Subgroups — C₃²×#5083 #13162523

Order 1728.

992 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#215	C₂	1	2	yes	maximal
3	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
4	C₃²×#835	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#461	—	3	1	no	maximal
7	C₃×#5083	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×D₃	—	2	1	no	—
11	C₃³×C₄⋊C₄	—	2	1	no	—
12	C₃³×C₄⋊C₄	—	2	1	no	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃²×#215	—	2	4	no	—
15	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃²×SD₁₆	—	3	2	no	—
19	C₃²×C₄⋊Q₈	—	2	4	no	—
20	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
21	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
22	C₃²×C₄×D₄	—	3	1	no	—
23	C₂×C₃×#215	C₂×C₃	1	8	yes	—
24	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
25	C₃²×C₄⋊C₈	—	3	1	no	—
26	C₃²×#103	—	3	2	no	—
27	C₃×#835	C₂×C₃	1	8	yes	complemented
28	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	complemented
29	C₂²×C₃²×D₃	—	2	1	no	—
30	C₃³×Q₈	—	2	2	no	—
31	C₃³×Q₈	—	2	2	no	—
32	C₂×C₃³×C₄	—	2	2	no	—
33	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
34	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
35	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₃²×Dic₃	—	2	1	no	—
37	C₃²×D₁₂	—	2	1	no	—
38	C₃²×D₁₂	D₄	1	2	yes	—
39	C₃²×C₄×D₃	—	4	2	no	—
40	C₃²×C₃⋊C₈	—	2	2	no	—
41	C₃×#461	—	3	4	no	—
42	#5083	C₃²	1	1	yes	characteristic, direct product factor
43	C₂×C₃²×Q₈	—	2	8	no	—
44	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
45	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
46	C₂×C₃²×D₄	—	3	1	no	—
47	C₂²×C₃²×C₄	—	6	1	no	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
49	C₂×C₃×C₄×D₃	—	2	4	no	—
50	C₃²×SD₁₆	—	6	4	no	—
51	C₂×C₃²×C₈	—	3	2	no	—
52	C₃²×C₄⋊C₄	—	2	8	no	—
53	C₃²×C₄⋊C₄	—	2	8	no	—
54	C₃²×C₄⋊C₄	—	2	1	no	—
55	C₃²×C₄⋊C₄	—	2	1	no	—
56	C₃²×C₄⋊C₄	—	2	4	no	—
57	C₃²×C₄⋊C₄	—	2	4	no	—
58	C₃²×C₄⋊C₄	—	3	1	no	—
59	C₃²×C₂²⋊C₄	—	6	1	no	—
60	C₃²×C₄²	—	2	4	no	—
61	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₃²×C₄²	C₂²×C₃	1	4	yes	—
63	C₃×#215	—	2	16	no	—
64	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
65	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
66	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
67	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃²×D₃	—	2	2	no	—
69	C₂×C₃²×D₃	—	4	1	no	—
70	C₂×C₃²×D₃	—	4	1	no	—
71	C₃³×C₄	—	2	1	no	—
72	C₃³×C₄	—	4	2	no	—
73	C₃³×C₄	SD₁₆	1	2	yes	complemented
74	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
75	C₃³×C₄	#41	1	1	yes	characteristic
76	C₃²×Dic₃	—	4	1	no	—
77	C₂×C₃×SD₁₆	—	3	8	no	—
78	C₃×C₄⋊Q₈	—	2	4	no	—
79	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
80	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₃×C₄×D₄	—	3	4	no	—
82	C₂×#215	C₂×C₃²	1	2	yes	—
83	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
84	C₃×C₄⋊C₈	—	3	4	no	—
85	C₃×#103	—	3	8	no	—
86	#835	C₂×C₃²	1	2	yes	complemented
87	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	1	yes	characteristic, complemented
88	C₂³×C₃²	—	6	1	no	—
89	C₂²×C₃×D₃	—	2	4	no	—
90	C₃²×Q₈	—	2	2	no	—
91	C₃²×Q₈	—	2	2	no	—
92	C₃²×Q₈	—	2	8	no	—
93	C₃²×Q₈	—	2	8	no	—
94	C₃²×Q₈	—	4	8	no	—
95	C₃²×Q₈	—	4	8	no	—
96	C₃²×D₄	—	3	2	no	—
97	C₃²×D₄	—	6	1	no	—
98	C₂×C₃²×C₄	—	2	16	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	—	2	8	no	—
101	C₂×C₃²×C₄	—	2	4	no	—
102	C₂×C₃²×C₄	—	2	4	no	—
103	C₂×C₃²×C₄	—	2	4	no	—
104	C₂×C₃²×C₄	—	6	1	no	—
