Subgroups — #13138422

Order 1728.

390 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	D₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	#11720906	C₂	1	1	yes	maximal, characteristic
4	#11720541	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63377	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720279	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720278	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590681	—	3	1	no	maximal
9	#10588805	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×D₃×Dic₉	—	2	1	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₉×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
19	(C₂×D₃)⋊Dic₉	—	2	1	no	—
20	Dic₃⋊Dic₉	—	2	1	no	—
21	Dic₃×Dic₉	—	2	1	no	—
22	C₂×C₃×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic
23	D₃×C₄⋊Dic₃	—	3	1	no	—
24	#63841	—	3	1	no	—
25	#63645	—	3	1	no	—
26	C₃×#817	—	3	1	no	—
27	#63575	—	3	1	no	—
28	#63573	—	3	1	no	—
29	C₂×C₉×M₁₆	—	3	1	no	—
30	#63488	—	3	1	no	—
31	C₂×C₄⋊Dic₉	—	3	1	no	—
32	#63378	—	3	1	no	—
33	#63378	—	3	1	no	—
34	#63378	—	6	1	no	—
35	#63377	—	3	1	no	—
36	#63377	D₃	1	1	yes	characteristic, complemented
37	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊Dic₉	—	2	1	no	—
40	C₂×C₃⋊Dic₉	—	2	1	no	—
41	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
42	C₂×C₃×Dic₉	—	2	1	no	—
43	C₂×C₃×Dic₉	—	2	1	no	—
44	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
45	C₄×D₃×C₉	Q₈	1	2	yes	—
46	D₃×Dic₉	—	4	2	no	—
47	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
48	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
49	#5121	—	9	1	no	—
50	#4861	—	9	1	no	—
51	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
52	C₂×D₃×Dic₃	—	6	1	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
55	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
56	C₃×C₄⋊Dic₃	—	3	1	no	—
57	C₃×C₄⋊Dic₃	—	3	1	no	—
58	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×ΓL(1,𝔽₂₅)	Dic₃	1	4	yes	—
60	Dic₃⋊Dic₃	—	6	1	no	—
61	(C₂×D₃)⋊Dic₃	—	6	1	no	—
62	Dic₃²	—	6	1	no	—
63	C₂²×C₄×C₉	—	3	1	no	—
64	C₂²×Dic₉	—	6	1	no	—
65	C₉×M₁₆	—	3	4	no	—
66	C₂×C₈×C₉	—	2	1	no	—
67	C₂×C₈×C₉	—	3	1	no	—
68	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
69	#3676	—	6	1	no	—
70	C₄⋊Dic₉	—	3	1	no	—
71	C₄⋊Dic₉	—	3	1	no	—
72	C₄⋊Dic₉	—	6	1	no	—
73	C₄⋊Dic₉	—	6	1	no	—
74	C₄⋊Dic₉	—	6	1	no	—
75	C₄⋊Dic₉	C₂×D₃	1	1	yes	characteristic
76	C₄⋊Dic₉	C₂×D₃	1	1	yes	characteristic
77	C₄×Dic₉	—	6	1	no	—
78	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×D₃×C₉	—	2	1	no	—
80	C₂×D₃×C₉	—	2	1	no	—
81	C₂×D₃×C₉	C₄⋊C₄	1	2	yes	—
82	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
84	C₃⋊Dic₉	—	4	1	no	—
85	C₃⋊Dic₉	—	4	1	no	—
86	C₉×Dic₃	C₄⋊C₄	1	2	yes	—
87	C₃×Dic₉	—	4	1	no	—
88	C₃×Dic₉	—	4	1	no	—
89	C₂×C₃×M₁₆	—	3	1	no	—
90	#926	—	9	1	no	—
91	C₂×C₄⋊Dic₃	—	9	1	no	—
92	C₂×ΓL(1,𝔽₂₅)	D₉	1	1	yes	characteristic
93	#892	—	9	1	no	—
94	D₃×C₄⋊C₄	—	9	1	no	—
95	C₃×#106	—	9	1	no	—
96	#818	—	9	1	no	—
97	#818	—	9	1	no	—
98	#818	—	9	1	no	—
99	#818	—	18	1	no	—
100	#817	—	3	1	no	—
101	#817	—	9	1	no	—
102	#808	—	9	1	no	—
103	#807	—	9	1	no	—
104	C₂²×C₃×D₃	D₁₂	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃⋊Dic₃	—	6	1	no	—
107	C₂×C₃⋊Dic₃	—	6	1	no	—
108	C₂×C₃×Dic₃	—	6	1	no	—
109	C₂×C₃×Dic₃	—	6	1	no	—
110	C₂×C₃×Dic₃	D₁₂	1	1	yes	characteristic
111	C₃×C₄×D₃	Dic₆	1	2	yes	—
112	C₃×C₄×D₃	C₂×Dic₃	1	2	yes	—
113	D₃×Dic₃	—	12	2	no	—
114	C₂³×C₉	—	3	1	no	—
115	C₃²×C₈	C₂×Dic₃	1	2	yes	—
