Subgroups — C₃²×#4872 #13162326

Order 1728.

948 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃²×#893	C₂	1	2	yes	maximal, complemented
3	C₃³×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#809	C₂	1	2	yes	maximal, complemented
6	C₃²×C₄×D₈	—	3	1	no	maximal
7	C₃×#4872	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
9	C₂×C₃²×C₄×D₃	—	2	2	no	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃²×(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃²×D₂₄	C₄	1	4	yes	complemented
16	C₂×C₃²×D₈	—	3	1	no	—
17	C₃²×C₄×D₄	—	3	2	no	—
18	C₂×C₃×D₂₄	C₂×C₃	1	4	yes	—
19	C₃×#893	C₂×C₃	1	8	yes	complemented
20	C₃²×#107	—	2	4	no	—
21	C₃²×#107	D₃	1	1	yes	characteristic, complemented
22	C₃²×#107	C₂×C₃	1	4	yes	complemented
23	C₃²×#102	—	3	2	no	—
24	C₃²×C₄×C₈	—	3	1	no	—
25	C₃×C₈×Dic₃	C₂×C₃	1	4	yes	complemented
26	C₃×#809	C₂×C₃	1	8	yes	complemented
27	C₂²×C₃²×D₃	—	2	2	no	—
28	C₂×C₃³×C₄	—	2	2	no	—
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
31	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
32	C₃²×D₁₂	—	2	2	no	—
33	C₃²×D₁₂	C₂×C₄	1	4	yes	complemented
34	C₃²×C₄×D₃	—	4	4	no	—
35	C₃³×C₈	C₂×C₄	1	2	yes	complemented
36	C₃²×C₃⋊C₈	—	2	1	no	—
37	C₃×C₄×D₈	—	3	4	no	—
38	#4872	C₃²	1	1	yes	characteristic, direct product factor
39	C₂×C₃²×D₄	—	3	2	no	—
40	C₂²×C₃²×C₄	—	6	2	no	—
41	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
42	C₂×C₃×C₄×D₃	—	2	8	no	—
43	C₃²×D₈	—	3	4	no	—
44	C₂×C₃²×C₈	—	2	4	no	—
45	C₂×C₃²×C₈	—	3	1	no	—
46	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
48	C₃²×C₄⋊C₄	—	2	8	no	—
49	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
51	C₃²×C₂²⋊C₄	—	6	2	no	—
52	C₃²×C₄²	—	3	1	no	—
53	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
54	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
55	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
56	C₃×D₂₄	C₃×C₄	1	16	yes	complemented
57	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃²×D₃	—	2	4	no	—
59	C₂×C₃²×D₃	—	4	2	no	—
60	C₂×C₃²×D₃	—	4	2	no	—
61	C₃³×C₄	—	4	2	no	—
62	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₃³×C₄	#41	1	1	yes	characteristic
64	C₃²×Dic₃	—	2	1	no	—
65	C₃²×Dic₃	D₈	1	2	yes	complemented
66	C₂×C₃×D₈	—	3	4	no	—
67	C₃×C₄×D₄	—	3	8	no	—
68	C₂×D₂₄	C₂×C₃²	1	1	yes	characteristic
69	#893	C₂×C₃²	1	2	yes	complemented
70	C₃×#107	—	2	4	no	—
71	C₃×#107	C₃×D₃	1	4	yes	complemented
72	C₃×#107	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₃×#102	—	3	8	no	—
74	C₃×C₄×C₈	—	3	4	no	—
75	C₈×Dic₃	C₂×C₃²	1	1	yes	characteristic, complemented
76	#809	C₂×C₃²	1	2	yes	complemented
77	C₂³×C₃²	—	6	2	no	—
78	C₂²×C₃×D₃	—	2	8	no	—
79	C₃²×D₄	—	3	4	no	—
80	C₃²×D₄	—	6	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	3	1	no	—
85	C₂×C₃²×C₄	—	3	1	no	—
86	C₂×C₃²×C₄	—	4	8	no	—
87	C₂×C₃²×C₄	—	12	4	no	—
88	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
90	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
91	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
92	C₃×D₁₂	—	2	8	no	—
93	C₃×D₁₂	C₂×C₃×C₄	1	16	yes	complemented
94	C₃×C₄×D₃	—	4	16	no	—
95	C₃²×C₈	—	2	8	no	—
96	C₃²×C₈	—	6	1	no	—
97	C₃²×C₈	C₄×D₃	1	2	yes	complemented
98	C₃²×C₈	C₂×C₃×C₄	1	8	yes	complemented
99	C₃×C₃⋊C₈	—	2	4	no	—
100	C₄×D₈	—	3	1	no	Sylow
101	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
102	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
103	C₂×C₃³	#135	1	1	yes	characteristic
104	C₃²×D₃	—	4	4	no	—
105	C₂×C₃×D₄	—	3	8	no	—
106	C₂²×C₃×C₄	—	6	8	no	—
107	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
108	C₂×C₄×D₃	—	2	2	no	—
109	C₃×D₈	—	3	16	no	—
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	—	3	4	no	—
112	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
113	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	2	8	no	—
115	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
116	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
117	C₃×C₂²⋊C₄	—	6	8	no	—
118	C₃×C₄²	—	3	4	no	—
119	(C₂×D₃)⋊C₄	—	2	2	no	—
120	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
122	D₂₄	C₃²×C₄	1	4	yes	complemented
123	C₂²×C₃²	—	2	4	no	—
124	C₂²×C₃²	—	6	4	no	—
125	C₂²×C₃²	—	12	2	no	—
126	C₂²×C₃²	—	12	2	no	—
127	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
