Subgroups — 3₋¹⁺²×#463 #13145149

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	3₋¹⁺²×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×#102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#463	C₃	1	1	yes	maximal, characteristic
9	C₉×#463	C₃	1	3	yes	maximal, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₂²×C₄×3₋¹⁺²	—	2	1	no	—
14	SD₁₆×3₋¹⁺²	—	2	2	no	—
15	D₈×3₋¹⁺²	—	2	2	no	—
16	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
17	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
20	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
21	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
22	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
23	C₂×C₃²×SD₁₆	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
25	C₃²×#124	C₂×C₃	1	1	yes	characteristic
26	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
27	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₃²×#103	C₂×C₃	1	1	yes	characteristic
29	C₃²×#102	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×SD₁₆	C₂×C₃	1	3	yes	complemented
31	C₂×C₉×D₈	C₂×C₃	1	3	yes	—
32	C₉×#124	C₂×C₃	1	3	yes	complemented
33	C₄×D₄×C₉	C₂×C₃	1	3	yes	complemented
34	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	complemented
35	C₉×#103	C₂×C₃	1	3	yes	complemented
36	C₉×#102	C₂×C₃	1	3	yes	complemented
37	C₂³×3₋¹⁺²	—	2	1	no	—
38	C₂³×3₋¹⁺²	—	2	1	no	—
39	Q₈×3₋¹⁺²	—	2	1	no	—
40	Q₈×3₋¹⁺²	—	2	1	no	—
41	D₄×3₋¹⁺²	—	2	1	no	—
42	D₄×3₋¹⁺²	—	2	1	no	—
43	D₄×3₋¹⁺²	—	2	1	no	—
44	D₄×3₋¹⁺²	D₄	1	2	yes	—
45	C₂×C₄×3₋¹⁺²	—	2	1	no	—
46	C₂×C₄×3₋¹⁺²	—	2	1	no	—
47	C₂×C₄×3₋¹⁺²	—	4	2	no	—
48	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
49	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
50	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
51	C₈×3₋¹⁺²	—	2	1	no	—
52	C₈×3₋¹⁺²	—	2	1	no	—
53	C₃×#463	—	3	1	no	—
54	C₃×#463	C₃²	1	1	yes	characteristic
55	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃²×C₄	—	2	1	no	—
59	C₃²×SD₁₆	—	2	2	no	—
60	C₃²×D₈	—	2	2	no	—
61	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₃²×C₂²⋊C₄	—	2	1	no	—
65	C₃²×C₂²⋊C₄	—	2	1	no	—
66	C₃²×C₂²⋊C₄	—	2	1	no	—
67	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
69	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
70	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
71	C₂²×C₄×C₉	—	2	3	no	—
72	C₉×SD₁₆	—	2	6	no	—
73	C₉×D₈	—	2	6	no	—
74	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
75	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
76	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
77	C₉×C₂²⋊C₄	—	2	3	no	—
78	C₉×C₂²⋊C₄	—	2	3	no	—
79	C₉×C₂²⋊C₄	—	2	3	no	—
80	C₄²×C₉	C₂²×C₃	1	3	yes	—
81	C₂²×3₋¹⁺²	—	2	2	no	—
82	C₂²×3₋¹⁺²	—	4	1	no	—
83	C₂²×3₋¹⁺²	—	4	1	no	—
84	C₂²×3₋¹⁺²	—	4	1	no	—
85	C₂²×3₋¹⁺²	—	4	1	no	—
86	C₂²×3₋¹⁺²	—	4	1	no	—
87	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
88	C₄×3₋¹⁺²	—	2	1	no	—
89	C₄×3₋¹⁺²	—	2	1	no	—
90	C₄×3₋¹⁺²	—	4	1	no	—
91	C₄×3₋¹⁺²	—	4	1	no	—
92	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
93	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
94	C₂×C₃×SD₁₆	—	3	1	no	—
95	C₂×C₃×SD₁₆	C₂×C₃²	1	1	yes	characteristic
96	C₂×C₃×D₈	—	3	1	no	—
97	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic
98	C₃×#124	—	3	1	no	—
99	C₃×#124	C₂×C₃²	1	1	yes	characteristic
100	C₃×C₄×D₄	—	3	1	no	—
101	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
102	C₃×C₄⋊C₈	—	3	1	no	—
103	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
104	C₃×#103	—	3	1	no	—
105	C₃×#103	C₂×C₃²	1	1	yes	characteristic
106	C₃×#102	—	3	1	no	—
