Subgroups — #10588482⋊C₃ #13141947

Order 1728.

471 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
2	C₂×#3663⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#63454⋊C₃	C₂	1	1	yes	maximal, characteristic
4	3₋¹⁺²×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	#63395⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#63376⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#63368⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4802	—	3	1	no	maximal
9	#10588482	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	Q₈×AGL(1,ℤ/9)	—	2	2	no	—
12	Q₈×AGL(1,ℤ/9)	C₄	1	4	yes	complemented
13	C₂×C₄×AGL(1,ℤ/9)	—	2	1	no	—
14	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
15	C₂×3₋¹⁺²⋊Q₈	—	2	1	no	—
16	C₂×3₋¹⁺²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
19	(C₂×AGL(1,ℤ/9))⋊C₄	—	2	1	no	—
20	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
21	C₄×3₋¹⁺²⋊C₄	—	2	1	no	—
22	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
23	#3663⋊C₃	—	2	2	no	—
24	C₂×C₃×D₃×Q₈	—	3	1	no	—
25	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
26	C₃×#892	—	3	1	no	—
27	C₂×Q₈×D₉	C₂×C₃	1	1	yes	characteristic
28	C₃²×#103	D₃	1	1	yes	characteristic
29	C₃×#835	—	3	1	no	—
30	C₃×#816	—	3	1	no	—
31	C₃×#810	—	3	1	no	—
32	C₂×#3663	C₂×C₃	1	1	yes	characteristic
33	#63454	C₂×C₃	1	1	yes	characteristic
34	C₉×#103	—	2	1	no	—
35	C₉×#103	C₂×C₃	1	1	yes	characteristic, complemented
36	#63395	C₂×C₃	1	1	yes	characteristic, complemented
37	#63376	C₂×C₃	1	1	yes	characteristic, complemented
38	#63368	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×AGL(1,ℤ/9)	D₄	1	1	yes	characteristic
40	Q₈×3₋¹⁺²	—	2	1	no	—
41	Q₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
42	C₂×C₄×3₋¹⁺²	—	2	1	no	—
43	C₂×C₄×3₋¹⁺²	—	2	1	no	—
44	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
45	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
46	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
47	C₂×3₋¹⁺²⋊C₄	D₄	1	1	yes	characteristic
48	C₄×AGL(1,ℤ/9)	—	2	4	no	—
49	C₄×AGL(1,ℤ/9)	C₂×C₄	1	2	yes	—
50	C₄×AGL(1,ℤ/9)	D₄	1	2	yes	—
51	3₋¹⁺²⋊Q₈	—	2	1	no	—
52	3₋¹⁺²⋊Q₈	—	2	4	no	—
53	3₋¹⁺²⋊Q₈	C₂×C₄	1	2	yes	—
54	C₈×3₋¹⁺²	—	2	1	no	—
55	3₋¹⁺²⋊C₈	—	2	1	no	—
56	C₃×#419	—	9	1	no	—
57	#4802	—	3	1	no	—
58	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₃×D₃×Q₈	—	3	4	no	—
60	C₃×D₃×Q₈	—	6	2	no	—
61	C₂×C₃×C₄×D₃	—	3	1	no	—
62	C₂×C₃×C₄×D₃	—	6	1	no	—
63	C₂×C₃×Dic₆	—	3	1	no	—
64	C₂×C₃×Dic₆	—	6	1	no	—
65	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
68	C₃×C₄⋊Dic₃	—	3	1	no	—
69	C₃×C₄×Dic₃	—	6	1	no	—
70	C₂×C₃×C₃⋊C₈	—	3	1	no	—
71	C₃×ΓL(1,𝔽₂₅)	—	6	2	no	—
72	C₂×Q₈×C₉	—	2	1	no	—
73	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
74	Q₈×D₉	—	2	2	no	—
75	Q₈×D₉	C₃×C₄	1	4	yes	complemented
76	C₂×C₄×D₉	—	2	1	no	—
77	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
78	C₂×Dic₁₈	—	2	1	no	—
