Subgroups — #10588521⋊C₃ #13142069

Order 1728.

397 conjugacy classes of subgroups.

The subgroup lattice has 286 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	2	yes	maximal
2	C₄²×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	(3₋¹⁺²⋊C₄)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4837	—	3	1	no	maximal
7	#10588521	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×AGL(1,ℤ/9)	C₄	1	1	yes	characteristic
9	C₂×C₄×AGL(1,ℤ/9)	C₄	1	1	yes	characteristic
10	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
11	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
12	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₄×3₋¹⁺²⋊C₄	C₄	1	1	yes	characteristic
14	C₄×3₋¹⁺²⋊C₄	C₄	1	1	yes	characteristic
15	C₄×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
17	#3663⋊C₃	—	2	4	no	—
18	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	2	no	—
19	C₃×C₄²×D₃	—	3	1	no	—
20	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
21	C₃×Dic₃⋊C₈	—	3	2	no	—
22	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×#3663	C₂×C₃	1	2	yes	—
24	C₄²×D₉	C₂×C₃	1	1	yes	characteristic
25	C₉×C₄⋊C₈	—	2	1	no	—
26	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	Dic₉⋊C₈	C₂×C₃	1	2	yes	complemented
28	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×AGL(1,ℤ/9)	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
33	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₄×AGL(1,ℤ/9)	—	2	2	no	—
37	C₄×AGL(1,ℤ/9)	—	2	2	no	—
38	C₄×AGL(1,ℤ/9)	D₄	1	2	yes	—
39	C₄×AGL(1,ℤ/9)	Q₈	1	2	yes	—
40	C₈×3₋¹⁺²	—	2	2	no	—
41	3₋¹⁺²⋊C₈	—	2	2	no	—
42	C₃×#423	—	9	1	no	—
43	#4837	—	3	1	no	—
44	C₂×C₃×C₄×D₃	—	3	1	no	—
45	C₂×C₃×C₄×D₃	—	3	1	no	—
46	C₂×C₃×C₄×D₃	—	3	1	no	—
47	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
48	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₃×C₄×Dic₃	—	3	1	no	—
50	C₃×C₄×Dic₃	—	3	1	no	—
51	C₃×C₄×Dic₃	—	3	1	no	—
52	C₂×C₃×C₃⋊C₈	—	3	2	no	—
53	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
54	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
56	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₈×C₉	—	2	2	no	—
58	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
61	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
62	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
63	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
65	#3663	—	2	4	no	—
66	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
67	C₂×AGL(1,ℤ/9)	—	2	1	no	—
68	C₂×AGL(1,ℤ/9)	—	2	1	no	—
69	C₂×AGL(1,ℤ/9)	C₄⋊C₄	1	2	yes	—
70	C₄×3₋¹⁺²	—	2	1	no	—
71	C₄×3₋¹⁺²	M₁₆	1	2	yes	complemented
72	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
73	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
74	3₋¹⁺²⋊C₄	—	2	1	no	—
75	3₋¹⁺²⋊C₄	C₄⋊C₄	1	2	yes	—
76	3₋¹⁺²⋊C₄	M₁₆	1	2	yes	complemented
77	C₂×C₃×M₁₆	—	9	2	no	—
78	C₂×C₃×C₄²	—	9	1	no	—
79	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
80	C₄²×D₃	—	3	1	no	—
81	C₃×C₄⋊C₈	—	3	1	no	—
82	C₃×C₄⋊C₈	—	9	1	no	—
83	C₃×C₄⋊C₈	—	9	2	no	—
84	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
85	Dic₃⋊C₈	—	3	2	no	—
86	C₄⋊(C₃⋊C₈)	—	3	1	no	—
87	C₂²×C₃×D₃	—	3	1	no	—
88	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃×Dic₃	—	3	1	no	—
