Subgroups — #10588504⋊C₃ #13141992

Order 1728.

340 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63513⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#11721693	C₂	1	1	yes	maximal, characteristic, complemented
3	#63469⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	#63381⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#63364⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4824	—	3	1	no	maximal
9	#10588504	C₃	1	1	yes	maximal, characteristic, complemented
10	#41×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	#3697⋊C₃	C₂²	1	1	yes	characteristic
12	M₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	#3675⋊C₃	C₄	1	1	yes	characteristic, complemented
15	#3674⋊C₃	C₄	1	1	yes	characteristic, complemented
16	#3673⋊C₃	C₄	1	1	yes	characteristic, complemented
17	#3672⋊C₃	C₄	1	1	yes	characteristic, complemented
18	#3667⋊C₃	—	2	1	no	—
19	#3667⋊C₃	C₂²	1	1	yes	characteristic
20	C₂×3₋¹⁺²⋊C₈	—	2	1	no	—
21	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
23	#3663⋊C₃	—	2	1	no	—
24	C₈×AGL(1,ℤ/9)	—	2	1	no	—
25	C₃×#950	—	3	1	no	—
26	C₃×#948	—	3	1	no	—
27	C₃×#907	—	3	1	no	—
28	C₃²×#104	D₃	1	1	yes	characteristic
29	C₃×#822	—	3	1	no	—
30	C₃×#805	—	3	1	no	—
31	C₃×C₄×C₃⋊C₈	—	3	1	no	—
32	#63513	C₂×C₃	1	1	yes	characteristic
33	#63510	C₂×C₃	1	1	yes	characteristic, complemented
34	#63469	C₂×C₃	1	1	yes	characteristic, complemented
35	C₉×#104	—	2	1	no	—
36	C₉×#104	C₂×C₃	1	1	yes	characteristic, complemented
37	#63381	C₂×C₃	1	1	yes	characteristic, complemented
38	#63364	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
40	Q₈×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
41	D₄×3₋¹⁺²	—	2	1	no	—
42	D₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×3₋¹⁺²	—	2	1	no	—
44	C₂×C₄×3₋¹⁺²	—	2	1	no	—
45	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
46	#611⋊C₃	—	2	1	no	—
47	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	1	yes	characteristic
48	C₄×AGL(1,ℤ/9)	—	2	1	no	—
49	3₋¹⁺²⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₈×3₋¹⁺²	—	2	1	no	—
51	3₋¹⁺²⋊C₈	—	2	1	no	—
52	3₋¹⁺²⋊C₈	C₂×C₄	1	1	yes	characteristic
53	3₋¹⁺²⋊C₈	C₂×C₄	1	1	yes	characteristic
54	3₋¹⁺²⋊C₈	D₄	1	1	yes	characteristic
55	3₋¹⁺²⋊C₈	D₄	1	1	yes	characteristic
56	C₃×#443	—	9	1	no	—
57	#4824	—	3	1	no	—
58	C₃²×#41	C₂×D₃	1	1	yes	characteristic
59	C₃×#235	—	3	1	no	—
60	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic
61	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₃×#216	—	3	1	no	—
63	C₃×#215	—	3	1	no	—
64	C₃×#214	—	3	1	no	—
65	C₃×#213	—	3	1	no	—
66	C₃×#208	—	3	1	no	—
67	C₃×#208	—	6	1	no	—
68	C₂×C₃×C₃⋊C₈	—	3	1	no	—
69	C₂×C₃×C₃⋊C₈	—	3	1	no	—
70	C₂×C₃×C₃⋊C₈	—	6	1	no	—
71	C₃×ΓL(1,𝔽₂₅)	—	6	1	no	—
72	C₃×D₃×C₈	—	6	1	no	—
73	C₉×#41	—	2	1	no	—
74	C₉×#41	C₂²×C₃	1	1	yes	characteristic
75	#3697	C₂²×C₃	1	1	yes	characteristic
76	C₉×M₁₆	—	2	1	no	—
