Subgroups — #13144853

Order 1728.

790 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	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	#11721815	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720582	C₂	1	2	yes	maximal, complemented
5	#11720576	C₂	1	2	yes	maximal, complemented
6	#10592932	—	3	1	no	maximal
7	#10588917	—	3	3	no	maximal
8	C₃×C₉×#41	—	2	2	no	—
9	C₃×C₉×#41	C₄	1	4	yes	complemented
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₄×C₉	—	2	1	no	—
12	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₉	—	2	1	no	—
14	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
15	#88551	—	2	1	no	—
16	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
17	C₄×C₃⋊Dic₉	—	2	1	no	—
18	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
19	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₃²×#41	D₃	1	1	yes	characteristic
21	#64135	—	3	1	no	—
22	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×C₉×#41	D₃	1	3	yes	—
24	#63662	—	3	2	no	—
25	#63659	—	3	2	no	—
26	#63488	—	3	3	no	—
27	C₂²×C₉⋊C₈	—	3	3	no	—
28	#63395	—	3	6	no	—
29	#63392	—	3	6	no	—
30	C₂³×C₃×C₉	—	2	1	no	—
31	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
32	C₃×Q₈×C₉	—	2	1	no	—
33	C₃×Q₈×C₉	C₂×C₄	1	2	yes	—
34	C₃×D₄×C₉	—	2	4	no	—
35	C₃×D₄×C₉	—	2	1	no	—
36	C₃×D₄×C₉	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	—	2	4	no	—
39	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₃⋊Dic₉	—	2	2	no	—
44	C₃⋊(C₉⋊C₈)	—	2	2	no	—
45	#5233	—	9	3	no	—
46	#5233	—	9	1	no	—
47	C₃²×#41	—	2	2	no	—
48	C₃²×#41	Dic₃	1	4	yes	—
49	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×D₄	—	2	1	no	—
51	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃²×C₄	—	2	1	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
54	#3757	—	6	1	no	—
55	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
56	C₄×C₃⋊Dic₃	—	6	1	no	—
57	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
59	C₉×#41	—	2	6	no	—
60	C₉×#41	Dic₃	1	12	yes	complemented
61	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
62	C₂×D₄×C₉	—	2	3	no	—
63	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
64	C₂²×C₄×C₉	—	2	3	no	—
65	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
66	#3676	—	6	3	no	—
67	C₄⋊Dic₉	—	3	6	no	—
68	C₄×Dic₉	—	6	3	no	—
69	C₂×C₉⋊C₈	—	3	6	no	—
70	C₂×C₉⋊C₈	—	6	6	no	—
71	C₂²×C₃×C₉	—	2	2	no	—
72	C₂²×C₃×C₉	—	2	2	no	—
73	C₂²×C₃×C₉	—	4	2	no	—
74	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×C₉	—	2	2	no	—
77	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
78	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
80	C₃⋊Dic₉	—	4	2	no	—
81	C₂×C₃×#41	D₉	1	3	yes	—
82	C₂×C₃×#41	C₃⋊D₃	1	1	yes	characteristic
83	#926	—	9	3	no	—
84	#926	—	9	1	no	—
85	C₂²×C₃⋊C₈	—	9	3	no	—
86	C₂²×C₃⋊C₈	—	9	1	no	—
87	#835	—	9	6	no	—
88	#835	—	9	2	no	—
89	#832	—	9	6	no	—
90	#832	—	9	2	no	—
91	C₂³×C₃²	—	2	1	no	—
92	C₂³×C₃²	#67	1	1	yes	characteristic
93	C₃²×Q₈	—	2	1	no	—
94	C₃²×Q₈	C₂×Dic₃	1	2	yes	—
95	C₃²×D₄	—	2	4	no	—
96	C₃²×D₄	—	2	1	no	—
97	C₃²×D₄	C₂×Dic₃	1	2	yes	—
98	C₂×C₃²×C₄	—	2	1	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