92	C₂×C₃×Dic₃	—	3	1	no	—
93	C₂×C₃×Dic₃	—	3	1	no	—
94	C₃×C₄×D₃	—	3	2	no	—
95	C₃×C₄×D₃	—	3	2	no	—
96	C₃×C₄×D₃	—	6	2	no	—
97	C₃×C₄×D₃	—	6	2	no	—
98	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
99	C₃²×C₈	—	2	2	no	—
100	C₃×C₃⋊C₈	—	6	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
107	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
110	C₄×D₉	—	2	2	no	—
111	C₄×D₉	—	2	2	no	—
112	C₄×D₉	C₃×D₄	1	2	yes	—
113	C₄×D₉	C₃×Q₈	1	2	yes	—
114	C₈×C₉	—	2	2	no	—
115	C₈×C₉	—	4	2	no	—
116	C₉⋊C₈	—	2	2	no	—
117	#423	—	9	1	no	Sylow
118	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×3₋¹⁺²	C₂×M₁₆	1	1	yes	characteristic
120	C₂×3₋¹⁺²	C₂×M₁₆	1	1	yes	characteristic
121	AGL(1,ℤ/9)	—	2	2	no	—
122	C₂²×C₃×C₄	—	9	1	no	—
123	C₂²×C₃×C₄	—	9	1	no	—
124	C₂²×C₃×C₄	—	9	1	no	—
125	C₂×C₄×D₃	—	3	1	no	—
126	C₂×C₄×D₃	—	3	1	no	—
127	C₂×C₄×D₃	—	3	1	no	—
128	C₃×M₁₆	—	18	4	no	—
129	C₂×C₃×C₈	—	3	2	no	—
130	C₂×C₃×C₈	—	9	2	no	—
131	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
132	C₃×C₄²	—	3	1	no	—
133	C₃×C₄²	—	9	1	no	—
134	C₃×C₄²	—	9	1	no	—
135	C₃×C₄²	—	9	1	no	—
136	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
137	C₄×Dic₃	—	3	1	no	—
138	C₄×Dic₃	—	3	1	no	—
139	C₄×Dic₃	—	3	1	no	—
140	C₂×C₃⋊C₈	—	3	2	no	—
141	ΓL(1,𝔽₂₅)	—	6	4	no	—
142	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×C₃×D₃	—	3	2	no	—
144	C₂×C₃×D₃	—	6	1	no	—
145	C₂×C₃×D₃	—	6	1	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
148	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
149	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
150	C₃×Dic₃	—	3	2	no	—
151	C₃×Dic₃	—	3	2	no	—
152	C₃×Dic₃	—	6	1	no	—
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×D₉	—	2	1	no	—
156	C₂×D₉	—	2	1	no	—
157	C₂×D₉	C₃×C₄⋊C₄	1	2	yes	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	4	1	no	—
163	C₄×C₉	C₃×M₁₆	1	2	yes	complemented
164	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
165	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
166	Dic₉	—	2	1	no	—
167	Dic₉	C₃×C₄⋊C₄	1	2	yes	—
168	Dic₉	C₃×M₁₆	1	2	yes	complemented
169	C₂×M₁₆	—	9	2	no	—
170	C₂×C₄²	—	9	1	no	—
171	C₄⋊C₈	—	9	2	no	—
172	C₄⋊C₈	—	9	1	no	—
173	C₄⋊C₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
174	3₋¹⁺²	#423	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	9	1	no	—
176	C₂²×D₃	—	3	1	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	3	1	no	—
180	C₂×C₃×C₄	—	9	2	no	—
181	C₂×C₃×C₄	—	9	2	no	—
182	C₂×C₃×C₄	—	9	1	no	—
183	C₂×C₃×C₄	—	9	1	no	—
184	C₂×C₃×C₄	—	9	1	no	—
185	C₂×C₃×C₄	—	18	2	no	—
186	C₂×C₃×C₄	—	18	2	no	—
187	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
190	C₂×Dic₃	—	3	1	no	—
191	C₂×Dic₃	—	3	1	no	—
192	C₂×Dic₃	—	3	1	no	—
193	C₄×D₃	—	3	2	no	—
194	C₄×D₃	—	3	2	no	—
195	C₄×D₃	—	6	2	no	—
196	C₄×D₃	—	6	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	6	2	no	—
199	C₃×C₈	—	18	2	no	—
200	C₃⋊C₈	—	6	2	no	—
201	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
203	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
204	C₃×D₃	—	6	2	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
210	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
211	D₉	—	2	2	no	—
