Subgroups — #13138410

Order 1728.

470 conjugacy classes of subgroups.

The subgroup lattice has 288 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	C₄×D₃×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#88420	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63375	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720274	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720271	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590667	—	3	1	no	maximal
9	M₁₆×Dic₉	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×D₃×Dic₉	C₄	1	2	yes	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₉×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
19	Dic₃×Dic₉	C₄	1	2	yes	—
20	#88420	C₄	1	4	yes	complemented
21	C₂×C₃×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic
22	C₄×D₃×Dic₃	—	3	1	no	—
23	C₂×#3711	—	3	1	no	—
24	C₈×C₃⋊Dic₃	—	3	1	no	—
25	C₃×#815	—	3	1	no	—
26	#63554	—	3	1	no	—
27	#63553	—	3	1	no	—
28	C₂×C₉×M₁₆	—	3	1	no	—
29	C₂×C₄×Dic₉	—	3	1	no	—
30	C₂×#3667	—	3	1	no	—
31	#63375	—	3	1	no	—
32	#63375	D₃	1	1	yes	characteristic, complemented
33	C₈×Dic₉	—	3	1	no	—
34	C₈×Dic₉	—	3	1	no	—
35	C₈×Dic₉	—	6	1	no	—
36	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
39	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
41	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
42	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
43	D₃×Dic₉	—	2	4	no	—
44	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
45	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	complemented
46	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
47	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	complemented
48	Dic₃×M₁₆	—	9	1	no	—
49	#4707	—	9	1	no	—
50	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
51	C₂×D₃×Dic₃	—	3	2	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₄×C₃⋊Dic₃	—	3	1	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
55	C₃×C₄×Dic₃	—	3	1	no	—
56	C₂×C₃×C₃⋊C₈	—	3	1	no	—
57	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×ΓL(1,𝔽₂₅)	Dic₃	1	4	yes	—
59	Dic₃²	—	3	2	no	—
60	#3711	—	3	4	no	—
61	C₂²×C₄×C₉	—	3	1	no	—
62	C₂²×Dic₉	—	3	2	no	—
63	C₉×M₁₆	—	3	4	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	—	3	1	no	—
66	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
67	C₄×Dic₉	—	3	1	no	—
68	C₄×Dic₉	—	3	2	no	—
69	C₄×Dic₉	—	6	1	no	—
70	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
71	#3667	—	3	4	no	—
72	C₂×C₉⋊C₈	—	3	1	no	—
73	C₂×C₉⋊C₈	—	6	1	no	—
74	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
75	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×D₃×C₉	—	2	2	no	—
77	C₂×D₃×C₉	C₄²	1	2	yes	—
78	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₃⋊Dic₉	M₁₆	1	4	yes	complemented
81	C₉×Dic₃	C₄²	1	2	yes	—
82	C₃×Dic₉	—	2	2	no	—
83	C₂×C₃×M₁₆	—	3	1	no	—
84	C₂×C₄×Dic₃	—	9	1	no	—
85	C₂×#208	—	9	1	no	—
86	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
87	C₂×ΓL(1,𝔽₂₅)	D₉	1	1	yes	characteristic
88	C₄²×D₃	—	9	1	no	—
89	C₃×#97	—	9	1	no	—
90	#815	—	3	1	no	—
91	#815	—	9	1	no	—
92	C₈×Dic₃	—	9	1	no	—
93	C₈×Dic₃	—	9	1	no	—
94	C₈×Dic₃	—	9	1	no	—
95	C₈×Dic₃	—	9	1	no	—
96	C₈×Dic₃	—	18	1	no	—
97	#803	—	9	1	no	—
98	C₂²×C₃×D₃	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	—	3	2	no	—
