Subgroups — C₂×C₃×#63481 #13153457

Order 1728.

858 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3701	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3675	C₂	1	2	yes	maximal, complemented
4	C₃×#63481	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×D₇₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×C₈×D₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#919	—	3	1	no	maximal
8	C₂×#63481	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
10	C₃×#3701	—	2	4	no	—
11	C₃×#3701	C₂²	1	8	yes	complemented
12	C₂×C₃×D₃₆	—	2	2	no	—
13	C₂×C₃×D₃₆	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₉	—	2	2	no	—
15	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
16	C₃×C₉×Q₁₆	C₂²	1	4	yes	complemented
17	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₃×#3675	C₂²	1	8	yes	complemented
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×D₇₂	C₂²	1	4	yes	complemented
21	C₃×C₈×D₉	C₂²	1	4	yes	complemented
22	C₂×C₃×#239	—	3	2	no	—
23	C₂×C₃²×Q₁₆	D₃	1	1	yes	characteristic
24	C₂×C₃×#215	—	3	2	no	—
25	C₃×#919	—	3	8	no	—
26	C₂×C₃×D₂₄	—	3	1	no	—
27	C₂×C₃×D₃×C₈	—	3	1	no	—
28	C₂×#3701	C₂×C₃	1	2	yes	complemented
29	C₂×C₉×Q₁₆	—	2	1	no	—
30	C₂×C₉×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#3675	C₂×C₃	1	2	yes	complemented
32	#63481	C₂×C₃	1	8	yes	direct product factor
33	C₂×D₇₂	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×D₉	—	2	2	no	—
36	C₂²×C₃×D₉	D₄	1	1	yes	characteristic
37	C₃×Q₈×C₉	—	2	2	no	—
38	C₃×Q₈×C₉	C₂³	1	4	yes	—
39	C₂×C₃×C₄×C₉	—	2	2	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
42	C₃×D₃₆	—	2	8	no	—
43	C₃×D₃₆	—	2	2	no	—
44	C₃×D₃₆	C₂³	1	4	yes	—
45	C₃×C₄×D₉	—	2	8	no	—
46	C₃×C₄×D₉	D₄	1	2	yes	—
47	C₃×C₄×D₉	C₂³	1	2	yes	—
48	C₃×C₈×C₉	C₂³	1	2	yes	—
49	C₃×C₉⋊C₈	C₂³	1	2	yes	—
50	C₂×C₃×#135	—	9	1	no	—
51	C₂×#919	—	3	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
53	C₃×#239	—	3	8	no	—
54	C₃×#239	—	6	4	no	—
55	C₂×C₃×D₁₂	—	3	2	no	—
56	C₂×C₃×D₁₂	—	6	2	no	—
57	C₂×C₃×C₄×D₃	—	3	1	no	—
58	C₂×C₃×C₄×D₃	—	6	2	no	—
59	C₃²×Q₁₆	C₂×D₃	1	4	yes	—
60	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#215	—	3	8	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₃×D₂₄	—	3	4	no	—
64	C₃×D₃×C₈	—	3	4	no	—
65	C₂×Q₈×C₉	—	2	2	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
67	#3701	—	2	4	no	—
68	#3701	C₂²×C₃	1	8	yes	complemented
69	C₂×D₃₆	—	2	2	no	—
70	C₂×D₃₆	C₂²×C₃	1	2	yes	—
71	C₂×C₄×D₉	—	2	2	no	—
72	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
73	C₉×Q₁₆	—	2	4	no	—
74	C₉×Q₁₆	C₂²×C₃	1	4	yes	complemented
75	C₂×C₈×C₉	—	2	1	no	—
76	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
77	#3675	C₂²×C₃	1	8	yes	complemented
78	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	D₇₂	C₂²×C₃	1	4	yes	complemented
80	C₈×D₉	C₂²×C₃	1	4	yes	complemented
