Subgroups — C₄⋊C₈×C₃⋊D₉ #13144050

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₃⋊D₉	C₂	1	2	yes	maximal
2	C₄²×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720537	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃⋊D₃×C₄⋊C₈	—	3	1	no	maximal
7	D₉×C₄⋊C₈	—	3	3	no	maximal
8	C₂×C₄×C₃⋊D₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₃⋊D₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊Dic₉	C₄	1	1	yes	characteristic
14	C₄×C₃⋊Dic₉	C₄	1	1	yes	characteristic
15	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
17	C₈×C₃⋊D₉	—	2	4	no	—
18	C₂×C₈×C₃⋊D₃	—	3	2	no	—
19	C₄²×C₃⋊D₃	—	3	1	no	—
20	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
21	#63641	—	3	2	no	—
22	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
23	C₂×C₈×D₉	—	3	6	no	—
24	C₄²×D₉	—	3	3	no	—
25	C₉×C₄⋊C₈	D₃	1	3	yes	complemented
26	Dic₉⋊C₈	—	3	6	no	—
27	C₄⋊(C₉⋊C₈)	—	3	3	no	—
28	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
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×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₃⋊D₉	—	2	2	no	—
36	C₄×C₃⋊D₉	C₈	1	4	yes	complemented
37	C₄×C₃⋊D₉	D₄	1	2	yes	—
38	C₄×C₃⋊D₉	Q₈	1	2	yes	—
39	C₃×C₈×C₉	—	2	2	no	—
40	C₃⋊(C₉⋊C₈)	—	2	2	no	—
41	D₃×C₄⋊C₈	—	9	3	no	—
42	D₃×C₄⋊C₈	—	9	1	no	—
43	C₂×C₄×C₃⋊D₃	—	3	1	no	—
44	C₂×C₄×C₃⋊D₃	—	3	1	no	—
45	C₂×C₄×C₃⋊D₃	—	3	1	no	—
46	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
47	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₄×C₃⋊Dic₃	—	3	1	no	—
49	C₄×C₃⋊Dic₃	—	3	1	no	—
50	C₄×C₃⋊Dic₃	—	3	1	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
52	C₈×C₃⋊D₃	—	6	4	no	—
53	C₂×C₄×D₉	—	3	3	no	—
54	C₂×C₄×D₉	—	3	3	no	—
55	C₂×C₄×D₉	—	3	3	no	—
56	C₂×C₈×C₉	C₂×D₃	1	6	yes	—
57	C₄²×C₉	C₂×D₃	1	3	yes	—
58	C₄×Dic₉	—	3	3	no	—
59	C₄×Dic₉	—	3	3	no	—
60	C₄×Dic₉	—	3	3	no	—
61	C₂×C₉⋊C₈	—	3	6	no	—
62	C₈×D₉	—	6	12	no	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃⋊D₉	C₄⋊C₄	1	2	yes	—
65	C₂×C₃⋊D₉	C₂×C₈	1	2	yes	—
66	C₂×C₃⋊D₉	M₁₆	1	2	yes	—
67	C₃×C₄×C₉	—	2	1	no	—
68	C₃×C₄×C₉	C₂×C₈	1	2	yes	complemented
69	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
71	C₃⋊Dic₉	—	2	1	no	—
72	C₃⋊Dic₉	C₄⋊C₄	1	2	yes	—
73	C₃⋊Dic₉	C₂×C₈	1	2	yes	complemented
74	C₂×D₃×C₈	—	9	6	no	—
75	C₂×D₃×C₈	—	9	2	no	—
76	C₄²×D₃	—	9	3	no	—
77	C₄²×D₃	—	9	1	no	—
78	C₃×C₄⋊C₈	D₉	1	3	yes	complemented
79	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
80	Dic₃⋊C₈	—	9	6	no	—
81	Dic₃⋊C₈	—	9	2	no	—
82	C₄⋊(C₃⋊C₈)	—	9	3	no	—
83	C₄⋊(C₃⋊C₈)	—	9	1	no	—
84	C₂²×C₃⋊D₃	—	3	1	no	—
85	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₃⋊Dic₃	—	3	1	no	—
89	C₂×C₃⋊Dic₃	—	3	1	no	—
90	C₂×C₃⋊Dic₃	—	3	1	no	—
91	C₄×C₃⋊D₃	—	3	4	no	—
92	C₄×C₃⋊D₃	—	3	2	no	—
93	C₄×C₃⋊D₃	—	3	2	no	—
94	C₄×C₃⋊D₃	—	6	2	no	—
95	C₂²×D₉	—	3	3	no	—
96	C₃²×C₈	—	2	2	no	—
97	C₃⋊(C₃⋊C₈)	—	6	2	no	—
98	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
99	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