105	C₂×C₃²×C₄	—	12	2	no	—
106	C₂×C₃²×C₄	#67	1	1	yes	characteristic
107	C₂×C₃²×C₄	#67	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
109	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
110	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
112	C₂×C₃×Dic₃	—	2	4	no	—
113	C₃×D₁₂	—	2	4	no	—
114	C₃×D₁₂	C₃×D₄	1	8	yes	—
115	C₃×C₄×D₃	—	4	8	no	—
116	C₃²×C₈	—	6	2	no	—
117	C₃×C₃⋊C₈	—	2	8	no	—
118	#461	—	3	1	no	Sylow
119	C₂×C₃³	#121	1	1	yes	characteristic
120	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₃³	#137	1	1	yes	characteristic
122	C₃²×D₃	—	4	2	no	—
123	C₂×C₃×Q₈	—	2	8	no	—
124	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
125	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
126	C₂×C₃×D₄	—	3	4	no	—
127	C₂²×C₃×C₄	—	6	4	no	—
128	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
129	C₂×C₄×D₃	—	2	1	no	—
130	C₃×SD₁₆	—	6	16	no	—
131	C₂×C₃×C₈	—	3	8	no	—
132	C₃×C₄⋊C₄	—	2	8	no	—
133	C₃×C₄⋊C₄	—	2	8	no	—
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₄⋊C₄	—	2	4	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	3	4	no	—
139	C₃×C₂²⋊C₄	—	6	4	no	—
140	C₃×C₄²	—	2	4	no	—
141	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
142	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
143	#215	—	2	4	no	—
144	(C₂×D₃)⋊C₄	—	2	1	no	—
145	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
146	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
147	C₂²×C₃²	—	2	4	no	—
148	C₂²×C₃²	—	6	2	no	—
149	C₂²×C₃²	—	12	1	no	—
150	C₂²×C₃²	—	12	1	no	—
151	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
152	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
153	C₂×C₃×D₃	—	2	8	no	—
154	C₂×C₃×D₃	—	4	4	no	—
155	C₂×C₃×D₃	—	4	4	no	—
156	C₃²×C₄	—	2	8	no	—
157	C₃²×C₄	—	2	1	no	—
158	C₃²×C₄	—	2	4	no	—
159	C₃²×C₄	—	2	8	no	—
160	C₃²×C₄	—	2	4	no	—
161	C₃²×C₄	—	2	4	no	—
162	C₃²×C₄	—	4	16	no	—
163	C₃²×C₄	—	4	2	no	—
164	C₃²×C₄	—	4	8	no	—
165	C₃²×C₄	—	12	1	no	—
166	C₃²×C₄	#215	1	2	yes	complemented
167	C₃²×C₄	C₃×SD₁₆	1	8	yes	complemented
168	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
169	C₃²×C₄	#237	1	1	yes	characteristic
170	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
171	C₃²×C₄	C₃×#41	1	4	yes	—
172	C₃×Dic₃	—	4	4	no	—
173	C₂×SD₁₆	—	3	2	no	—
174	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
175	C₄×D₄	—	3	1	no	—
176	C₄⋊C₈	—	3	1	no	—
177	#103	—	3	2	no	—
178	C₃³	#461	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	6	4	no	—
180	C₂²×D₃	—	2	1	no	—
181	C₃×Q₈	—	2	8	no	—
182	C₃×Q₈	—	2	8	no	—
183	C₃×Q₈	—	2	2	no	—
184	C₃×Q₈	—	2	2	no	—
185	C₃×Q₈	—	4	8	no	—
186	C₃×Q₈	—	4	8	no	—
187	C₃×D₄	—	3	8	no	—
188	C₃×D₄	—	6	4	no	—
189	C₂×C₃×C₄	—	2	16	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	6	4	no	—
196	C₂×C₃×C₄	—	12	8	no	—
197	C₂×C₃×C₄	C₃×#67	1	4	yes	—
198	C₂×C₃×C₄	C₃×#67	1	4	yes	—
199	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
202	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
203	C₂×Dic₃	—	2	1	no	—
204	D₁₂	—	2	1	no	—
205	D₁₂	C₃²×D₄	1	2	yes	—
206	C₄×D₃	—	4	2	no	—
207	C₃×C₈	—	6	8	no	—
208	C₃⋊C₈	—	2	2	no	—
209	C₂×C₃²	—	2	4	no	—
210	C₂×C₃²	—	2	4	no	—
211	C₂×C₃²	—	2	4	no	—
212	C₂×C₃²	—	12	2	no	—
213	C₂×C₃²	#938	1	1	yes	characteristic
214	C₂×C₃²	C₂×#215	1	1	yes	characteristic
215	C₂×C₃²	#942	1	1	yes	characteristic
216	C₂×C₃²	C₃×#121	1	4	yes	—
217	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
218	C₂×C₃²	C₃×#137	1	4	yes	—
219	C₃×D₃	—	4	8	no	—
