Subgroups — C₈×C₉×D₁₂ #13141427

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×C₈×C₉	C₂	1	2	yes	maximal, complemented
2	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×D₁₂	C₃	1	1	yes	maximal, characteristic
7	C₈×D₄×C₉	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
14	C₉×C₄⋊Dic₃	C₄	1	1	yes	characteristic
15	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
16	D₃×C₈×C₉	—	2	4	no	—
17	C₂×C₃×D₃×C₈	C₂×C₃	1	2	yes	—
18	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
19	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	2	yes	—
21	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₈×C₉	—	3	2	no	—
23	C₄×D₄×C₉	—	3	1	no	—
24	C₉×C₄⋊C₈	—	3	1	no	—
25	C₉×C₂²⋊C₈	—	3	2	no	—
26	C₄×C₈×C₉	—	2	1	no	—
27	C₄×C₈×C₉	D₃	1	1	yes	characteristic, complemented
28	C₂²×D₃×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
33	C₉×D₁₂	C₈	1	4	yes	direct product factor
34	C₄×D₃×C₉	—	2	2	no	—
35	C₄×D₃×C₉	—	2	2	no	—
36	C₃×C₈×C₉	—	2	1	no	—
37	C₃×C₈×C₉	D₄	1	2	yes	complemented
38	C₉×C₃⋊C₈	—	2	2	no	—
39	C₃×C₈×D₄	—	3	1	no	—
40	C₈×D₁₂	C₉	1	1	yes	characteristic, direct product factor
41	C₂×C₃×D₁₂	C₃×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×(C₂×D₃)⋊C₄	C₃×C₄	1	2	yes	—
47	C₃×C₄⋊Dic₃	C₃×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
49	C₃×D₃×C₈	—	2	4	no	—
50	C₂×D₄×C₉	—	3	1	no	—
51	C₂²×C₄×C₉	—	3	2	no	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₂×C₈×C₉	—	2	1	no	—
54	C₂×C₈×C₉	—	3	2	no	—
55	C₂×C₈×C₉	—	6	4	no	—
56	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₄⋊C₄	—	3	1	no	—
59	C₉×C₂²⋊C₄	—	3	2	no	—
60	C₄²×C₉	—	2	1	no	—
61	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×D₃×C₉	—	2	2	no	—
64	C₂×D₃×C₉	—	2	2	no	—
65	C₂×D₃×C₉	C₂×C₈	1	4	yes	complemented
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	C₂×C₈	1	2	yes	complemented
68	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄×C₉	#41	1	1	yes	characteristic
70	C₉×Dic₃	—	2	2	no	—
71	C₂²×C₃×C₈	—	3	2	no	—
72	C₃×C₄×D₄	—	3	1	no	—
73	C₂×D₃×C₈	C₂×C₉	1	2	yes	complemented
74	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic
75	C₃×C₄⋊C₈	—	3	1	no	—
76	C₃×C₂²⋊C₈	—	3	2	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
79	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
80	(C₂×D₃)⋊C₈	C₂×C₉	1	2	yes	complemented
81	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
82	C₂²×C₃×D₃	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
87	C₃×D₁₂	C₃×C₈	1	4	yes	—
88	C₃×C₄×D₃	—	2	2	no	—
89	C₃×C₄×D₃	—	2	2	no	—
90	C₂³×C₉	—	3	2	no	—
91	C₃²×C₈	—	2	1	no	—
92	C₃²×C₈	C₃×D₄	1	2	yes	—
93	C₃×C₃⋊C₈	—	2	2	no	—
94	D₄×C₉	—	3	4	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	3	2	no	—
99	C₂×C₄×C₉	—	6	2	no	—
100	C₂×C₄×C₉	—	6	2	no	—
101	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
102	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
104	C₈×C₉	—	2	2	no	—
105	C₈×C₉	—	2	1	no	—
106	C₈×C₉	—	2	2	no	—
107	C₈×C₉	—	6	2	no	—
108	C₈×C₉	D₁₂	1	2	yes	direct product factor
109	C₈×D₄	—	3	1	no	Sylow
110	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₃×C₉	#131	1	1	yes	characteristic
113	D₃×C₉	—	2	4	no	—
114	C₂×C₃×D₄	—	3	1	no	—
115	C₂²×C₃×C₄	—	3	2	no	—
116	C₂×D₁₂	C₄×C₉	1	1	yes	characteristic
117	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	3	2	no	—
121	C₂×C₃×C₈	—	6	4	no	—
122	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	3	1	no	—
127	C₃×C₂²⋊C₄	—	3	2	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
130	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
131	(C₂×D₃)⋊C₄	C₄×C₉	1	2	yes	—
132	C₄⋊Dic₃	C₄×C₉	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
134	D₃×C₈	—	2	4	no	—
135	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₂×C₃×D₃	—	2	2	no	—
137	C₂×C₃×D₃	—	2	2	no	—
138	C₂×C₃×D₃	C₂×C₃×C₈	1	4	yes	—
139	C₃²×C₄	—	2	1	no	—
140	C₃²×C₄	C₂×C₃×C₈	1	2	yes	—
141	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₃²×C₄	C₃×#41	1	1	yes	characteristic
143	C₃×Dic₃	—	2	2	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	3	4	no	—
146	C₂²×C₉	—	6	2	no	—
147	C₂²×C₉	—	6	2	no	—
148	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
149	C₄×C₉	—	2	2	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	2	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	2	1	no	—
154	C₄×C₉	—	6	2	no	—
155	C₄×C₉	D₃×C₈	1	2	yes	complemented
156	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
157	C₄×C₉	#235	1	1	yes	characteristic
158	C₂²×C₈	—	3	2	no	—
159	C₄×D₄	—	3	1	no	—
160	C₄⋊C₈	—	3	1	no	—
161	C₂²⋊C₈	—	3	2	no	—
162	C₄×C₈	D₃×C₉	1	1	yes	characteristic, complemented
163	C₃×C₉	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₃	—	3	2	no	—
