Subgroups — #10588400⋊C₃ #13141625

Order 1728.

313 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63488⋊C₃	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#63389⋊C₃	C₂	1	1	yes	maximal, characteristic
4	#63375⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	(3₋¹⁺²⋊C₄)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	(3₋¹⁺²⋊C₄)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4720	—	3	1	no	maximal
9	#10588400	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	#3676⋊C₃	C₄	1	1	yes	characteristic
14	#3676⋊C₃	C₄	1	1	yes	characteristic
15	C₄⋊(3₋¹⁺²⋊C₄)	C₄	1	1	yes	characteristic
16	C₄⋊(3₋¹⁺²⋊C₄)	C₄	1	1	yes	characteristic
17	C₄×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
18	C₄×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#926	—	3	1	no	—
22	C₃²×C₂²⋊C₈	D₃	1	1	yes	characteristic
23	C₃×#830	—	3	1	no	—
24	C₃×#815	—	3	1	no	—
25	C₃×Dic₃⋊C₈	—	3	1	no	—
26	C₃×Dic₃⋊C₈	—	3	1	no	—
27	C₃×C₈×Dic₃	—	3	1	no	—
28	#63488	C₂×C₃	1	1	yes	characteristic
29	C₉×C₂²⋊C₈	—	2	1	no	—
30	C₉×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
31	#63389	C₂×C₃	1	1	yes	characteristic
32	#63375	C₂×C₃	1	1	yes	characteristic, complemented
33	Dic₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
34	Dic₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
35	C₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×3₋¹⁺²	—	2	1	no	—
38	C₂×C₄×3₋¹⁺²	—	2	1	no	—
39	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
41	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₈×3₋¹⁺²	—	2	1	no	—
46	C₈×3₋¹⁺²	—	2	1	no	—
47	3₋¹⁺²⋊C₈	—	2	1	no	—
48	3₋¹⁺²⋊C₈	—	2	1	no	—
49	C₃×#433	—	9	1	no	—
50	#4720	—	3	1	no	—
51	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃×#217	—	3	1	no	—
55	C₃×#217	—	3	1	no	—
56	C₃×C₄⋊Dic₃	—	3	1	no	—
57	C₃×C₄⋊Dic₃	—	3	1	no	—
58	C₃×C₄×Dic₃	—	3	1	no	—
59	C₃×C₄×Dic₃	—	3	1	no	—
60	C₂×C₃×C₃⋊C₈	—	3	1	no	—
61	C₂×C₃×C₃⋊C₈	—	3	1	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	—	2	1	no	—
66	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
68	#3676	C₃×C₄	1	1	yes	characteristic
69	#3676	C₃×C₄	1	1	yes	characteristic
70	C₄⋊Dic₉	C₃×C₄	1	1	yes	characteristic
71	C₄⋊Dic₉	C₃×C₄	1	1	yes	characteristic
72	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
73	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
76	C₂²×3₋¹⁺²	—	2	1	no	—
77	C₂²×3₋¹⁺²	—	2	1	no	—
78	C₂²×3₋¹⁺²	—	2	1	no	—
79	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
80	C₄×3₋¹⁺²	—	2	1	no	—
81	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
82	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
83	3₋¹⁺²⋊C₄	—	2	1	no	—
84	3₋¹⁺²⋊C₄	—	2	1	no	—
85	3₋¹⁺²⋊C₄	—	2	1	no	—
86	3₋¹⁺²⋊C₄	—	2	1	no	—
87	C₃×#117	—	9	1	no	—
88	#926	—	3	1	no	—
89	C₃×C₄⋊C₈	—	9	1	no	—
90	C₃×C₄⋊C₈	—	9	1	no	—
91	C₃×C₂²⋊C₈	—	3	1	no	—
92	C₃×C₂²⋊C₈	—	9	1	no	—
93	C₃×C₂²⋊C₈	C₃×D₃	1	1	yes	characteristic
94	C₃×#97	—	9	1	no	—
95	C₃×C₄×C₈	—	9	1	no	—
96	#830	—	3	1	no	—
97	#815	—	3	1	no	—
98	Dic₃⋊C₈	—	3	1	no	—
99	Dic₃⋊C₈	—	3	1	no	—
100	C₈×Dic₃	—	3	1	no	—