100	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
101	C₂×Dic₉	—	3	3	no	—
102	C₂×Dic₉	—	3	3	no	—
103	C₂×Dic₉	—	3	3	no	—
104	C₄×D₉	—	3	12	no	—
105	C₄×D₉	—	3	6	no	—
106	C₄×D₉	—	3	6	no	—
107	C₄×D₉	—	6	6	no	—
108	C₈×C₉	—	2	6	no	—
109	C₉⋊C₈	—	6	6	no	—
110	C₂×C₄⋊C₈	—	27	1	no	Sylow
111	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
114	C₃⋊D₉	C₄⋊C₈	1	4	yes	direct product factor
115	C₂×C₄×D₃	—	9	3	no	—
116	C₂×C₄×D₃	—	9	3	no	—
117	C₂×C₄×D₃	—	9	1	no	—
118	C₂×C₄×D₃	—	9	1	no	—
119	C₂×C₄×D₃	—	9	3	no	—
120	C₂×C₄×D₃	—	9	1	no	—
121	C₂×C₃×C₈	C₂×D₉	1	6	yes	—
122	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
123	C₃×C₄²	C₂×D₉	1	3	yes	—
124	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
125	C₄×Dic₃	—	9	3	no	—
126	C₄×Dic₃	—	9	3	no	—
127	C₄×Dic₃	—	9	3	no	—
128	C₄×Dic₃	—	9	1	no	—
129	C₄×Dic₃	—	9	1	no	—
130	C₄×Dic₃	—	9	1	no	—
131	C₂×C₃⋊C₈	—	9	6	no	—
132	C₂×C₃⋊C₈	—	9	2	no	—
133	D₃×C₈	—	18	4	no	—
134	D₃×C₈	—	18	12	no	—
135	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₃⋊D₃	—	3	2	no	—
137	C₂×C₃⋊D₃	—	3	2	no	—
138	C₂×C₃⋊D₃	—	3	2	no	—
139	C₃²×C₄	—	2	1	no	—
140	C₃²×C₄	D₃×C₈	1	2	yes	—
141	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
142	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
143	C₃⋊Dic₃	—	3	2	no	—
144	C₃⋊Dic₃	—	3	2	no	—
145	C₃⋊Dic₃	—	6	1	no	—
146	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
147	C₂×D₉	—	3	6	no	—
148	C₂×D₉	—	3	6	no	—
149	C₂×D₉	—	3	6	no	—
150	C₄×C₉	—	2	3	no	—
151	C₄×C₉	D₃×C₈	1	6	yes	complemented
152	C₄×C₉	D₃×D₄	1	3	yes	—
153	C₄×C₉	D₃×Q₈	1	3	yes	—
154	Dic₉	—	3	6	no	—
155	Dic₉	—	3	6	no	—
156	Dic₉	—	6	3	no	—
157	C₂²×C₈	—	27	2	no	—
158	C₂×C₄²	—	27	1	no	—
159	C₄⋊C₈	—	27	2	no	—
160	C₄⋊C₈	—	27	1	no	—
161	C₄⋊C₈	C₃⋊D₉	1	1	yes	characteristic, direct product factor
162	C₃×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₃	—	9	3	no	—
164	C₂²×D₃	—	9	1	no	—
165	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
166	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
167	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
168	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
171	C₂×Dic₃	—	9	3	no	—
172	C₂×Dic₃	—	9	3	no	—
173	C₂×Dic₃	—	9	1	no	—
174	C₂×Dic₃	—	9	1	no	—
175	C₂×Dic₃	—	9	3	no	—
176	C₂×Dic₃	—	9	1	no	—
177	C₄×D₃	—	9	12	no	—
178	C₄×D₃	—	9	6	no	—
179	C₄×D₃	—	9	6	no	—
180	C₄×D₃	—	9	4	no	—
181	C₄×D₃	—	9	2	no	—
182	C₄×D₃	—	9	2	no	—
183	C₄×D₃	—	18	2	no	—
184	C₄×D₃	—	18	6	no	—
185	C₃×C₈	—	2	6	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃⋊C₈	—	18	6	no	—
188	C₃⋊C₈	—	18	2	no	—
189	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
191	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
192	C₃⋊D₃	—	3	4	no	—
193	C₂×C₉	D₃×C₄⋊C₄	1	3	yes	—
194	C₂×C₉	C₂×D₃×C₈	1	3	yes	—
195	C₂×C₉	D₃×M₁₆	1	3	yes	—
196	D₉	—	3	12	no	—
197	C₂²×C₄	—	27	1	no	—
198	C₂²×C₄	—	27	1	no	—
199	C₂²×C₄	—	27	1	no	—