165	C₂²×D₃	C₂×C₄×C₉	1	2	yes	—
166	C₃×D₄	—	3	4	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	3	2	no	—
171	C₂×C₃×C₄	—	6	2	no	—
172	C₂×C₃×C₄	—	6	2	no	—
173	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
179	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
180	D₁₂	C₈×C₉	1	4	yes	direct product factor
181	C₄×D₃	—	2	2	no	—
182	C₄×D₃	—	2	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	—	2	1	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	6	2	no	—
188	C₃×C₈	D₄×C₉	1	2	yes	complemented
189	C₃×C₈	C₃×D₁₂	1	2	yes	—
190	C₃⋊C₈	—	2	2	no	—
191	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
192	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
193	C₂×C₃²	C₃×#131	1	1	yes	characteristic
194	C₃×D₃	—	2	4	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	6	4	no	—
199	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
200	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
201	C₂×C₉	#901	1	1	yes	characteristic
202	C₂×D₄	—	3	1	no	—
203	C₂²×C₄	—	3	2	no	—
204	C₂×C₈	—	3	2	no	—
205	C₂×C₈	—	6	4	no	—
206	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
207	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
208	C₄⋊C₄	—	3	1	no	—
209	C₂²⋊C₄	—	3	2	no	—
210	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	3	4	no	—
213	C₂²×C₃	—	6	2	no	—
214	C₂²×C₃	—	6	2	no	—
215	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
216	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
217	C₂×D₃	—	2	2	no	—
218	C₂×D₃	—	2	2	no	—
219	C₂×D₃	C₂×C₈×C₉	1	4	yes	complemented
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	6	2	no	—
227	C₃×C₄	C₂×C₈×C₉	1	2	yes	complemented
228	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
229	C₃×C₄	C₉×#41	1	1	yes	characteristic
230	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
231	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
232	C₃×C₄	C₃×#235	1	1	yes	characteristic
233	Dic₃	—	2	2	no	—
234	C₃²	C₃×C₈×D₄	1	1	yes	characteristic
235	C₉	—	2	1	no	—
236	C₉	C₈×D₁₂	1	1	yes	characteristic, direct product factor
237	C₂³	—	3	2	no	—
238	D₄	—	3	4	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	6	2	no	—
241	C₂×C₄	—	6	2	no	—
242	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
243	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	6	2	no	—
247	C₈	C₉×D₁₂	1	2	yes	direct product factor
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	6	4	no	—
252	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
253	C₂×C₃	C₂²×C₈×C₉	1	1	yes	characteristic
254	C₂×C₃	C₉×#131	1	1	yes	characteristic
255	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
256	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
257	C₂×C₃	C₃×#901	1	1	yes	characteristic
258	D₃	—	2	4	no	—
259	C₂²	—	3	4	no	—
260	C₂²	—	6	2	no	—
261	C₂²	—	6	2	no	—
262	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	6	2	no	—
265	C₄	D₃×C₈×C₉	1	2	yes	complemented
266	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
267	C₄	C₉×#235	1	1	yes	characteristic
268	C₃	—	2	1	no	—
269	C₃	C₈×D₄×C₉	1	1	yes	characteristic, complemented
270	C₃	C₃×C₈×D₁₂	1	1	yes	characteristic
271	C₂	—	6	4	no	—
272	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic
273	C₂	C₂×D₃×C₈×C₉	1	1	yes	characteristic
274	C₂	C₉×#901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₉	C₂×D₃
commutator	C₂×C₃	C₂²×C₈×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂×C₃×D₁₂	C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₈×D₄×C₉
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×C₈×C₉	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₉×(C₂×D₃)⋊C₈	C₂	semidirect
5. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
27. C₄×C₈×C₉	D₃	semidirect
33. C₉×D₁₂	C₈	direct (N × H)
37. C₃×C₈×C₉	D₄	semidirect
40. C₈×D₁₂	C₉	direct (N × H)
65. C₂×D₃×C₉	C₂×C₈	semidirect
67. C₃×C₄×C₉	C₂×C₈	semidirect
73. C₂×D₃×C₈	C₂×C₉	semidirect
78. C₃×C₄×C₈	C₂×C₉	semidirect
80. (C₂×D₃)⋊C₈	C₂×C₉	semidirect
81. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
108. C₈×C₉	D₁₂	direct (N × H)
155. C₄×C₉	D₃×C₈	semidirect
162. C₄×C₈	D₃×C₉	semidirect
163. C₃×C₉	C₈×D₄	semidirect
180. D₁₂	C₈×C₉	direct (N × H)
188. C₃×C₈	D₄×C₉	semidirect
219. C₂×D₃	C₂×C₈×C₉	semidirect
227. C₃×C₄	C₂×C₈×C₉	semidirect
236. C₉	C₈×D₁₂	direct (N × H)
247. C₈	C₉×D₁₂	direct (N × H)
265. C₄	D₃×C₈×C₉	semidirect
269. C₃	C₈×D₄×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×C₈×C₉	2
C₄×C₉×D₁₂	2
C₃×C₄×C₈×C₉	2
C₉×(C₂×D₃)⋊C₈	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×C₈×D₁₂	3
C₈×D₄×C₉	3

Subgroups — C8×C9×D12 #13141427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₉×D₁₂ #13141427