220	C₂×Q₈	C₂×C₃²×D₃	1	2	yes	—
221	C₂×D₄	—	3	1	no	—
222	C₂²×C₄	—	6	1	no	—
223	SD₁₆	—	6	4	no	—
224	C₂×C₈	—	3	2	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₂²⋊C₄	—	6	1	no	—
229	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	6	8	no	—
232	C₂²×C₃	—	12	4	no	—
233	C₂²×C₃	—	12	4	no	—
234	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
235	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
236	C₂×D₃	—	2	2	no	—
237	C₂×D₃	—	4	1	no	—
238	C₂×D₃	—	4	1	no	—
239	C₃×C₄	—	2	8	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	8	no	—
245	C₃×C₄	—	4	16	no	—
246	C₃×C₄	—	4	8	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	—	12	4	no	—
249	C₃×C₄	C₃×#215	1	8	yes	complemented
250	C₃×C₄	C₃²×SD₁₆	1	2	yes	complemented
251	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
252	C₃×C₄	C₃×#237	1	4	yes	—
253	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
254	C₃×C₄	C₃²×#41	1	1	yes	characteristic
255	Dic₃	—	4	1	no	—
256	C₃²	—	2	4	no	—
257	C₃²	#5083	1	1	yes	characteristic, direct product factor
258	C₃²	C₃×#461	1	4	yes	complemented
259	C₂³	—	6	1	no	—
260	Q₈	—	2	2	no	—
261	Q₈	—	2	2	no	—
262	D₄	—	3	2	no	—
263	D₄	—	6	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	6	1	no	—
266	C₂×C₄	—	12	2	no	—
267	C₂×C₄	C₃²×#67	1	1	yes	characteristic
268	C₂×C₄	C₃²×#67	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
270	C₈	—	6	2	no	—
271	C₂×C₃	—	2	4	no	—
272	C₂×C₃	—	2	4	no	—
273	C₂×C₃	—	2	4	no	—
274	C₂×C₃	—	12	8	no	—
275	C₂×C₃	C₃×#938	1	4	yes	—
276	C₂×C₃	C₂×C₃×#215	1	4	yes	—
277	C₂×C₃	C₃×#942	1	4	yes	—
278	C₂×C₃	C₃²×#121	1	1	yes	characteristic
279	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
280	C₂×C₃	C₃²×#137	1	1	yes	characteristic
281	D₃	—	4	2	no	—
282	C₂²	—	6	2	no	—
283	C₂²	—	12	1	no	—
284	C₂²	—	12	1	no	—
285	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	12	1	no	—
289	C₄	C₃²×#215	1	2	yes	complemented
290	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
291	C₄	C₃²×#237	1	1	yes	characteristic
292	C₃	—	2	4	no	—
293	C₃	C₃×#5083	1	4	yes	direct product factor
294	C₃	C₃²×#461	1	1	yes	characteristic, complemented
295	C₂	—	12	2	no	—
296	C₂	C₃²×#938	1	1	yes	characteristic
297	C₂	C₂×C₃²×#215	1	1	yes	characteristic
298	C₂	C₃²×#942	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×#461
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊Q₈	C₂	semidirect
4. C₃²×#835	C₂	semidirect
5. C₃²×C₄⋊(C₃⋊C₈)	C₂	semidirect
7. C₃×#5083	C₃	direct (N × H)
20. C₃²×C₄⋊Q₈	D₃	semidirect
21. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
27. C₃×#835	C₂×C₃	semidirect
28. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
42. #5083	C₃²	direct (N × H)
73. C₃³×C₄	SD₁₆	semidirect
79. C₃×C₄⋊Q₈	C₃×D₃	semidirect
80. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
86. #835	C₂×C₃²	semidirect
87. C₄⋊(C₃⋊C₈)	C₂×C₃²	semidirect
166. C₃²×C₄	#215	semidirect
167. C₃²×C₄	C₃×SD₁₆	semidirect
174. C₄⋊Q₈	C₃²×D₃	semidirect
178. C₃³	#461	semidirect
249. C₃×C₄	C₃×#215	semidirect
250. C₃×C₄	C₃²×SD₁₆	semidirect
257. C₃²	#5083	direct (N × H)
258. C₃²	C₃×#461	semidirect
289. C₄	C₃²×#215	semidirect
293. C₃	C₃×#5083	direct (N × H)
294. C₃	C₃²×#461	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₂×C₃²×#215	2
C₃²×C₄×D₁₂	2
C₃²×#835	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₃²×#461	3
C₃×#5083	3

Subgroups — C32×#5083 #13162523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5083 #13162523