101	C₂×C₃×Dic₃	—	3	2	no	—
102	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
103	C₃×C₄×D₃	C₄×D₃	1	2	yes	—
104	C₃×C₄×D₃	C₂×Dic₃	1	2	yes	—
105	D₃×Dic₃	—	6	4	no	—
106	C₂³×C₉	—	3	1	no	—
107	C₃²×C₈	C₂×Dic₃	1	2	yes	—
108	C₃⋊(C₃⋊C₈)	—	3	2	no	—
109	C₃×C₃⋊C₈	—	3	2	no	—
110	C₃×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	3	1	no	—
113	C₂×C₄×C₉	—	3	2	no	—
114	C₂×C₄×C₉	—	3	2	no	—
115	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
116	C₂×Dic₉	—	3	2	no	—
117	C₂×Dic₉	—	6	2	no	—
118	C₂×Dic₉	—	6	4	no	—
119	C₂×Dic₉	C₄×D₃	1	2	yes	—
120	C₈×C₉	—	2	2	no	—
121	C₈×C₉	—	3	2	no	—
122	C₈×C₉	C₄×D₃	1	2	yes	complemented
123	C₉⋊C₈	—	3	2	no	—
124	C₉⋊C₈	—	6	2	no	—
125	C₉⋊C₈	C₄×D₃	1	2	yes	complemented
126	C₄×M₁₆	—	27	1	no	Sylow
127	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
129	D₃×C₉	—	2	2	no	—
130	C₂²×C₃×C₄	—	3	1	no	—
131	C₂²×Dic₃	—	9	2	no	—
132	C₂×C₄×D₃	—	9	2	no	—
133	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
134	C₃×M₁₆	—	3	4	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	3	1	no	—
137	C₂×C₃×C₈	—	9	1	no	—
138	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
139	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
140	C₃×C₄²	—	9	1	no	—
141	C₄×Dic₃	—	3	1	no	—
142	C₄×Dic₃	—	9	2	no	—
143	C₄×Dic₃	—	9	1	no	—
144	C₄×Dic₃	—	9	1	no	—
145	C₄×Dic₃	—	9	2	no	—
146	C₄×Dic₃	—	18	1	no	—
147	#208	—	9	4	no	—
148	C₂×C₃⋊C₈	—	3	1	no	—
149	C₂×C₃⋊C₈	—	9	1	no	—
150	C₂×C₃⋊C₈	—	9	1	no	—
151	C₂×C₃⋊C₈	—	18	1	no	—
152	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
153	ΓL(1,𝔽₂₅)	—	9	4	no	—
154	ΓL(1,𝔽₂₅)	Dic₉	1	4	yes	complemented
155	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₃×D₃	—	2	2	no	—
157	C₂×C₃×D₃	C₄×Dic₃	1	2	yes	—
158	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
159	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
160	C₃⋊Dic₃	—	3	4	no	—
161	C₃×Dic₃	—	6	2	no	—
162	C₃×Dic₃	C₄×Dic₃	1	2	yes	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	3	2	no	—
165	C₂²×C₉	—	6	2	no	—
166	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	3	2	no	—
170	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
171	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
172	Dic₉	—	2	2	no	—
173	Dic₉	—	3	4	no	—
174	Dic₉	—	6	4	no	—
175	C₂×M₁₆	—	3	1	no	—
176	C₂×M₁₆	—	27	1	no	—
177	C₂×C₄²	—	27	1	no	—
178	#97	—	9	1	no	—
179	#97	—	27	1	no	—
180	C₄×C₈	—	27	1	no	—
181	C₄×C₈	—	27	1	no	—
182	C₃×C₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₃	—	3	1	no	—
184	C₂²×D₃	C₄×D₉	1	1	yes	characteristic
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	3	1	no	—
187	C₂×C₃×C₄	—	3	2	no	—
188	C₂×C₃×C₄	—	3	2	no	—
189	C₂×C₃×C₄	—	9	2	no	—
190	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
192	C₂×Dic₃	—	3	2	no	—
193	C₂×Dic₃	—	9	2	no	—
194	C₂×Dic₃	—	9	2	no	—
195	C₂×Dic₃	—	18	2	no	—
196	C₂×Dic₃	—	18	4	no	—
197	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
198	C₄×D₃	—	18	4	no	—
199	C₄×D₃	C₄×D₉	1	2	yes	—
200	C₄×D₃	C₂×Dic₉	1	2	yes	—
201	C₃×C₈	—	2	2	no	—
202	C₃×C₈	—	3	2	no	—
203	C₃×C₈	—	9	2	no	—
204	C₃×C₈	C₂×Dic₉	1	2	yes	complemented
205	C₃×C₈	D₃×Dic₃	1	2	yes	—
206	C₃⋊C₈	—	3	2	no	—
207	C₃⋊C₈	—	9	2	no	—
208	C₃⋊C₈	—	9	2	no	—
209	C₃⋊C₈	—	18	2	no	—
210	C₃⋊C₈	C₂×Dic₉	1	2	yes	complemented
211	C₂×C₃²	D₃×M₁₆	1	2	yes	—
