Subgroups — C₂×C₃×C₄⋊(C₉⋊C₈) #13142367

Order 1728.

522 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₉⋊C₈	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₉⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	maximal
5	C₂×C₄⋊(C₉⋊C₈)	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₄	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	2	yes	—
11	C₂×C₃×C₉⋊C₈	—	2	4	no	—
12	C₂×C₃×C₉⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	—	3	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
16	C₂×C₄²×C₉	—	2	1	no	—
17	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₉⋊C₈	C₂×C₃	1	2	yes	—
19	C₄⋊(C₉⋊C₈)	C₂×C₃	1	4	yes	direct product factor
20	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₉	—	2	2	no	—
22	C₂×C₃×C₄×C₉	C₈	1	4	yes	complemented
23	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₃×C₉⋊C₈	—	2	4	no	—
30	C₂×C₃×C₄⋊C₈	—	9	1	no	—
31	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
32	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃²×C₄²	Dic₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₃⋊C₈	—	3	4	no	—
38	C₂×C₃×C₃⋊C₈	—	6	4	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
45	C₄²×C₉	—	2	2	no	—
46	C₄²×C₉	—	2	2	no	—
47	C₄²×C₉	C₃×C₄	1	2	yes	—
48	C₄²×C₉	C₂²×C₃	1	2	yes	—
49	C₂×C₉⋊C₈	—	2	4	no	—
50	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	—
51	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
53	C₂²×C₃×C₉	M₁₆	1	2	yes	—
54	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₄×C₉	—	2	2	no	—
56	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
57	C₃×C₄×C₉	C₂×C₈	1	4	yes	complemented
58	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
59	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₈	—	9	2	no	—
61	C₂×C₃×C₄²	—	2	1	no	—
62	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	—	3	2	no	—
65	C₃×C₄⋊C₈	—	9	4	no	—
66	C₄⋊(C₃⋊C₈)	—	3	4	no	—
67	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
68	C₂×C₃²×C₄	—	2	2	no	—
69	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	—
70	C₂×C₃²×C₄	Dic₆	1	2	yes	—
71	C₂×C₃²×C₄	D₁₂	1	2	yes	—
72	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
73	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
74	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂³×C₉	—	2	1	no	—
77	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
78	C₃×C₃⋊C₈	—	6	4	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	C₃×C₈	1	4	yes	complemented
89	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
90	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
93	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
94	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
95	C₉⋊C₈	—	2	4	no	—
96	C₂×C₄⋊C₈	—	9	1	no	Sylow
97	C₂×C₃×C₉	C₄⋊C₈	1	4	yes	complemented
98	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
105	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
110	C₂×C₃×C₈	—	9	4	no	—
111	C₂×C₃×C₈	—	18	4	no	—
112	C₃×C₄²	—	2	2	no	—
113	C₃×C₄²	—	2	2	no	—
114	C₃×C₄²	Dic₉	1	2	yes	—
115	C₃×C₄²	C₂×D₉	1	2	yes	—
116	C₃×C₄²	C₃×Dic₃	1	2	yes	—
117	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
118	C₂×C₃⋊C₈	—	3	4	no	—
119	C₂×C₃⋊C₈	—	6	4	no	—
120	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
121	C₂²×C₃²	#208	1	2	yes	—
122	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
123	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
124	C₃²×C₄	—	2	2	no	—
125	C₃²×C₄	C₂×C₃⋊C₈	1	4	yes	—
126	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
127	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
128	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
129	C₂²×C₉	—	2	2	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
134	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
135	C₂²×C₉	C₃×M₁₆	1	2	yes	—
136	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₄×C₉	—	2	4	no	—
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	4	no	—
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
144	C₄×C₉	C₂×C₃×C₈	1	4	yes	complemented
145	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
146	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
147	C₂²×C₈	—	9	2	no	—
148	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
149	C₄⋊C₈	—	9	4	no	—
150	C₃×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
153	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	C₉⋊C₈	1	4	yes	complemented
165	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
166	C₂×C₃×C₄	D₃₆	1	2	yes	—
