Subgroups — #13144238

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
4	C₂×#88534	C₂	1	2	yes	maximal
5	#11720572	C₂	1	2	yes	maximal
6	#10592627	—	3	1	no	maximal
7	#10588608	—	3	3	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
12	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
13	C₃×C₄²×C₉	—	2	2	no	—
14	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
15	C₄⋊(C₃⋊Dic₉)	C₄	1	4	yes	complemented
16	#88534	—	2	4	no	—
17	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
18	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
19	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
20	C₂×#3750	—	3	2	no	—
21	#63658	—	3	2	no	—
22	C₂×C₄²×C₉	D₃	1	3	yes	—
23	C₂×C₄⋊Dic₉	—	3	3	no	—
24	C₂×C₄⋊Dic₉	—	3	3	no	—
25	C₂×Dic₉⋊C₄	—	3	6	no	—
26	C₉⋊#95	—	3	6	no	—
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	—	2	1	no	—
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
36	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
37	C₂×C₃⋊Dic₉	—	2	2	no	—
38	C₂×C₃⋊Dic₉	—	2	2	no	—
39	C₂×C₃⋊Dic₉	—	2	2	no	—
40	C₂×C₃⋊Dic₉	—	2	2	no	—
41	C₂×C₃⋊Dic₉	—	2	2	no	—
42	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
43	#4928	—	9	3	no	—
44	#4928	—	9	1	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃⋊Dic₃	—	3	2	no	—
49	C₂²×C₃⋊Dic₃	—	3	2	no	—
50	C₃²×C₄²	—	2	2	no	—
51	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
52	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
53	#3750	—	6	4	no	—
54	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
55	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
56	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
57	C₂²×Dic₉	—	3	6	no	—
58	C₂²×Dic₉	—	3	6	no	—
59	C₄²×C₉	—	2	6	no	—
60	C₄⋊Dic₉	—	3	12	no	—
61	C₄⋊Dic₉	—	6	6	no	—
62	Dic₉⋊C₄	—	6	12	no	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₃×C₉	#41	1	1	yes	characteristic
69	C₂²×C₃×C₉	#41	1	1	yes	characteristic
70	C₃×C₄×C₉	—	2	2	no	—
71	C₃×C₄×C₉	—	2	2	no	—
72	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
73	C₃⋊Dic₉	—	2	4	no	—
74	C₃⋊Dic₉	—	4	2	no	—
75	C₂×C₃×C₄²	D₉	1	3	yes	—
76	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
77	C₂×C₄⋊Dic₃	—	9	3	no	—
78	C₂×C₄⋊Dic₃	—	9	1	no	—
79	C₂×C₄⋊Dic₃	—	9	3	no	—
80	C₂×C₄⋊Dic₃	—	9	1	no	—
81	C₂×Dic₃⋊C₄	—	9	6	no	—
82	C₂×Dic₃⋊C₄	—	9	2	no	—
83	C₃⋊#95	—	9	6	no	—
84	C₃⋊#95	—	9	2	no	—
85	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	1	no	—
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	Dic₆	1	2	yes	—
91	C₂×C₃²×C₄	Dic₆	1	2	yes	—
92	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
93	C₂×C₃²×C₄	D₁₂	1	2	yes	—
94	C₂×C₃²×C₄	#67	1	2	yes	—
95	C₂×C₃⋊Dic₃	—	3	4	no	—
96	C₂×C₃⋊Dic₃	—	6	2	no	—
97	C₂×C₃⋊Dic₃	—	6	2	no	—
98	C₂×C₃⋊Dic₃	—	6	2	no	—
99	C₂×C₃⋊Dic₃	—	6	2	no	—
100	C₂×C₃⋊Dic₃	—	6	2	no	—
101	C₂³×C₉	C₂²×D₃	1	3	yes	—
102	C₂×C₄×C₉	—	2	3	no	—
103	C₂×C₄×C₉	—	2	3	no	—
104	C₂×C₄×C₉	—	2	3	no	—
105	C₂×C₄×C₉	—	2	3	no	—
106	C₂×C₄×C₉	Dic₆	1	6	yes	—
107	C₂×C₄×C₉	Dic₆	1	6	yes	—
108	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
109	C₂×C₄×C₉	D₁₂	1	6	yes	—
110	C₂×C₄×C₉	#67	1	6	yes	—
111	C₂×Dic₉	—	3	12	no	—
112	C₂×Dic₉	—	6	6	no	—
113	C₂×Dic₉	—	6	6	no	—
114	C₂×Dic₉	—	6	6	no	—
115	C₂×Dic₉	—	6	6	no	—
