Subgroups — #10589686⋊C₃ #13152745

Order 1728.

671 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	#63508⋊C₃	C₂	1	2	yes	maximal, complemented
3	#63460⋊C₃	C₂	1	4	yes	maximal, complemented
4	#63454⋊C₃	C₂	1	2	yes	maximal, complemented
5	#63437⋊C₃	C₂	1	2	yes	maximal
6	C₄×C₄⋊AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
7	#63431⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×3₋¹⁺²⋊Q₈	C₂	1	2	yes	maximal, complemented
9	C₃×#5584	—	3	1	no	maximal
10	#10589686	C₃	1	1	yes	maximal, characteristic, complemented
11	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
13	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	2	yes	—
14	C₂×3₋¹⁺²⋊Q₈	C₂²	1	2	yes	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
17	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
18	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
19	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	4	yes	—
20	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	2	yes	—
21	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	4	yes	—
22	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	—
23	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	—
24	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
25	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	4	yes	—
26	C₄×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
27	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
28	C₃×#946	—	3	2	no	—
29	C₃×#898	—	3	4	no	—
30	C₃×#892	—	3	2	no	—
31	C₃×#875	—	3	2	no	—
32	C₃×C₄×D₁₂	—	3	1	no	—
33	C₃×#869	—	3	1	no	—
34	C₃×C₄×Dic₆	—	3	2	no	—
35	C₄×Q₈×C₉	—	2	1	no	—
36	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
37	#63508	C₂×C₃	1	2	yes	complemented
38	#63460	C₂×C₃	1	4	yes	complemented
39	#63454	C₂×C₃	1	2	yes	complemented
40	#63437	C₂×C₃	1	2	yes	—
41	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic
42	#63431	C₂×C₃	1	1	yes	characteristic, complemented
43	C₄×Dic₁₈	C₂×C₃	1	2	yes	complemented
44	C₂²×AGL(1,ℤ/9)	C₂³	1	2	yes	—
45	Q₈×3₋¹⁺²	—	2	2	no	—
46	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
47	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
48	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
49	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
50	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
51	C₂×3₋¹⁺²⋊C₄	C₂³	1	4	yes	—
52	C₂×3₋¹⁺²⋊C₄	C₂³	1	2	yes	—
53	C₄⋊AGL(1,ℤ/9)	—	2	2	no	—
54	C₄×AGL(1,ℤ/9)	—	2	4	no	—
55	3₋¹⁺²⋊Q₈	—	2	4	no	—
56	C₃×#556	—	9	1	no	—
57	#5584	—	3	1	no	—
58	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×D₁₂	—	3	1	no	—
60	C₂×C₃×C₄×D₃	—	3	2	no	—
61	C₂×C₃×Dic₆	—	3	2	no	—
62	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
63	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
64	C₃²×C₄²	C₂×D₃	1	2	yes	—
65	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
66	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
67	C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
68	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