101	C₂×C₃²×C₄	#67	1	1	yes	characteristic
102	C₂×C₃²×C₄	#67	1	2	yes	—
103	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₃⋊Dic₃	—	6	2	no	—
105	C₂³×C₉	—	2	3	no	—
106	C₂³×C₉	#67	1	3	yes	—
107	C₃⋊(C₃⋊C₈)	—	6	2	no	—
108	Q₈×C₉	—	2	3	no	—
109	Q₈×C₉	C₂×Dic₃	1	6	yes	—
110	D₄×C₉	—	2	12	no	—
111	D₄×C₉	—	2	3	no	—
112	D₄×C₉	C₂×Dic₃	1	6	yes	—
113	C₂×C₄×C₉	—	2	3	no	—
114	C₂×C₄×C₉	—	2	12	no	—
115	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
116	C₂×C₄×C₉	#67	1	3	yes	—
117	C₂×C₄×C₉	#67	1	6	yes	—
118	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
119	C₂×Dic₉	—	6	6	no	—
120	C₉⋊C₈	—	6	6	no	—
121	#416	—	27	1	no	Sylow
122	C₂×C₃×C₉	—	2	2	no	—
123	C₂×C₃×C₉	—	4	2	no	—
124	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₃×C₉	#135	1	2	yes	—
126	C₃×#41	—	2	6	no	—
127	C₃×#41	—	2	2	no	—
128	C₃×#41	Dic₉	1	12	yes	complemented
129	C₃×#41	C₃⋊Dic₃	1	4	yes	—
130	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
131	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
132	C₂×C₃×D₄	—	2	3	no	—
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
135	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
136	C₂²×C₃×C₄	—	2	3	no	—
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
139	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
140	#217	—	18	3	no	—
141	#217	—	18	1	no	—
142	C₄⋊Dic₃	—	9	6	no	—
143	C₄⋊Dic₃	—	9	2	no	—
144	C₄×Dic₃	—	18	3	no	—
145	C₄×Dic₃	—	18	1	no	—
146	C₂×C₃⋊C₈	—	9	6	no	—
147	C₂×C₃⋊C₈	—	9	2	no	—
148	C₂×C₃⋊C₈	—	18	6	no	—
149	C₂×C₃⋊C₈	—	18	2	no	—
150	C₂²×C₃²	—	2	2	no	—
151	C₂²×C₃²	—	2	2	no	—
152	C₂²×C₃²	—	4	2	no	—
153	C₂²×C₃²	#217	1	2	yes	—
154	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
155	C₃²×C₄	—	2	2	no	—
156	C₃²×C₄	#217	1	2	yes	—
157	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
158	C₃²×C₄	C₂×#67	1	1	yes	characteristic
159	C₃⋊Dic₃	—	12	2	no	—
160	C₂²×C₉	—	2	6	no	—
161	C₂²×C₉	—	2	6	no	—
162	C₂²×C₉	—	4	6	no	—
163	C₂²×C₉	#217	1	6	yes	—
164	C₂²×C₉	C₂×#67	1	3	yes	—
165	C₄×C₉	—	2	6	no	—
166	C₄×C₉	#217	1	6	yes	—
167	C₄×C₉	C₂²×Dic₃	1	3	yes	—
168	C₄×C₉	C₂×#67	1	3	yes	—
169	Dic₉	—	12	6	no	—
170	C₂×#41	C₃⋊D₉	1	1	yes	characteristic
171	C₂²×C₈	—	27	1	no	—
172	#117	—	27	1	no	—
173	#103	—	27	2	no	—
174	#102	—	27	2	no	—
175	C₃×C₉	#416	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₃	—	2	3	no	—
177	C₂³×C₃	—	2	1	no	—
178	C₂³×C₃	#611	1	3	yes	—
179	C₂³×C₃	#638	1	1	yes	characteristic
180	C₃×Q₈	—	2	3	no	—
181	C₃×Q₈	—	2	1	no	—
182	C₃×Q₈	C₂×Dic₉	1	6	yes	—
183	C₃×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
184	C₃×D₄	—	2	12	no	—
185	C₃×D₄	—	2	4	no	—
186	C₃×D₄	—	2	3	no	—
187	C₃×D₄	—	2	1	no	—
188	C₃×D₄	C₂×Dic₉	1	6	yes	—
189	C₃×D₄	C₂×C₃⋊Dic₃	1	2	yes	—
190	C₂×C₃×C₄	—	2	3	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	12	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
195	C₂×C₃×C₄	#611	1	6	yes	—
196	C₂×C₃×C₄	#611	1	3	yes	—
197	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
198	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
199	C₂×C₃×C₄	#638	1	2	yes	—
200	C₂×C₃×C₄	#638	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
