Subgroups — #10588561⋊C₃ #13142242

Order 1728.

372 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3663⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#63454⋊C₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
4	3₋¹⁺²×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	#63377⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#63367⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#63366⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4881	—	3	1	no	maximal
9	#10588561	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×AGL(1,ℤ/9)	—	2	1	no	—
11	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
12	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	(C₂×AGL(1,ℤ/9))⋊C₄	—	2	1	no	—
16	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
17	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
18	(3₋¹⁺²⋊C₄)⋊C₄	—	2	1	no	—
19	C₄×3₋¹⁺²⋊C₄	—	2	1	no	—
20	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
21	#3663⋊C₃	C₄	1	4	yes	complemented
22	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
23	C₃×#892	—	3	1	no	—
24	C₃×D₃×C₄⋊C₄	—	3	1	no	—
25	C₃²×#107	D₃	1	1	yes	characteristic
26	C₃×#817	—	3	1	no	—
27	C₃×#808	—	3	1	no	—
28	C₃×#807	—	3	1	no	—
29	C₂×#3663	C₂×C₃	1	1	yes	characteristic
30	#63454	C₂×C₃	1	1	yes	characteristic
31	C₄⋊C₄×D₉	C₂×C₃	1	1	yes	characteristic
32	C₉×#107	—	2	1	no	—
33	C₉×#107	C₂×C₃	1	1	yes	characteristic, complemented
34	#63377	C₂×C₃	1	1	yes	characteristic, complemented
35	#63367	C₂×C₃	1	1	yes	characteristic, complemented
36	#63366	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×AGL(1,ℤ/9)	D₄	1	1	yes	characteristic
38	C₂×C₄×3₋¹⁺²	—	2	1	no	—
39	C₂×C₄×3₋¹⁺²	—	2	1	no	—
40	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
41	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
42	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
43	C₂×3₋¹⁺²⋊C₄	D₄	1	1	yes	characteristic
44	C₄×AGL(1,ℤ/9)	—	4	2	no	—
45	C₄×AGL(1,ℤ/9)	C₂×C₄	1	2	yes	—
46	C₄×AGL(1,ℤ/9)	Q₈	1	2	yes	—
47	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
48	3₋¹⁺²⋊C₈	C₂×C₄	1	2	yes	complemented
49	C₃×#428	—	9	1	no	—
50	#4881	—	3	1	no	—
51	C₂×C₃×C₄×D₃	—	3	1	no	—
52	C₂×C₃×C₄×D₃	—	6	1	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
56	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
57	C₃×C₄⋊Dic₃	—	3	1	no	—
58	C₃×C₄⋊Dic₃	—	3	1	no	—
59	C₃×Dic₃⋊C₄	—	6	1	no	—
60	C₃×C₄×Dic₃	—	6	1	no	—
61	C₂×C₃×C₃⋊C₈	—	3	1	no	—
62	C₃×ΓL(1,𝔽₂₅)	—	3	4	no	—
63	C₂×C₄×D₉	—	2	1	no	—
64	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₈×C₉	—	2	1	no	—
66	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₉×C₄⋊C₄	—	2	1	no	—
68	C₉×C₄⋊C₄	—	2	1	no	—
69	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	(C₂×D₉)⋊C₄	—	2	1	no	—
72	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
73	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
74	Dic₉⋊C₄	—	2	1	no	—
75	C₄×Dic₉	—	2	1	no	—
76	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
77	#3663	C₃×C₄	1	4	yes	complemented
78	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
79	C₂×AGL(1,ℤ/9)	—	2	1	no	—
80	C₂×AGL(1,ℤ/9)	—	2	1	no	—
81	C₂×AGL(1,ℤ/9)	C₄⋊C₄	1	2	yes	—
82	C₄×3₋¹⁺²	—	4	1	no	—
83	C₄×3₋¹⁺²	—	4	1	no	—
84	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
85	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
86	3₋¹⁺²⋊C₄	—	4	1	no	—
87	3₋¹⁺²⋊C₄	—	4	1	no	—
88	3₋¹⁺²⋊C₄	C₄⋊C₄	1	2	yes	—
89	C₂×C₃×M₁₆	—	9	1	no	—
90	C₃×#117	—	9	1	no	—
91	C₂×C₃×C₄⋊C₄	—	9	1	no	—
92	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