69	C₃×C₄⋊Dic₃	—	3	2	no	—
70	C₃×C₄⋊Dic₃	—	3	2	no	—
71	C₃×C₄⋊Dic₃	—	3	1	no	—
72	C₃×Dic₃⋊C₄	—	3	4	no	—
73	C₃×C₄×Dic₃	—	3	4	no	—
74	C₂×Q₈×C₉	—	2	1	no	—
75	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
76	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×D₉	C₂²×C₃	1	2	yes	—
78	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
79	C₉×C₄⋊C₄	—	2	1	no	—
80	C₉×C₄⋊C₄	—	2	2	no	—
81	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₄²×C₉	—	2	2	no	—
84	C₄²×C₉	—	2	1	no	—
85	C₄²×C₉	C₂²×C₃	1	2	yes	—
86	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
87	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	—
88	(C₂×D₉)⋊C₄	C₂²×C₃	1	2	yes	—
89	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	—
90	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
91	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
92	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
93	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
94	C₄×Dic₉	C₂²×C₃	1	4	yes	—
95	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
96	C₂×AGL(1,ℤ/9)	—	2	2	no	—
97	C₂×AGL(1,ℤ/9)	—	2	2	no	—
98	C₂×AGL(1,ℤ/9)	—	2	2	no	—
99	C₄×3₋¹⁺²	—	2	2	no	—
100	C₄×3₋¹⁺²	—	2	2	no	—
101	C₄×3₋¹⁺²	—	2	1	no	—
102	C₄×3₋¹⁺²	#41	1	2	yes	—
103	C₄×3₋¹⁺²	#41	1	2	yes	—
104	3₋¹⁺²⋊C₄	—	2	4	no	—
105	3₋¹⁺²⋊C₄	—	2	2	no	—
106	C₃×C₄⋊Q₈	—	9	1	no	—
107	C₃×#126	—	9	4	no	—
108	C₃×#124	—	9	2	no	—
109	C₃×C₂²⋊Q₈	—	9	2	no	—
110	C₃×C₄×Q₈	—	3	1	no	—
111	C₃×C₄×Q₈	—	9	2	no	—
112	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
113	C₃×C₄×D₄	—	9	1	no	—
114	C₃×#117	—	9	2	no	—
115	#946	—	3	2	no	—
116	#898	—	3	4	no	—
117	#892	—	3	2	no	—
118	#875	—	3	2	no	—
119	C₄×D₁₂	—	3	1	no	—
120	#869	—	3	1	no	—
121	C₄×Dic₆	—	3	2	no	—
122	C₂²×C₃×D₃	—	3	2	no	—
123	C₃²×Q₈	—	2	2	no	—
124	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
125	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
126	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₃×Dic₃	—	3	2	no	—
130	C₂×C₃×Dic₃	—	3	4	no	—
131	C₃×D₁₂	—	6	2	no	—
132	C₃×C₄×D₃	—	6	4	no	—
133	C₃×Dic₆	—	6	4	no	—
134	C₂²×D₉	C₂³×C₃	1	2	yes	—
135	Q₈×C₉	—	2	4	no	—
136	Q₈×C₉	—	2	2	no	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	—	2	2	no	—
139	C₂×C₄×C₉	—	2	1	no	—
140	C₂×C₄×C₉	—	2	2	no	—
141	C₂×C₄×C₉	—	2	1	no	—
142	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
143	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
144	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
147	C₂×Dic₉	C₂³×C₃	1	2	yes	—
148	C₂×Dic₉	C₂³×C₃	1	4	yes	—
149	D₃₆	—	2	2	no	—
150	C₄×D₉	—	2	4	no	—
151	Dic₁₈	—	2	4	no	—
152	#556	—	9	1	no	Sylow
153	C₂×3₋¹⁺²	C₂×#41	1	1	yes	characteristic
154	C₂×3₋¹⁺²	C₂×#41	1	1	yes	characteristic
155	C₂×3₋¹⁺²	2₋¹⁺⁴	1	1	yes	characteristic
156	AGL(1,ℤ/9)	—	4	2	no	—
157	C₂×C₃×Q₈	—	3	1	no	—
158	C₂×C₃×Q₈	—	9	2	no	—
159	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
160	C₂×C₃×D₄	—	9	1	no	—
161	C₂²×C₃×C₄	—	9	2	no	—
162	C₂×D₁₂	—	3	1	no	—
163	C₂×C₄×D₃	—	3	2	no	—