167	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
168	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
169	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
170	C₂×C₃×C₄	C₃×C₃⋊C₈	1	4	yes	—
171	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
172	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
173	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
175	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₃×C₈	—	18	4	no	—
179	C₃⋊C₈	—	6	4	no	—
180	C₂×C₃²	C₄⋊(C₃⋊C₈)	1	4	yes	—
181	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
182	C₂×C₃²	C₂×#208	1	1	yes	characteristic
183	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	2	4	no	—
188	C₂×C₉	C₃×C₄⋊C₈	1	4	yes	complemented
189	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
191	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
192	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
193	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
195	C₂×C₈	—	9	4	no	—
196	C₂×C₈	—	18	4	no	—
197	C₄²	C₃×Dic₉	1	2	yes	—
198	C₄²	C₂×C₃×D₉	1	2	yes	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	C₂×C₉⋊C₈	1	2	yes	—
204	C₂²×C₃	#3667	1	2	yes	—
205	C₂²×C₃	C₄⋊Dic₉	1	2	yes	—
206	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
207	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	2	yes	—
208	C₂²×C₃	C₃×#208	1	2	yes	—
209	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
210	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	C₂×C₉⋊C₈	1	4	yes	complemented
219	C₃×C₄	C₄⋊Dic₉	1	2	yes	—
220	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
221	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
222	C₃×C₄	C₂×C₃×C₃⋊C₈	1	4	yes	—
223	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
224	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
225	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
226	C₃²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
227	C₉	—	2	1	no	—
228	C₉	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic, complemented
229	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	C₃×C₉⋊C₈	1	4	yes	complemented
232	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
233	C₂×C₄	C₃×D₃₆	1	2	yes	—
234	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
235	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
236	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
238	C₈	—	18	4	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	4	no	—
243	C₂×C₃	C₄⋊(C₉⋊C₈)	1	4	yes	direct product factor
244	C₂×C₃	C₂²×C₉⋊C₈	1	1	yes	characteristic
245	C₂×C₃	C₂×#3667	1	1	yes	characteristic
246	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
247	C₂×C₃	C₃×C₄⋊(C₃⋊C₈)	1	4	yes	—
248	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
249	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
250	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
251	C₂²	C₂×C₃×C₉⋊C₈	1	2	yes	—
252	C₂²	C₃×#3667	1	2	yes	—
253	C₂²	C₃×C₄⋊Dic₉	1	2	yes	—
254	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	C₂×C₃×C₉⋊C₈	1	4	yes	complemented
257	C₄	C₃×C₄⋊Dic₉	1	2	yes	—
258	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
259	C₄	C₂×C₃×D₃₆	1	1	yes	characteristic
260	C₃	—	2	1	no	—
261	C₃	C₂×C₄⋊(C₉⋊C₈)	1	1	yes	characteristic, direct product factor
262	C₃	C₂×C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
263	C₂	C₃×C₄⋊(C₉⋊C₈)	1	4	yes	direct product factor
264	C₂	C₂²×C₃×C₉⋊C₈	1	1	yes	characteristic
265	C₂	C₂×C₃×#3667	1	1	yes	characteristic
266	C₂	C₂×C₃×C₄⋊Dic₉	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	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₂×C₃×C₄⋊C₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊(C₉⋊C₈)	C₂	direct (N × H)
5. C₂×C₄⋊(C₉⋊C₈)	C₃	direct (N × H)
19. C₄⋊(C₉⋊C₈)	C₂×C₃	direct (N × H)
22. C₂×C₃×C₄×C₉	C₈	semidirect
57. C₃×C₄×C₉	C₂×C₈	semidirect
88. C₂×C₄×C₉	C₃×C₈	semidirect
97. C₂×C₃×C₉	C₄⋊C₈	semidirect
144. C₄×C₉	C₂×C₃×C₈	semidirect
150. C₃×C₉	C₂×C₄⋊C₈	semidirect
164. C₂×C₃×C₄	C₉⋊C₈	semidirect
188. C₂×C₉	C₃×C₄⋊C₈	semidirect
218. C₃×C₄	C₂×C₉⋊C₈	semidirect
228. C₉	C₂×C₃×C₄⋊C₈	semidirect
231. C₂×C₄	C₃×C₉⋊C₈	semidirect
243. C₂×C₃	C₄⋊(C₉⋊C₈)	direct (N × H)
256. C₄	C₂×C₃×C₉⋊C₈	semidirect
261. C₃	C₂×C₄⋊(C₉⋊C₈)	direct (N × H)
263. C₂	C₃×C₄⋊(C₉⋊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
C₂×C₃×C₄²×C₉	2
C₂²×C₃×C₉⋊C₈	2
C₃×C₄⋊(C₉⋊C₈)	2
C₂×C₃×C₄⋊(C₃⋊C₈)	3
C₂×C₄⋊(C₉⋊C₈)	3

Subgroups — C2×C3×C4⋊(C9⋊C8) #13142367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄⋊(C₉⋊C₈) #13142367