Subgroups — C₃²×#4834 #13162273

Order 1728.

956 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×C₈	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₈×D₄	—	3	1	no	maximal
7	C₃×#4834	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	2	yes	—
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃²×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
13	C₃²×C₄⋊Dic₃	C₄	1	1	yes	characteristic
14	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×D₃×C₈	—	2	4	no	—
17	C₂²×C₃²×C₈	—	3	2	no	—
18	C₃²×C₄×D₄	—	3	1	no	—
19	C₂×C₃×D₃×C₈	C₂×C₃	1	8	yes	complemented
20	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
21	C₃²×C₄⋊C₈	—	2	4	no	—
22	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
23	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
24	C₃²×C₂²⋊C₈	—	3	2	no	—
25	C₃²×C₄×C₈	—	3	1	no	—
26	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	8	yes	complemented
27	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
28	C₂²×C₃²×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
33	C₃²×D₁₂	C₈	1	4	yes	complemented
34	C₃²×C₄×D₃	—	2	2	no	—
35	C₃²×C₄×D₃	—	2	2	no	—
36	C₃³×C₈	—	2	2	no	—
37	C₃²×C₃⋊C₈	—	2	1	no	—
38	C₃²×C₃⋊C₈	D₄	1	2	yes	complemented
39	C₃×C₈×D₄	—	3	4	no	—
40	#4834	C₃²	1	1	yes	characteristic, direct product factor
41	C₂×C₃²×D₄	—	3	1	no	—
42	C₂²×C₃²×C₄	—	3	2	no	—
43	C₂×C₃×D₁₂	C₃×C₄	1	4	yes	—
44	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
45	C₂×C₃²×C₈	—	2	8	no	—
46	C₂×C₃²×C₈	—	3	1	no	—
47	C₂×C₃²×C₈	—	3	1	no	—
48	C₂×C₃²×C₈	—	6	4	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
50	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
51	C₃²×C₄⋊C₄	—	3	1	no	—
52	C₃²×C₂²⋊C₄	—	3	2	no	—
53	C₃²×C₄²	—	2	4	no	—
54	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₃²×C₄²	C₂²×C₃	1	4	yes	—
56	C₃×(C₂×D₃)⋊C₄	C₃×C₄	1	8	yes	—
57	C₃×C₄⋊Dic₃	C₃×C₄	1	4	yes	—
58	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
59	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
60	C₃×D₃×C₈	—	2	16	no	—
61	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×D₃	—	2	2	no	—
63	C₂×C₃²×D₃	—	2	2	no	—
64	C₂×C₃²×D₃	C₂×C₈	1	4	yes	complemented
65	C₃³×C₄	—	2	1	no	—
66	C₃³×C₄	C₂×C₈	1	2	yes	complemented
67	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃³×C₄	#41	1	1	yes	characteristic
69	C₃²×Dic₃	—	2	2	no	—
70	C₂²×C₃×C₈	—	3	8	no	—
71	C₃×C₄×D₄	—	3	4	no	—
72	C₂×D₃×C₈	C₂×C₃²	1	2	yes	complemented
73	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
74	C₃×C₄⋊C₈	—	2	4	no	—
75	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
76	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
77	C₃×C₂²⋊C₈	—	3	8	no	—
78	C₃×C₄×C₈	—	3	4	no	—
79	(C₂×D₃)⋊C₈	C₂×C₃²	1	2	yes	complemented
80	C₄×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₂³×C₃²	—	3	2	no	—
82	C₂²×C₃×D₃	C₂×C₃×C₄	1	8	yes	—
83	C₃²×D₄	—	3	4	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	3	2	no	—
88	C₂×C₃²×C₄	—	6	2	no	—
89	C₂×C₃²×C₄	—	6	2	no	—
90	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
96	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
97	C₃×D₁₂	C₃×C₈	1	16	yes	complemented
98	C₃×C₄×D₃	—	2	8	no	—
99	C₃×C₄×D₃	—	2	8	no	—
100	C₃²×C₈	—	2	2	no	—
101	C₃²×C₈	—	2	8	no	—
102	C₃²×C₈	—	3	2	no	—
103	C₃²×C₈	—	4	8	no	—
104	C₃²×C₈	—	6	1	no	—
105	C₃×C₃⋊C₈	—	2	4	no	—