93	#892	—	3	1	no	—
94	D₃×C₄⋊C₄	—	3	1	no	—
95	C₃×#107	—	3	1	no	—
96	C₃×#107	—	9	1	no	—
97	C₃×#107	C₃×D₃	1	1	yes	characteristic
98	C₃×#106	—	9	1	no	—
99	C₃×#106	—	9	1	no	—
100	#817	—	3	1	no	—
101	#808	—	3	1	no	—
102	#807	—	3	1	no	—
103	C₂²×C₃×D₃	—	3	1	no	—
104	C₂×C₃²×C₄	—	2	1	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	—	3	1	no	—
108	C₂×C₃×Dic₃	—	6	1	no	—
109	C₂×C₃×Dic₃	—	6	1	no	—
110	C₃×C₄×D₃	—	3	2	no	—
111	C₃×C₄×D₃	—	3	2	no	—
112	C₃×C₄×D₃	—	12	2	no	—
113	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
114	C₃²×C₈	C₄×D₃	1	2	yes	—
115	C₃×C₃⋊C₈	—	3	2	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	—	4	1	no	—
120	C₂×C₄×C₉	—	4	1	no	—
121	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
122	C₂×Dic₉	—	2	1	no	—
123	C₂×Dic₉	—	2	1	no	—
124	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
125	C₄×D₉	—	4	2	no	—
126	C₄×D₉	C₂×C₃×C₄	1	2	yes	—
127	C₄×D₉	C₃×Q₈	1	2	yes	—
128	C₈×C₉	—	2	2	no	—
129	C₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
130	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	complemented
131	#428	—	9	1	no	Sylow
132	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
135	AGL(1,ℤ/9)	—	2	2	no	—
136	C₂²×C₃×C₄	—	9	1	no	—
137	C₂²×C₃×C₄	—	18	1	no	—
138	C₂×C₄×D₃	—	3	1	no	—
139	C₂×C₄×D₃	—	6	1	no	—
140	C₃×M₁₆	—	9	4	no	—
141	C₂×C₃×C₈	—	3	1	no	—
142	C₂×C₃×C₈	—	9	1	no	—
143	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	3	1	no	—
145	C₃×C₄⋊C₄	—	3	1	no	—
146	C₃×C₄⋊C₄	—	9	1	no	—
147	C₃×C₄⋊C₄	—	9	1	no	—
148	C₃×C₄⋊C₄	—	18	1	no	—
149	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
150	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
151	C₃×C₂²⋊C₄	—	18	1	no	—
152	C₃×C₄²	—	18	1	no	—
153	(C₂×D₃)⋊C₄	—	6	1	no	—
154	C₄⋊Dic₃	—	3	1	no	—
155	C₄⋊Dic₃	—	3	1	no	—
156	Dic₃⋊C₄	—	6	1	no	—
157	C₄×Dic₃	—	6	1	no	—
158	C₂×C₃⋊C₈	—	3	1	no	—
159	ΓL(1,𝔽₂₅)	—	3	4	no	—
160	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
161	C₂×C₃×D₃	—	3	2	no	—
162	C₂×C₃×D₃	—	6	1	no	—
163	C₂×C₃×D₃	—	6	1	no	—
164	C₃²×C₄	—	4	1	no	—
165	C₃²×C₄	—	4	1	no	—
166	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
168	C₃×Dic₃	—	3	2	no	—
169	C₃×Dic₃	—	12	1	no	—
170	C₃×Dic₃	—	12	1	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×D₉	—	2	1	no	—
174	C₂×D₉	—	2	1	no	—
175	C₂×D₉	C₃×C₄⋊C₄	1	2	yes	—
176	C₄×C₉	—	2	1	no	—
177	C₄×C₉	—	2	1	no	—
178	C₄×C₉	—	4	1	no	—
179	C₄×C₉	—	4	1	no	—
180	C₄×C₉	—	8	1	no	—
181	C₄×C₉	—	8	1	no	—
182	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
183	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
184	Dic₉	—	4	1	no	—
185	Dic₉	—	4	1	no	—
186	Dic₉	C₃×C₄⋊C₄	1	2	yes	—
187	C₂×M₁₆	—	9	1	no	—
188	#117	—	9	1	no	—
189	C₂×C₄⋊C₄	—	9	1	no	—
190	#107	—	9	1	no	—
191	#107	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
192	#106	—	9	1	no	—
193	#106	—	9	1	no	—
194	3₋¹⁺²	#428	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₃	—	9	1	no	—
196	C₂²×D₃	—	3	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	3	1	no	—
200	C₂×C₃×C₄	—	6	1	no	—
201	C₂×C₃×C₄	—	6	1	no	—
202	C₂×C₃×C₄	—	9	2	no	—
203	C₂×C₃×C₄	—	9	2	no	—
204	C₂×C₃×C₄	—	9	1	no	—
205	C₂×C₃×C₄	—	18	1	no	—
206	C₂×C₃×C₄	—	18	1	no	—
207	C₂×C₃×C₄	—	36	2	no	—
208	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
209	C₂×Dic₃	—	3	1	no	—
210	C₂×Dic₃	—	6	1	no	—
211	C₂×Dic₃	—	6	1	no	—
212	C₄×D₃	—	3	2	no	—
213	C₄×D₃	—	3	2	no	—
214	C₄×D₃	—	12	2	no	—
215	C₃×C₈	—	3	2	no	—
216	C₃×C₈	—	9	2	no	—
217	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
218	C₃⋊C₈	—	3	2	no	—