116	C₃×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	3	1	no	—
119	C₂×C₄×C₉	—	3	2	no	—
120	C₂×C₄×C₉	—	3	2	no	—
121	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
122	C₂×Dic₉	—	2	1	no	—
123	C₂×Dic₉	—	2	1	no	—
124	C₂×Dic₉	—	6	1	no	—
125	C₂×Dic₉	—	6	1	no	—
126	C₂×Dic₉	—	12	1	no	—
127	C₂×Dic₉	—	12	1	no	—
128	C₂×Dic₉	—	12	2	no	—
129	C₈×C₉	—	2	2	no	—
130	C₈×C₉	—	3	2	no	—
131	C₈×C₉	C₄×D₃	1	2	yes	complemented
132	#428	—	27	1	no	Sylow
133	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₃×C₉	#137	1	1	yes	characteristic
136	D₃×C₉	—	2	2	no	—
137	C₂²×C₃×C₄	—	3	1	no	—
138	C₂²×Dic₃	—	18	1	no	—
139	C₂×C₄×D₃	—	18	1	no	—
140	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
141	C₃×M₁₆	—	3	4	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	3	1	no	—
144	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
145	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	9	1	no	—
147	C₃×C₄⋊C₄	—	9	1	no	—
148	#217	—	18	1	no	—
149	(C₂×D₃)⋊C₄	—	18	1	no	—
150	C₄⋊Dic₃	—	3	1	no	—
151	C₄⋊Dic₃	—	3	1	no	—
152	C₄⋊Dic₃	—	9	1	no	—
153	C₄⋊Dic₃	—	9	1	no	—
154	C₄⋊Dic₃	—	9	1	no	—
155	C₄⋊Dic₃	—	9	1	no	—
156	C₄⋊Dic₃	—	18	1	no	—
157	C₄⋊Dic₃	—	18	1	no	—
158	C₄⋊Dic₃	—	18	1	no	—
159	Dic₃⋊C₄	—	18	1	no	—
160	C₄×Dic₃	—	18	1	no	—
161	C₄×Dic₃	—	18	1	no	—
162	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
163	ΓL(1,𝔽₂₅)	Dic₉	1	4	yes	complemented
164	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
165	C₂×C₃×D₃	—	2	1	no	—
166	C₂×C₃×D₃	—	2	1	no	—
167	C₂×C₃×D₃	C₄⋊Dic₃	1	2	yes	—
168	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
169	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
170	C₃⋊Dic₃	—	12	1	no	—
171	C₃⋊Dic₃	—	12	1	no	—
172	C₃×Dic₃	—	12	1	no	—
173	C₃×Dic₃	—	12	1	no	—
174	C₃×Dic₃	C₄⋊Dic₃	1	2	yes	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	3	2	no	—
177	C₂²×C₉	—	6	1	no	—
178	C₂²×C₉	—	6	1	no	—
179	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	2	1	no	—
182	C₄×C₉	—	3	2	no	—
183	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
184	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
185	Dic₉	—	4	1	no	—
186	Dic₉	—	4	1	no	—
187	Dic₉	—	12	1	no	—
188	Dic₉	—	12	1	no	—
189	Dic₉	—	24	1	no	—
190	Dic₉	—	24	1	no	—
191	C₂×M₁₆	—	3	1	no	—
192	#117	—	27	1	no	—
193	C₂×C₄⋊C₄	—	27	1	no	—
194	#107	—	27	1	no	—
195	#107	—	27	1	no	—
196	#106	—	9	1	no	—
197	#106	—	27	1	no	—
198	C₃×C₉	#428	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	3	1	no	—
200	C₂²×D₃	D₃₆	1	1	yes	characteristic
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	3	1	no	—
203	C₂×C₃×C₄	—	3	2	no	—
204	C₂×C₃×C₄	—	3	2	no	—
205	C₂×C₃×C₄	—	18	1	no	—
206	C₂×C₃×C₄	—	18	1	no	—
207	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
209	C₂×Dic₃	—	6	1	no	—
210	C₂×Dic₃	—	6	1	no	—
211	C₂×Dic₃	—	18	1	no	—
212	C₂×Dic₃	—	18	1	no	—
213	C₂×Dic₃	—	18	1	no	—
214	C₂×Dic₃	—	18	1	no	—
215	C₂×Dic₃	—	36	1	no	—
216	C₂×Dic₃	—	36	1	no	—
217	C₂×Dic₃	—	36	2	no	—
218	C₂×Dic₃	D₃₆	1	1	yes	characteristic
219	C₄×D₃	—	36	2	no	—
220	C₄×D₃	Dic₁₈	1	2	yes	—
221	C₄×D₃	C₂×Dic₉	1	2	yes	—
222	C₃×C₈	—	2	2	no	—
223	C₃×C₈	—	3	2	no	—
224	C₃×C₈	C₂×Dic₉	1	2	yes	complemented
225	C₃×C₈	D₃×Dic₃	1	2	yes	—
226	C₃⋊C₈	C₂×Dic₉	1	2	yes	complemented
227	C₂×C₃²	#908	1	1	yes	characteristic
228	C₂×C₃²	#909	1	1	yes	characteristic
229	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
230	C₃×D₃	—	2	2	no	—
231	C₂×C₉	—	2	1	no	—
232	C₂×C₉	—	2	1	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	6	2	no	—