79	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₈×C₉	—	2	1	no	—
81	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
82	C₉×C₄⋊C₄	—	2	1	no	—
83	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	(C₂×D₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
86	C₄×Dic₉	—	2	1	no	—
87	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
88	#3663	—	2	2	no	—
89	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
90	C₂×AGL(1,ℤ/9)	—	2	1	no	—
91	C₂×AGL(1,ℤ/9)	—	2	1	no	—
92	C₂×AGL(1,ℤ/9)	C₂²⋊C₄	1	2	yes	—
93	C₄×3₋¹⁺²	—	2	2	no	—
94	C₄×3₋¹⁺²	—	4	1	no	—
95	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
96	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
97	3₋¹⁺²⋊C₄	—	2	2	no	—
98	3₋¹⁺²⋊C₄	—	4	1	no	—
99	3₋¹⁺²⋊C₄	C₂²⋊C₄	1	2	yes	—
100	C₂²×C₃×Q₈	—	9	1	no	—
101	C₂×D₃×Q₈	—	3	1	no	—
102	C₂×C₃×M₁₆	—	9	1	no	—
103	C₃×#117	—	9	1	no	—
104	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
105	#892	—	3	1	no	—
106	C₃×#103	—	3	1	no	—
107	C₃×#103	—	9	1	no	—
108	C₃×#103	—	9	1	no	—
109	C₃×#103	—	9	1	no	—
110	C₃×#103	C₃×D₃	1	1	yes	characteristic
111	#835	—	3	1	no	—
112	#816	—	3	1	no	—
113	#810	—	3	1	no	—
114	C₂²×C₃×D₃	—	3	1	no	—
115	C₃²×Q₈	—	2	1	no	—
116	C₃²×Q₈	C₄×D₃	1	2	yes	—
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₃×Dic₃	—	3	1	no	—
121	C₂×C₃×Dic₃	—	6	1	no	—
122	C₂×C₃×Dic₃	—	6	1	no	—
123	C₃×C₄×D₃	—	3	2	no	—
124	C₃×C₄×D₃	—	3	2	no	—
125	C₃×C₄×D₃	—	6	4	no	—
126	C₃×Dic₆	—	3	2	no	—
127	C₃×Dic₆	—	6	4	no	—
128	C₃×Dic₆	—	6	1	no	—
129	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
130	C₃²×C₈	—	2	1	no	—
131	C₃×C₃⋊C₈	—	6	1	no	—
132	Q₈×C₉	—	2	1	no	—
133	Q₈×C₉	—	2	2	no	—
134	Q₈×C₉	—	4	1	no	—
135	Q₈×C₉	C₂×C₃×C₄	1	2	yes	—
136	C₂×C₄×C₉	—	2	1	no	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	—	2	1	no	—
139	C₂×C₄×C₉	—	4	1	no	—
140	C₂×C₄×C₉	—	4	1	no	—
141	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
142	C₂×Dic₉	—	2	1	no	—
143	C₂×Dic₉	—	2	1	no	—
144	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
145	C₄×D₉	—	2	4	no	—
146	C₄×D₉	C₂×C₃×C₄	1	2	yes	—
147	C₄×D₉	C₃×D₄	1	2	yes	—
148	Dic₁₈	—	2	1	no	—
149	Dic₁₈	—	2	4	no	—
150	Dic₁₈	C₂×C₃×C₄	1	2	yes	—
151	C₈×C₉	—	2	1	no	—
152	C₈×C₉	—	4	1	no	—
153	C₉⋊C₈	—	2	1	no	—
154	#419	—	9	1	no	Sylow
155	C₂×3₋¹⁺²	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
157	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
158	AGL(1,ℤ/9)	—	2	2	no	—
159	C₂×C₃×Q₈	—	3	1	no	—
160	C₂×C₃×Q₈	—	9	1	no	—
161	C₂×C₃×Q₈	—	9	4	no	—
162	C₂×C₃×Q₈	—	18	1	no	—
163	C₂×C₃×Q₈	—	18	2	no	—
164	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
165	C₂²×C₃×C₄	—	9	1	no	—
166	C₂²×C₃×C₄	—	18	1	no	—
167	D₃×Q₈	—	3	4	no	—
168	D₃×Q₈	—	6	2	no	—
169	C₂×C₄×D₃	—	3	1	no	—
170	C₂×C₄×D₃	—	6	1	no	—
171	C₂×Dic₆	—	3	1	no	—
172	C₂×Dic₆	—	6	1	no	—
173	C₃×M₁₆	—	18	2	no	—
174	C₂×C₃×C₈	—	3	1	no	—
175	C₂×C₃×C₈	—	9	1	no	—
176	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
177	C₃×C₄⋊C₄	—	3	1	no	—
