Subgroups — C₃²×#4833 #13162272

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
4	C₃²×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#431	—	3	1	no	maximal
7	C₃×#4833	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄×D₃	C₄	1	1	yes	characteristic
10	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₃²×C₄×Dic₃	C₄	1	1	yes	characteristic
15	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃²×C₄²	—	3	1	no	—
18	C₃×C₄²×D₃	C₂×C₃	1	4	yes	—
19	C₃²×C₄⋊C₈	—	2	4	no	—
20	C₃²×C₄⋊C₈	—	3	1	no	—
21	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
22	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
23	C₃²×C₂²⋊C₈	—	3	2	no	—
24	C₃²×C₄×C₈	—	3	2	no	—
25	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	8	yes	—
26	C₃×C₈×Dic₃	C₂×C₃	1	8	yes	complemented
27	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	complemented
28	C₂²×C₃²×D₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
33	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₃²×C₄×D₃	—	2	1	no	—
36	C₃²×C₄×D₃	—	2	1	no	—
37	C₃²×C₄×D₃	—	2	2	no	—
38	C₃²×C₄×D₃	C₈	1	4	yes	complemented
39	C₃³×C₈	—	2	2	no	—
40	C₃²×C₃⋊C₈	—	2	2	no	—
41	C₃×#431	—	3	4	no	—
42	#4833	C₃²	1	1	yes	characteristic, direct product factor
43	C₂²×C₃²×C₄	—	3	1	no	—
44	C₂²×C₃²×C₄	—	3	1	no	—
45	C₂²×C₃²×C₄	—	3	1	no	—
46	C₂×C₃×C₄×D₃	C₃×C₄	1	4	yes	—
47	C₂×C₃×C₄×D₃	C₃×C₄	1	4	yes	—
48	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
49	C₂×C₃²×C₈	—	2	8	no	—
50	C₂×C₃²×C₈	—	3	2	no	—
51	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
52	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
53	C₃²×C₄²	—	2	4	no	—
54	C₃²×C₄²	—	3	1	no	—
55	C₃²×C₄²	—	3	1	no	—
56	C₃²×C₄²	—	3	1	no	—
57	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₃²×C₄²	C₂²×C₃	1	4	yes	—
59	C₃×C₄×Dic₃	C₃×C₄	1	4	yes	—
60	C₃×C₄×Dic₃	C₃×C₄	1	4	yes	—
61	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
62	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
63	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×D₃	—	2	1	no	—
65	C₂×C₃²×D₃	—	2	1	no	—
66	C₂×C₃²×D₃	C₂×C₈	1	2	yes	—
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	C₂×C₈	1	2	yes	—
69	C₃³×C₄	#41	1	1	yes	characteristic
70	C₃³×C₄	#41	1	1	yes	characteristic
71	C₃²×Dic₃	—	2	1	no	—
72	C₃²×Dic₃	C₂×C₈	1	2	yes	—
73	C₃²×Dic₃	M₁₆	1	2	yes	—
74	C₂×C₃×C₄²	—	3	4	no	—
75	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic
76	C₃×C₄⋊C₈	—	2	4	no	—
77	C₃×C₄⋊C₈	—	3	4	no	—
78	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
79	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
80	C₃×C₂²⋊C₈	—	3	8	no	—
81	C₃×C₄×C₈	—	3	8	no	—
82	(C₂×D₃)⋊C₈	C₂×C₃²	1	2	yes	—
83	C₈×Dic₃	C₂×C₃²	1	2	yes	complemented
84	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic, complemented
85	C₂³×C₃²	—	3	1	no	—
86	C₂²×C₃×D₃	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	3	1	no	—
91	C₂×C₃²×C₄	—	3	1	no	—
92	C₂×C₃²×C₄	—	3	4	no	—
93	C₂×C₃²×C₄	—	3	1	no	—
94	C₂×C₃²×C₄	—	6	2	no	—
95	C₂×C₃²×C₄	—	6	1	no	—
96	C₂×C₃²×C₄	—	6	1	no	—
97	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
100	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
101	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
103	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
104	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
105	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
106	C₃×C₄×D₃	—	2	4	no	—