81	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×D₉	—	2	2	no	—
83	C₂×C₃×D₉	—	2	4	no	—
84	C₂×C₃×D₉	—	4	4	no	—
85	C₂×C₃×D₉	C₂×D₄	1	2	yes	—
86	C₃×C₄×C₉	—	2	4	no	—
87	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
88	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
89	C₃×Dic₉	C₂×D₄	1	2	yes	—
90	C₂×C₃×#41	—	9	2	no	—
91	C₂×#239	—	3	2	no	—
92	C₃×#135	—	9	8	no	—
93	C₂×C₃×Q₁₆	—	2	1	no	—
94	C₂×C₃×Q₁₆	D₉	1	1	yes	characteristic, complemented
95	C₂×C₃×Q₁₆	C₃×D₃	1	1	yes	characteristic
96	C₂×C₃×SD₁₆	—	9	2	no	—
97	C₂×C₃×D₈	—	9	1	no	—
98	C₂²×C₃×C₈	—	9	1	no	—
99	C₂×#215	—	3	2	no	—
100	#919	—	3	8	no	—
101	C₂×D₂₄	—	3	1	no	—
102	C₂×D₃×C₈	—	3	1	no	—
103	C₂²×C₃×D₃	—	3	1	no	—
104	C₂²×C₃×D₃	—	6	2	no	—
105	C₃²×Q₈	—	2	2	no	—
106	C₃²×Q₈	C₂²×D₃	1	4	yes	—
107	C₂×C₃²×C₄	—	2	2	no	—
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	—	3	1	no	—
110	C₃×D₁₂	—	3	4	no	—
111	C₃×D₁₂	—	6	8	no	—
112	C₃×D₁₂	—	6	2	no	—
113	C₃×C₄×D₃	—	3	2	no	—
114	C₃×C₄×D₃	—	3	2	no	—
115	C₃×C₄×D₃	—	6	8	no	—
116	C₂²×D₉	—	2	2	no	—
117	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
118	C₃²×C₈	C₂²×D₃	1	2	yes	—
119	C₃×C₃⋊C₈	—	3	2	no	—
120	Q₈×C₉	—	2	2	no	—
121	Q₈×C₉	—	2	4	no	—
122	Q₈×C₉	—	4	2	no	—
123	Q₈×C₉	C₂³×C₃	1	4	yes	—
124	C₂×C₄×C₉	—	2	2	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	4	2	no	—
127	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
129	D₃₆	—	2	8	no	—
130	D₃₆	—	2	2	no	—
131	D₃₆	C₂³×C₃	1	4	yes	—
132	C₄×D₉	—	2	8	no	—
133	C₄×D₉	C₃×D₄	1	2	yes	—
134	C₄×D₉	C₂³×C₃	1	2	yes	—
135	C₈×C₉	—	2	2	no	—
136	C₈×C₉	C₂³×C₃	1	2	yes	—
137	C₉⋊C₈	C₂³×C₃	1	2	yes	—
138	C₂×#135	—	9	1	no	Sylow
139	C₂×C₃×C₉	#135	1	2	yes	complemented
140	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
141	C₃×D₉	—	2	2	no	—
142	C₃×D₉	—	4	4	no	—
143	C₃×#41	—	9	8	no	—
144	C₃×#41	—	18	4	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
147	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
148	C₂×C₃×D₄	—	9	2	no	—
149	C₂×C₃×D₄	—	18	2	no	—
150	C₂²×C₃×C₄	—	9	1	no	—
151	C₂²×C₃×C₄	—	18	2	no	—
152	#239	—	3	8	no	—
153	#239	—	6	4	no	—
154	C₂×D₁₂	—	3	2	no	—
155	C₂×D₁₂	—	6	2	no	—
156	C₂×C₄×D₃	—	3	1	no	—
157	C₂×C₄×D₃	—	6	2	no	—
158	C₃×Q₁₆	—	2	4	no	—
159	C₃×Q₁₆	C₂×D₉	1	4	yes	complemented
160	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	—
161	C₃×SD₁₆	—	9	8	no	—
162	C₃×D₈	—	9	4	no	—
163	C₂×C₃×C₈	—	2	1	no	—
164	C₂×C₃×C₈	—	9	1	no	—
165	C₂×C₃×C₈	—	9	4	no	—
166	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
167	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
168	#215	—	3	8	no	—
169	C₂×C₃⋊C₈	—	3	1	no	—
170	D₂₄	—	3	4	no	—
171	D₃×C₈	—	3	4	no	—
172	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×D₃	—	3	2	no	—
174	C₂×C₃×D₃	—	6	2	no	—
175	C₂×C₃×D₃	—	6	4	no	—
176	C₂×C₃×D₃	—	12	4	no	—
177	C₃²×C₄	—	2	4	no	—
178	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