200	C₂×C₈	—	27	2	no	—
201	C₂×C₈	—	54	4	no	—
202	C₂×C₈	C₂×C₃⋊D₉	1	2	yes	—
203	C₄²	—	27	1	no	—
204	C₄²	—	27	1	no	—
205	C₄²	—	27	1	no	—
206	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
207	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
208	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
209	C₂×D₃	—	9	2	no	—
210	C₂×D₃	—	9	2	no	—
211	C₂×D₃	—	9	2	no	—
212	C₂×D₃	—	9	6	no	—
213	C₂×D₃	—	9	6	no	—
214	C₂×D₃	—	9	6	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	3	no	—
217	C₃×C₄	C₈×D₉	1	6	yes	complemented
218	C₃×C₄	D₄×D₉	1	3	yes	—
219	C₃×C₄	Q₈×D₉	1	3	yes	—
220	C₃×C₄	C₈×C₃⋊D₃	1	2	yes	—
221	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
222	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
223	Dic₃	—	9	2	no	—
224	Dic₃	—	9	2	no	—
225	Dic₃	—	9	6	no	—
226	Dic₃	—	9	6	no	—
227	Dic₃	—	18	1	no	—
228	Dic₃	—	18	3	no	—
229	C₃²	D₃×C₄⋊C₈	1	1	yes	characteristic
230	C₉	D₃×C₄⋊C₈	1	3	yes	complemented
231	C₂³	—	27	1	no	—
232	C₂×C₄	—	27	1	no	—
233	C₂×C₄	—	27	1	no	—
234	C₂×C₄	—	27	2	no	—
235	C₂×C₄	—	27	2	no	—
236	C₂×C₄	—	27	4	no	—
237	C₂×C₄	—	27	1	no	—
238	C₂×C₄	—	54	2	no	—
239	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
240	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	54	2	no	—
244	C₂×C₃	C₄⋊C₄×D₉	1	3	yes	—
245	C₂×C₃	C₂×C₈×D₉	1	3	yes	—
246	C₂×C₃	M₁₆×D₉	1	3	yes	—
247	C₂×C₃	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
248	C₂×C₃	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
249	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
250	D₃	—	9	4	no	—
251	D₃	—	9	12	no	—
252	C₂²	—	27	2	no	—
253	C₂²	—	27	2	no	—
254	C₂²	—	27	2	no	—
255	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	27	2	no	—
258	C₄	—	27	2	no	—
259	C₄	—	54	1	no	—
260	C₄	C₈×C₃⋊D₉	1	2	yes	complemented
261	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
262	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
263	C₃	D₉×C₄⋊C₈	1	3	yes	complemented
264	C₃	C₃⋊D₃×C₄⋊C₈	1	1	yes	characteristic
265	C₂	—	27	4	no	—
266	C₂	C₄⋊C₄×C₃⋊D₉	1	1	yes	characteristic
267	C₂	C₂×C₈×C₃⋊D₉	1	1	yes	characteristic
268	C₂	M₁₆×C₃⋊D₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
4. #11720537	C₂	semidirect
5. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
25. C₉×C₄⋊C₈	D₃	semidirect
36. C₄×C₃⋊D₉	C₈	semidirect
68. C₃×C₄×C₉	C₂×C₈	semidirect
73. C₃⋊Dic₉	C₂×C₈	semidirect
78. C₃×C₄⋊C₈	D₉	semidirect
114. C₃⋊D₉	C₄⋊C₈	direct (N × H)
151. C₄×C₉	D₃×C₈	semidirect
161. C₄⋊C₈	C₃⋊D₉	direct (N × H)
162. C₃×C₉	C₂×C₄⋊C₈	semidirect
217. C₃×C₄	C₈×D₉	semidirect
230. C₉	D₃×C₄⋊C₈	semidirect
260. C₄	C₈×C₃⋊D₉	semidirect
263. C₃	D₉×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×C₃⋊D₉	2
C₄²×C₃⋊D₉	2
C₃×C₉×C₄⋊C₈	2
#11720537	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₃⋊D₃×C₄⋊C₈	3
D₉×C₄⋊C₈	3

Subgroups — C4⋊C8×C3⋊D9 #13144050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₈×C₃⋊D₉ #13144050