107	C₃×#102	C₂×C₃²	1	1	yes	characteristic
108	C₂³×C₃²	—	2	1	no	—
109	C₂³×C₃²	—	2	1	no	—
110	C₃²×Q₈	—	2	1	no	—
111	C₃²×Q₈	—	2	1	no	—
112	C₃²×D₄	—	2	1	no	—
113	C₃²×D₄	—	2	1	no	—
114	C₃²×D₄	—	2	1	no	—
115	C₃²×D₄	C₃×D₄	1	2	yes	—
116	C₂×C₃²×C₄	—	2	1	no	—
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	—	4	2	no	—
119	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
121	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂³×C₉	—	2	3	no	—
123	C₂³×C₉	—	2	3	no	—
124	C₃²×C₈	—	2	1	no	—
125	C₃²×C₈	—	2	1	no	—
126	Q₈×C₉	—	2	3	no	—
127	Q₈×C₉	—	2	3	no	—
128	D₄×C₉	—	2	3	no	—
129	D₄×C₉	—	2	3	no	—
130	D₄×C₉	—	2	3	no	—
131	D₄×C₉	C₃×D₄	1	6	yes	—
132	C₂×C₄×C₉	—	2	3	no	—
133	C₂×C₄×C₉	—	2	3	no	—
134	C₂×C₄×C₉	—	4	6	no	—
135	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
136	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
137	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
138	C₈×C₉	—	2	3	no	—
139	C₈×C₉	—	2	3	no	—
140	#463	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×3₋¹⁺²	—	4	2	no	—
142	C₂×3₋¹⁺²	—	8	1	no	—
143	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
144	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
145	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₃×Q₈	—	3	1	no	—
147	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
148	C₂×C₃×D₄	—	3	1	no	—
149	C₂×C₃×D₄	—	3	1	no	—
150	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
151	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂²×C₃×C₄	—	6	1	no	—
154	C₃×SD₁₆	—	2	2	no	—
155	C₃×SD₁₆	—	6	2	no	—
156	C₃×D₈	—	2	2	no	—
157	C₃×D₈	—	6	2	no	—
158	C₂×C₃×C₈	—	3	1	no	—
159	C₂×C₃×C₈	—	3	1	no	—
160	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	3	1	no	—
163	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
164	C₃×C₂²⋊C₄	—	2	1	no	—
165	C₃×C₂²⋊C₄	—	2	1	no	—
166	C₃×C₂²⋊C₄	—	2	1	no	—
167	C₃×C₂²⋊C₄	—	6	1	no	—
168	C₃×C₂²⋊C₄	—	6	1	no	—
169	C₃×C₂²⋊C₄	—	6	1	no	—
170	C₃×C₄²	—	3	1	no	—
171	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
172	C₂²×C₃²	—	2	2	no	—
173	C₂²×C₃²	—	4	1	no	—
174	C₂²×C₃²	—	4	1	no	—
175	C₂²×C₃²	—	4	1	no	—
176	C₂²×C₃²	—	4	1	no	—
177	C₂²×C₃²	—	4	1	no	—
178	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
179	C₃²×C₄	—	2	1	no	—
180	C₃²×C₄	—	2	1	no	—
181	C₃²×C₄	—	4	1	no	—
182	C₃²×C₄	—	4	1	no	—
183	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₃²×C₄	C₃×#41	1	1	yes	characteristic
185	C₂²×C₉	—	2	6	no	—
186	C₂²×C₉	—	4	3	no	—
187	C₂²×C₉	—	4	3	no	—
188	C₂²×C₉	—	4	3	no	—
189	C₂²×C₉	—	4	3	no	—
190	C₂²×C₉	—	4	3	no	—
191	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
192	C₄×C₉	—	2	3	no	—
193	C₄×C₉	—	2	3	no	—
194	C₄×C₉	—	4	3	no	—
195	C₄×C₉	—	4	3	no	—
196	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
197	C₄×C₉	C₃×#41	1	3	yes	—
198	C₂×SD₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
199	C₂×D₈	C₂×3₋¹⁺²	1	1	yes	characteristic
200	#124	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
201	C₄×D₄	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
202	C₄⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
203	#103	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
204	#102	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
205	3₋¹⁺²	#463	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³×C₃	—	2	1	no	—
207	C₂³×C₃	—	2	1	no	—
208	C₂³×C₃	—	6	1	no	—
209	C₂³×C₃	—	6	1	no	—
210	C₃×Q₈	—	2	1	no	—
211	C₃×Q₈	—	2	1	no	—
212	C₃×Q₈	—	6	1	no	—
213	C₃×Q₈	—	6	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₃×D₄	—	2	1	no	—
217	C₃×D₄	—	3	2	no	—
218	C₃×D₄	—	6	1	no	—