128	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
129	C₂×C₃×D₃	—	2	16	no	—
130	C₂×C₃×D₃	—	4	8	no	—
131	C₂×C₃×D₃	—	4	8	no	—
132	C₃²×C₄	—	2	4	no	—
133	C₃²×C₄	—	2	4	no	—
134	C₃²×C₄	—	3	2	no	—
135	C₃²×C₄	—	4	2	no	—
136	C₃²×C₄	—	4	8	no	—
137	C₃²×C₄	—	6	1	no	—
138	C₃²×C₄	—	8	8	no	—
139	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
140	C₃²×C₄	#239	1	1	yes	characteristic
141	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
142	C₃²×C₄	C₃×#41	1	4	yes	—
143	C₃×Dic₃	—	2	4	no	—
144	C₃×Dic₃	C₃×D₈	1	8	yes	complemented
145	C₂×D₈	—	3	1	no	—
146	C₄×D₄	—	3	2	no	—
147	#107	C₃²×D₃	1	1	yes	characteristic, complemented
148	#102	—	3	2	no	—
149	C₄×C₈	—	3	1	no	—
150	C₃³	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₃	—	6	8	no	—
152	C₂²×D₃	—	2	2	no	—
153	C₃×D₄	—	3	16	no	—
154	C₃×D₄	—	6	8	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	4	no	—
160	C₂×C₃×C₄	—	4	8	no	—
161	C₂×C₃×C₄	—	12	16	no	—
162	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
163	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
164	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
165	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
166	D₁₂	—	2	2	no	—
167	D₁₂	C₂×C₃²×C₄	1	4	yes	complemented
168	C₄×D₃	—	4	4	no	—
169	C₃×C₈	—	2	8	no	—
170	C₃×C₈	—	6	4	no	—
171	C₃×C₈	C₃×C₄×D₃	1	8	yes	complemented
172	C₃×C₈	C₂×C₃²×C₄	1	2	yes	complemented
173	C₃⋊C₈	—	2	1	no	—
174	C₂×C₃²	—	2	4	no	—
175	C₂×C₃²	—	2	4	no	—
176	C₂×C₃²	—	2	4	no	—
177	C₂×C₃²	—	12	4	no	—
178	C₂×C₃²	#893	1	1	yes	characteristic
179	C₂×C₃²	D₃×D₈	1	1	yes	characteristic
180	C₂×C₃²	#919	1	1	yes	characteristic
181	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
182	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
183	C₂×C₃²	C₃×#135	1	4	yes	—
184	C₃×D₃	—	4	16	no	—
185	C₂×D₄	—	3	2	no	—
186	C₂²×C₄	—	6	2	no	—
187	D₈	—	3	4	no	—
188	C₂×C₈	—	3	1	no	—
189	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
191	C₂²⋊C₄	—	6	2	no	—
192	C₄²	—	3	1	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	6	16	no	—
195	C₂²×C₃	—	12	8	no	—
196	C₂²×C₃	—	12	8	no	—
197	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
198	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
199	C₂×D₃	—	2	4	no	—
200	C₂×D₃	—	4	2	no	—
201	C₂×D₃	—	4	2	no	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	3	8	no	—
205	C₃×C₄	—	4	8	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	—	6	4	no	—
208	C₃×C₄	—	8	8	no	—
209	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
210	C₃×C₄	C₃×#239	1	4	yes	—
211	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
212	C₃×C₄	C₃²×#41	1	1	yes	characteristic
213	Dic₃	—	2	1	no	—
214	Dic₃	C₃²×D₈	1	2	yes	complemented
215	C₃²	—	2	4	no	—
216	C₃²	#4872	1	1	yes	characteristic, direct product factor
217	C₃²	C₃×C₄×D₈	1	4	yes	complemented
218	C₂³	—	6	2	no	—
219	D₄	—	3	4	no	—
220	D₄	—	6	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	12	4	no	—
225	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
226	C₈	—	6	1	no	—
227	C₈	C₃²×C₄×D₃	1	2	yes	complemented
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	12	16	no	—
232	C₂×C₃	C₃×#893	1	4	yes	—
233	C₂×C₃	C₃×D₃×D₈	1	4	yes	—
234	C₂×C₃	C₃×#919	1	4	yes	—
235	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
236	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
237	C₂×C₃	C₃²×#135	1	1	yes	characteristic
238	D₃	—	4	4	no	—
239	C₂²	—	6	4	no	—
240	C₂²	—	12	2	no	—
241	C₂²	—	12	2	no	—
242	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
243	C₄	—	3	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	6	1	no	—
246	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
247	C₄	C₃²×#239	1	1	yes	characteristic
248	C₃	—	2	4	no	—
249	C₃	C₃×#4872	1	4	yes	direct product factor
250	C₃	C₃²×C₄×D₈	1	1	yes	characteristic, complemented
251	C₂	—	12	4	no	—
252	C₂	C₃²×#893	1	1	yes	characteristic
253	C₂	C₃²×D₃×D₈	1	1	yes	characteristic
254	C₂	C₃²×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	D₃×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#107	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₂₄	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#107	D₃
hypocenter	C₃	C₃²×C₄×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#107

