Subgroups — #10588709⋊C₃ #13142809

Order 1728.

395 conjugacy classes of subgroups.

The subgroup lattice has 324 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	C₂×#3675⋊C₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
4	#63395⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#63369⋊C₃	C₂	1	1	yes	maximal, characteristic
6	#63366⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#63362⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5026	—	3	1	no	maximal
9	#10588709	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
12	C₂×C₄×AGL(1,ℤ/9)	—	2	1	no	—
13	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
14	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	C₄²×3₋¹⁺²	—	2	1	no	—
16	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
17	#3675⋊C₃	C₄	1	4	yes	complemented
18	(C₂×AGL(1,ℤ/9))⋊C₄	—	2	1	no	—
19	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
20	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
22	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
23	C₂×C₃×#215	—	3	1	no	—
24	C₃×C₄×D₁₂	—	3	1	no	—
25	C₃×#835	—	3	1	no	—
26	C₃×#809	—	3	1	no	—
27	C₃×#807	—	3	1	no	—
28	C₃×#803	—	3	1	no	—
29	C₄×Q₈×C₉	—	2	1	no	—
30	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#3675	C₂×C₃	1	1	yes	characteristic
32	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic
33	#63395	C₂×C₃	1	1	yes	characteristic, complemented
34	#63369	C₂×C₃	1	1	yes	characteristic
35	#63366	C₂×C₃	1	1	yes	characteristic, complemented
36	#63362	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×AGL(1,ℤ/9)	—	2	1	no	—
38	Q₈×3₋¹⁺²	—	2	1	no	—
39	Q₈×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
40	C₂×C₄×3₋¹⁺²	—	2	1	no	—
41	C₂×C₄×3₋¹⁺²	—	2	1	no	—
42	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
43	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
44	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
45	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
46	C₄⋊AGL(1,ℤ/9)	—	2	1	no	—
47	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	2	yes	—
48	C₄×AGL(1,ℤ/9)	—	4	2	no	—
49	3₋¹⁺²⋊C₈	—	2	1	no	—
50	3₋¹⁺²⋊C₈	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	9	1	no	—
52	#5026	—	3	1	no	—
53	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×D₁₂	—	3	1	no	—
55	C₂×C₃×C₄×D₃	—	6	1	no	—
56	C₃²×C₄⋊C₄	—	2	1	no	—
57	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄²	—	2	1	no	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃×#215	—	3	4	no	—
61	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
62	C₃×C₄⋊Dic₃	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₂×C₃×C₃⋊C₈	—	3	1	no	—
65	C₂×Q₈×C₉	—	2	1	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×D₉	—	2	1	no	—
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	2	1	no	—
72	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₄²×C₉	—	2	2	no	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	—	2	1	no	—
76	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
77	#3675	C₃×C₄	1	4	yes	complemented
78	(C₂×D₉)⋊C₄	—	2	1	no	—
79	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
83	C₂×AGL(1,ℤ/9)	—	2	2	no	—
84	C₂×AGL(1,ℤ/9)	—	4	1	no	—
85	C₂×AGL(1,ℤ/9)	—	4	1	no	—
86	C₄×3₋¹⁺²	—	2	2	no	—
87	C₄×3₋¹⁺²	—	2	1	no	—
88	C₄×3₋¹⁺²	—	2	1	no	—
89	C₄×3₋¹⁺²	—	4	1	no	—
90	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
91	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
92	3₋¹⁺²⋊C₄	—	4	1	no	—
93	C₂×C₃×SD₁₆	—	9	1	no	—
94	C₃×C₄×Q₈	—	3	1	no	—
95	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
96	C₃×C₄×D₄	—	9	1	no	—
97	C₂×#215	—	3	1	no	—
98	C₄×D₁₂	—	3	1	no	—
99	C₃×#107	—	9	1	no	—
100	C₃×#103	—	9	1	no	—
101	C₃×#102	—	9	1	no	—
102	C₃×#97	—	9	1	no	—