107	C₃×C₄×D₃	—	2	4	no	—
108	C₃×C₄×D₃	—	2	8	no	—
109	C₃×C₄×D₃	C₃×C₈	1	16	yes	complemented
110	C₃²×C₈	—	2	2	no	—
111	C₃²×C₈	—	2	8	no	—
112	C₃²×C₈	—	4	8	no	—
113	C₃²×C₈	—	6	2	no	—
114	C₃×C₃⋊C₈	—	2	8	no	—
115	#431	—	3	1	no	Sylow
116	C₂×C₃³	#117	1	1	yes	characteristic
117	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
119	C₃²×D₃	—	2	2	no	—
120	C₂²×C₃×C₄	—	3	4	no	—
121	C₂²×C₃×C₄	—	3	4	no	—
122	C₂²×C₃×C₄	—	3	4	no	—
123	C₂×C₄×D₃	C₃²×C₄	1	1	yes	characteristic
124	C₂×C₄×D₃	C₃²×C₄	1	1	yes	characteristic
125	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
126	C₂×C₃×C₈	—	2	8	no	—
127	C₂×C₃×C₈	—	3	8	no	—
128	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
129	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
130	C₃×C₄²	—	2	4	no	—
131	C₃×C₄²	—	3	4	no	—
132	C₃×C₄²	—	3	4	no	—
133	C₃×C₄²	—	3	4	no	—
134	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
135	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
136	C₄×Dic₃	C₃²×C₄	1	1	yes	characteristic
137	C₄×Dic₃	C₃²×C₄	1	1	yes	characteristic
138	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
139	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
140	C₂²×C₃²	—	2	4	no	—
141	C₂²×C₃²	—	3	2	no	—
142	C₂²×C₃²	—	6	1	no	—
143	C₂²×C₃²	—	6	1	no	—
144	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
146	C₂×C₃×D₃	—	2	4	no	—
147	C₂×C₃×D₃	—	2	4	no	—
148	C₂×C₃×D₃	C₂×C₃×C₈	1	8	yes	—
149	C₃²×C₄	—	2	1	no	—
150	C₃²×C₄	—	2	8	no	—
151	C₃²×C₄	—	2	4	no	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	—	2	4	no	—
154	C₃²×C₄	—	3	2	no	—
155	C₃²×C₄	—	3	2	no	—
156	C₃²×C₄	—	4	4	no	—
157	C₃²×C₄	—	6	1	no	—
158	C₃²×C₄	D₃×C₈	1	2	yes	—
159	C₃²×C₄	C₂×C₃×C₈	1	8	yes	—
160	C₃²×C₄	#237	1	1	yes	characteristic
161	C₃²×C₄	#239	1	1	yes	characteristic
162	C₃²×C₄	C₃×#41	1	4	yes	—
163	C₃²×C₄	C₃×#41	1	4	yes	—
164	C₃×Dic₃	—	2	4	no	—
165	C₃×Dic₃	C₂×C₃×C₈	1	8	yes	—
166	C₃×Dic₃	C₃×M₁₆	1	8	yes	—
167	C₂×C₄²	—	3	1	no	—
168	C₄⋊C₈	—	3	1	no	—
169	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
170	C₂²⋊C₈	—	3	2	no	—
171	C₄×C₈	—	3	2	no	—
172	C₃³	#431	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₃	—	3	4	no	—
174	C₂²×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	3	4	no	—
179	C₂×C₃×C₄	—	3	4	no	—
180	C₂×C₃×C₄	—	3	16	no	—
181	C₂×C₃×C₄	—	3	4	no	—
182	C₂×C₃×C₄	—	6	8	no	—
183	C₂×C₃×C₄	—	6	4	no	—
184	C₂×C₃×C₄	—	6	4	no	—
185	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
186	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
187	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
190	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
191	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
192	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
193	C₂×Dic₃	C₂×C₃²×C₄	1	1	yes	characteristic
194	C₄×D₃	—	2	1	no	—
195	C₄×D₃	—	2	1	no	—
196	C₄×D₃	—	2	2	no	—
197	C₄×D₃	C₃²×C₈	1	4	yes	complemented
198	C₃×C₈	—	2	8	no	—
199	C₃×C₈	—	2	2	no	—
200	C₃×C₈	—	4	8	no	—
201	C₃×C₈	—	6	8	no	—
202	C₃⋊C₈	—	2	2	no	—
203	C₂×C₃²	—	2	4	no	—
204	C₂×C₃²	—	2	4	no	—
205	C₂×C₃²	—	2	4	no	—
206	C₂×C₃²	—	6	2	no	—
207	C₂×C₃²	#892	1	1	yes	characteristic
208	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
209	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
210	C₂×C₃²	C₃×#117	1	4	yes	—
211	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
212	C₂×C₃²	C₂×C₃×M₁₆	1	4	yes	—
213	C₃×D₃	—	2	8	no	—
214	C₂²×C₄	—	3	1	no	—