202	C₂×Dic₃	—	18	6	no	—
203	C₂×Dic₃	—	18	2	no	—
204	C₃⋊C₈	—	18	6	no	—
205	C₃⋊C₈	—	18	2	no	—
206	C₂×C₃²	—	2	2	no	—
207	C₂×C₃²	—	4	2	no	—
208	C₂×C₃²	#950	1	2	yes	—
209	C₂×C₃²	C₂×#217	1	1	yes	characteristic
210	C₂×C₉	—	2	6	no	—
211	C₂×C₉	—	4	6	no	—
212	C₂×C₉	#950	1	6	yes	—
213	C₂×C₉	C₂×#217	1	3	yes	—
214	#41	—	2	2	no	—
215	#41	C₃⋊Dic₉	1	4	yes	complemented
216	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	C₂×C₃⋊D₉	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
221	C₂×C₈	—	27	2	no	—
222	C₂×C₈	—	54	2	no	—
223	C₄⋊C₄	—	27	2	no	—
224	C₂²⋊C₄	—	54	1	no	—
225	C₄²	—	54	1	no	—
226	C₂²×C₃	—	2	6	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	6	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	4	6	no	—
231	C₂²×C₃	—	4	2	no	—
232	C₂²×C₃	#3676	1	6	yes	—
233	C₂²×C₃	C₂×#611	1	3	yes	—
234	C₂²×C₃	#3757	1	2	yes	—
235	C₂²×C₃	C₂×#638	1	1	yes	characteristic
236	C₃×C₄	—	2	6	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	#3676	1	6	yes	—
239	C₃×C₄	C₂²×Dic₉	1	3	yes	—
240	C₃×C₄	C₂×#611	1	3	yes	—
241	C₃×C₄	#3757	1	2	yes	—
242	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
243	C₃×C₄	C₂×#638	1	1	yes	characteristic
244	Dic₃	—	36	6	no	—
245	Dic₃	—	36	2	no	—
246	C₃²	#5233	1	1	yes	characteristic
247	C₉	#5233	1	3	yes	complemented
248	C₂³	—	2	1	no	—
249	C₂³	#6233	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₂×C₃⋊Dic₉	1	2	yes	—
252	D₄	—	2	4	no	—
253	D₄	—	2	1	no	—
254	D₄	C₂×C₃⋊Dic₉	1	2	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	54	2	no	—
258	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
259	C₂×C₄	#6233	1	2	yes	—
260	C₂×C₄	#6233	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
262	C₈	—	54	2	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	6	no	—
265	C₂×C₃	—	4	2	no	—
266	C₂×C₃	—	4	6	no	—
267	C₂×C₃	#63513	1	6	yes	—
268	C₂×C₃	C₂×#3676	1	3	yes	—
269	C₂×C₃	#64159	1	2	yes	—
270	C₂×C₃	C₂×#3757	1	1	yes	characteristic
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	#88551	1	2	yes	—
275	C₂²	C₂×#6233	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	108	2	no	—
278	C₄	#88551	1	2	yes	—
279	C₄	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
280	C₄	C₂×#6233	1	1	yes	characteristic
281	C₃	#10588917	1	3	yes	complemented
282	C₃	#10592932	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	4	2	no	—
285	C₂	#11721868	1	2	yes	—
286	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃²×#41	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×#41	C₃⋊D₉
hypocenter	C₃×C₉	#416
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #11721815	C₂	semidirect
3. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
4. #11720582	C₂	semidirect
5. #11720576	C₂	semidirect
9. C₃×C₉×#41	C₄	semidirect
60. C₉×#41	Dic₃	semidirect
128. C₃×#41	Dic₉	semidirect
175. C₃×C₉	#416	semidirect
215. #41	C₃⋊Dic₉	semidirect
247. C₉	#5233	semidirect
281. C₃	#10588917	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×#41	2
#11721815	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720582	2
#11720576	2
#10592932	3
#10588917	3