Subgroups — C₂×C₉×#875 #13152293

Order 1728.

1098 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	C₂²×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×(C₂×D₃)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₉×#875	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#875	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#124	—	3	1	no	maximal
8	C₂³×D₃×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
12	C₂×C₉×D₁₂	—	2	4	no	—
13	C₂×C₉×D₁₂	C₂²	1	4	yes	—
14	C₂×C₉×Dic₆	—	2	4	no	—
15	C₂×C₉×Dic₆	C₂²	1	4	yes	complemented
16	C₃×C₄²×C₉	C₂²	1	4	yes	complemented
17	C₉×(C₂×D₃)⋊C₄	C₂²	1	16	yes	complemented
18	C₂²×C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×(C₂×D₃)⋊C₄	C₂×C₃	1	4	yes	—
22	C₃×#875	C₂×C₃	1	8	yes	—
23	C₂²×Q₈×C₉	—	3	1	no	—
24	C₂²×D₄×C₉	—	3	1	no	—
25	C₉×#124	—	3	8	no	—
26	C₂×C₉×C₂²⋊C₄	—	3	4	no	—
27	C₂×C₄²×C₉	—	2	1	no	—
28	C₂×C₄²×C₉	D₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
30	C₂²×D₃×C₉	—	2	4	no	—
31	C₂²×D₃×C₉	—	2	8	no	—
32	C₂²×D₃×C₉	C₂³	1	4	yes	—
33	C₂×C₃×C₄×C₉	—	2	4	no	—
34	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
37	C₂×C₉×Dic₃	—	2	4	no	—
38	C₂×C₉×Dic₃	C₂³	1	4	yes	—
39	C₉×D₁₂	—	2	8	no	—
40	C₉×Dic₆	—	2	8	no	—
41	C₂×C₃×#124	—	3	1	no	—
42	C₂×#875	C₉	1	1	yes	characteristic, direct product factor
43	C₂³×C₃×D₃	C₂²×C₃	1	2	yes	—
44	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
45	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
47	C₂×C₃×D₁₂	—	2	4	no	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
49	C₂×C₃×Dic₆	—	2	4	no	—
50	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
51	C₂⁴×C₉	—	3	2	no	—
52	C₃²×C₄²	C₂²×C₃	1	4	yes	—
53	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	—
54	C₂×Q₈×C₉	—	3	4	no	—
55	C₂×Q₈×C₉	—	6	4	no	—
56	C₂×D₄×C₉	—	3	4	no	—
57	C₂×D₄×C₉	—	6	4	no	—
58	C₂²×C₄×C₉	—	2	2	no	—
59	C₂²×C₄×C₉	—	2	1	no	—
60	C₂²×C₄×C₉	—	3	2	no	—
61	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
62	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
63	C₉×C₂²⋊C₄	—	3	16	no	—
64	C₄²×C₉	—	2	4	no	—
65	C₄²×C₉	C₂×D₃	1	4	yes	complemented
66	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
67	C₂²×C₃×C₉	#41	1	4	yes	—
68	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
69	C₂×D₃×C₉	—	2	4	no	—
70	C₂×D₃×C₉	—	2	8	no	—
71	C₂×D₃×C₉	—	4	8	no	—
72	C₃×C₄×C₉	—	2	4	no	—
73	C₃×C₄×C₉	C₂×D₄	1	4	yes	—
74	C₉×Dic₃	—	2	4	no	—
75	C₂²×C₃×Q₈	—	3	1	no	—
76	C₂²×C₃×D₄	—	3	1	no	—
77	C₂²×D₁₂	C₂×C₉	1	1	yes	characteristic
78	C₂²×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
79	C₃×#124	—	3	8	no	—
80	C₂×C₃×C₂²⋊C₄	—	3	4	no	—
81	C₂×C₃×C₄²	—	2	1	no	—
82	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic, complemented
83	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
84	C₂×(C₂×D₃)⋊C₄	C₂×C₉	1	4	yes	complemented
85	#875	C₂×C₉	1	8	yes	direct product factor
86	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
87	C₂²×C₃×D₃	—	2	4	no	—
88	C₂²×C₃×D₃	—	2	8	no	—
89	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
92	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
93	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
94	C₂×C₃×Dic₃	—	2	4	no	—
95	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
96	C₃×D₁₂	—	2	8	no	—
97	C₃×Dic₆	—	2	8	no	—
98	C₂³×C₉	—	2	1	no	—
99	C₂³×C₉	—	3	4	no	—
100	C₂³×C₉	—	6	4	no	—
101	C₂³×C₉	—	6	8	no	—
102	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