212	C₂²×C₄	—	9	1	no	—
213	C₂²×C₄	—	9	1	no	—
214	C₂²×C₄	—	9	1	no	—
215	M₁₆	—	18	4	no	—
216	C₂×C₈	—	9	2	no	—
217	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
218	C₄²	—	9	1	no	—
219	C₄²	—	9	1	no	—
220	C₄²	—	9	1	no	—
221	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
222	C₂²×C₃	—	3	1	no	—
223	C₂²×C₃	—	9	2	no	—
224	C₂²×C₃	—	18	1	no	—
225	C₂²×C₃	—	18	1	no	—
226	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
227	C₂×D₃	—	3	2	no	—
228	C₂×D₃	—	6	1	no	—
229	C₂×D₃	—	6	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	3	1	no	—
232	C₃×C₄	—	3	1	no	—
233	C₃×C₄	—	3	2	no	—
234	C₃×C₄	—	6	1	no	—
235	C₃×C₄	—	9	2	no	—
236	C₃×C₄	—	9	2	no	—
237	C₃×C₄	—	18	1	no	—
238	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
239	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
240	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
241	Dic₃	—	3	2	no	—
242	Dic₃	—	3	2	no	—
243	Dic₃	—	6	1	no	—
244	C₃²	#4837	1	1	yes	characteristic
245	C₉	—	2	1	no	—
246	C₉	C₃×#423	1	1	yes	characteristic, complemented
247	C₂³	—	9	1	no	—
248	C₂×C₄	—	9	1	no	—
249	C₂×C₄	—	9	1	no	—
250	C₂×C₄	—	9	2	no	—
251	C₂×C₄	—	9	2	no	—
252	C₂×C₄	—	9	1	no	—
253	C₂×C₄	—	18	2	no	—
254	C₂×C₄	—	18	2	no	—
255	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
256	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
257	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	18	2	no	—
260	C₂×C₃	—	3	1	no	—
261	C₂×C₃	—	3	1	no	—
262	C₂×C₃	—	3	1	no	—
263	C₂×C₃	—	18	2	no	—
264	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
266	C₂×C₃	C₃×D₃×M₁₆	1	1	yes	characteristic
267	D₃	—	6	2	no	—
268	C₂²	—	9	2	no	—
269	C₂²	—	18	1	no	—
270	C₂²	—	18	1	no	—
271	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	9	2	no	—
274	C₄	—	9	2	no	—
275	C₄	—	18	1	no	—
276	C₄	#3663⋊C₃	1	2	yes	complemented
277	C₄	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
278	C₄	Q₈×AGL(1,ℤ/9)	1	1	yes	characteristic
279	C₃	—	3	1	no	—
280	C₃	C₃×#4837	1	1	yes	characteristic
281	C₂	—	18	2	no	—
282	C₂	C₄⋊C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
283	C₂	C₂×#3663⋊C₃	1	1	yes	characteristic
284	C₂	M₁₆×AGL(1,ℤ/9)	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×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂×C₃×C₄×D₃
prime radical	C₂²×AGL(1,ℤ/9)	C₂×C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₄⋊C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#423
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×C₄⋊C₈	C₂	semidirect
4. (3₋¹⁺²⋊C₄)⋊C₈	C₂	semidirect
5. C₄⋊(3₋¹⁺²⋊C₈)	C₂	semidirect
7. #10588521	C₃	semidirect
26. C₉×C₄⋊C₈	C₂×C₃	semidirect
27. Dic₉⋊C₈	C₂×C₃	semidirect
28. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
71. C₄×3₋¹⁺²	M₁₆	semidirect
76. 3₋¹⁺²⋊C₄	M₁₆	semidirect
163. C₄×C₉	C₃×M₁₆	semidirect
168. Dic₉	C₃×M₁₆	semidirect
173. C₄⋊C₈	AGL(1,ℤ/9)	semidirect
174. 3₋¹⁺²	#423	semidirect
246. C₉	C₃×#423	semidirect
276. C₄	#3663⋊C₃	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
C₄²×AGL(1,ℤ/9)	2
3₋¹⁺²×C₄⋊C₈	2
(3₋¹⁺²⋊C₄)⋊C₈	2
C₄⋊(3₋¹⁺²⋊C₈)	2
C₃×#4837	3
#10588521	3

Subgroups — #10588521⋊C3 #13142069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588521⋊C₃ #13142069