101	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃²×C₄	—	2	1	no	—
103	C₂×C₃²×C₄	—	2	1	no	—
104	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃×Dic₃	—	3	1	no	—
107	C₂×C₃×Dic₃	—	3	1	no	—
108	C₂×C₃×Dic₃	—	3	1	no	—
109	C₂×C₃×Dic₃	—	3	1	no	—
110	C₂³×C₉	—	2	1	no	—
111	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
112	C₃²×C₈	—	2	1	no	—
113	C₃²×C₈	—	2	1	no	—
114	C₃×C₃⋊C₈	—	6	1	no	—
115	C₃×C₃⋊C₈	—	6	1	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	4	1	no	—
121	C₂×C₄×C₉	—	4	1	no	—
122	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
128	C₈×C₉	—	2	1	no	—
129	C₈×C₉	—	2	1	no	—
130	C₈×C₉	—	4	1	no	—
131	C₈×C₉	—	4	1	no	—
132	C₉⋊C₈	—	2	1	no	—
133	C₉⋊C₈	—	2	1	no	—
134	#433	—	9	1	no	Sylow
135	C₂×3₋¹⁺²	—	4	1	no	—
136	C₂×3₋¹⁺²	#117	1	1	yes	characteristic
137	C₂×3₋¹⁺²	#131	1	1	yes	characteristic
138	C₂×3₋¹⁺²	#131	1	1	yes	characteristic
139	C₂²×C₃×C₄	—	3	1	no	—
140	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
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	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	9	1	no	—
148	C₃×C₄⋊C₄	—	9	1	no	—
149	C₃×C₂²⋊C₄	—	9	1	no	—
150	C₃×C₂²⋊C₄	—	9	1	no	—
151	C₃×C₄²	—	9	1	no	—
152	C₃×C₄²	—	9	1	no	—
153	#217	—	3	1	no	—
154	#217	—	3	1	no	—
155	C₄⋊Dic₃	—	3	1	no	—
156	C₄⋊Dic₃	—	3	1	no	—
157	C₄×Dic₃	—	3	1	no	—
158	C₄×Dic₃	—	3	1	no	—
159	C₂×C₃⋊C₈	—	3	1	no	—
160	C₂×C₃⋊C₈	—	3	1	no	—
161	C₂²×C₃²	—	2	1	no	—
162	C₂²×C₃²	—	2	1	no	—
163	C₂²×C₃²	—	2	1	no	—
164	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	#237	1	1	yes	characteristic
167	C₃²×C₄	#237	1	1	yes	characteristic
168	C₃×Dic₃	—	6	1	no	—
169	C₃×Dic₃	—	6	1	no	—
170	C₃×Dic₃	—	6	1	no	—
171	C₃×Dic₃	—	6	1	no	—
172	C₂²×C₉	—	2	2	no	—
173	C₂²×C₉	—	2	1	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	2	1	no	—
177	C₂²×C₉	—	4	1	no	—
178	C₂²×C₉	—	4	1	no	—
179	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	2	1	no	—
182	C₄×C₉	—	2	1	no	—
183	C₄×C₉	—	4	1	no	—
184	C₄×C₉	C₃×#41	1	1	yes	characteristic
185	C₄×C₉	C₃×#41	1	1	yes	characteristic
186	Dic₉	—	2	1	no	—
187	Dic₉	—	2	1	no	—
188	Dic₉	—	2	1	no	—
189	Dic₉	—	2	1	no	—
190	#117	—	9	1	no	—
191	C₄⋊C₈	—	9	1	no	—
192	C₄⋊C₈	—	9	1	no	—
193	C₂²⋊C₈	—	9	1	no	—
194	C₂²⋊C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
195	#97	—	9	1	no	—
196	C₄×C₈	—	9	1	no	—
197	3₋¹⁺²	#433	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₃	—	3	1	no	—
199	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	3	1	no	—
203	C₂×C₃×C₄	—	3	1	no	—
204	C₂×C₃×C₄	—	6	1	no	—
205	C₂×C₃×C₄	—	6	1	no	—
206	C₂×C₃×C₄	—	9	1	no	—
207	C₂×C₃×C₄	—	9	1	no	—
208	C₂×C₃×C₄	—	9	1	no	—
209	C₂×C₃×C₄	—	9	1	no	—
210	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×Dic₃	—	3	1	no	—
213	C₂×Dic₃	—	3	1	no	—
214	C₂×Dic₃	—	3	1	no	—
215	C₂×Dic₃	—	3	1	no	—
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	6	1	no	—
219	C₃×C₈	—	6	1	no	—
220	C₃×C₈	—	18	1	no	—
221	C₃×C₈	—	18	1	no	—
222	C₃⋊C₈	—	6	1	no	—
223	C₃⋊C₈	—	6	1	no	—
224	C₂×C₃²	—	4	1	no	—
225	C₂×C₃²	#877	1	1	yes	characteristic
