Subgroups — C₂×C₃×#63437 #13152284

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×(C₂×D₉)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₃×#63437	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#875	—	3	1	no	maximal
7	C₂×#63437	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃×D₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
12	C₂×C₃×D₃₆	—	2	4	no	—
13	C₂×C₃×D₃₆	C₂²	1	4	yes	—
14	C₂×C₃×Dic₁₈	—	2	4	no	—
15	C₂×C₃×Dic₁₈	C₂²	1	4	yes	complemented
16	C₃×C₄²×C₉	C₂²	1	4	yes	complemented
17	C₃×(C₂×D₉)⋊C₄	C₂²	1	16	yes	complemented
18	C₂²×C₃×D₁₂	—	3	1	no	—
19	C₂²×C₃×Dic₆	—	3	1	no	—
20	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
21	C₂×C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
22	C₃×#875	—	3	8	no	—
23	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic
24	C₂²×Dic₁₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₄²×C₉	—	2	1	no	—
26	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	4	yes	complemented
28	#63437	C₂×C₃	1	8	yes	direct product factor
29	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
30	C₂²×C₃×D₉	—	2	4	no	—
31	C₂²×C₃×D₉	—	2	8	no	—
32	C₂²×C₃×D₉	C₂³	1	4	yes	—
33	C₂×C₃×C₄×C₉	—	2	4	no	—
34	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
37	C₂×C₃×Dic₉	—	2	4	no	—
38	C₂×C₃×Dic₉	C₂³	1	4	yes	—
39	C₃×D₃₆	—	2	8	no	—
40	C₃×Dic₁₈	—	2	8	no	—
41	C₂×C₃×#124	—	9	1	no	—
42	C₂×#875	—	3	1	no	—
43	C₂³×C₃×D₃	—	3	2	no	—
44	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×Dic₃	—	3	2	no	—
47	C₂×C₃×D₁₂	—	3	4	no	—
48	C₂×C₃×D₁₂	—	6	4	no	—
49	C₂×C₃×Dic₆	—	3	4	no	—
50	C₂×C₃×Dic₆	—	6	4	no	—
51	C₂³×D₉	C₂²×C₃	1	2	yes	—
52	C₃²×C₄²	C₂×D₃	1	4	yes	—
53	C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
54	C₂²×C₄×C₉	—	2	2	no	—
55	C₂²×C₄×C₉	—	2	1	no	—
56	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
58	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
59	C₂×D₃₆	—	2	4	no	—
60	C₂×D₃₆	C₂²×C₃	1	4	yes	—
61	C₂×Dic₁₈	—	2	4	no	—
62	C₂×Dic₁₈	C₂²×C₃	1	4	yes	complemented
63	C₄²×C₉	—	2	4	no	—
64	C₄²×C₉	C₂²×C₃	1	4	yes	complemented
65	(C₂×D₉)⋊C₄	C₂²×C₃	1	16	yes	complemented
66	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
67	C₂²×C₃×C₉	#41	1	4	yes	—
68	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₃×D₉	—	2	4	no	—
70	C₂×C₃×D₉	—	2	8	no	—
71	C₂×C₃×D₉	—	4	8	no	—
72	C₃×C₄×C₉	—	2	4	no	—
73	C₃×C₄×C₉	C₂×D₄	1	4	yes	—
74	C₃×Dic₉	—	2	4	no	—
75	C₂²×C₃×Q₈	—	9	1	no	—
76	C₂²×C₃×D₄	—	9	1	no	—
77	C₂²×D₁₂	—	3	1	no	—
78	C₂²×Dic₆	—	3	1	no	—
79	C₃×#124	—	9	8	no	—
80	C₂×C₃×C₂²⋊C₄	—	9	4	no	—
81	C₂×C₃×C₄²	—	2	1	no	—
82	C₂×C₃×C₄²	D₉	1	1	yes	characteristic, complemented
83	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
84	C₂×(C₂×D₃)⋊C₄	—	3	4	no	—
85	#875	—	3	8	no	—
86	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₃×D₃	—	3	4	no	—
88	C₂²×C₃×D₃	—	6	4	no	—
89	C₂²×C₃×D₃	—	6	8	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	D₁₂	1	4	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
94	C₂×C₃×Dic₃	—	3	4	no	—
95	C₂×C₃×Dic₃	—	6	4	no	—
96	C₃×D₁₂	—	6	8	no	—
97	C₃×Dic₆	—	6	8	no	—
98	C₂³×C₉	—	2	1	no	—
99	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
100	C₂²×D₉	—	2	4	no	—
101	C₂²×D₉	—	2	8	no	—
102	C₂²×D₉	C₂³×C₃	1	4	yes	—
103	C₂×C₄×C₉	—	2	8	no	—
104	C₂×C₄×C₉	—	2	4	no	—
105	C₂×C₄×C₉	—	2	4	no	—
