1	no	—
158	C₃²×C₄	—	2	2	no	—
159	C₃²×C₄	—	2	1	no	—
160	C₃²×C₄	—	2	1	no	—
161	C₃²×C₄	—	4	1	no	—
162	C₃²×C₄	D₃×C₈	1	2	yes	complemented
163	C₃²×C₄	C₂×C₃×C₈	1	2	yes	complemented
164	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
165	C₃²×C₄	#239	1	1	yes	characteristic
166	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₃²×C₄	C₃×#41	1	1	yes	characteristic
168	C₃⋊Dic₃	—	2	2	no	—
169	C₃×Dic₃	—	6	2	no	—
170	C₂²×C₈	—	9	2	no	—
171	C₄×D₄	—	9	1	no	—
172	C₄⋊C₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
173	C₂²⋊C₈	—	9	2	no	—
174	C₄×C₈	—	9	1	no	—
175	H(3,𝔽₃)	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₃	—	9	2	no	—
177	C₂²×D₃	—	3	2	no	—
178	C₂²×D₃	—	9	2	no	—
179	C₃×D₄	—	9	4	no	—
180	C₂×C₃×C₄	—	3	1	no	—
181	C₂×C₃×C₄	—	3	1	no	—
182	C₂×C₃×C₄	—	3	1	no	—
183	C₂×C₃×C₄	—	3	1	no	—
184	C₂×C₃×C₄	—	3	1	no	—
185	C₂×C₃×C₄	—	3	1	no	—
186	C₂×C₃×C₄	—	6	1	no	—
187	C₂×C₃×C₄	—	6	1	no	—
188	C₂×C₃×C₄	—	6	1	no	—
189	C₂×C₃×C₄	—	9	2	no	—
190	C₂×C₃×C₄	—	18	2	no	—
191	C₂×C₃×C₄	—	18	2	no	—
192	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
195	C₂×Dic₃	—	3	2	no	—
196	C₂×Dic₃	—	9	2	no	—
197	D₁₂	—	3	4	no	—
198	D₁₂	—	9	4	no	—
199	C₄×D₃	—	6	2	no	—
200	C₄×D₃	—	6	2	no	—
201	C₄×D₃	—	18	2	no	—
202	C₄×D₃	—	18	2	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	6	2	no	—
205	C₃×C₈	—	6	2	no	—
206	C₃×C₈	—	9	2	no	—
207	C₃×C₈	—	12	2	no	—
208	C₃×C₈	—	18	1	no	—
209	C₃⋊C₈	—	3	2	no	—
210	C₃⋊C₈	—	6	1	no	—
211	C₃⋊C₈	—	9	2	no	—
212	C₃⋊C₈	—	18	1	no	—
213	C₂×C₃²	—	2	1	no	—
214	C₂×C₃²	—	2	1	no	—
215	C₂×C₃²	—	2	1	no	—
216	C₂×C₃²	#893	1	1	yes	characteristic
217	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
218	C₂×C₃²	#907	1	1	yes	characteristic
219	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
220	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
221	C₂×C₃²	C₃×#131	1	1	yes	characteristic
222	C₃⋊D₃	—	2	4	no	—
223	C₃×D₃	—	6	4	no	—
224	C₂×D₄	—	9	1	no	—
225	C₂²×C₄	—	9	2	no	—
226	C₂×C₈	—	9	1	no	—
227	C₂×C₈	—	9	1	no	—
228	C₂×C₈	—	18	4	no	—
229	C₂×C₈	C₂×(C₃⋊D₃)⋊C₃	1	2	yes	—
230	C₄⋊C₄	—	9	1	no	—
231	C₂²⋊C₄	—	9	2	no	—
232	C₄²	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
233	C₂²×C₃	—	3	1	no	—
234	C₂²×C₃	—	3	1	no	—
235	C₂²×C₃	—	6	1	no	—
236	C₂²×C₃	—	9	4	no	—
237	C₂²×C₃	—	18	2	no	—
238	C₂²×C₃	—	18	2	no	—
239	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
240	C₂×D₃	—	3	4	no	—
241	C₂×D₃	—	6	2	no	—
242	C₂×D₃	—	6	2	no	—
243	C₂×D₃	—	9	4	no	—
244	C₂×D₃	—	18	2	no	—
245	C₂×D₃	—	18	2	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	3	1	no	—
248	C₃×C₄	—	3	1	no	—
249	C₃×C₄	—	3	1	no	—
250	C₃×C₄	—	3	1	no	—
251	C₃×C₄	—	3	2	no	—
252	C₃×C₄	—	3	2	no	—
253	C₃×C₄	—	6	1	no	—
254	C₃×C₄	—	6	1	no	—
255	C₃×C₄	—	6	1	no	—
256	C₃×C₄	—	6	1	no	—
257	C₃×C₄	—	6	2	no	—
258	C₃×C₄	—	12	1	no	—
259	C₃×C₄	—	18	2	no	—
260	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
261	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
