Subgroups — C₂×C₃×Dic₉⋊C₈ #13143074

Order 1728.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₉⋊C₈	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×Dic₃⋊C₈	—	3	1	no	maximal
6	C₂×Dic₉⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×Dic₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×Dic₉	C₄	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	—	2	2	no	—
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄×Dic₉	C₄	1	2	yes	—
13	C₃×C₄×Dic₉	C₂²	1	2	yes	—
14	C₂×C₃×C₉⋊C₈	—	2	2	no	—
15	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
17	C₂×C₃×C₄×Dic₃	—	3	1	no	—
18	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
19	C₃×Dic₃⋊C₈	—	3	4	no	—
20	C₂²×C₈×C₉	—	2	1	no	—
21	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
24	Dic₉⋊C₈	C₂×C₃	1	4	yes	direct product factor
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃×Dic₉	—	2	2	no	—
31	C₂×C₃×Dic₉	C₈	1	4	yes	complemented
32	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
34	C₃×C₈×C₉	—	2	2	no	—
35	C₃×C₉⋊C₈	—	2	2	no	—
36	C₂×C₃×C₄⋊C₈	—	9	1	no	—
37	C₂×Dic₃⋊C₈	—	3	1	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	3	1	no	—
40	C₂²×C₃×Dic₃	—	3	1	no	—
41	C₂×C₃²×C₈	—	2	2	no	—
42	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
43	C₃×C₄×Dic₃	—	3	2	no	—
44	C₃×C₄×Dic₃	—	3	2	no	—
45	C₂×C₃×C₃⋊C₈	—	3	2	no	—
46	C₂×C₃×C₃⋊C₈	—	6	2	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₂²×Dic₉	C₃×C₄	1	1	yes	characteristic
50	C₂²×Dic₉	C₃×C₄	1	1	yes	characteristic
51	C₂×C₈×C₉	—	2	4	no	—
52	C₂×C₈×C₉	—	2	2	no	—
53	C₂×C₈×C₉	—	2	2	no	—
54	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
55	C₄×Dic₉	C₃×C₄	1	2	yes	—
56	C₄×Dic₉	C₂²×C₃	1	2	yes	—
57	C₂×C₉⋊C₈	—	2	2	no	—
58	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
59	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
61	C₂²×C₃×C₉	M₁₆	1	2	yes	—
62	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
64	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₃×Dic₉	—	2	2	no	—
67	C₃×Dic₉	C₂×C₈	1	4	yes	complemented
68	C₂²×C₃×C₈	—	2	1	no	—
69	C₂²×C₃×C₈	—	9	1	no	—
70	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
72	C₂×C₃×C₄²	—	9	1	no	—
73	C₂×C₄×Dic₃	—	3	1	no	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	C₃×C₄⋊C₈	—	9	4	no	—
76	Dic₃⋊C₈	—	3	4	no	—
77	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	Dic₆	1	2	yes	—
79	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	#67	1	2	yes	—
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	3	4	no	—
83	C₂×C₃×Dic₃	—	3	2	no	—
84	C₂×C₃×Dic₃	—	3	2	no	—
85	C₂×C₃×Dic₃	—	6	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
88	C₃²×C₈	—	2	2	no	—
89	C₃×C₃⋊C₈	—	6	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
97	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
98	C₂×Dic₉	—	2	2	no	—
99	C₂×Dic₉	C₃×C₈	1	4	yes	complemented
100	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
101	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
102	C₈×C₉	—	2	4	no	—
103	C₈×C₉	—	2	2	no	—
104	C₉⋊C₈	—	2	2	no	—
105	C₂×C₄⋊C₈	—	9	1	no	Sylow
106	C₂×C₃×C₉	C₄⋊C₈	1	4	yes	complemented
107	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	9	1	no	—
112	C₂²×C₃×C₄	—	9	1	no	—
113	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
115	C₂²×Dic₃	—	3	1	no	—
116	C₂²×Dic₃	—	3	1	no	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	9	2	no	—
122	C₂×C₃×C₈	—	18	2	no	—
123	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
125	C₃×C₄²	—	9	2	no	—
126	C₃×C₄²	—	9	2	no	—
127	C₄×Dic₃	—	3	2	no	—
128	C₄×Dic₃	—	3	2	no	—
129	C₂×C₃⋊C₈	—	3	2	no	—
130	C₂×C₃⋊C₈	—	6	2	no	—
131	C₂²×C₃²	D₃×C₈	1	2	yes	—
132	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	2	yes	—
133	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
134	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
136	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
137	C₃²×C₄	C₂×#67	1	1	yes	characteristic
138	C₃×Dic₃	—	3	4	no	—
139	C₃×Dic₃	—	6	2	no	—
140	C₂²×C₉	—	2	2	no	—
141	C₂²×C₉	—	2	2	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
145	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
146	C₂²×C₉	C₃×M₁₆	1	2	yes	—
147	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₄×C₉	—	2	2	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
152	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
153	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
154	Dic₉	—	2	2	no	—
155	Dic₉	C₂×C₃×C₈	1	4	yes	complemented
156	C₂²×C₈	—	9	1	no	—
157	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