116	C₂×Dic₉	—	6	6	no	—
117	#389	—	27	1	no	Sylow
118	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
121	C₂×C₃×C₉	#121	1	1	yes	characteristic
122	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₃×C₉	#125	1	1	yes	characteristic
124	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
126	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
127	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
128	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
131	C₂²×Dic₃	—	9	6	no	—
132	C₂²×Dic₃	—	9	2	no	—
133	C₂²×Dic₃	—	9	6	no	—
134	C₂²×Dic₃	—	9	2	no	—
135	C₃×C₄²	—	2	6	no	—
136	C₃×C₄²	—	2	2	no	—
137	C₄⋊Dic₃	—	9	12	no	—
138	C₄⋊Dic₃	—	9	4	no	—
139	C₄⋊Dic₃	—	18	6	no	—
140	C₄⋊Dic₃	—	18	2	no	—
141	Dic₃⋊C₄	—	18	12	no	—
142	Dic₃⋊C₄	—	18	4	no	—
143	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
144	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
145	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
147	C₂²×C₃²	#235	1	1	yes	characteristic
148	C₂²×C₃²	#235	1	1	yes	characteristic
149	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
150	C₃²×C₄	—	2	2	no	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
153	C₃⋊Dic₃	—	6	4	no	—
154	C₃⋊Dic₃	—	12	2	no	—
155	C₂²×C₉	C₂×Dic₆	1	3	yes	—
156	C₂²×C₉	C₂×Dic₆	1	3	yes	—
157	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
158	C₂²×C₉	C₂×D₁₂	1	3	yes	—
159	C₂²×C₉	#235	1	3	yes	—
160	C₂²×C₉	#235	1	3	yes	—
161	C₂²×C₉	C₂×#67	1	3	yes	—
162	C₄×C₉	—	2	6	no	—
163	C₄×C₉	—	2	6	no	—
164	C₄×C₉	Dic₃⋊C₄	1	12	yes	complemented
165	Dic₉	—	6	12	no	—
166	Dic₉	—	12	6	no	—
167	C₂×C₄⋊C₄	—	27	1	no	—
168	C₂×C₄⋊C₄	—	27	2	no	—
169	C₂×C₄⋊C₄	—	27	1	no	—
170	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
171	#95	—	27	2	no	—
172	C₃×C₉	#389	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₃	C₂²×D₉	1	3	yes	—
174	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	3	no	—
176	C₂×C₃×C₄	—	2	3	no	—
177	C₂×C₃×C₄	—	2	3	no	—
178	C₂×C₃×C₄	—	2	3	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
184	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
185	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
186	C₂×C₃×C₄	D₃₆	1	6	yes	—
187	C₂×C₃×C₄	#611	1	6	yes	—
188	C₂×C₃×C₄	#634	1	2	yes	—
189	C₂×C₃×C₄	#634	1	2	yes	—
190	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
191	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
192	C₂×C₃×C₄	#638	1	2	yes	—
193	C₂×Dic₃	—	9	12	no	—
194	C₂×Dic₃	—	9	4	no	—
195	C₂×Dic₃	—	18	6	no	—
196	C₂×Dic₃	—	18	6	no	—
197	C₂×Dic₃	—	18	2	no	—
198	C₂×Dic₃	—	18	2	no	—
199	C₂×Dic₃	—	18	6	no	—
200	C₂×Dic₃	—	18	6	no	—
201	C₂×Dic₃	—	18	6	no	—
202	C₂×Dic₃	—	18	2	no	—
203	C₂×Dic₃	—	18	2	no	—
204	C₂×Dic₃	—	18	2	no	—
205	C₂×C₃²	C₄×Dic₆	1	1	yes	characteristic
206	C₂×C₃²	#869	1	1	yes	characteristic
207	C₂×C₃²	#870	1	1	yes	characteristic
208	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
209	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
210	C₂×C₃²	#924	1	1	yes	characteristic
211	C₂×C₃²	#930	1	1	yes	characteristic
212	C₂×C₉	C₄×Dic₆	1	3	yes	—
213	C₂×C₉	#869	1	3	yes	—
214	C₂×C₉	#870	1	3	yes	—
215	C₂×C₉	C₄×D₁₂	1	3	yes	—
216	C₂×C₉	C₂×Dic₃⋊C₄	1	3	yes	—
217	C₂×C₉	#924	1	3	yes	—