215	C₂²×C₄	—	3	1	no	—
216	C₂²×C₄	—	3	1	no	—
217	C₂×C₈	—	3	2	no	—
218	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
219	C₄²	—	3	1	no	—
220	C₄²	—	3	1	no	—
221	C₄²	—	3	1	no	—
222	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	3	8	no	—
225	C₂²×C₃	—	6	4	no	—
226	C₂²×C₃	—	6	4	no	—
227	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
228	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
229	C₂×D₃	—	2	1	no	—
230	C₂×D₃	—	2	1	no	—
231	C₂×D₃	C₂×C₃²×C₈	1	2	yes	—
232	C₃×C₄	—	2	4	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	8	no	—
237	C₃×C₄	—	3	8	no	—
238	C₃×C₄	—	3	8	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	—	6	4	no	—
241	C₃×C₄	C₃×D₃×C₈	1	8	yes	—
242	C₃×C₄	C₂×C₃²×C₈	1	2	yes	—
243	C₃×C₄	C₃×#237	1	4	yes	—
244	C₃×C₄	C₃×#239	1	4	yes	—
245	C₃×C₄	C₃²×#41	1	1	yes	characteristic
246	C₃×C₄	C₃²×#41	1	1	yes	characteristic
247	Dic₃	—	2	1	no	—
248	Dic₃	C₂×C₃²×C₈	1	2	yes	—
249	Dic₃	C₃²×M₁₆	1	2	yes	—
250	C₃²	—	2	4	no	—
251	C₃²	#4833	1	1	yes	characteristic, direct product factor
252	C₃²	C₃×#431	1	4	yes	complemented
253	C₂³	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	4	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	—	6	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
262	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	6	2	no	—
266	C₂×C₃	—	2	4	no	—
267	C₂×C₃	—	2	4	no	—
268	C₂×C₃	—	2	4	no	—
269	C₂×C₃	—	6	8	no	—
270	C₂×C₃	C₃×#892	1	4	yes	—
271	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
272	C₂×C₃	C₃×D₃×M₁₆	1	4	yes	—
273	C₂×C₃	C₃²×#117	1	1	yes	characteristic
274	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
275	C₂×C₃	C₂×C₃²×M₁₆	1	1	yes	characteristic
276	D₃	—	2	2	no	—
277	C₂²	—	3	2	no	—
278	C₂²	—	6	1	no	—
279	C₂²	—	6	1	no	—
280	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	3	2	no	—
283	C₄	—	3	2	no	—
284	C₄	—	6	1	no	—
285	C₄	C₃²×D₃×C₈	1	2	yes	—
286	C₄	C₃²×#237	1	1	yes	characteristic
287	C₄	C₃²×#239	1	1	yes	characteristic
288	C₃	—	2	4	no	—
289	C₃	C₃×#4833	1	4	yes	direct product factor
290	C₃	C₃²×#431	1	1	yes	characteristic, complemented
291	C₂	—	6	2	no	—
292	C₂	C₃²×#892	1	1	yes	characteristic
293	C₂	C₂×C₃²×D₃×C₈	1	1	yes	characteristic
294	C₂	C₃²×D₃×M₁₆	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	C₂²×C₃²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#431
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
2. C₃³×C₄⋊C₈	C₂	semidirect
4. C₃²×C₈×Dic₃	C₂	semidirect
5. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
7. C₃×#4833	C₃	direct (N × H)
21. C₃²×C₄⋊C₈	D₃	semidirect
22. C₃²×C₄⋊C₈	C₂×C₃	semidirect
26. C₃×C₈×Dic₃	C₂×C₃	semidirect
27. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
38. C₃²×C₄×D₃	C₈	semidirect
42. #4833	C₃²	direct (N × H)
78. C₃×C₄⋊C₈	C₃×D₃	semidirect
79. C₃×C₄⋊C₈	C₂×C₃²	semidirect
83. C₈×Dic₃	C₂×C₃²	semidirect
84. C₄⋊(C₃⋊C₈)	C₂×C₃²	semidirect
109. C₃×C₄×D₃	C₃×C₈	semidirect
169. C₄⋊C₈	C₃²×D₃	semidirect
172. C₃³	#431	semidirect
197. C₄×D₃	C₃²×C₈	semidirect
251. C₃²	#4833	direct (N × H)
252. C₃²	C₃×#431	semidirect
289. C₃	C₃×#4833	direct (N × H)
290. C₃	C₃²×#431	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄²×D₃	2
C₃³×C₄⋊C₈	2
C₃²×(C₂×D₃)⋊C₈	2
C₃²×C₈×Dic₃	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#431	3
C₃×#4833	3

Subgroups — C32×#4833 #13162272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4833 #13162272