103	Q₈×C₉	—	6	8	no	—
104	D₄×C₉	—	6	8	no	—
105	C₂×C₄×C₉	—	2	8	no	—
106	C₂×C₄×C₉	—	2	4	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	4	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	3	4	no	—
111	C₂×C₄×C₉	—	6	4	no	—
112	C₂×C₄×C₉	D₁₂	1	4	yes	—
113	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
114	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
115	C₂×#124	—	3	1	no	Sylow
116	C₂×C₃×C₉	#124	1	4	yes	complemented
117	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×#41	1	2	yes	—
119	D₃×C₉	—	4	4	no	—
120	C₂⁴×C₃	—	3	2	no	—
121	C₂³×D₃	C₂²×C₉	1	2	yes	—
122	C₂×C₃×Q₈	—	3	4	no	—
123	C₂×C₃×Q₈	—	6	4	no	—
124	C₂×C₃×D₄	—	3	4	no	—
125	C₂×C₃×D₄	—	6	4	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	3	2	no	—
129	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
130	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
132	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
133	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
134	C₂×D₁₂	—	2	4	no	—
135	C₂×D₁₂	C₂²×C₉	1	4	yes	—
136	C₂×Dic₆	—	2	4	no	—
137	C₂×Dic₆	C₂²×C₉	1	4	yes	complemented
138	C₃×C₂²⋊C₄	—	3	16	no	—
139	C₃×C₄²	—	2	4	no	—
140	C₃×C₄²	C₂²×C₉	1	4	yes	complemented
141	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
142	(C₂×D₃)⋊C₄	C₂²×C₉	1	16	yes	complemented
143	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
144	C₂²×C₃²	C₃×#41	1	4	yes	—
145	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
146	C₂×C₃×D₃	—	2	4	no	—
147	C₂×C₃×D₃	—	2	8	no	—
148	C₂×C₃×D₃	—	4	8	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
151	C₃×Dic₃	—	2	4	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	—	6	4	no	—
156	C₂²×C₉	—	6	8	no	—
157	C₂²×C₉	—	12	8	no	—
158	C₂²×C₉	C₂×D₁₂	1	2	yes	—
159	C₂²×C₉	#235	1	4	yes	—
160	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
161	C₄×C₉	—	2	8	no	—
162	C₄×C₉	—	2	4	no	—
163	C₄×C₉	—	2	4	no	—
164	C₄×C₉	—	6	4	no	—
165	C₄×C₉	C₂×D₁₂	1	4	yes	—
166	C₂²×Q₈	—	3	1	no	—
167	C₂²×D₄	—	3	1	no	—
168	#124	—	3	8	no	—
169	C₂×C₂²⋊C₄	—	3	4	no	—
170	C₂×C₄²	D₃×C₉	1	1	yes	characteristic, complemented
171	C₃×C₉	C₂×#124	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	3	4	no	—
174	C₂³×C₃	—	6	4	no	—
175	C₂³×C₃	—	6	8	no	—
176	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
177	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₂²×D₃	—	2	4	no	—
179	C₂²×D₃	—	2	8	no	—
180	C₂²×D₃	C₂³×C₉	1	4	yes	—
181	C₃×Q₈	—	6	8	no	—
182	C₃×D₄	—	6	8	no	—
183	C₂×C₃×C₄	—	2	8	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	3	4	no	—
190	C₂×C₃×C₄	—	6	4	no	—
191	C₂×C₃×C₄	D₄×C₉	1	4	yes	—
192	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
193	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
194	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
195	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
196	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
197	C₂×Dic₃	—	2	4	no	—
198	C₂×Dic₃	C₂³×C₉	1	4	yes	—
199	D₁₂	—	2	8	no	—
200	Dic₆	—	2	8	no	—
201	C₂×C₃²	C₃×#124	1	4	yes	—
202	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
203	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
204	C₃×D₃	—	4	4	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	2	no	—
207	C₂×C₉	—	2	4	no	—
208	C₂×C₉	—	12	4	no	—
209	C₂×C₉	#875	1	4	yes	direct product factor
210	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
211	C₂×C₉	C₂×#235	1	2	yes	—
