Subgroups — C₃²×#4848 #13162285

Order 1728.

920 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	C₂×C₃²×#204	C₂	1	2	yes	maximal, complemented
2	C₃²×#874	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#821	C₂	1	2	yes	maximal, complemented
6	C₃²×#460	—	3	1	no	maximal
7	C₃×#4848	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	—	2	1	no	—
9	C₂×C₃²×D₁₂	—	2	1	no	—
10	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
11	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	2	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₃²×Dic₃⋊C₄	—	2	1	no	—
16	C₃²×C₄×Dic₃	—	2	1	no	—
17	C₃²×#204	—	2	4	no	—
18	C₂×C₃²×SD₁₆	—	3	2	no	—
19	C₃²×#127	—	3	1	no	—
20	C₃²×C₄×Q₈	—	3	1	no	—
21	C₂×C₃×#204	C₂×C₃	1	8	yes	complemented
22	C₃×#874	C₂×C₃	1	4	yes	—
23	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	complemented
24	C₃²×C₄⋊C₈	—	2	4	no	—
25	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
27	C₃²×#102	—	3	2	no	—
28	C₃×#821	C₂×C₃	1	8	yes	complemented
29	C₂²×C₃²×D₃	—	2	2	no	—
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	—	2	1	no	—
34	C₂×C₃²×Dic₃	—	2	1	no	—
35	C₃²×D₁₂	—	2	2	no	—
36	C₃²×D₁₂	—	2	2	no	—
37	C₃²×D₁₂	—	4	2	no	—
38	C₃²×D₁₂	—	4	2	no	—
39	C₃²×Dic₆	—	2	1	no	—
40	C₃²×Dic₆	D₄	1	2	yes	complemented
41	C₃³×C₈	—	2	2	no	—
42	C₃×#460	—	3	4	no	—
43	#4848	C₃²	1	1	yes	characteristic, direct product factor
44	C₂×C₃²×Q₈	—	3	1	no	—
45	C₂×C₃²×D₄	—	3	2	no	—
46	C₂×C₃²×D₄	—	6	1	no	—
47	C₂×C₃²×D₄	—	6	1	no	—
48	C₂×C₃×D₁₂	—	2	4	no	—
49	C₂×C₃×D₁₂	—	2	4	no	—
50	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
51	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
52	C₃²×SD₁₆	—	6	4	no	—
53	C₂×C₃²×C₈	—	2	8	no	—
54	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
55	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
56	C₃²×C₄⋊C₄	—	3	1	no	—
57	C₃²×C₄⋊C₄	—	6	1	no	—
58	C₃²×C₄²	—	2	4	no	—
59	C₃²×C₄²	—	6	1	no	—
60	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
61	C₃²×C₄²	C₂²×C₃	1	4	yes	—
62	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
63	C₃×Dic₃⋊C₄	—	2	4	no	—
64	C₃×C₄×Dic₃	—	2	4	no	—
65	C₃×#204	—	2	16	no	—
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×D₃	—	4	2	no	—
68	C₂×C₃²×D₃	—	4	2	no	—
69	C₂×C₃²×D₃	—	4	2	no	—
70	C₃³×C₄	—	2	1	no	—
71	C₃³×C₄	SD₁₆	1	2	yes	complemented
72	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
73	C₃³×C₄	#41	1	1	yes	characteristic
74	C₃²×Dic₃	—	2	2	no	—
75	C₃²×Dic₃	—	4	1	no	—
76	C₂×C₃×SD₁₆	—	3	8	no	—
77	C₃×#127	—	3	4	no	—
78	C₃×C₄×Q₈	—	3	4	no	—
79	C₂×#204	C₂×C₃²	1	2	yes	complemented
80	#874	C₂×C₃²	1	1	yes	characteristic
81	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic, complemented
82	C₃×C₄⋊C₈	—	2	4	no	—
83	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
84	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
85	C₃×#102	—	3	8	no	—
86	#821	C₂×C₃²	1	2	yes	complemented
87	C₂³×C₃²	—	6	2	no	—
88	C₂²×C₃×D₃	—	2	8	no	—
89	C₃²×Q₈	—	3	2	no	—
90	C₃²×Q₈	—	6	1	no	—
91	C₃²×D₄	—	6	2	no	—
92	C₃²×D₄	—	6	2	no	—
93	C₃²×D₄	—	12	2	no	—
94	C₃²×D₄	—	12	2	no	—
95	C₂×C₃²×C₄	—	2	4	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	—	6	1	no	—
99	C₂×C₃²×C₄	—	6	1	no	—
100	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
101	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
103	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
106	C₂×C₃×Dic₃	—	2	4	no	—
107	C₂×C₃×Dic₃	—	2	4	no	—
108	C₃×D₁₂	—	2	8	no	—