103	#835	—	3	1	no	—
104	#809	—	3	1	no	—
105	#807	—	3	1	no	—
106	#803	—	3	1	no	—
107	C₂²×C₃×D₃	—	6	1	no	—
108	C₃²×Q₈	—	2	1	no	—
109	C₃²×Q₈	C₄×D₃	1	2	yes	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	#67	1	1	yes	characteristic
113	C₂×C₃²×C₄	#67	1	1	yes	characteristic
114	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₃×Dic₃	—	6	1	no	—
116	C₃×D₁₂	—	3	2	no	—
117	C₃×D₁₂	—	6	1	no	—
118	C₃×C₄×D₃	—	12	2	no	—
119	C₂²×D₉	—	2	1	no	—
120	C₃×C₃⋊C₈	—	3	2	no	—
121	C₃×C₃⋊C₈	—	6	1	no	—
122	Q₈×C₉	—	2	2	no	—
123	Q₈×C₉	—	2	1	no	—
124	Q₈×C₉	—	2	2	no	—
125	Q₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
126	C₂×C₄×C₉	—	2	2	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	2	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	2	1	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	2	1	no	—
133	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
134	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
135	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
136	C₂×Dic₉	—	2	1	no	—
137	D₃₆	—	2	1	no	—
138	D₃₆	C₂×C₃×C₄	1	2	yes	—
139	C₄×D₉	—	4	2	no	—
140	C₉⋊C₈	—	2	1	no	—
141	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	complemented
142	#440	—	9	1	no	Sylow
143	C₂×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic
144	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
146	AGL(1,ℤ/9)	—	4	2	no	—
147	C₂×C₃×Q₈	—	3	1	no	—
148	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
149	C₂×C₃×D₄	—	9	1	no	—
150	C₂²×C₃×C₄	—	18	1	no	—
151	C₂×D₁₂	—	3	1	no	—
152	C₂×C₄×D₃	—	6	1	no	—
153	C₃×SD₁₆	—	9	4	no	—
154	C₂×C₃×C₈	—	9	1	no	—
155	C₂×C₃×C₈	—	9	1	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	3	1	no	—
158	C₃×C₄⋊C₄	—	6	1	no	—
159	C₃×C₄⋊C₄	—	9	1	no	—
160	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
161	C₃×C₂²⋊C₄	—	18	1	no	—
162	C₃×C₄²	—	2	1	no	—
163	C₃×C₄²	—	3	1	no	—
164	C₃×C₄²	—	6	1	no	—
165	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
166	#215	—	3	4	no	—
167	(C₂×D₃)⋊C₄	—	6	1	no	—
168	C₄⋊Dic₃	—	3	1	no	—
169	C₂×C₃⋊C₈	—	3	1	no	—
170	C₂×C₃⋊C₈	—	3	1	no	—
171	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
172	C₂×C₃×D₃	—	6	2	no	—
173	C₂×C₃×D₃	—	12	1	no	—
174	C₂×C₃×D₃	—	12	1	no	—
175	C₃²×C₄	—	2	2	no	—
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	—	2	1	no	—
178	C₃²×C₄	—	4	1	no	—
179	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₃²×C₄	#235	1	1	yes	characteristic
181	C₃×Dic₃	—	12	1	no	—
182	C₂²×C₉	—	2	1	no	—
183	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×D₉	—	2	2	no	—
185	C₂×D₉	—	4	1	no	—
186	C₂×D₉	—	4	1	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	2	4	no	—
189	C₄×C₉	—	2	2	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	—	4	1	no	—
195	C₄×C₉	—	4	2	no	—
196	C₄×C₉	—	4	1	no	—
197	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
198	C₄×C₉	C₃×#41	1	1	yes	characteristic
199	Dic₉	—	4	1	no	—
200	C₂×SD₁₆	—	9	1	no	—
201	C₄×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
202	C₄×D₄	—	9	1	no	—
203	#107	—	9	1	no	—
204	#103	—	9	1	no	—
205	#102	—	9	1	no	—
206	#97	—	9	1	no	—
207	3₋¹⁺²	#440	1	1	yes	characteristic, complemented, Sylow
208	C₂³×C₃	—	18	1	no	—
209	C₂²×D₃	—	6	1	no	—
210	C₃×Q₈	—	2	1	no	—
211	C₃×Q₈	—	3	2	no	—
212	C₃×Q₈	—	6	1	no	—
213	C₃×Q₈	C₃×C₄×D₃	1	2	yes	—
214	C₃×D₄	—	9	2	no	—
215	C₃×D₄	—	18	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	3	1	no	—
219	C₂×C₃×C₄	—	3	1	no	—
220	C₂×C₃×C₄	—	3	1	no	—
221	C₂×C₃×C₄	—	6	1	no	—
222	C₂×C₃×C₄	—	6	1	no	—
223	C₂×C₃×C₄	—	18	1	no	—
224	C₂×C₃×C₄	—	36	2	no	—
225	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
226	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
228	C₂×Dic₃	—	6	1	no	—
229	D₁₂	—	3	2	no	—
230	D₁₂	—	6	1	no	—