179	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
180	C₃×Dic₃	—	3	2	no	—
181	C₂²×C₉	—	2	1	no	—
182	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×D₉	—	2	2	no	—
184	C₂×D₉	—	2	4	no	—
185	C₂×D₉	—	4	4	no	—
186	C₂×D₉	C₂×C₃×D₄	1	2	yes	—
187	C₄×C₉	—	2	4	no	—
188	C₄×C₉	—	2	1	no	—
189	C₄×C₉	—	2	1	no	—
190	C₄×C₉	—	4	4	no	—
191	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
192	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
193	Dic₉	C₂×C₃×D₄	1	2	yes	—
194	C₂×#41	—	9	2	no	—
195	#135	—	9	8	no	—
196	C₂×Q₁₆	C₃×D₉	1	1	yes	characteristic, complemented
197	C₂×SD₁₆	—	9	2	no	—
198	C₂×D₈	—	9	1	no	—
199	C₂²×C₈	—	9	1	no	—
200	C₃×C₉	C₂×#135	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₃	—	9	1	no	—
202	C₂³×C₃	—	18	2	no	—
203	C₂²×D₃	—	3	1	no	—
204	C₂²×D₃	—	6	2	no	—
205	C₃×Q₈	—	2	2	no	—
206	C₃×Q₈	—	2	2	no	—
207	C₃×Q₈	—	2	4	no	—
208	C₃×Q₈	—	4	2	no	—
209	C₃×Q₈	C₂²×D₉	1	4	yes	—
210	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
211	C₃×D₄	—	9	4	no	—
212	C₃×D₄	—	18	8	no	—
213	C₃×D₄	—	18	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	4	2	no	—
218	C₂×C₃×C₄	—	9	1	no	—
219	C₂×C₃×C₄	—	9	2	no	—
220	C₂×C₃×C₄	—	9	2	no	—
221	C₂×C₃×C₄	—	18	8	no	—
222	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
224	C₂×Dic₃	—	3	1	no	—
225	D₁₂	—	3	4	no	—
226	D₁₂	—	6	8	no	—
227	D₁₂	—	6	2	no	—
228	C₄×D₃	—	3	2	no	—
229	C₄×D₃	—	3	2	no	—
230	C₄×D₃	—	6	8	no	—
231	C₃×C₈	—	2	2	no	—
232	C₃×C₈	—	9	2	no	—
233	C₃×C₈	C₂²×D₉	1	2	yes	—
234	C₃×C₈	C₂²×C₃×D₃	1	2	yes	—
235	C₃⋊C₈	—	3	2	no	—
236	C₂×C₃²	#919	1	2	yes	—
237	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
238	C₃×D₃	—	6	2	no	—
239	C₃×D₃	—	12	4	no	—
240	C₂×C₉	—	2	2	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	C₃×#135	1	2	yes	complemented
243	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
244	D₉	—	2	2	no	—
245	D₉	—	4	4	no	—
246	#41	—	9	8	no	—
247	#41	—	18	4	no	—
248	C₂×Q₈	C₂×C₃×D₉	1	2	yes	—
249	C₂×D₄	—	9	2	no	—
250	C₂×D₄	—	18	2	no	—
251	C₂²×C₄	—	9	1	no	—
252	C₂²×C₄	—	18	2	no	—
253	Q₁₆	C₂×C₃×D₉	1	4	yes	complemented
254	SD₁₆	—	9	8	no	—
255	D₈	—	9	4	no	—
256	C₂×C₈	—	9	1	no	—
257	C₂×C₈	—	9	4	no	—
258	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	9	2	no	—
261	C₂²×C₃	—	18	2	no	—
262	C₂²×C₃	—	18	4	no	—
263	C₂²×C₃	—	36	4	no	—
264	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
265	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
266	C₂×D₃	—	3	2	no	—
267	C₂×D₃	—	6	2	no	—
268	C₂×D₃	—	6	4	no	—
269	C₂×D₃	—	12	4	no	—
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	2	4	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	4	4	no	—
275	C₃×C₄	—	9	2	no	—
276	C₃×C₄	D₄×D₉	1	1	yes	characteristic
277	C₃×C₄	C₂³×D₉	1	1	yes	characteristic
278	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