109	C₃×D₁₂	—	2	8	no	—
110	C₃×D₁₂	—	4	8	no	—
111	C₃×D₁₂	—	4	8	no	—
112	C₃×Dic₆	—	2	4	no	—
113	C₃×Dic₆	C₃×D₄	1	8	yes	complemented
114	C₃²×C₈	—	2	2	no	—
115	C₃²×C₈	—	2	8	no	—
116	C₃²×C₈	—	4	8	no	—
117	#460	—	3	1	no	Sylow
118	C₂×C₃³	#121	1	1	yes	characteristic
119	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₃²×D₃	—	8	2	no	—
122	C₂×C₃×Q₈	—	3	4	no	—
123	C₂×C₃×D₄	—	3	8	no	—
124	C₂×C₃×D₄	—	6	4	no	—
125	C₂×C₃×D₄	—	6	4	no	—
126	C₂×D₁₂	—	2	1	no	—
127	C₂×D₁₂	—	2	1	no	—
128	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
129	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
130	C₃×SD₁₆	—	6	16	no	—
131	C₂×C₃×C₈	—	2	8	no	—
132	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
133	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
134	C₃×C₄⋊C₄	—	3	4	no	—
135	C₃×C₄⋊C₄	—	6	4	no	—
136	C₃×C₄²	—	2	4	no	—
137	C₃×C₄²	—	6	4	no	—
138	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
139	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
140	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
141	Dic₃⋊C₄	—	2	1	no	—
142	C₄×Dic₃	—	2	1	no	—
143	#204	—	2	4	no	—
144	C₂²×C₃²	—	2	4	no	—
145	C₂²×C₃²	—	12	2	no	—
146	C₂²×C₃²	—	12	2	no	—
147	C₂²×C₃²	—	12	2	no	—
148	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
149	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
150	C₂×C₃×D₃	—	4	8	no	—
151	C₂×C₃×D₃	—	4	8	no	—
152	C₂×C₃×D₃	—	4	8	no	—
153	C₃²×C₄	—	2	1	no	—
154	C₃²×C₄	—	2	8	no	—
155	C₃²×C₄	—	2	4	no	—
156	C₃²×C₄	—	2	4	no	—
157	C₃²×C₄	—	2	4	no	—
158	C₃²×C₄	—	4	4	no	—
159	C₃²×C₄	—	6	2	no	—
160	C₃²×C₄	—	12	1	no	—
161	C₃²×C₄	#204	1	2	yes	complemented
162	C₃²×C₄	C₃×SD₁₆	1	8	yes	complemented
163	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
164	C₃²×C₄	#239	1	1	yes	characteristic
165	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
166	C₃²×C₄	C₃×#41	1	4	yes	—
167	C₃×Dic₃	—	2	8	no	—
168	C₃×Dic₃	—	4	4	no	—
169	C₂×SD₁₆	—	3	2	no	—
170	#127	—	3	1	no	—
171	C₄×Q₈	—	3	1	no	—
172	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
173	#102	—	3	2	no	—
174	C₃³	#460	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	6	8	no	—
176	C₂²×D₃	—	2	2	no	—
177	C₃×Q₈	—	3	8	no	—
178	C₃×Q₈	—	6	4	no	—
179	C₃×D₄	—	6	8	no	—
180	C₃×D₄	—	6	8	no	—
181	C₃×D₄	—	12	8	no	—
182	C₃×D₄	—	12	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	6	4	no	—
187	C₂×C₃×C₄	—	6	4	no	—
188	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
189	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
190	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
193	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
194	C₂×Dic₃	—	2	1	no	—
195	C₂×Dic₃	—	2	1	no	—
196	D₁₂	—	2	2	no	—
197	D₁₂	—	2	2	no	—
198	D₁₂	—	4	2	no	—
199	D₁₂	—	4	2	no	—
200	Dic₆	—	2	1	no	—
201	Dic₆	C₃²×D₄	1	2	yes	complemented
202	C₃×C₈	—	2	8	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	4	8	no	—
205	C₂×C₃²	—	2	4	no	—
206	C₂×C₃²	—	2	4	no	—
207	C₂×C₃²	—	2	4	no	—
208	C₂×C₃²	—	24	2	no	—
209	C₂×C₃²	#895	1	1	yes	characteristic
210	C₂×C₃²	C₂×#204	1	1	yes	characteristic
211	C₂×C₃²	#908	1	1	yes	characteristic
212	C₂×C₃²	C₃×#121	1	4	yes	—
213	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
214	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
215	C₃×D₃	—	8	8	no	—
216	C₂×Q₈	—	3	1	no	—
217	C₂×D₄	—	3	2	no	—
218	C₂×D₄	—	6	1	no	—
219	C₂×D₄	—	6	1	no	—
220	SD₁₆	—	6	4	no	—
221	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄²	—	6	1	no	—