262	C₃×C₄	C₃×#239	1	1	yes	characteristic
263	Dic₃	—	6	2	no	—
264	Dic₃	—	18	2	no	—
265	C₃²	—	2	1	no	—
266	C₃²	#4834	1	1	yes	characteristic, complemented
267	C₃²	C₃×C₈×D₄	1	1	yes	characteristic, complemented
268	C₂³	—	9	2	no	—
269	D₄	—	9	4	no	—
270	C₂×C₄	—	9	2	no	—
271	C₂×C₄	—	18	2	no	—
272	C₂×C₄	—	18	2	no	—
273	C₂×C₄	C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
274	C₂×C₄	C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	9	2	no	—
278	C₈	—	18	1	no	—
279	C₂×C₃	—	3	1	no	—
280	C₂×C₃	—	3	1	no	—
281	C₂×C₃	—	3	1	no	—
282	C₂×C₃	—	3	1	no	—
283	C₂×C₃	—	3	1	no	—
284	C₂×C₃	—	3	1	no	—
285	C₂×C₃	—	6	1	no	—
286	C₂×C₃	—	6	1	no	—
287	C₂×C₃	—	6	1	no	—
288	C₂×C₃	—	18	4	no	—
289	C₂×C₃	C₃×#893	1	1	yes	characteristic
290	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
291	C₂×C₃	C₃×#907	1	1	yes	characteristic
292	D₃	—	6	4	no	—
293	D₃	—	18	4	no	—
294	C₂²	—	9	4	no	—
295	C₂²	—	18	2	no	—
296	C₂²	—	18	2	no	—
297	C₂²	C₂×C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	18	2	no	—
300	C₄	C₈×(C₃⋊D₃)⋊C₃	1	2	yes	complemented
301	C₄	D₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
302	C₄	#88722	1	1	yes	characteristic
303	C₃	—	3	1	no	—
304	C₃	—	3	1	no	—
305	C₃	—	6	1	no	—
306	C₃	C₃×#4834	1	1	yes	characteristic
307	C₂	—	18	4	no	—
308	C₂	#64102⋊C₃	1	1	yes	characteristic
309	C₂	C₂×C₈×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
310	C₂	#64116⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃
commutator	C₂×C₃²	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	H(3,𝔽₃)×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂×C₃×C₄×D₃
prime radical	C₂×C₄⋊((C₃⋊D₃)⋊C₃)	C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄⋊C₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×C₈×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×(C₃⋊D₃)⋊C₃	C₂	semidirect
3. H(3,𝔽₃)×C₄⋊C₈	C₂	semidirect
4. #63647⋊C₃	C₂	semidirect
5. C₄×(C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
6. #10592533	C₃	semidirect
17. C₂×C₈×C₃⋊D₃	C₂×C₃	semidirect
22. C₃²×C₄⋊C₈	D₃	semidirect
23. C₃²×C₄⋊C₈	C₂×C₃	semidirect
24. #63647	C₂×C₃	semidirect
25. C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
33. C₄⋊((C₃⋊D₃)⋊C₃)	C₈	semidirect
38. (C₃⋊(C₃⋊C₈))⋊C₃	D₄	semidirect
65. C₂×(C₃⋊D₃)⋊C₃	C₂×C₈	semidirect
67. C₄×H(3,𝔽₃)	C₂×C₈	semidirect
99. C₄⋊(C₃⋊D₃)	C₃×C₈	semidirect
110. C₃⋊(C₃⋊C₈)	C₃×D₄	semidirect
152. C₂×C₃⋊D₃	C₂×C₃×C₈	semidirect
162. C₃²×C₄	D₃×C₈	semidirect
163. C₃²×C₄	C₂×C₃×C₈	semidirect
172. C₄⋊C₈	(C₃⋊D₃)⋊C₃	semidirect
175. H(3,𝔽₃)	C₈×D₄	semidirect
266. C₃²	#4834	semidirect
267. C₃²	C₃×C₈×D₄	semidirect
300. C₄	C₈×(C₃⋊D₃)⋊C₃	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⋊D₃)⋊C₃	2
C₄×C₄⋊((C₃⋊D₃)⋊C₃)	2
H(3,𝔽₃)×C₄⋊C₈	2
#63647⋊C₃	2
C₄×(C₃⋊(C₃⋊C₈))⋊C₃	2
#10592533	3
C₃×#4834	3

Subgroups — #10592533⋊C3 #13142056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10592533⋊C₃ #13142056