164	C₂×Dic₆	—	3	2	no	—
165	C₃×C₄⋊C₄	—	3	1	no	—
166	C₃×C₄⋊C₄	—	3	2	no	—
167	C₃×C₄⋊C₄	—	9	4	no	—
168	C₃×C₄⋊C₄	—	9	2	no	—
169	C₃×C₄⋊C₄	—	9	1	no	—
170	C₃×C₄⋊C₄	—	9	2	no	—
171	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
172	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
173	C₃×C₂²⋊C₄	—	9	4	no	—
174	C₃×C₂²⋊C₄	—	9	2	no	—
175	C₃×C₂²⋊C₄	—	9	4	no	—
176	C₃×C₄²	—	3	1	no	—
177	C₃×C₄²	—	3	2	no	—
178	C₃×C₄²	—	9	4	no	—
179	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
180	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
181	(C₂×D₃)⋊C₄	—	3	2	no	—
182	(C₂×D₃)⋊C₄	—	3	4	no	—
183	(C₂×D₃)⋊C₄	—	3	4	no	—
184	C₄⋊Dic₃	—	3	1	no	—
185	C₄⋊Dic₃	—	3	2	no	—
186	C₄⋊Dic₃	—	3	2	no	—
187	Dic₃⋊C₄	—	3	4	no	—
188	C₄×Dic₃	—	3	4	no	—
189	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
190	C₂×C₃×D₃	—	6	2	no	—
191	C₂×C₃×D₃	—	6	2	no	—
192	C₂×C₃×D₃	—	6	2	no	—
193	C₃²×C₄	—	2	2	no	—
194	C₃²×C₄	—	2	2	no	—
195	C₃²×C₄	—	2	1	no	—
196	C₃²×C₄	#235	1	2	yes	—
197	C₃²×C₄	#239	1	2	yes	—
198	C₃×Dic₃	—	6	2	no	—
199	C₃×Dic₃	—	6	4	no	—
200	C₂²×C₉	—	2	1	no	—
201	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
202	C₂×D₉	—	2	2	no	—
203	C₂×D₉	—	2	2	no	—
204	C₂×D₉	—	2	2	no	—
205	C₄×C₉	—	2	4	no	—
206	C₄×C₉	—	2	2	no	—
207	C₄×C₉	—	2	2	no	—
208	C₄×C₉	—	2	1	no	—
209	C₄×C₉	—	2	2	no	—
210	C₄×C₉	—	2	2	no	—
211	C₄×C₉	—	4	2	no	—
212	C₄×C₉	—	4	1	no	—
213	C₄×C₉	C₃×#41	1	2	yes	—
214	C₄×C₉	C₃×#41	1	2	yes	—
215	Dic₉	—	2	4	no	—
216	Dic₉	—	2	2	no	—
217	C₄⋊Q₈	—	9	1	no	—
218	#126	—	9	4	no	—
219	#124	—	9	2	no	—
220	C₂²⋊Q₈	—	9	2	no	—
221	C₄×Q₈	—	9	2	no	—
222	C₄×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
223	C₄×D₄	—	9	1	no	—
224	#117	—	9	2	no	—
225	3₋¹⁺²	#556	1	1	yes	characteristic, complemented, Sylow
226	C₂³×C₃	—	9	2	no	—
227	C₂²×D₃	—	3	2	no	—
228	C₃×Q₈	—	2	2	no	—
229	C₃×Q₈	—	6	2	no	—
230	C₃×Q₈	—	18	4	no	—
231	C₃×D₄	—	18	2	no	—
232	C₂×C₃×C₄	—	3	1	no	—
233	C₂×C₃×C₄	—	3	1	no	—
234	C₂×C₃×C₄	—	3	2	no	—
235	C₂×C₃×C₄	—	3	1	no	—
236	C₂×C₃×C₄	—	3	2	no	—
237	C₂×C₃×C₄	—	9	2	no	—
238	C₂×C₃×C₄	—	9	4	no	—
239	C₂×C₃×C₄	—	18	4	no	—
240	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
241	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
242	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₂×Dic₃	—	3	2	no	—
246	C₂×Dic₃	—	3	4	no	—
247	D₁₂	—	6	2	no	—
248	C₄×D₃	—	6	4	no	—
249	Dic₆	—	6	4	no	—
250	C₂×C₃²	C₂×#235	1	1	yes	characteristic
251	C₂×C₃²	C₂×#239	1	1	yes	characteristic
252	C₂×C₃²	#1010	1	1	yes	characteristic
253	C₃×D₃	—	12	2	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	—	2	1	no	—
257	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
258	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
259	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
260	D₉	—	4	2	no	—
261	C₂×Q₈	—	9	2	no	—
262	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
263	C₂×D₄	—	9	1	no	—
264	C₂²×C₄	—	9	2	no	—
265	C₄⋊C₄	—	9	2	no	—
266	C₄⋊C₄	—	9	1	no	—
267	C₄⋊C₄	—	9	4	no	—
268	C₄⋊C₄	—	9	2	no	—
