Subgroups — C₄×C₉×D₂₄ #13141433

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×D₁₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	2	yes	maximal, complemented
5	C₉×#818	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×D₂₄	C₃	1	1	yes	maximal, characteristic
7	C₄×C₉×D₈	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	2	yes	—
9	C₂×C₄×D₃×C₉	—	2	2	no	—
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	—	2	2	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₉×D₂₄	C₄	1	4	yes	direct product factor
16	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄×D₁₂	C₂×C₃	1	2	yes	—
18	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×#821	C₂×C₃	1	2	yes	—
20	C₃×#818	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×D₈	—	3	1	no	—
22	C₄×D₄×C₉	—	3	2	no	—
23	C₉×#107	—	3	1	no	—
24	C₉×#102	—	3	2	no	—
25	C₄×C₈×C₉	—	2	1	no	—
26	C₄×C₈×C₉	D₃	1	1	yes	characteristic, complemented
27	C₂²×D₃×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₉×Dic₃	—	2	2	no	—
32	C₉×D₁₂	—	2	2	no	—
33	C₉×D₁₂	C₂×C₄	1	4	yes	complemented
34	C₄×D₃×C₉	—	4	4	no	—
35	C₃×C₈×C₉	—	2	1	no	—
36	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
37	C₃×C₄×D₈	—	3	1	no	—
38	C₄×D₂₄	C₉	1	1	yes	characteristic, direct product factor
39	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
40	C₂×C₃×C₄×D₃	—	2	2	no	—
41	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
44	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
46	C₃×D₂₄	C₃×C₄	1	4	yes	—
47	C₂×D₄×C₉	—	3	2	no	—
48	C₂²×C₄×C₉	—	6	2	no	—
49	C₉×D₈	—	3	4	no	—
50	C₂×C₈×C₉	—	2	1	no	—
51	C₂×C₈×C₉	—	2	1	no	—
52	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
53	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
54	C₉×C₄⋊C₄	—	3	2	no	—
55	C₉×C₂²⋊C₄	—	6	2	no	—
56	C₄²×C₉	—	2	1	no	—
57	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×D₃×C₉	—	2	4	no	—
60	C₂×D₃×C₉	—	4	2	no	—
61	C₂×D₃×C₉	—	4	2	no	—
62	C₃×C₄×C₉	—	2	1	no	—
63	C₃×C₄×C₉	D₈	1	2	yes	complemented
64	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	#41	1	1	yes	characteristic
66	C₉×Dic₃	—	4	2	no	—
67	C₂×C₃×D₈	—	3	1	no	—
68	C₃×C₄×D₄	—	3	2	no	—
69	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic
70	C₄×D₁₂	C₂×C₉	1	2	yes	complemented
71	C₃×#107	—	3	1	no	—
72	C₃×#102	—	3	2	no	—
73	C₃×C₄×C₈	—	2	1	no	—
74	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
75	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
76	#821	C₂×C₉	1	2	yes	complemented
77	#818	C₂×C₉	1	1	yes	characteristic, complemented
78	C₂²×C₃×D₃	—	2	2	no	—
79	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	2	2	no	—
83	C₃×D₁₂	—	2	2	no	—
84	C₃×D₁₂	C₂×C₃×C₄	1	4	yes	—
85	C₃×C₄×D₃	—	4	4	no	—
86	C₂³×C₉	—	6	2	no	—
87	C₃²×C₈	—	2	1	no	—
88	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
89	D₄×C₉	—	3	4	no	—
90	D₄×C₉	—	6	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	6	2	no	—
95	C₂×C₄×C₉	—	12	4	no	—
96	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
97	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	1	no	—
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	C₄×D₃	1	2	yes	complemented
103	C₄×D₈	—	3	1	no	Sylow
104	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₃×C₉	#135	1	1	yes	characteristic
107	D₃×C₉	—	4	4	no	—
108	C₂×C₃×D₄	—	3	2	no	—
109	C₂²×C₃×C₄	—	6	2	no	—
110	C₂×D₁₂	C₂²×C₉	1	2	yes	—
111	C₂×C₄×D₃	—	2	2	no	—
112	C₃×D₈	—	3	4	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	3	2	no	—
120	C₃×C₂²⋊C₄	—	6	2	no	—
121	C₃×C₄²	—	2	1	no	—
122	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
123	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	—	2	2	no	—
125	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
126	D₂₄	C₄×C₉	1	4	yes	direct product factor
127	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂×C₃×D₃	—	2	4	no	—
129	C₂×C₃×D₃	—	4	2	no	—
130	C₂×C₃×D₃	—	4	2	no	—
131	C₃²×C₄	—	2	1	no	—
132	C₃²×C₄	C₃×D₈	1	2	yes	—
133	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₃²×C₄	C₃×#41	1	1	yes	characteristic
135	C₃×Dic₃	—	4	2	no	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	6	4	no	—
138	C₂²×C₉	—	12	2	no	—
139	C₂²×C₉	—	12	2	no	—
140	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
141	C₄×C₉	—	2	2	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	12	2	no	—
147	C₄×C₉	D₂₄	1	2	yes	direct product factor
148	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
149	C₄×C₉	#235	1	1	yes	characteristic
150	C₂×D₈	—	3	1	no	—