77	C₉×M₁₆	C₂²×C₃	1	1	yes	characteristic
78	C₄²×C₉	—	2	1	no	—
79	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
80	#3675	C₃×C₄	1	1	yes	characteristic, complemented
81	#3674	C₃×C₄	1	1	yes	characteristic, complemented
82	#3673	C₃×C₄	1	1	yes	characteristic, complemented
83	#3672	C₃×C₄	1	1	yes	characteristic, complemented
84	#3667	—	2	1	no	—
85	#3667	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₉⋊C₈	—	2	1	no	—
87	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
89	#3663	—	2	1	no	—
90	C₈×D₉	—	2	1	no	—
91	C₂²×3₋¹⁺²	—	2	1	no	—
92	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
93	C₂×AGL(1,ℤ/9)	—	2	1	no	—
94	C₄×3₋¹⁺²	—	2	2	no	—
95	C₄×3₋¹⁺²	—	2	1	no	—
96	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
97	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
98	3₋¹⁺²⋊C₄	—	2	1	no	—
99	C₃×#135	—	9	1	no	—
100	C₃×#131	—	9	1	no	—
101	C₃×#131	—	9	1	no	—
102	#950	—	3	1	no	—
103	#948	—	3	1	no	—
104	#907	—	3	1	no	—
105	C₃×#108	—	9	1	no	—
106	C₃×#104	—	3	1	no	—
107	C₃×#104	—	9	1	no	—
108	C₃×#104	C₃×D₃	1	1	yes	characteristic
109	C₃×C₄×C₈	—	9	1	no	—
110	#822	—	3	1	no	—
111	#805	—	3	1	no	—
112	C₄×C₃⋊C₈	—	3	1	no	—
113	C₃²×Q₈	C₄×D₃	1	1	yes	characteristic
114	C₃²×D₄	—	2	1	no	—
115	C₃²×D₄	C₄×D₃	1	1	yes	characteristic
116	C₂×C₃²×C₄	—	2	1	no	—
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₃×#67	—	6	1	no	—
120	C₃×D₁₂	—	3	1	no	—
121	C₃×C₄×D₃	—	6	1	no	—
122	C₃×Dic₆	—	3	1	no	—
123	C₃²×C₈	—	2	1	no	—
124	C₃×C₃⋊C₈	—	3	1	no	—
125	C₃×C₃⋊C₈	—	3	1	no	—
126	C₃×C₃⋊C₈	—	3	1	no	—
127	C₃×C₃⋊C₈	—	3	1	no	—
128	C₃×C₃⋊C₈	—	6	1	no	—
129	Q₈×C₉	—	2	1	no	—
130	Q₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
131	D₄×C₉	—	2	1	no	—
132	D₄×C₉	—	2	1	no	—
133	D₄×C₉	—	4	1	no	—
134	D₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂×C₄×C₉	—	2	1	no	—
136	C₂×C₄×C₉	—	2	1	no	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	—	4	1	no	—
139	C₂×C₄×C₉	—	4	1	no	—
140	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
141	#611	—	2	1	no	—
142	D₃₆	C₂×C₃×C₄	1	1	yes	characteristic
143	C₄×D₉	—	2	1	no	—
144	Dic₁₈	C₂×C₃×C₄	1	1	yes	characteristic
145	C₈×C₉	—	2	1	no	—
146	C₈×C₉	—	4	1	no	—
147	C₉⋊C₈	—	2	1	no	—
148	C₉⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
149	C₉⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
150	C₉⋊C₈	C₃×D₄	1	1	yes	characteristic
151	C₉⋊C₈	C₃×D₄	1	1	yes	characteristic
152	#443	—	9	1	no	Sylow
153	C₂×3₋¹⁺²	—	2	1	no	—
154	C₂×3₋¹⁺²	—	4	1	no	—
155	C₂×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic
156	AGL(1,ℤ/9)	—	4	1	no	—
157	C₃×#41	—	3	1	no	—
158	C₃×#41	—	9	1	no	—
159	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
160	#235	—	3	1	no	—
161	C₃×Q₁₆	—	9	1	no	—
162	C₃×SD₁₆	—	9	1	no	—
163	C₃×SD₁₆	—	9	1	no	—
164	C₃×D₈	—	9	1	no	—
165	C₃×M₁₆	—	3	1	no	—
166	C₃×M₁₆	—	9	1	no	—
167	C₃×M₁₆	—	18	1	no	—
168	C₃×M₁₆	—	18	1	no	—