106	C₃×C₃⋊C₈	C₃×D₄	1	8	yes	complemented
107	C₈×D₄	—	3	1	no	Sylow
108	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₃³	#131	1	1	yes	characteristic
111	C₃²×D₃	—	2	4	no	—
112	C₂×C₃×D₄	—	3	4	no	—
113	C₂²×C₃×C₄	—	3	8	no	—
114	C₂×D₁₂	C₃²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
116	C₂×C₃×C₈	—	2	8	no	—
117	C₂×C₃×C₈	—	3	4	no	—
118	C₂×C₃×C₈	—	3	4	no	—
119	C₂×C₃×C₈	—	6	16	no	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
121	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
122	C₃×C₄⋊C₄	—	3	4	no	—
123	C₃×C₂²⋊C₄	—	3	8	no	—
124	C₃×C₄²	—	2	4	no	—
125	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
126	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	C₃²×C₄	1	2	yes	—
128	C₄⋊Dic₃	C₃²×C₄	1	1	yes	characteristic
129	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
130	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
131	D₃×C₈	—	2	4	no	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	—	3	4	no	—
134	C₂²×C₃²	—	6	2	no	—
135	C₂²×C₃²	—	6	2	no	—
136	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
138	C₂×C₃×D₃	—	2	8	no	—
139	C₂×C₃×D₃	—	2	8	no	—
140	C₂×C₃×D₃	C₂×C₃×C₈	1	16	yes	complemented
141	C₃²×C₄	—	2	1	no	—
142	C₃²×C₄	—	2	8	no	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	4	4	no	—
147	C₃²×C₄	—	6	2	no	—
148	C₃²×C₄	D₃×C₈	1	2	yes	complemented
149	C₃²×C₄	C₂×C₃×C₈	1	8	yes	complemented
150	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
151	C₃²×C₄	#239	1	1	yes	characteristic
152	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
153	C₃²×C₄	C₃×#41	1	4	yes	—
154	C₃×Dic₃	—	2	8	no	—
155	C₂²×C₈	—	3	2	no	—
156	C₄×D₄	—	3	1	no	—
157	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
158	C₂²⋊C₈	—	3	2	no	—
159	C₄×C₈	—	3	1	no	—
160	C₃³	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	3	8	no	—
162	C₂²×D₃	C₂×C₃²×C₄	1	2	yes	—
163	C₃×D₄	—	3	16	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	3	8	no	—
168	C₂×C₃×C₄	—	6	8	no	—
169	C₂×C₃×C₄	—	6	8	no	—
170	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
171	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
175	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
176	C₂×Dic₃	C₂×C₃²×C₄	1	2	yes	—
177	D₁₂	C₃²×C₈	1	4	yes	complemented
178	C₄×D₃	—	2	2	no	—
179	C₄×D₃	—	2	2	no	—
180	C₃×C₈	—	2	8	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	3	8	no	—
183	C₃×C₈	—	4	8	no	—
184	C₃×C₈	—	6	4	no	—
185	C₃⋊C₈	—	2	1	no	—
186	C₃⋊C₈	C₃²×D₄	1	2	yes	complemented
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	—	2	4	no	—
189	C₂×C₃²	—	2	4	no	—
190	C₂×C₃²	—	6	4	no	—
191	C₂×C₃²	#893	1	1	yes	characteristic
192	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
193	C₂×C₃²	#907	1	1	yes	characteristic
194	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
195	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
196	C₂×C₃²	C₃×#131	1	4	yes	—
197	C₃×D₃	—	2	16	no	—
198	C₂×D₄	—	3	1	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	—	3	1	no	—
202	C₂×C₈	—	6	4	no	—
203	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
204	C₄⋊C₄	—	3	1	no	—
205	C₂²⋊C₄	—	3	2	no	—
206	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	3	16	no	—
209	C₂²×C₃	—	6	8	no	—
210	C₂²×C₃	—	6	8	no	—
211	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