178	C₃×C₄⋊C₄	—	9	1	no	—
179	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
180	C₃×C₂²⋊C₄	—	18	1	no	—
181	C₃×C₄²	—	18	1	no	—
182	(C₂×D₃)⋊C₄	—	6	1	no	—
183	C₄⋊Dic₃	—	3	1	no	—
184	C₄×Dic₃	—	6	1	no	—
185	C₂×C₃⋊C₈	—	3	1	no	—
186	ΓL(1,𝔽₂₅)	—	6	2	no	—
187	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
188	C₂×C₃×D₃	—	3	2	no	—
189	C₂×C₃×D₃	—	6	1	no	—
190	C₂×C₃×D₃	—	6	1	no	—
191	C₃²×C₄	—	2	2	no	—
192	C₃²×C₄	—	4	1	no	—
193	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
194	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
195	C₃×Dic₃	—	3	2	no	—
196	C₃×Dic₃	—	6	2	no	—
197	C₃×Dic₃	—	12	1	no	—
198	C₂²×C₉	—	2	1	no	—
199	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×D₉	—	2	1	no	—
201	C₂×D₉	—	2	1	no	—
202	C₂×D₉	C₃×C₂²⋊C₄	1	2	yes	—
203	C₄×C₉	—	2	2	no	—
204	C₄×C₉	—	2	1	no	—
205	C₄×C₉	—	2	1	no	—
206	C₄×C₉	—	4	2	no	—
207	C₄×C₉	—	4	1	no	—
208	C₄×C₉	—	8	1	no	—
209	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
210	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
211	Dic₉	—	2	2	no	—
212	Dic₉	—	4	1	no	—
213	Dic₉	C₃×C₂²⋊C₄	1	2	yes	—
214	C₂²×Q₈	—	9	1	no	—
215	C₂×M₁₆	—	9	1	no	—
216	#117	—	9	1	no	—
217	#103	—	9	1	no	—
218	#103	—	9	1	no	—
219	#103	—	9	1	no	—
220	#103	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
221	3₋¹⁺²	#419	1	1	yes	characteristic, complemented, Sylow
222	C₂³×C₃	—	9	1	no	—
223	C₂²×D₃	—	3	1	no	—
224	C₃×Q₈	—	2	1	no	—
225	C₃×Q₈	—	3	2	no	—
226	C₃×Q₈	—	6	1	no	—
227	C₃×Q₈	—	9	2	no	—
228	C₃×Q₈	—	18	1	no	—
229	C₃×Q₈	—	18	4	no	—
230	C₃×Q₈	C₃×C₄×D₃	1	2	yes	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	—	3	1	no	—
234	C₂×C₃×C₄	—	6	1	no	—
235	C₂×C₃×C₄	—	6	1	no	—
236	C₂×C₃×C₄	—	9	2	no	—
237	C₂×C₃×C₄	—	9	2	no	—
238	C₂×C₃×C₄	—	9	1	no	—
239	C₂×C₃×C₄	—	18	4	no	—
240	C₂×C₃×C₄	—	18	1	no	—
241	C₂×C₃×C₄	—	18	1	no	—
242	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
243	C₂×Dic₃	—	3	1	no	—
244	C₂×Dic₃	—	6	1	no	—
245	C₂×Dic₃	—	6	1	no	—
246	C₄×D₃	—	3	2	no	—
247	C₄×D₃	—	3	2	no	—
248	C₄×D₃	—	6	4	no	—
249	Dic₆	—	3	2	no	—
250	Dic₆	—	6	4	no	—
251	Dic₆	—	6	1	no	—
252	C₃×C₈	—	2	1	no	—
253	C₃×C₈	—	6	1	no	—
254	C₃×C₈	—	18	1	no	—
255	C₃⋊C₈	—	6	1	no	—
256	C₂×C₃²	D₃×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₃²	#915	1	1	yes	characteristic
258	C₂×C₃²	#918	1	1	yes	characteristic
259	C₃×D₃	—	6	2	no	—
260	C₂×C₉	—	2	1	no	—
261	C₂×C₉	—	2	1	no	—
262	C₂×C₉	—	2	1	no	—
263	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₉	C₃×#137	1	1	yes	characteristic
265	C₂×C₉	C₃×#137	1	1	yes	characteristic
266	D₉	—	2	2	no	—
267	C₂×Q₈	—	9	4	no	—
268	C₂×Q₈	—	9	1	no	—
269	C₂×Q₈	—	18	1	no	—
270	C₂×Q₈	—	18	2	no	—
271	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
272	C₂²×C₄	—	9	1	no	—
273	C₂²×C₄	—	18	1	no	—
274	M₁₆	—	18	2	no	—
275	C₂×C₈	—	9	1	no	—
276	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
277	C₄⋊C₄	—	9	1	no	—