231	C₄×D₃	—	12	2	no	—
232	C₃×C₈	—	9	2	no	—
233	C₃×C₈	—	18	1	no	—
234	C₃⋊C₈	—	3	2	no	—
235	C₃⋊C₈	—	6	1	no	—
236	C₂×C₃²	C₄×#67	1	1	yes	characteristic
237	C₂×C₃²	#942	1	1	yes	characteristic
238	C₂×C₃²	#949	1	1	yes	characteristic
239	C₃×D₃	—	12	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
244	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂×C₉	C₃×#137	1	1	yes	characteristic
246	D₉	—	4	2	no	—
247	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
248	C₂×D₄	—	9	1	no	—
249	C₂²×C₄	—	18	1	no	—
250	SD₁₆	—	9	4	no	—
251	C₂×C₈	—	9	1	no	—
252	C₂×C₈	—	9	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	9	1	no	—
255	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
256	C₂²⋊C₄	—	18	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
259	C₂²×C₃	—	3	1	no	—
260	C₂²×C₃	—	18	2	no	—
261	C₂²×C₃	—	36	1	no	—
262	C₂²×C₃	—	36	1	no	—
263	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
264	C₂×D₃	—	6	2	no	—
265	C₂×D₃	—	12	1	no	—
266	C₂×D₃	—	12	1	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	3	1	no	—
271	C₃×C₄	—	3	1	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	—	6	1	no	—
275	C₃×C₄	—	6	2	no	—
276	C₃×C₄	—	12	1	no	—
277	C₃×C₄	—	36	1	no	—
278	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
279	C₃×C₄	C₃×#235	1	1	yes	characteristic
280	Dic₃	—	12	1	no	—
281	C₃²	#5026	1	1	yes	characteristic
282	C₉	—	2	1	no	—
283	C₉	C₃×#440	1	1	yes	characteristic, complemented
284	C₂³	—	18	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	C₄×AGL(1,ℤ/9)	1	2	yes	complemented
287	D₄	—	9	2	no	—
288	D₄	—	18	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	18	1	no	—
292	C₂×C₄	—	36	2	no	—
293	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
294	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
296	C₈	—	9	2	no	—
297	C₈	—	18	1	no	—
298	C₂×C₃	—	3	1	no	—
299	C₂×C₃	—	3	1	no	—
300	C₂×C₃	—	3	1	no	—
301	C₂×C₃	—	36	2	no	—
302	C₂×C₃	C₃×C₄×#67	1	1	yes	characteristic
303	C₂×C₃	C₃×#942	1	1	yes	characteristic
304	C₂×C₃	C₃×#949	1	1	yes	characteristic
305	D₃	—	12	2	no	—
306	C₂²	—	18	2	no	—
307	C₂²	—	36	1	no	—
308	C₂²	—	36	1	no	—
309	C₂²	C₂×#611⋊C₃	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	36	1	no	—
315	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
316	C₄	#3697⋊C₃	1	1	yes	characteristic
317	C₃	—	3	1	no	—
318	C₃	C₃×#5026	1	1	yes	characteristic
319	C₂	—	36	2	no	—
320	C₂	C₄×#611⋊C₃	1	1	yes	characteristic
321	C₂	#63505⋊C₃	1	1	yes	characteristic
322	C₂	#63512⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#611⋊C₃
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×Q₈×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×#67
prime radical	C₂×C₄⋊AGL(1,ℤ/9)	C₂²
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₄×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#440
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ 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
4. #63395⋊C₃	C₂	semidirect
6. #63366⋊C₃	C₂	semidirect
7. #63362⋊C₃	C₂	semidirect
9. #10588709	C₃	semidirect
17. #3675⋊C₃	C₄	semidirect
30. C₄×Q₈×C₉	C₂×C₃	semidirect
33. #63395	C₂×C₃	semidirect
35. #63366	C₂×C₃	semidirect
36. #63362	C₂×C₃	semidirect
39. Q₈×3₋¹⁺²	C₂×C₄	semidirect
50. 3₋¹⁺²⋊C₈	C₂×C₄	semidirect
77. #3675	C₃×C₄	semidirect
125. Q₈×C₉	C₂×C₃×C₄	semidirect
141. C₉⋊C₈	C₂×C₃×C₄	semidirect
201. C₄×Q₈	AGL(1,ℤ/9)	semidirect
207. 3₋¹⁺²	#440	semidirect
283. C₉	C₃×#440	semidirect
286. Q₈	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
C₄×Q₈×3₋¹⁺²	2
C₂×#3675⋊C₃	2
C₄×C₄⋊AGL(1,ℤ/9)	2
#63395⋊C₃	2
#63369⋊C₃	2
#63366⋊C₃	2
#63362⋊C₃	2
C₃×#5026	3
#10588709	3

Subgroups — #10588709⋊C3 #13142809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588709⋊C₃ #13142809