219	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₃²	#911	1	1	yes	characteristic
221	C₂×C₃²	#918	1	1	yes	characteristic
222	C₃×D₃	—	6	2	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂×C₉	C₃×#137	1	1	yes	characteristic
229	D₉	—	2	2	no	—
230	C₂²×C₄	—	9	1	no	—
231	C₂²×C₄	—	18	1	no	—
232	M₁₆	—	9	4	no	—
233	C₂×C₈	—	9	1	no	—
234	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
235	C₄⋊C₄	—	9	1	no	—
236	C₄⋊C₄	—	9	1	no	—
237	C₄⋊C₄	—	18	1	no	—
238	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
239	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
240	C₂²⋊C₄	—	18	1	no	—
241	C₄²	—	18	1	no	—
242	C₂²×C₃	—	3	1	no	—
243	C₂²×C₃	—	9	2	no	—
244	C₂²×C₃	—	18	1	no	—
245	C₂²×C₃	—	18	1	no	—
246	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
247	C₂×D₃	—	3	2	no	—
248	C₂×D₃	—	6	1	no	—
249	C₂×D₃	—	6	1	no	—
250	C₃×C₄	—	3	1	no	—
251	C₃×C₄	—	3	1	no	—
252	C₃×C₄	—	4	1	no	—
253	C₃×C₄	—	4	1	no	—
254	C₃×C₄	—	9	2	no	—
255	C₃×C₄	—	12	1	no	—
256	C₃×C₄	—	12	1	no	—
257	C₃×C₄	—	36	1	no	—
258	C₃×C₄	—	36	1	no	—
259	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
260	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
261	Dic₃	—	3	2	no	—
262	Dic₃	—	12	1	no	—
263	Dic₃	—	12	1	no	—
264	C₃²	#4881	1	1	yes	characteristic
265	C₉	—	2	1	no	—
266	C₉	C₃×#428	1	1	yes	characteristic, complemented
267	C₂³	—	9	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	9	1	no	—
271	C₂×C₄	—	9	2	no	—
272	C₂×C₄	—	9	2	no	—
273	C₂×C₄	—	18	1	no	—
274	C₂×C₄	—	18	1	no	—
275	C₂×C₄	—	36	2	no	—
276	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
277	C₈	—	9	2	no	—
278	C₈	C₄×AGL(1,ℤ/9)	1	2	yes	complemented
279	C₂×C₃	—	3	1	no	—
280	C₂×C₃	—	3	1	no	—
281	C₂×C₃	—	3	1	no	—
282	C₂×C₃	—	18	2	no	—
283	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
284	C₂×C₃	C₃×#911	1	1	yes	characteristic
285	C₂×C₃	C₃×#918	1	1	yes	characteristic
286	D₃	—	6	2	no	—
287	C₂²	—	9	2	no	—
288	C₂²	—	18	1	no	—
289	C₂²	—	18	1	no	—
290	C₂²	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	9	2	no	—
294	C₄	—	36	1	no	—
295	C₄	—	36	1	no	—
296	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
297	C₄	Q₈×AGL(1,ℤ/9)	1	1	yes	characteristic
298	C₃	—	3	1	no	—
299	C₃	C₃×#4881	1	1	yes	characteristic
300	C₂	—	18	2	no	—
301	C₂	C₄⋊C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
302	C₂	#63473⋊C₃	1	1	yes	characteristic
303	C₂	#63480⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,ℤ/9)
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#107	C₂
socle	C₂²×C₃	C₃×D₃×D₄
prime radical	C₂²×AGL(1,ℤ/9)	D₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	#107	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#428
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
4. 3₋¹⁺²×#107	C₂	semidirect
5. #63377⋊C₃	C₂	semidirect
6. #63367⋊C₃	C₂	semidirect
7. #63366⋊C₃	C₂	semidirect
9. #10588561	C₃	semidirect
21. #3663⋊C₃	C₄	semidirect
33. C₉×#107	C₂×C₃	semidirect
34. #63377	C₂×C₃	semidirect
35. #63367	C₂×C₃	semidirect
36. #63366	C₂×C₃	semidirect
47. C₈×3₋¹⁺²	C₂×C₄	semidirect
48. 3₋¹⁺²⋊C₈	C₂×C₄	semidirect
77. #3663	C₃×C₄	semidirect
129. C₈×C₉	C₂×C₃×C₄	semidirect
130. C₉⋊C₈	C₂×C₃×C₄	semidirect
191. #107	AGL(1,ℤ/9)	semidirect
194. 3₋¹⁺²	#428	semidirect
266. C₉	C₃×#428	semidirect
278. C₈	C₄×AGL(1,ℤ/9)	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3663⋊C₃	2
#63454⋊C₃	2
C₄⋊C₄×AGL(1,ℤ/9)	2
3₋¹⁺²×#107	2
#63377⋊C₃	2
#63367⋊C₃	2
#63366⋊C₃	2
C₃×#4881	3
#10588561	3

Subgroups — #10588561⋊C3 #13142242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588561⋊C₃ #13142242