235	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₉	#914	1	1	yes	characteristic
237	C₂×C₉	#915	1	1	yes	characteristic
238	C₂²×C₄	—	3	1	no	—
239	C₂²×C₄	—	54	1	no	—
240	M₁₆	—	3	4	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	D₃×D₉	1	1	yes	characteristic
243	C₄⋊C₄	—	9	1	no	—
244	C₄⋊C₄	—	9	1	no	—
245	C₄⋊C₄	—	27	1	no	—
246	C₄⋊C₄	—	27	1	no	—
247	C₄⋊C₄	—	54	1	no	—
248	C₂²⋊C₄	—	54	1	no	—
249	C₄²	—	54	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	3	2	no	—
252	C₂²×C₃	—	6	1	no	—
253	C₂²×C₃	—	6	1	no	—
254	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
255	C₂²×C₃	D₃×D₁₂	1	1	yes	characteristic
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	2	1	no	—
258	C₂×D₃	C₄⋊Dic₉	1	2	yes	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	3	2	no	—
262	C₃×C₄	—	36	1	no	—
263	C₃×C₄	—	36	1	no	—
264	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
265	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
266	C₃×C₄	D₃×Dic₆	1	1	yes	characteristic
267	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
268	Dic₃	—	12	1	no	—
269	Dic₃	—	12	1	no	—
270	Dic₃	—	36	1	no	—
271	Dic₃	—	36	1	no	—
272	Dic₃	—	36	1	no	—
273	Dic₃	—	36	1	no	—
274	Dic₃	—	72	1	no	—
275	Dic₃	—	72	1	no	—
276	Dic₃	C₄⋊Dic₉	1	2	yes	—
277	C₃²	#5121	1	1	yes	characteristic
278	C₉	—	2	1	no	—
279	C₉	#4861	1	1	yes	characteristic, complemented
280	C₂³	—	3	1	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	2	no	—
283	C₂×C₄	—	3	2	no	—
284	C₂×C₄	—	18	1	no	—
285	C₂×C₄	—	18	1	no	—
286	C₂×C₄	—	54	1	no	—
287	C₂×C₄	—	54	1	no	—
288	C₂×C₄	—	108	2	no	—
289	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
290	C₈	—	3	2	no	—
291	C₈	D₃×Dic₉	1	2	yes	complemented
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	6	2	no	—
296	C₂×C₃	#63470	1	1	yes	characteristic
297	C₂×C₃	#63471	1	1	yes	characteristic
298	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
299	C₂×C₃	#63794	1	1	yes	characteristic
300	C₂×C₃	#63800	1	1	yes	characteristic
301	C₂×C₃	D₃×C₄⋊Dic₃	1	1	yes	characteristic
302	D₃	—	2	2	no	—
303	C₂²	—	3	2	no	—
304	C₂²	—	6	1	no	—
305	C₂²	—	6	1	no	—
306	C₂²	D₃×D₃₆	1	1	yes	characteristic
307	C₄	—	3	2	no	—
308	C₄	—	36	1	no	—
309	C₄	—	36	1	no	—
310	C₄	—	108	1	no	—
311	C₄	—	108	1	no	—
312	C₄	D₃×Dic₁₈	1	1	yes	characteristic
313	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
314	C₃	—	2	1	no	—
315	C₃	#10588805	1	1	yes	characteristic, complemented
316	C₃	#10590681	1	1	yes	characteristic
317	C₂	—	6	2	no	—
318	C₂	#11720796	1	1	yes	characteristic
319	C₂	#11720804	1	1	yes	characteristic
320	C₂	D₃×C₄⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₃₆
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×D₁₂
prime radical	C₂²×C₃×D₃	D₁₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	#428
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #11720541	C₂	semidirect
5. C₃×#63377	C₂	semidirect
6. #11720279	C₂	semidirect
7. #11720278	C₂	semidirect
18. C₉×ΓL(1,𝔽₂₅)	C₄	semidirect
36. #63377	D₃	semidirect
47. C₃×C₈×C₉	C₂×C₄	semidirect
48. C₉×C₃⋊C₈	C₂×C₄	semidirect
131. C₈×C₉	C₄×D₃	semidirect
163. ΓL(1,𝔽₂₅)	Dic₉	semidirect
198. C₃×C₉	#428	semidirect
224. C₃×C₈	C₂×Dic₉	semidirect
226. C₃⋊C₈	C₂×Dic₉	semidirect
279. C₉	#4861	semidirect
291. C₈	D₃×Dic₉	semidirect
315. C₃	#10588805	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×ΓL(1,𝔽₂₅)	2
D₃×C₄⋊Dic₉	2
#11720906	2
#11720541	2
C₃×#63377	2
#11720279	2
#11720278	2
#10590681	3
#10588805	3