225	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	12	8	no	—
228	C₂²×C₃	—	12	8	no	—
229	C₂²×C₃	—	12	8	no	—
230	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
231	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
232	C₂×D₃	—	4	2	no	—
233	C₂×D₃	—	4	2	no	—
234	C₂×D₃	—	4	2	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	8	no	—
240	C₃×C₄	—	4	4	no	—
241	C₃×C₄	—	6	8	no	—
242	C₃×C₄	—	12	4	no	—
243	C₃×C₄	C₃×#204	1	8	yes	complemented
244	C₃×C₄	C₃²×SD₁₆	1	2	yes	complemented
245	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
246	C₃×C₄	C₃×#239	1	4	yes	—
247	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
248	C₃×C₄	C₃²×#41	1	1	yes	characteristic
249	Dic₃	—	2	2	no	—
250	Dic₃	—	4	1	no	—
251	C₃²	—	2	4	no	—
252	C₃²	#4848	1	1	yes	characteristic, direct product factor
253	C₃²	C₃×#460	1	4	yes	complemented
254	C₂³	—	6	2	no	—
255	Q₈	—	3	2	no	—
256	Q₈	—	6	1	no	—
257	D₄	—	6	2	no	—
258	D₄	—	6	2	no	—
259	D₄	—	12	2	no	—
260	D₄	—	12	2	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	6	1	no	—
263	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
264	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₂×C₃	—	2	4	no	—
268	C₂×C₃	—	2	4	no	—
269	C₂×C₃	—	2	4	no	—
270	C₂×C₃	—	24	8	no	—
271	C₂×C₃	C₃×#895	1	4	yes	—
272	C₂×C₃	C₂×C₃×#204	1	4	yes	—
273	C₂×C₃	C₃×#908	1	4	yes	—
274	C₂×C₃	C₃²×#121	1	1	yes	characteristic
275	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
276	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
277	D₃	—	8	2	no	—
278	C₂²	—	12	2	no	—
279	C₂²	—	12	2	no	—
280	C₂²	—	12	2	no	—
281	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	6	2	no	—
284	C₄	—	12	1	no	—
285	C₄	C₃²×#204	1	2	yes	complemented
286	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
287	C₄	C₃²×#239	1	1	yes	characteristic
288	C₃	—	2	4	no	—
289	C₃	C₃×#4848	1	4	yes	direct product factor
290	C₃	C₃²×#460	1	1	yes	characteristic, complemented
291	C₂	—	24	2	no	—
292	C₂	C₃²×#895	1	1	yes	characteristic
293	C₂	C₂×C₃²×#204	1	1	yes	characteristic
294	C₂	C₃²×#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃²×#874	C₂
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#460
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#204	C₂	semidirect
3. C₃²×C₄×Dic₆	C₂	semidirect
4. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃²×#821	C₂	semidirect
7. C₃×#4848	C₃	direct (N × H)
21. C₂×C₃×#204	C₂×C₃	semidirect
23. C₃×C₄×Dic₆	C₂×C₃	semidirect
25. C₃²×C₄⋊C₈	D₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
28. C₃×#821	C₂×C₃	semidirect
40. C₃²×Dic₆	D₄	semidirect
43. #4848	C₃²	direct (N × H)
71. C₃³×C₄	SD₁₆	semidirect
79. C₂×#204	C₂×C₃²	semidirect
81. C₄×Dic₆	C₂×C₃²	semidirect
83. C₃×C₄⋊C₈	C₃×D₃	semidirect
84. C₃×C₄⋊C₈	C₂×C₃²	semidirect
86. #821	C₂×C₃²	semidirect
113. C₃×Dic₆	C₃×D₄	semidirect
161. C₃²×C₄	#204	semidirect
162. C₃²×C₄	C₃×SD₁₆	semidirect
172. C₄⋊C₈	C₃²×D₃	semidirect
174. C₃³	#460	semidirect
201. Dic₆	C₃²×D₄	semidirect
243. C₃×C₄	C₃×#204	semidirect
244. C₃×C₄	C₃²×SD₁₆	semidirect
252. C₃²	#4848	direct (N × H)
253. C₃²	C₃×#460	semidirect
285. C₄	C₃²×#204	semidirect
289. C₃	C₃×#4848	direct (N × H)
290. C₃	C₃²×#460	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#204	2
C₃²×#874	2
C₃²×C₄×Dic₆	2
C₃³×C₄⋊C₈	2
C₃²×#821	2
C₃²×#460	3
C₃×#4848	3

Subgroups — C32×#4848 #13162285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4848 #13162285