278	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
279	C₂²⋊C₄	—	18	1	no	—
280	C₄²	—	18	1	no	—
281	C₂²×C₃	—	3	1	no	—
282	C₂²×C₃	—	9	2	no	—
283	C₂²×C₃	—	18	1	no	—
284	C₂²×C₃	—	18	1	no	—
285	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
286	C₂×D₃	—	3	2	no	—
287	C₂×D₃	—	6	1	no	—
288	C₂×D₃	—	6	1	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	3	1	no	—
291	C₃×C₄	—	3	1	no	—
292	C₃×C₄	—	4	1	no	—
293	C₃×C₄	—	6	2	no	—
294	C₃×C₄	—	9	2	no	—
295	C₃×C₄	—	12	1	no	—
296	C₃×C₄	—	18	2	no	—
297	C₃×C₄	—	36	1	no	—
298	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
299	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
300	Dic₃	—	3	2	no	—
301	Dic₃	—	6	2	no	—
302	Dic₃	—	12	1	no	—
303	C₃²	#4802	1	1	yes	characteristic
304	C₉	—	2	1	no	—
305	C₉	C₃×#419	1	1	yes	characteristic, complemented
306	C₂³	—	9	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	9	2	no	—
309	Q₈	—	18	4	no	—
310	Q₈	—	18	1	no	—
311	Q₈	C₄×AGL(1,ℤ/9)	1	2	yes	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	9	1	no	—
315	C₂×C₄	—	9	2	no	—
316	C₂×C₄	—	9	2	no	—
317	C₂×C₄	—	18	4	no	—
318	C₂×C₄	—	18	1	no	—
319	C₂×C₄	—	18	1	no	—
320	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	18	1	no	—
323	C₂×C₃	—	3	1	no	—
324	C₂×C₃	—	3	1	no	—
325	C₂×C₃	—	3	1	no	—
326	C₂×C₃	—	18	2	no	—
327	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
328	C₂×C₃	C₃×#915	1	1	yes	characteristic
329	C₂×C₃	C₃×#918	1	1	yes	characteristic
330	D₃	—	6	2	no	—
331	C₂²	—	9	2	no	—
332	C₂²	—	18	1	no	—
333	C₂²	—	18	1	no	—
334	C₂²	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	9	2	no	—
338	C₄	—	18	2	no	—
339	C₄	—	36	1	no	—
340	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
341	C₄	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
342	C₃	—	3	1	no	—
343	C₃	C₃×#4802	1	1	yes	characteristic
344	C₂	—	18	2	no	—
345	C₂	C₂²⋊C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
346	C₂	#63477⋊C₃	1	1	yes	characteristic
347	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×#103	C₂
socle	C₂²×C₃	C₃×D₃×D₄
prime radical	C₂²×AGL(1,ℤ/9)	D₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	#103	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#419
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
4. 3₋¹⁺²×#103	C₂	semidirect
5. #63395⋊C₃	C₂	semidirect
6. #63376⋊C₃	C₂	semidirect
7. #63368⋊C₃	C₂	semidirect
9. #10588482	C₃	semidirect
12. Q₈×AGL(1,ℤ/9)	C₄	semidirect
35. C₉×#103	C₂×C₃	semidirect
36. #63395	C₂×C₃	semidirect
37. #63376	C₂×C₃	semidirect
38. #63368	C₂×C₃	semidirect
75. Q₈×D₉	C₃×C₄	semidirect
220. #103	AGL(1,ℤ/9)	semidirect
221. 3₋¹⁺²	#419	semidirect
305. C₉	C₃×#419	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×AGL(1,ℤ/9)	2
C₂×#3663⋊C₃	2
#63454⋊C₃	2
3₋¹⁺²×#103	2
#63395⋊C₃	2
#63376⋊C₃	2
#63368⋊C₃	2
C₃×#4802	3
#10588482	3

Subgroups — #10588482⋊C3 #13141947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588482⋊C₃ #13141947