218	C₂×C₉	#930	1	3	yes	—
219	C₂²×C₄	—	27	2	no	—
220	C₂²×C₄	—	27	2	no	—
221	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
224	C₄⋊C₄	—	27	4	no	—
225	C₄⋊C₄	—	54	4	no	—
226	C₄⋊C₄	—	54	2	no	—
227	C₄²	—	2	2	no	—
228	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
229	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
230	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
231	C₂²×C₃	C₂×D₃₆	1	3	yes	—
232	C₂²×C₃	#3697	1	3	yes	—
233	C₂²×C₃	#3697	1	3	yes	—
234	C₂²×C₃	C₂×#611	1	3	yes	—
235	C₂²×C₃	C₂×#634	1	1	yes	characteristic
236	C₂²×C₃	C₂×#634	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
239	C₂²×C₃	#3828	1	1	yes	characteristic
240	C₂²×C₃	#3828	1	1	yes	characteristic
241	C₂²×C₃	C₂×#638	1	1	yes	characteristic
242	C₃×C₄	—	2	6	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	6	no	—
246	C₃×C₄	Dic₉⋊C₄	1	12	yes	complemented
247	C₃×C₄	#3750	1	4	yes	—
248	Dic₃	—	18	4	no	—
249	Dic₃	—	18	12	no	—
250	Dic₃	—	36	6	no	—
251	Dic₃	—	36	2	no	—
252	C₃²	#4928	1	1	yes	characteristic
253	C₉	#4928	1	3	yes	complemented
254	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	27	4	no	—
260	C₂×C₄	—	54	2	no	—
261	C₂×C₄	—	54	2	no	—
262	C₂×C₄	—	54	2	no	—
263	C₂×C₄	—	54	2	no	—
264	C₂×C₄	—	54	2	no	—
265	C₂×C₄	#6226	1	2	yes	—
266	C₂×C₄	#6226	1	2	yes	—
267	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
268	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
269	C₂×C₄	#6233	1	2	yes	—
270	C₂×C₃	C₄×Dic₁₈	1	3	yes	—
271	C₂×C₃	#63431	1	3	yes	—
272	C₂×C₃	#63432	1	3	yes	—
273	C₂×C₃	C₄×D₃₆	1	3	yes	—
274	C₂×C₃	C₂×Dic₉⋊C₄	1	3	yes	—
275	C₂×C₃	#63486	1	3	yes	—
276	C₂×C₃	#63492	1	3	yes	—
277	C₂×C₃	C₄×#634	1	1	yes	characteristic
278	C₂×C₃	#64078	1	1	yes	characteristic
279	C₂×C₃	#64079	1	1	yes	characteristic
280	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
281	C₂×C₃	C₂×#3750	1	1	yes	characteristic
282	C₂×C₃	#64133	1	1	yes	characteristic
283	C₂×C₃	#64139	1	1	yes	characteristic
284	C₂²	C₂×#6226	1	1	yes	characteristic
285	C₂²	C₂×#6226	1	1	yes	characteristic
286	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
287	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
288	C₂²	#88736	1	1	yes	characteristic
289	C₂²	#88736	1	1	yes	characteristic
290	C₂²	C₂×#6233	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	54	4	no	—
294	C₄	—	108	2	no	—
295	C₄	#88534	1	4	yes	complemented
296	C₃	#10588608	1	3	yes	complemented
297	C₃	#10592627	1	1	yes	characteristic
298	C₂	C₄×#6226	1	1	yes	characteristic
299	C₂	#11721706	1	1	yes	characteristic
300	C₂	#11721707	1	1	yes	characteristic
301	C₂	C₄×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
302	C₂	C₂×#88534	1	1	yes	characteristic
303	C₂	#11721813	1	1	yes	characteristic
304	C₂	#11721819	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	#389
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. C₄⋊(C₃⋊Dic₉)	C₄	semidirect
72. C₃×C₄×C₉	C₄⋊C₄	semidirect
164. C₄×C₉	Dic₃⋊C₄	semidirect
172. C₃×C₉	#389	semidirect
246. C₃×C₄	Dic₉⋊C₄	semidirect
253. C₉	#4928	semidirect
295. C₄	#88534	semidirect
296. C₃	#10588608	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂×#88534	2
#11720572	2
#10592627	3
#10588608	3