169	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
170	C₂×C₃×C₈	—	9	1	no	—
171	C₂×C₃×C₈	—	9	1	no	—
172	C₂×C₃×C₈	—	18	1	no	—
173	C₂×C₃×C₈	—	18	1	no	—
174	C₃×C₄²	—	3	1	no	—
175	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
176	#216	—	3	1	no	—
177	#215	—	3	1	no	—
178	#214	—	3	1	no	—
179	#213	—	3	1	no	—
180	#208	—	3	1	no	—
181	#208	—	6	1	no	—
182	C₂×C₃⋊C₈	—	3	1	no	—
183	C₂×C₃⋊C₈	—	3	1	no	—
184	C₂×C₃⋊C₈	—	6	1	no	—
185	ΓL(1,𝔽₂₅)	—	6	1	no	—
186	D₃×C₈	—	6	1	no	—
187	C₂²×C₃²	—	2	1	no	—
188	C₂²×C₃²	#237	1	1	yes	characteristic
189	C₂×C₃×D₃	—	6	1	no	—
190	C₃²×C₄	—	2	2	no	—
191	C₃²×C₄	—	2	1	no	—
192	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
194	C₃×Dic₃	—	6	1	no	—
195	C₂²×C₉	—	2	1	no	—
196	C₂²×C₉	—	2	1	no	—
197	C₂²×C₉	—	4	1	no	—
198	C₂²×C₉	C₃×#41	1	1	yes	characteristic
199	C₂×D₉	—	2	1	no	—
200	C₄×C₉	—	2	2	no	—
201	C₄×C₉	—	2	1	no	—
202	C₄×C₉	—	2	1	no	—
203	C₄×C₉	—	2	1	no	—
204	C₄×C₉	—	4	1	no	—
205	C₄×C₉	—	4	2	no	—
206	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
207	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
208	Dic₉	—	2	1	no	—
209	#135	—	9	1	no	—
210	#131	—	9	1	no	—
211	#131	—	9	1	no	—
212	#108	—	9	1	no	—
213	#104	—	9	1	no	—
214	#104	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
215	C₄×C₈	—	9	1	no	—
216	3₋¹⁺²	#443	1	1	yes	characteristic, complemented, Sylow
217	C₃×Q₈	—	3	1	no	—
218	C₃×Q₈	—	9	1	no	—
219	C₃×Q₈	C₃×C₄×D₃	1	1	yes	characteristic
220	C₃×D₄	—	2	1	no	—
221	C₃×D₄	—	3	1	no	—
222	C₃×D₄	—	6	1	no	—
223	C₃×D₄	—	9	1	no	—
224	C₃×D₄	—	18	1	no	—
225	C₃×D₄	C₃×C₄×D₃	1	1	yes	characteristic
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	3	1	no	—
229	C₂×C₃×C₄	—	6	1	no	—
230	C₂×C₃×C₄	—	6	1	no	—
231	C₂×C₃×C₄	—	18	1	no	—
232	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
233	#67	—	6	1	no	—
234	D₁₂	—	3	1	no	—
235	C₄×D₃	—	6	1	no	—
236	Dic₆	—	3	1	no	—
237	C₃×C₈	—	2	1	no	—
238	C₃×C₈	—	6	1	no	—
239	C₃×C₈	—	9	1	no	—
240	C₃×C₈	—	9	1	no	—
241	C₃×C₈	—	9	1	no	—
242	C₃×C₈	—	9	1	no	—
243	C₃×C₈	—	18	1	no	—
244	C₃⋊C₈	—	3	1	no	—
245	C₃⋊C₈	—	3	1	no	—
246	C₃⋊C₈	—	3	1	no	—
247	C₃⋊C₈	—	3	1	no	—
248	C₃⋊C₈	—	6	1	no	—
249	C₂×C₃²	—	2	1	no	—
250	C₂×C₃²	—	4	1	no	—
251	C₂×C₃²	#67⋊C₄	1	1	yes	characteristic
252	C₃×D₃	—	12	1	no	—
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	4	1	no	—
256	C₂×C₉	—	4	1	no	—
257	C₂×C₉	—	8	1	no	—
258	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
259	D₉	—	4	1	no	—
260	#41	—	9	1	no	—
261	#41	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
262	Q₁₆	—	9	1	no	—
263	SD₁₆	—	9	1	no	—
264	SD₁₆	—	9	1	no	—
265	D₈	—	9	1	no	—
266	M₁₆	—	9	1	no	—
267	M₁₆	—	18	1	no	—
268	M₁₆	—	18	1	no	—
269	M₁₆	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
270	C₂×C₈	—	9	1	no	—
271	C₂×C₈	—	9	1	no	—
272	C₂×C₈	—	18	1	no	—
273	C₂×C₈	—	18	1	no	—
274	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	3	1	no	—