151	C₄×D₄	—	3	2	no	—
152	#107	—	3	1	no	—
153	#102	—	3	2	no	—
154	C₄×C₈	D₃×C₉	1	1	yes	characteristic, complemented
155	C₃×C₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	6	2	no	—
157	C₂²×D₃	—	2	2	no	—
158	C₃×D₄	—	3	4	no	—
159	C₃×D₄	—	6	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×C₃×C₄	—	12	4	no	—
165	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
166	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₂×Dic₃	—	2	2	no	—
172	D₁₂	—	2	2	no	—
173	D₁₂	C₂×C₄×C₉	1	4	yes	complemented
174	C₄×D₃	—	4	4	no	—
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	2	no	—
179	C₃×C₈	C₂×C₄×C₉	1	2	yes	complemented
180	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
181	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
182	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
183	C₂×C₃²	C₃×#135	1	1	yes	characteristic
184	C₃×D₃	—	4	4	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	2	1	no	—
188	C₂×C₉	—	12	4	no	—
189	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
190	C₂×C₉	C₂×D₂₄	1	1	yes	characteristic
191	C₂×C₉	#904	1	1	yes	characteristic
192	C₂×D₄	—	3	2	no	—
193	C₂²×C₄	—	6	2	no	—
194	D₈	—	3	4	no	—
195	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
196	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
197	C₄⋊C₄	—	3	2	no	—
198	C₂²⋊C₄	—	6	2	no	—
199	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	6	4	no	—
202	C₂²×C₃	—	12	2	no	—
203	C₂²×C₃	—	12	2	no	—
204	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
205	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
206	C₂×D₃	—	2	4	no	—
207	C₂×D₃	—	4	2	no	—
208	C₂×D₃	—	4	2	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	12	2	no	—
216	C₃×C₄	C₉×D₈	1	2	yes	complemented
217	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
218	C₃×C₄	C₉×#41	1	1	yes	characteristic
219	C₃×C₄	C₃×D₂₄	1	2	yes	—
220	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
221	C₃×C₄	C₃×#235	1	1	yes	characteristic
222	Dic₃	—	4	2	no	—
223	C₃²	C₃×C₄×D₈	1	1	yes	characteristic
224	C₉	—	2	1	no	—
225	C₉	C₄×D₂₄	1	1	yes	characteristic, direct product factor
226	C₂³	—	6	2	no	—
227	D₄	—	3	4	no	—
228	D₄	—	6	2	no	—
229	C₂×C₄	—	6	2	no	—
230	C₂×C₄	—	12	4	no	—
231	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
232	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	C₄×D₃×C₉	1	2	yes	complemented
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	12	4	no	—
240	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
241	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
242	C₂×C₃	C₉×#135	1	1	yes	characteristic
243	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
244	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
245	C₂×C₃	C₃×#904	1	1	yes	characteristic
246	D₃	—	4	4	no	—
247	C₂²	—	6	4	no	—
248	C₂²	—	12	2	no	—
249	C₂²	—	12	2	no	—
250	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	12	2	no	—
253	C₄	C₉×D₂₄	1	2	yes	direct product factor
254	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
255	C₄	C₉×#235	1	1	yes	characteristic
256	C₃	—	2	1	no	—
257	C₃	C₄×C₉×D₈	1	1	yes	characteristic, complemented
258	C₃	C₃×C₄×D₂₄	1	1	yes	characteristic
259	C₂	—	12	4	no	—
260	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic
261	C₂	C₂×C₉×D₂₄	1	1	yes	characteristic
262	C₂	C₉×#904	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₂₄	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₄×C₉×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
2. C₄×C₉×D₁₂	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₉×#821	C₂	semidirect
5. C₉×#818	C₂	semidirect
15. C₉×D₂₄	C₄	direct (N × H)
26. C₄×C₈×C₉	D₃	semidirect
33. C₉×D₁₂	C₂×C₄	semidirect
36. C₃×C₈×C₉	C₂×C₄	semidirect
38. C₄×D₂₄	C₉	direct (N × H)
63. C₃×C₄×C₉	D₈	semidirect
70. C₄×D₁₂	C₂×C₉	semidirect
74. C₃×C₄×C₈	C₂×C₉	semidirect
76. #821	C₂×C₉	semidirect
77. #818	C₂×C₉	semidirect
102. C₈×C₉	C₄×D₃	semidirect
126. D₂₄	C₄×C₉	direct (N × H)
147. C₄×C₉	D₂₄	direct (N × H)
154. C₄×C₈	D₃×C₉	semidirect
155. C₃×C₉	C₄×D₈	semidirect
173. D₁₂	C₂×C₄×C₉	semidirect
179. C₃×C₈	C₂×C₄×C₉	semidirect
216. C₃×C₄	C₉×D₈	semidirect
225. C₉	C₄×D₂₄	direct (N × H)
235. C₈	C₄×D₃×C₉	semidirect
253. C₄	C₉×D₂₄	direct (N × H)
257. C₃	C₄×C₉×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×D₂₄	2
C₄×C₉×D₁₂	2
C₃×C₄×C₈×C₉	2
C₉×#821	2
C₉×#818	2
C₃×C₄×D₂₄	3
C₄×C₉×D₈	3

Subgroups — C4×C9×D24 #13141433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₉×D₂₄ #13141433