212	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
213	C₃×D₃	—	2	2	no	—
214	C₂×C₉	—	2	2	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	6	2	no	—
217	C₂×C₉	C₄²×D₃	1	1	yes	characteristic
218	C₂×C₉	D₃×M₁₆	1	2	yes	—
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	27	2	no	—
221	M₁₆	—	3	4	no	—
222	M₁₆	—	27	4	no	—
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	9	1	no	—
225	C₂×C₈	—	27	1	no	—
226	C₂×C₈	D₃×D₉	1	1	yes	characteristic
227	C₄²	—	9	1	no	—
228	C₄²	—	27	2	no	—
229	C₄²	—	27	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	3	2	no	—
232	C₂²×C₃	—	6	2	no	—
233	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
234	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
235	C₂×D₃	—	2	2	no	—
236	C₂×D₃	C₄×Dic₉	1	2	yes	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	3	2	no	—
240	C₃×C₄	—	18	2	no	—
241	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
242	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
243	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
244	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
245	Dic₃	—	6	2	no	—
246	Dic₃	—	9	4	no	—
247	Dic₃	—	9	4	no	—
248	Dic₃	—	18	4	no	—
249	Dic₃	C₄×Dic₉	1	2	yes	—
250	C₃²	Dic₃×M₁₆	1	1	yes	characteristic
251	C₉	—	2	1	no	—
252	C₉	#4707	1	1	yes	characteristic, complemented
253	C₂³	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	2	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	9	2	no	—
258	C₂×C₄	—	27	2	no	—
259	C₂×C₄	—	54	4	no	—
260	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
261	C₈	—	3	2	no	—
262	C₈	—	9	2	no	—
263	C₈	—	27	2	no	—
264	C₈	D₃×Dic₉	1	2	yes	complemented
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	—	6	2	no	—
268	C₂×C₃	M₁₆×D₉	1	2	yes	—
269	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
270	C₂×C₃	#63791	1	2	yes	—
271	C₂×C₃	C₄×D₃×Dic₃	1	1	yes	characteristic
272	D₃	—	2	2	no	—
273	C₂²	—	3	2	no	—
274	C₂²	—	6	2	no	—
275	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
276	C₄	—	3	2	no	—
277	C₄	—	18	2	no	—
278	C₄	—	27	4	no	—
279	C₄	C₄×D₃×D₉	1	1	yes	characteristic
280	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
281	C₃	—	2	1	no	—
282	C₃	M₁₆×Dic₉	1	1	yes	characteristic, complemented
283	C₃	#10590667	1	1	yes	characteristic
284	C₂	—	6	2	no	—
285	C₂	#11720801	1	2	yes	—
286	C₂	C₄×D₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₃	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	C₄×M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₃⋊Dic₉	C₂	semidirect
5. C₃×#63375	C₂	semidirect
6. #11720274	C₂	semidirect
7. #11720271	C₂	semidirect
18. C₉×ΓL(1,𝔽₂₅)	C₄	semidirect
20. #88420	C₄	semidirect
32. #63375	D₃	semidirect
44. C₃×C₈×C₉	C₂×C₄	semidirect
45. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
46. C₉×C₃⋊C₈	C₂×C₄	semidirect
47. C₃×C₉⋊C₈	C₂×C₄	semidirect
80. C₃⋊Dic₉	M₁₆	semidirect
122. C₈×C₉	C₄×D₃	semidirect
125. C₉⋊C₈	C₄×D₃	semidirect
154. ΓL(1,𝔽₂₅)	Dic₉	semidirect
182. C₃×C₉	C₄×M₁₆	semidirect
204. C₃×C₈	C₂×Dic₉	semidirect
210. C₃⋊C₈	C₂×Dic₉	semidirect
252. C₉	#4707	semidirect
264. C₈	D₃×Dic₉	semidirect
282. C₃	M₁₆×Dic₉	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
C₄×D₃×Dic₉	2
C₂×#88420	2
C₈×C₃⋊Dic₉	2
C₃×#63375	2
#11720274	2
#11720271	2
#10590667	3
M₁₆×Dic₉	3