106	C₂×C₄×C₉	—	2	4	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	C₃×D₄	1	4	yes	—
109	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
110	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
111	C₂×Dic₉	—	2	4	no	—
112	C₂×Dic₉	C₂³×C₃	1	4	yes	—
113	D₃₆	—	2	8	no	—
114	Dic₁₈	—	2	8	no	—
115	C₂×#124	—	9	1	no	Sylow
116	C₂×C₃×C₉	#124	1	4	yes	complemented
117	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×#41	1	2	yes	—
119	C₃×D₉	—	4	4	no	—
120	C₂⁴×C₃	—	9	2	no	—
121	C₂³×D₃	—	3	2	no	—
122	C₂×C₃×Q₈	—	9	4	no	—
123	C₂×C₃×Q₈	—	18	4	no	—
124	C₂×C₃×D₄	—	9	4	no	—
125	C₂×C₃×D₄	—	18	4	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	9	2	no	—
129	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
130	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
132	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
133	C₂²×Dic₃	—	3	2	no	—
134	C₂×D₁₂	—	3	4	no	—
135	C₂×D₁₂	—	6	4	no	—
136	C₂×Dic₆	—	3	4	no	—
137	C₂×Dic₆	—	6	4	no	—
138	C₃×C₂²⋊C₄	—	9	16	no	—
139	C₃×C₄²	—	2	4	no	—
140	C₃×C₄²	C₂×D₉	1	4	yes	complemented
141	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
142	(C₂×D₃)⋊C₄	—	3	16	no	—
143	C₂²×C₃²	C₂×D₁₂	1	2	yes	—
144	C₂²×C₃²	#235	1	4	yes	—
145	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
146	C₂×C₃×D₃	—	6	4	no	—
147	C₂×C₃×D₃	—	6	8	no	—
148	C₂×C₃×D₃	—	12	8	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	C₂×D₁₂	1	4	yes	—
151	C₃×Dic₃	—	6	4	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	C₂×C₃×D₄	1	2	yes	—
156	C₂²×C₉	C₃×#41	1	4	yes	—
157	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
158	C₂×D₉	—	2	4	no	—
159	C₂×D₉	—	2	8	no	—
160	C₂×D₉	—	4	8	no	—
161	C₄×C₉	—	2	8	no	—
162	C₄×C₉	—	2	4	no	—
163	C₄×C₉	—	2	4	no	—
164	C₄×C₉	C₂×C₃×D₄	1	4	yes	—
165	Dic₉	—	2	4	no	—
166	C₂²×Q₈	—	9	1	no	—
167	C₂²×D₄	—	9	1	no	—
168	#124	—	9	8	no	—
169	C₂×C₂²⋊C₄	—	9	4	no	—
170	C₂×C₄²	C₃×D₉	1	1	yes	characteristic, complemented
171	C₃×C₉	C₂×#124	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	9	4	no	—
174	C₂³×C₃	—	18	4	no	—
175	C₂³×C₃	—	18	8	no	—
176	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
177	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₂²×D₃	—	3	4	no	—
179	C₂²×D₃	—	6	4	no	—
180	C₂²×D₃	—	6	8	no	—
181	C₃×Q₈	—	18	8	no	—
182	C₃×D₄	—	18	8	no	—
183	C₂×C₃×C₄	—	2	8	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	9	4	no	—
190	C₂×C₃×C₄	—	18	4	no	—
191	C₂×C₃×C₄	D₃₆	1	4	yes	—
192	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
193	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
194	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
195	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
196	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
197	C₂×Dic₃	—	3	4	no	—
198	C₂×Dic₃	—	6	4	no	—
199	D₁₂	—	6	8	no	—
200	Dic₆	—	6	8	no	—
201	C₂×C₃²	#875	1	4	yes	—
202	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
203	C₂×C₃²	C₂×#235	1	2	yes	—
204	C₃×D₃	—	12	4	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	2	no	—
207	C₂×C₉	—	2	4	no	—
208	C₂×C₉	C₃×#124	1	4	yes	complemented
209	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₂×C₉	C₂×C₃×#41	1	2	yes	—
211	D₉	—	4	4	no	—
212	C₂⁴	—	9	2	no	—
213	C₂×Q₈	—	9	4	no	—
214	C₂×Q₈	—	18	4	no	—
215	C₂×D₄	—	9	4	no	—
216	C₂×D₄	—	18	4	no	—