212	C₂⁴	—	3	2	no	—
213	C₂×Q₈	—	3	4	no	—
214	C₂×Q₈	—	6	4	no	—
215	C₂×D₄	—	3	4	no	—
216	C₂×D₄	—	6	4	no	—
217	C₂²×C₄	—	3	2	no	—
218	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
219	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
220	C₂²⋊C₄	—	3	16	no	—
221	C₄²	C₂×D₃×C₉	1	4	yes	complemented
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	6	4	no	—
226	C₂²×C₃	—	6	8	no	—
227	C₂²×C₃	—	12	8	no	—
228	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
229	C₂²×C₃	C₉×#41	1	4	yes	—
230	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
232	C₂²×C₃	C₃×#235	1	4	yes	—
233	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
234	C₂×D₃	—	2	4	no	—
235	C₂×D₃	—	2	8	no	—
236	C₂×D₃	—	4	8	no	—
237	C₃×C₄	—	2	8	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	6	4	no	—
242	C₃×C₄	C₂×D₄×C₉	1	4	yes	—
243	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
244	Dic₃	—	2	4	no	—
245	C₃²	C₂×C₃×#124	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	C₂×#875	1	1	yes	characteristic, direct product factor
248	C₂³	—	3	4	no	—
249	C₂³	—	6	4	no	—
250	C₂³	—	6	8	no	—
251	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
252	Q₈	—	6	8	no	—
253	D₄	—	6	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	3	4	no	—
256	C₂×C₄	—	6	4	no	—
257	C₂×C₄	C₉×D₁₂	1	4	yes	—
258	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
259	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	12	4	no	—
264	C₂×C₃	C₉×#124	1	4	yes	complemented
265	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
266	C₂×C₃	C₂×C₉×#41	1	2	yes	—
267	C₂×C₃	C₃×#875	1	4	yes	—
268	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×#235	1	2	yes	—
270	D₃	—	4	4	no	—
271	C₂²	—	6	4	no	—
272	C₂²	—	6	8	no	—
273	C₂²	—	12	8	no	—
274	C₂²	C₂×C₉×D₁₂	1	2	yes	—
275	C₂²	C₉×#235	1	4	yes	—
276	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	6	4	no	—
279	C₄	C₂×C₉×D₁₂	1	4	yes	—
280	C₃	—	2	1	no	—
281	C₃	C₂×C₉×#124	1	1	yes	characteristic, complemented
282	C₃	C₂×C₃×#875	1	1	yes	characteristic
283	C₂	—	12	4	no	—
284	C₂	C₉×#875	1	4	yes	direct product factor
285	C₂	C₂²×C₉×D₁₂	1	1	yes	characteristic
286	C₂	C₂×C₉×#235	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂²×C₃×D₁₂	C₂×C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂×C₄²×C₉	D₃
hypocenter	C₃	C₂×C₉×#124
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×Dic₆	C₂	semidirect
3. C₂×C₃×C₄²×C₉	C₂	semidirect
4. C₂×C₉×(C₂×D₃)⋊C₄	C₂	semidirect
5. C₉×#875	C₂	direct (N × H)
15. C₂×C₉×Dic₆	C₂²	semidirect
16. C₃×C₄²×C₉	C₂²	semidirect
17. C₉×(C₂×D₃)⋊C₄	C₂²	semidirect
28. C₂×C₄²×C₉	D₃	semidirect
42. C₂×#875	C₉	direct (N × H)
65. C₄²×C₉	C₂×D₃	semidirect
78. C₂²×Dic₆	C₂×C₉	semidirect
82. C₂×C₃×C₄²	C₂×C₉	semidirect
84. C₂×(C₂×D₃)⋊C₄	C₂×C₉	semidirect
85. #875	C₂×C₉	direct (N × H)
116. C₂×C₃×C₉	#124	semidirect
137. C₂×Dic₆	C₂²×C₉	semidirect
140. C₃×C₄²	C₂²×C₉	semidirect
142. (C₂×D₃)⋊C₄	C₂²×C₉	semidirect
170. C₂×C₄²	D₃×C₉	semidirect
171. C₃×C₉	C₂×#124	semidirect
209. C₂×C₉	#875	direct (N × H)
221. C₄²	C₂×D₃×C₉	semidirect
247. C₉	C₂×#875	direct (N × H)
264. C₂×C₃	C₉×#124	semidirect
281. C₃	C₂×C₉×#124	semidirect
284. C₂	C₉×#875	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₁₂	2
C₂²×C₉×Dic₆	2
C₂×C₃×C₄²×C₉	2
C₂×C₉×(C₂×D₃)⋊C₄	2
C₉×#875	2
C₂×C₃×#875	3
C₂×C₉×#124	3

Subgroups — C2×C9×#875 #13152293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#875 #13152293