158	C₂×C₄²	—	9	1	no	—
159	C₄⋊C₈	—	9	4	no	—
160	C₃×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
163	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	9	4	no	—
169	C₂×C₃×C₄	—	9	2	no	—
170	C₂×C₃×C₄	—	9	2	no	—
171	C₂×C₃×C₄	—	18	2	no	—
172	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
173	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
174	C₂×C₃×C₄	#611	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
177	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₃×#67	1	2	yes	—
179	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
180	C₂×Dic₃	—	3	4	no	—
181	C₂×Dic₃	—	3	2	no	—
182	C₂×Dic₃	—	3	2	no	—
183	C₂×Dic₃	—	6	2	no	—
184	C₃×C₈	—	2	4	no	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	18	2	no	—
188	C₃⋊C₈	—	6	2	no	—
189	C₂×C₃²	Dic₃⋊C₈	1	4	yes	—
190	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
191	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
192	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	4	no	—
197	C₂×C₉	C₃×C₄⋊C₈	1	4	yes	complemented
198	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
200	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
201	C₂²×C₄	—	9	1	no	—
202	C₂²×C₄	—	9	1	no	—
203	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	9	2	no	—
206	C₂×C₈	—	18	2	no	—
207	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
208	C₄²	—	9	2	no	—
209	C₄²	—	9	2	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	C₈×D₉	1	2	yes	—
215	C₂²×C₃	#3663	1	2	yes	—
216	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
217	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
218	C₂²×C₃	C₃×D₃×C₈	1	2	yes	—
219	C₂²×C₃	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
220	C₂²×C₃	C₃×Dic₃⋊C₄	1	2	yes	—
221	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	9	4	no	—
226	C₃×C₄	—	18	2	no	—
227	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
228	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
229	C₃×C₄	C₂×#611	1	1	yes	characteristic
230	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
231	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
232	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
233	Dic₃	—	3	4	no	—
234	Dic₃	—	6	2	no	—
235	C₃²	C₂×Dic₃⋊C₈	1	1	yes	characteristic
236	C₉	—	2	1	no	—
237	C₉	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic, complemented
238	C₂³	C₃×C₄×D₉	1	1	yes	characteristic
239	C₂×C₄	—	9	2	no	—
240	C₂×C₄	—	9	4	no	—
241	C₂×C₄	—	9	2	no	—
242	C₂×C₄	—	18	2	no	—
243	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
244	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
245	C₂×C₄	C₃×#611	1	2	yes	—
246	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	18	2	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	Dic₉⋊C₈	1	4	yes	direct product factor
254	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
255	C₂×C₃	C₂×#3663	1	1	yes	characteristic
256	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
257	C₂×C₃	C₃×Dic₃⋊C₈	1	4	yes	—
258	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
259	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
260	C₂×C₃	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
261	C₂²	C₃×C₈×D₉	1	2	yes	—
262	C₂²	C₃×#3663	1	2	yes	—
263	C₂²	C₃×Dic₉⋊C₄	1	2	yes	—
264	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
265	C₄	—	9	4	no	—
266	C₄	—	18	2	no	—
267	C₄	C₃×Dic₉⋊C₄	1	2	yes	—
268	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
269	C₄	C₂×C₃×#611	1	1	yes	characteristic
270	C₃	—	2	1	no	—
271	C₃	C₂×Dic₉⋊C₈	1	1	yes	characteristic, direct product factor
272	C₃	C₂×C₃×Dic₃⋊C₈	1	1	yes	characteristic
273	C₂	C₃×Dic₉⋊C₈	1	4	yes	direct product factor
274	C₂	C₂×C₃×C₈×D₉	1	1	yes	characteristic
275	C₂	C₂×C₃×#3663	1	1	yes	characteristic
276	C₂	C₂×C₃×Dic₉⋊C₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×Dic₉⋊C₈	C₂	direct (N × H)
6. C₂×Dic₉⋊C₈	C₃	direct (N × H)
24. Dic₉⋊C₈	C₂×C₃	direct (N × H)
31. C₂×C₃×Dic₉	C₈	semidirect
67. C₃×Dic₉	C₂×C₈	semidirect
99. C₂×Dic₉	C₃×C₈	semidirect
106. C₂×C₃×C₉	C₄⋊C₈	semidirect
155. Dic₉	C₂×C₃×C₈	semidirect
160. C₃×C₉	C₂×C₄⋊C₈	semidirect
197. C₂×C₉	C₃×C₄⋊C₈	semidirect
237. C₉	C₂×C₃×C₄⋊C₈	semidirect
253. C₂×C₃	Dic₉⋊C₈	direct (N × H)
271. C₃	C₂×Dic₉⋊C₈	direct (N × H)
273. C₂	C₃×Dic₉⋊C₈	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×C₃×C₈×C₉	2
C₂×C₃×C₄×Dic₉	2
C₂²×C₃×C₉⋊C₈	2
C₃×Dic₉⋊C₈	2
C₂×C₃×Dic₃⋊C₈	3
C₂×Dic₉⋊C₈	3

Subgroups — C2×C3×Dic9⋊C8 #13143074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Dic₉⋊C₈ #13143074