277	C₂²×C₃	—	6	1	no	—
278	C₂²×C₃	—	18	1	no	—
279	C₂²×C₃	C₃×#237	1	1	yes	characteristic
280	C₂×D₃	—	6	1	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	3	1	no	—
284	C₃×C₄	—	3	1	no	—
285	C₃×C₄	—	6	1	no	—
286	C₃×C₄	—	6	2	no	—
287	C₃×C₄	—	18	1	no	—
288	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
289	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
290	Dic₃	—	6	1	no	—
291	C₃²	#4824	1	1	yes	characteristic
292	C₉	—	2	1	no	—
293	C₉	C₃×#443	1	1	yes	characteristic, complemented
294	Q₈	—	9	1	no	—
295	Q₈	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
296	D₄	—	2	1	no	—
297	D₄	—	9	1	no	—
298	D₄	—	18	1	no	—
299	D₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	18	1	no	—
303	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	9	1	no	—
306	C₈	—	9	1	no	—
307	C₈	—	9	1	no	—
308	C₈	—	9	1	no	—
309	C₈	—	18	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	3	1	no	—
312	C₂×C₃	—	4	1	no	—
313	C₂×C₃	—	6	1	no	—
314	C₂×C₃	—	12	1	no	—
315	C₂×C₃	—	36	1	no	—
316	C₂×C₃	C₃×#67⋊C₄	1	1	yes	characteristic
317	D₃	—	12	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	18	1	no	—
320	C₂²	#3699⋊C₃	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	1	no	—
323	C₄	—	18	1	no	—
324	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
325	C₄	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
326	C₃	—	3	1	no	—
327	C₃	C₃×#4824	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	4	1	no	—
330	C₂	—	36	1	no	—
331	C₂	(#611⋊C₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×AGL(1,ℤ/9)
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#104	C₂
socle	C₂×C₃	C₃×#67⋊C₄
prime radical	#63513⋊C₃	C₂
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	#104	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#443
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
2. #11721693	C₂	semidirect
3. #63469⋊C₃	C₂	semidirect
4. 3₋¹⁺²×#104	C₂	semidirect
5. #63381⋊C₃	C₂	semidirect
6. #63364⋊C₃	C₂	semidirect
7. C₄×3₋¹⁺²⋊C₈	C₂	semidirect
9. #10588504	C₃	semidirect
14. #3675⋊C₃	C₄	semidirect
15. #3674⋊C₃	C₄	semidirect
16. #3673⋊C₃	C₄	semidirect
17. #3672⋊C₃	C₄	semidirect
33. #63510	C₂×C₃	semidirect
34. #63469	C₂×C₃	semidirect
36. C₉×#104	C₂×C₃	semidirect
37. #63381	C₂×C₃	semidirect
38. #63364	C₂×C₃	semidirect
39. C₄×C₉⋊C₈	C₂×C₃	semidirect
80. #3675	C₃×C₄	semidirect
81. #3674	C₃×C₄	semidirect
82. #3673	C₃×C₄	semidirect
83. #3672	C₃×C₄	semidirect
214. #104	AGL(1,ℤ/9)	semidirect
216. 3₋¹⁺²	#443	semidirect
293. C₉	C₃×#443	semidirect

Maximal subgroups

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

subgroup	index
#63513⋊C₃	2
#11721693	2
#63469⋊C₃	2
3₋¹⁺²×#104	2
#63381⋊C₃	2
#63364⋊C₃	2
C₄×3₋¹⁺²⋊C₈	2
C₃×#4824	3
#10588504	3

Subgroups — #10588504⋊C3 #13141992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588504⋊C₃ #13141992