279	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
280	Dic₃	—	3	2	no	—
281	C₃²	C₂×#919	1	1	yes	characteristic
282	C₉	—	2	1	no	—
283	C₉	C₂×C₃×#135	1	1	yes	characteristic, complemented
284	C₂³	—	9	1	no	—
285	C₂³	—	18	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	C₂²×C₃×D₉	1	4	yes	—
288	D₄	—	9	4	no	—
289	D₄	—	18	8	no	—
290	D₄	—	18	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	9	1	no	—
293	C₂×C₄	—	9	2	no	—
294	C₂×C₄	—	9	2	no	—
295	C₂×C₄	—	18	8	no	—
296	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
297	C₈	—	9	2	no	—
298	C₈	C₂²×C₃×D₉	1	2	yes	—
299	C₂×C₃	—	2	2	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	18	2	no	—
302	C₂×C₃	—	36	4	no	—
303	C₂×C₃	#63481	1	2	yes	direct product factor
304	C₂×C₃	C₂×D₄×D₉	1	1	yes	characteristic
305	C₂×C₃	C₃×#919	1	2	yes	—
306	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
307	D₃	—	6	2	no	—
308	D₃	—	12	4	no	—
309	C₂²	—	9	2	no	—
310	C₂²	—	18	2	no	—
311	C₂²	—	18	4	no	—
312	C₂²	—	36	4	no	—
313	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	9	2	no	—
316	C₄	C₃×D₄×D₉	1	1	yes	characteristic
317	C₄	C₂³×C₃×D₉	1	1	yes	characteristic
318	C₃	—	2	1	no	—
319	C₃	C₂×#63481	1	1	yes	characteristic, direct product factor
320	C₃	C₂×C₃×#919	1	1	yes	characteristic
321	C₂	—	18	2	no	—
322	C₂	—	36	4	no	—
323	C₂	C₃×#63481	1	2	yes	direct product factor
324	C₂	C₂×C₃×D₄×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×Q₁₆	C₂
socle	C₂²×C₃²	D₃×D₄
hypercenter	C₂×C₃×Q₁₆	D₉
hypocenter	C₉	C₂×C₃×#135
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3701	C₂	semidirect
2. C₂×C₃×C₉×Q₁₆	C₂	semidirect
3. C₂×C₃×#3675	C₂	semidirect
4. C₃×#63481	C₂	direct (N × H)
5. C₂×C₃×D₇₂	C₂	semidirect
6. C₂×C₃×C₈×D₉	C₂	semidirect
8. C₂×#63481	C₃	direct (N × H)
11. C₃×#3701	C₂²	semidirect
16. C₃×C₉×Q₁₆	C₂²	semidirect
18. C₃×#3675	C₂²	semidirect
20. C₃×D₇₂	C₂²	semidirect
21. C₃×C₈×D₉	C₂²	semidirect
28. C₂×#3701	C₂×C₃	semidirect
30. C₂×C₉×Q₁₆	C₂×C₃	semidirect
31. C₂×#3675	C₂×C₃	semidirect
32. #63481	C₂×C₃	direct (N × H)
33. C₂×D₇₂	C₂×C₃	semidirect
34. C₂×C₈×D₉	C₂×C₃	semidirect
68. #3701	C₂²×C₃	semidirect
74. C₉×Q₁₆	C₂²×C₃	semidirect
77. #3675	C₂²×C₃	semidirect
79. D₇₂	C₂²×C₃	semidirect
80. C₈×D₉	C₂²×C₃	semidirect
94. C₂×C₃×Q₁₆	D₉	semidirect
139. C₂×C₃×C₉	#135	semidirect
159. C₃×Q₁₆	C₂×D₉	semidirect
196. C₂×Q₁₆	C₃×D₉	semidirect
200. C₃×C₉	C₂×#135	semidirect
242. C₂×C₉	C₃×#135	semidirect
253. Q₁₆	C₂×C₃×D₉	semidirect
283. C₉	C₂×C₃×#135	semidirect
303. C₂×C₃	#63481	direct (N × H)
319. C₃	C₂×#63481	direct (N × H)
323. C₂	C₃×#63481	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3701	2
C₂×C₃×C₉×Q₁₆	2
C₂×C₃×#3675	2
C₃×#63481	2
C₂×C₃×D₇₂	2
C₂×C₃×C₈×D₉	2
C₂×C₃×#919	3
C₂×#63481	3

Subgroups — C2×C3×#63481 #13153457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63481 #13153457