Complements

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

N	H ≅ G/N	type
2. C₃²×#893	C₂	semidirect
3. C₃³×#107	C₂	semidirect
4. C₃²×C₈×Dic₃	C₂	semidirect
5. C₃²×#809	C₂	semidirect
7. C₃×#4872	C₃	direct (N × H)
15. C₃²×D₂₄	C₄	semidirect
19. C₃×#893	C₂×C₃	semidirect
21. C₃²×#107	D₃	semidirect
22. C₃²×#107	C₂×C₃	semidirect
25. C₃×C₈×Dic₃	C₂×C₃	semidirect
26. C₃×#809	C₂×C₃	semidirect
33. C₃²×D₁₂	C₂×C₄	semidirect
35. C₃³×C₈	C₂×C₄	semidirect
38. #4872	C₃²	direct (N × H)
56. C₃×D₂₄	C₃×C₄	semidirect
65. C₃²×Dic₃	D₈	semidirect
69. #893	C₂×C₃²	semidirect
71. C₃×#107	C₃×D₃	semidirect
72. C₃×#107	C₂×C₃²	semidirect
75. C₈×Dic₃	C₂×C₃²	semidirect
76. #809	C₂×C₃²	semidirect
93. C₃×D₁₂	C₂×C₃×C₄	semidirect
97. C₃²×C₈	C₄×D₃	semidirect
98. C₃²×C₈	C₂×C₃×C₄	semidirect
122. D₂₄	C₃²×C₄	semidirect
144. C₃×Dic₃	C₃×D₈	semidirect
147. #107	C₃²×D₃	semidirect
150. C₃³	C₄×D₈	semidirect
167. D₁₂	C₂×C₃²×C₄	semidirect
171. C₃×C₈	C₃×C₄×D₃	semidirect
172. C₃×C₈	C₂×C₃²×C₄	semidirect
214. Dic₃	C₃²×D₈	semidirect
216. C₃²	#4872	direct (N × H)
217. C₃²	C₃×C₄×D₈	semidirect
227. C₈	C₃²×C₄×D₃	semidirect
249. C₃	C₃×#4872	direct (N × H)
250. C₃	C₃²×C₄×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₂₄	2
C₃²×#893	2
C₃³×#107	2
C₃²×C₈×Dic₃	2
C₃²×#809	2
C₃²×C₄×D₈	3
C₃×#4872	3

Subgroups — C32×#4872 #13162326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4872 #13162326