226	C₂×C₃²	#901	1	1	yes	characteristic
227	C₂×C₃²	#907	1	1	yes	characteristic
228	C₂×C₉	—	2	1	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	—	4	2	no	—
232	C₂×C₉	—	4	1	no	—
233	C₂×C₉	C₃×#117	1	1	yes	characteristic
234	C₂×C₉	C₃×#131	1	1	yes	characteristic
235	C₂×C₉	C₃×#131	1	1	yes	characteristic
236	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
237	C₂×C₈	—	9	1	no	—
238	C₂×C₈	—	9	1	no	—
239	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
240	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
241	C₄⋊C₄	—	9	1	no	—
242	C₄⋊C₄	—	9	1	no	—
243	C₂²⋊C₄	—	9	1	no	—
244	C₂²⋊C₄	—	9	1	no	—
245	C₄²	—	9	1	no	—
246	C₄²	—	9	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	3	1	no	—
251	C₂²×C₃	—	6	1	no	—
252	C₂²×C₃	—	6	1	no	—
253	C₂²×C₃	—	6	1	no	—
254	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	3	1	no	—
257	C₃×C₄	—	3	1	no	—
258	C₃×C₄	—	6	1	no	—
259	C₃×C₄	—	18	1	no	—
260	C₃×C₄	—	18	1	no	—
261	C₃×C₄	—	18	1	no	—
262	C₃×C₄	—	18	1	no	—
263	C₃×C₄	C₃×#237	1	1	yes	characteristic
264	C₃×C₄	C₃×#237	1	1	yes	characteristic
265	Dic₃	—	6	1	no	—
266	Dic₃	—	6	1	no	—
267	Dic₃	—	6	1	no	—
268	Dic₃	—	6	1	no	—
269	C₃²	#4720	1	1	yes	characteristic
270	C₉	—	2	1	no	—
271	C₉	C₃×#433	1	1	yes	characteristic, complemented
272	C₂³	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	9	1	no	—
276	C₂×C₄	—	9	1	no	—
277	C₂×C₄	—	9	1	no	—
278	C₂×C₄	—	9	1	no	—
279	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
280	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	18	1	no	—
284	C₈	—	18	1	no	—
285	C₂×C₃	—	3	1	no	—
286	C₂×C₃	—	3	1	no	—
287	C₂×C₃	—	3	1	no	—
288	C₂×C₃	—	4	1	no	—
289	C₂×C₃	—	12	1	no	—
290	C₂×C₃	C₃×#877	1	1	yes	characteristic
291	C₂×C₃	C₃×#901	1	1	yes	characteristic
292	C₂×C₃	C₃×#907	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	18	1	no	—
299	C₄	—	18	1	no	—
300	C₄	—	18	1	no	—
301	C₄	—	18	1	no	—
302	C₄	#3699⋊C₃	1	1	yes	characteristic
303	C₄	#3699⋊C₃	1	1	yes	characteristic
304	C₃	—	3	1	no	—
305	C₃	C₃×#4720	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	#63439⋊C₃	1	1	yes	characteristic
308	C₂	#63464⋊C₃	1	1	yes	characteristic
309	C₂	#63469⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	3₋¹⁺²×C₂²⋊C₈	C₂
socle	C₂²×C₃	C₂×C₃×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²⋊C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#433
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. #63375⋊C₃	C₂	semidirect
5. (3₋¹⁺²⋊C₄)⋊C₈	C₂	semidirect
6. (3₋¹⁺²⋊C₄)⋊C₈	C₂	semidirect
7. C₈×3₋¹⁺²⋊C₄	C₂	semidirect
9. #10588400	C₃	semidirect
32. #63375	C₂×C₃	semidirect
33. Dic₉⋊C₈	C₂×C₃	semidirect
34. Dic₉⋊C₈	C₂×C₃	semidirect
35. C₈×Dic₉	C₂×C₃	semidirect
197. 3₋¹⁺²	#433	semidirect
271. C₉	C₃×#433	semidirect

Maximal subgroups

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

subgroup	index
#63488⋊C₃	2
3₋¹⁺²×C₂²⋊C₈	2
#63389⋊C₃	2
#63375⋊C₃	2
(3₋¹⁺²⋊C₄)⋊C₈	2
C₈×3₋¹⁺²⋊C₄	2
C₃×#4720	3
#10588400	3

Subgroups — #10588400⋊C3 #13141625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588400⋊C₃ #13141625