212	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
213	C₂×D₃	—	2	2	no	—
214	C₂×D₃	—	2	2	no	—
215	C₂×D₃	C₂×C₃²×C₈	1	4	yes	complemented
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	8	no	—
221	C₃×C₄	—	4	4	no	—
222	C₃×C₄	—	6	8	no	—
223	C₃×C₄	C₃×D₃×C₈	1	8	yes	complemented
224	C₃×C₄	C₂×C₃²×C₈	1	2	yes	complemented
225	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
226	C₃×C₄	C₃×#239	1	4	yes	—
227	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
228	C₃×C₄	C₃²×#41	1	1	yes	characteristic
229	Dic₃	—	2	2	no	—
230	C₃²	—	2	4	no	—
231	C₃²	#4834	1	1	yes	characteristic, direct product factor
232	C₃²	C₃×C₈×D₄	1	4	yes	complemented
233	C₂³	—	3	2	no	—
234	D₄	—	3	4	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
239	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	3	2	no	—
243	C₈	—	6	1	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	2	4	no	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	—	6	16	no	—
248	C₂×C₃	C₃×#893	1	4	yes	—
249	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
250	C₂×C₃	C₃×#907	1	4	yes	—
251	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
252	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
253	C₂×C₃	C₃²×#131	1	1	yes	characteristic
254	D₃	—	2	4	no	—
255	C₂²	—	3	4	no	—
256	C₂²	—	6	2	no	—
257	C₂²	—	6	2	no	—
258	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	6	2	no	—
261	C₄	C₃²×D₃×C₈	1	2	yes	complemented
262	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
263	C₄	C₃²×#239	1	1	yes	characteristic
264	C₃	—	2	4	no	—
265	C₃	C₃×#4834	1	4	yes	direct product factor
266	C₃	C₃²×C₈×D₄	1	1	yes	characteristic, complemented
267	C₂	—	6	4	no	—
268	C₂	C₃²×#893	1	1	yes	characteristic
269	C₂	C₂×C₃²×D₃×C₈	1	1	yes	characteristic
270	C₂	C₃²×#907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂×C₃²×D₁₂	C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×C₈×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₃×C₈	C₂	semidirect
3. C₃³×C₄⋊C₈	C₂	semidirect
4. C₃²×(C₂×D₃)⋊C₈	C₂	semidirect
5. C₃²×C₄×C₃⋊C₈	C₂	semidirect
7. C₃×#4834	C₃	direct (N × H)
19. C₂×C₃×D₃×C₈	C₂×C₃	semidirect
22. C₃²×C₄⋊C₈	D₃	semidirect
23. C₃²×C₄⋊C₈	C₂×C₃	semidirect
26. C₃×(C₂×D₃)⋊C₈	C₂×C₃	semidirect
27. C₃×C₄×C₃⋊C₈	C₂×C₃	semidirect
33. C₃²×D₁₂	C₈	semidirect
38. C₃²×C₃⋊C₈	D₄	semidirect
40. #4834	C₃²	direct (N × H)
64. C₂×C₃²×D₃	C₂×C₈	semidirect
66. C₃³×C₄	C₂×C₈	semidirect
72. C₂×D₃×C₈	C₂×C₃²	semidirect
75. C₃×C₄⋊C₈	C₃×D₃	semidirect
76. C₃×C₄⋊C₈	C₂×C₃²	semidirect
79. (C₂×D₃)⋊C₈	C₂×C₃²	semidirect
80. C₄×C₃⋊C₈	C₂×C₃²	semidirect
97. C₃×D₁₂	C₃×C₈	semidirect
106. C₃×C₃⋊C₈	C₃×D₄	semidirect
140. C₂×C₃×D₃	C₂×C₃×C₈	semidirect
148. C₃²×C₄	D₃×C₈	semidirect
149. C₃²×C₄	C₂×C₃×C₈	semidirect
157. C₄⋊C₈	C₃²×D₃	semidirect
160. C₃³	C₈×D₄	semidirect
177. D₁₂	C₃²×C₈	semidirect
186. C₃⋊C₈	C₃²×D₄	semidirect
215. C₂×D₃	C₂×C₃²×C₈	semidirect
223. C₃×C₄	C₃×D₃×C₈	semidirect
224. C₃×C₄	C₂×C₃²×C₈	semidirect
231. C₃²	#4834	direct (N × H)
232. C₃²	C₃×C₈×D₄	semidirect
261. C₄	C₃²×D₃×C₈	semidirect
265. C₃	C₃×#4834	direct (N × H)
266. C₃	C₃²×C₈×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₃×C₈	2
C₃²×C₄×D₁₂	2
C₃³×C₄⋊C₈	2
C₃²×(C₂×D₃)⋊C₈	2
C₃²×C₄×C₃⋊C₈	2
C₃²×C₈×D₄	3
C₃×#4834	3

Subgroups — C32×#4834 #13162273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4834 #13162273