219	C₃×D₄	—	6	1	no	—
220	C₃×D₄	—	6	1	no	—
221	C₃×D₄	C₃²×D₄	1	2	yes	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	3	1	no	—
225	C₂×C₃×C₄	—	3	1	no	—
226	C₂×C₃×C₄	—	3	1	no	—
227	C₂×C₃×C₄	—	4	2	no	—
228	C₂×C₃×C₄	—	6	1	no	—
229	C₂×C₃×C₄	—	6	1	no	—
230	C₂×C₃×C₄	—	12	2	no	—
231	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
234	C₃×C₈	—	2	1	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	—	6	1	no	—
237	C₃×C₈	—	6	1	no	—
238	C₂×C₃²	—	4	2	no	—
239	C₂×C₃²	—	8	1	no	—
240	C₂×C₃²	C₃×#121	1	1	yes	characteristic
241	C₂×C₃²	C₃×#135	1	1	yes	characteristic
242	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
243	C₂×C₉	—	4	6	no	—
244	C₂×C₉	—	8	3	no	—
245	C₂×C₉	C₃×#121	1	3	yes	—
246	C₂×C₉	C₃×#135	1	3	yes	—
247	C₂×C₉	C₃×AGL(1,ℤ/8)	1	3	yes	—
248	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
249	C₂×D₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
250	C₂×D₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	SD₁₆	—	2	2	no	—
253	D₈	—	2	2	no	—
254	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
255	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	3	1	no	—
263	C₂²×C₃	—	4	1	no	—
264	C₂²×C₃	—	4	1	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	—	6	2	no	—
269	C₂²×C₃	—	12	1	no	—
270	C₂²×C₃	—	12	1	no	—
271	C₂²×C₃	—	12	1	no	—
272	C₂²×C₃	—	12	1	no	—
273	C₂²×C₃	—	12	1	no	—
274	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	3	1	no	—
278	C₃×C₄	—	3	1	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	6	1	no	—
282	C₃×C₄	—	6	1	no	—
283	C₃×C₄	—	12	1	no	—
284	C₃×C₄	—	12	1	no	—
285	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
286	C₃×C₄	C₃²×#41	1	1	yes	characteristic
287	C₃²	C₃×#463	1	1	yes	characteristic
288	C₉	C₃×#463	1	3	yes	complemented
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	D₄×3₋¹⁺²	1	2	yes	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
301	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
302	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₂×C₃	—	3	1	no	—
306	C₂×C₃	—	3	1	no	—
307	C₂×C₃	—	3	1	no	—
308	C₂×C₃	—	4	2	no	—
309	C₂×C₃	—	8	1	no	—
310	C₂×C₃	—	12	2	no	—
311	C₂×C₃	—	24	1	no	—
312	C₂×C₃	C₃²×#121	1	1	yes	characteristic
313	C₂×C₃	C₃²×#135	1	1	yes	characteristic
314	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
327	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
328	C₃	—	3	1	no	—
329	C₃	C₃²×#463	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
333	C₂	3₋¹⁺²×#135	1	1	yes	characteristic
334	C₂	3₋¹⁺²×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂×C₃²×D₈	C₂×C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×SD₁₆×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²×#124	C₂	semidirect
4. C₄×D₄×3₋¹⁺²	C₂	semidirect
5. 3₋¹⁺²×C₄⋊C₈	C₂	semidirect
6. 3₋¹⁺²×#103	C₂	semidirect
7. 3₋¹⁺²×#102	C₂	semidirect
9. C₉×#463	C₃	semidirect
30. C₂×C₉×SD₁₆	C₂×C₃	semidirect
32. C₉×#124	C₂×C₃	semidirect
33. C₄×D₄×C₉	C₂×C₃	semidirect
34. C₉×C₄⋊C₈	C₂×C₃	semidirect
35. C₉×#103	C₂×C₃	semidirect
36. C₉×#102	C₂×C₃	semidirect
140. #463	3₋¹⁺²	direct (N × H)
198. C₂×SD₁₆	C₂×3₋¹⁺²	semidirect
200. #124	C₂×3₋¹⁺²	semidirect
201. C₄×D₄	C₂×3₋¹⁺²	semidirect
202. C₄⋊C₈	C₂×3₋¹⁺²	semidirect
203. #103	C₂×3₋¹⁺²	semidirect
204. #102	C₂×3₋¹⁺²	semidirect
205. 3₋¹⁺²	#463	direct (N × H)
288. C₉	C₃×#463	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×3₋¹⁺²	2
C₂×D₈×3₋¹⁺²	2
3₋¹⁺²×#124	2
C₄×D₄×3₋¹⁺²	2
3₋¹⁺²×C₄⋊C₈	2
3₋¹⁺²×#103	2
3₋¹⁺²×#102	2
C₃²×#463	3
C₉×#463	3

Subgroups — 3−1+2×#463 #13145149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#463 #13145149