269	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
270	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
271	C₂²⋊C₄	—	9	4	no	—
272	C₂²⋊C₄	—	9	2	no	—
273	C₂²⋊C₄	—	9	4	no	—
274	C₄²	—	9	4	no	—
275	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
276	C₄²	C₂×AGL(1,ℤ/9)	1	2	yes	—
277	C₂²×C₃	—	3	1	no	—
278	C₂²×C₃	—	18	2	no	—
279	C₂²×C₃	—	18	2	no	—
280	C₂²×C₃	—	18	2	no	—
281	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
282	C₂×D₃	—	6	2	no	—
283	C₂×D₃	—	6	2	no	—
284	C₂×D₃	—	6	2	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	3	2	no	—
289	C₃×C₄	—	3	2	no	—
290	C₃×C₄	—	6	2	no	—
291	C₃×C₄	—	6	2	no	—
292	C₃×C₄	—	6	1	no	—
293	C₃×C₄	—	18	2	no	—
294	C₃×C₄	—	18	4	no	—
295	C₃×C₄	C₃×#235	1	2	yes	—
296	C₃×C₄	C₃×#239	1	2	yes	—
297	Dic₃	—	6	2	no	—
298	Dic₃	—	6	4	no	—
299	C₃²	#5584	1	1	yes	characteristic
300	C₉	—	2	1	no	—
301	C₉	C₃×#556	1	1	yes	characteristic, complemented
302	C₂³	—	9	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	18	4	no	—
305	D₄	—	18	2	no	—
306	C₂×C₄	—	9	2	no	—
307	C₂×C₄	—	9	4	no	—
308	C₂×C₄	—	18	4	no	—
309	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	2	yes	—
310	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
311	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	2	yes	—
312	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
313	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
314	C₂×C₃	—	3	1	no	—
315	C₂×C₃	—	3	1	no	—
316	C₂×C₃	—	3	1	no	—
317	C₂×C₃	—	36	2	no	—
318	C₂×C₃	C₂×C₃×#235	1	1	yes	characteristic
319	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
320	C₂×C₃	C₃×#1010	1	1	yes	characteristic
321	D₃	—	12	2	no	—
322	C₂²	—	18	2	no	—
323	C₂²	—	18	2	no	—
324	C₂²	—	18	2	no	—
325	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	18	2	no	—
330	C₄	—	18	4	no	—
331	C₄	#3697⋊C₃	1	2	yes	—
332	C₄	#88724	1	2	yes	—
333	C₃	—	3	1	no	—
334	C₃	C₃×#5584	1	1	yes	characteristic
335	C₂	—	36	2	no	—
336	C₂	C₂×#3697⋊C₃	1	1	yes	characteristic
337	C₂	C₂×#88724	1	1	yes	characteristic
338	C₂	#11722642	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₄×Q₈×3₋¹⁺²	C₂
prime radical	C₂×C₄⋊AGL(1,ℤ/9)	C₂²
hypercenter	C₄×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#556
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×Q₈×3₋¹⁺²	C₂	semidirect
2. #63508⋊C₃	C₂	semidirect
3. #63460⋊C₃	C₂	semidirect
4. #63454⋊C₃	C₂	semidirect
7. #63431⋊C₃	C₂	semidirect
8. C₄×3₋¹⁺²⋊Q₈	C₂	semidirect
10. #10589686	C₃	semidirect
36. C₄×Q₈×C₉	C₂×C₃	semidirect
37. #63508	C₂×C₃	semidirect
38. #63460	C₂×C₃	semidirect
39. #63454	C₂×C₃	semidirect
42. #63431	C₂×C₃	semidirect
43. C₄×Dic₁₈	C₂×C₃	semidirect
222. C₄×Q₈	AGL(1,ℤ/9)	semidirect
225. 3₋¹⁺²	#556	semidirect
301. C₉	C₃×#556	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×3₋¹⁺²	2
#63508⋊C₃	2
#63460⋊C₃	2
#63454⋊C₃	2
#63437⋊C₃	2
C₄×C₄⋊AGL(1,ℤ/9)	2
#63431⋊C₃	2
C₄×3₋¹⁺²⋊Q₈	2
C₃×#5584	3
#10589686	3

Subgroups — #10589686⋊C3 #13152745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10589686⋊C₃ #13152745