217	C₂²×C₄	—	9	2	no	—
218	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
219	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
220	C₂²⋊C₄	—	9	16	no	—
221	C₄²	C₂×C₃×D₉	1	4	yes	complemented
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	18	4	no	—
226	C₂²×C₃	—	18	8	no	—
227	C₂²×C₃	—	36	8	no	—
228	C₂²×C₃	C₂×D₃₆	1	2	yes	—
229	C₂²×C₃	#3697	1	4	yes	—
230	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
232	C₂²×C₃	C₃×#235	1	4	yes	—
233	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
234	C₂×D₃	—	6	4	no	—
235	C₂×D₃	—	6	8	no	—
236	C₂×D₃	—	12	8	no	—
237	C₃×C₄	—	2	8	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	18	4	no	—
242	C₃×C₄	C₂×D₃₆	1	4	yes	—
243	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
244	Dic₃	—	6	4	no	—
245	C₃²	C₂×#875	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	C₂×C₃×#124	1	1	yes	characteristic, complemented
248	C₂³	—	9	4	no	—
249	C₂³	—	18	4	no	—
250	C₂³	—	18	8	no	—
251	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
252	Q₈	—	18	8	no	—
253	D₄	—	18	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	9	4	no	—
256	C₂×C₄	—	18	4	no	—
257	C₂×C₄	C₃×D₃₆	1	4	yes	—
258	C₂×C₄	C₂²×C₃×D₉	1	4	yes	—
259	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	36	4	no	—
264	C₂×C₃	#63437	1	4	yes	direct product factor
265	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
266	C₂×C₃	C₂×#3697	1	2	yes	—
267	C₂×C₃	C₃×#875	1	4	yes	—
268	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×#235	1	2	yes	—
270	D₃	—	12	4	no	—
271	C₂²	—	18	4	no	—
272	C₂²	—	18	8	no	—
273	C₂²	—	36	8	no	—
274	C₂²	C₂×C₃×D₃₆	1	2	yes	—
275	C₂²	C₃×#3697	1	4	yes	—
276	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	18	4	no	—
279	C₄	C₂×C₃×D₃₆	1	4	yes	—
280	C₃	—	2	1	no	—
281	C₃	C₂×#63437	1	1	yes	characteristic, direct product factor
282	C₃	C₂×C₃×#875	1	1	yes	characteristic
283	C₂	—	36	4	no	—
284	C₂	C₃×#63437	1	4	yes	direct product factor
285	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic
286	C₂	C₂×C₃×#3697	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂²×C₃×D₃₆	C₂
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₂×C₃×#124
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₁₈	C₂	semidirect
3. C₂×C₃×C₄²×C₉	C₂	semidirect
4. C₂×C₃×(C₂×D₉)⋊C₄	C₂	semidirect
5. C₃×#63437	C₂	direct (N × H)
7. C₂×#63437	C₃	direct (N × H)
15. C₂×C₃×Dic₁₈	C₂²	semidirect
16. C₃×C₄²×C₉	C₂²	semidirect
17. C₃×(C₂×D₉)⋊C₄	C₂²	semidirect
24. C₂²×Dic₁₈	C₂×C₃	semidirect
26. C₂×C₄²×C₉	C₂×C₃	semidirect
27. C₂×(C₂×D₉)⋊C₄	C₂×C₃	semidirect
28. #63437	C₂×C₃	direct (N × H)
62. C₂×Dic₁₈	C₂²×C₃	semidirect
64. C₄²×C₉	C₂²×C₃	semidirect
65. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
82. C₂×C₃×C₄²	D₉	semidirect
116. C₂×C₃×C₉	#124	semidirect
140. C₃×C₄²	C₂×D₉	semidirect
170. C₂×C₄²	C₃×D₉	semidirect
171. C₃×C₉	C₂×#124	semidirect
208. C₂×C₉	C₃×#124	semidirect
221. C₄²	C₂×C₃×D₉	semidirect
247. C₉	C₂×C₃×#124	semidirect
264. C₂×C₃	#63437	direct (N × H)
281. C₃	C₂×#63437	direct (N × H)
284. C₂	C₃×#63437	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₃₆	2
C₂²×C₃×Dic₁₈	2
C₂×C₃×C₄²×C₉	2
C₂×C₃×(C₂×D₉)⋊C₄	2
C₃×#63437	2
C₂×C₃×#875	3
C₂×#63437	3

Subgroups — C2×C3×#63437 #13152284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63437 #13152284