Subgroups — #13144855

Order 1728.

790 conjugacy classes of subgroups.

The subgroup lattice has 252 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	2	yes	maximal, complemented
3	#11720577	C₂	1	4	yes	maximal, complemented
4	#10592939	—	3	1	no	maximal
5	#10588920	—	3	3	no	maximal
6	C₃×C₉×#41	—	2	2	no	—
7	C₃×C₉×#41	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₄	1	2	yes	complemented
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	#88551	C₂²	1	4	yes	—
14	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
15	C₄×C₃⋊Dic₉	—	2	2	no	—
16	C₂×C₃²×#41	D₃	1	1	yes	characteristic
17	#64135	—	3	2	no	—
18	C₂×C₉×#41	D₃	1	3	yes	—
19	#63661	—	3	4	no	—
20	#63488	—	3	6	no	—
21	#63393	—	3	12	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₃×Q₈×C₉	—	2	2	no	—
25	C₃×D₄×C₉	—	2	4	no	—
26	C₃×D₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
34	C₂×C₃⋊Dic₉	—	2	4	no	—
35	#5240	—	9	3	no	—
36	#5240	—	9	1	no	—
37	C₃²×#41	—	2	2	no	—
38	C₃²×#41	—	2	2	no	—
39	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
40	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
41	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
42	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	#3757	—	3	4	no	—
45	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
46	C₄×C₃⋊Dic₃	—	6	2	no	—
47	C₉×#41	—	2	6	no	—
48	C₉×#41	—	2	6	no	—
49	C₂×Q₈×C₉	Dic₃	1	3	yes	complemented
50	C₂×D₄×C₉	Dic₃	1	3	yes	complemented
51	C₂×D₄×C₉	C₂×D₃	1	6	yes	—
52	C₂²×C₄×C₉	Dic₃	1	6	yes	complemented
53	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
54	#3676	—	3	12	no	—
55	C₄⋊Dic₉	—	6	6	no	—
56	C₄×Dic₉	—	6	6	no	—
57	C₂²×C₃×C₉	—	2	2	no	—
58	C₂²×C₃×C₉	—	4	1	no	—
59	C₂²×C₃×C₉	—	4	2	no	—
60	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
66	C₃⋊Dic₉	—	4	4	no	—
67	C₂×C₃×#41	D₉	1	3	yes	—
68	C₂×C₃×#41	C₃⋊D₃	1	1	yes	characteristic
69	#926	—	9	6	no	—
70	#926	—	9	2	no	—
71	#834	—	9	12	no	—
72	#834	—	9	4	no	—
73	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
74	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
75	C₃²×Q₈	—	2	2	no	—
76	C₃²×D₄	—	2	4	no	—
77	C₃²×D₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
81	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	#67	1	2	yes	—
84	C₂×C₃²×C₄	#67	1	2	yes	—
85	C₂×C₃⋊Dic₃	—	6	4	no	—
86	C₂³×C₉	C₂×Dic₃	1	6	yes	—
87	C₂³×C₉	C₂²×D₃	1	3	yes	—
88	Q₈×C₉	—	2	6	no	—
89	D₄×C₉	—	2	12	no	—
90	D₄×C₉	—	2	6	no	—
91	C₂×C₄×C₉	—	2	6	no	—
92	C₂×C₄×C₉	—	2	6	no	—
93	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
94	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
95	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
96	C₂×C₄×C₉	#67	1	6	yes	—
97	C₂×C₄×C₉	#67	1	6	yes	—
98	C₂×Dic₉	—	6	12	no	—
99	#410	—	27	1	no	Sylow
100	C₂×C₃×C₉	—	2	2	no	—
101	C₂×C₃×C₉	—	2	1	no	—
102	C₂×C₃×C₉	—	4	2	no	—
103	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₃×#41	—	2	6	no	—
105	C₃×#41	—	2	6	no	—
106	C₃×#41	—	2	2	no	—
107	C₃×#41	—	2	2	no	—
108	C₂×C₃×Q₈	Dic₉	1	3	yes	complemented
109	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
110	C₂×C₃×D₄	Dic₉	1	3	yes	complemented
111	C₂×C₃×D₄	C₂×D₉	1	6	yes	—
112	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
113	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
114	C₂²×C₃×C₄	Dic₉	1	6	yes	complemented
115	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
116	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
117	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
118	#217	—	9	12	no	—
119	#217	—	9	4	no	—
120	C₄⋊Dic₃	—	18	6	no	—
121	C₄⋊Dic₃	—	18	2	no	—
122	C₄×Dic₃	—	18	6	no	—
123	C₄×Dic₃	—	18	2	no	—
124	C₂²×C₃²	—	2	2	no	—
125	C₂²×C₃²	—	4	1	no	—
126	C₂²×C₃²	—	4	2	no	—
127	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
128	C₂²×C₃²	C₂×#67	1	2	yes	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	#217	1	1	yes	characteristic
131	C₃²×C₄	#217	1	1	yes	characteristic
132	C₃²×C₄	#217	1	2	yes	—
133	C₃⋊Dic₃	—	12	4	no	—
134	C₂²×C₉	—	2	6	no	—
135	C₂²×C₉	—	4	3	no	—
136	C₂²×C₉	—	4	6	no	—
137	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
138	C₂²×C₉	C₂×#67	1	6	yes	—
139	C₄×C₉	—	2	6	no	—
140	C₄×C₉	#217	1	3	yes	—
141	C₄×C₉	#217	1	3	yes	—
142	C₄×C₉	#217	1	6	yes	—
143	Dic₉	—	12	12	no	—
144	C₂×#41	C₃⋊D₉	1	1	yes	characteristic
145	#117	—	27	2	no	—
146	#99	—	27	4	no	—
147	C₃×C₉	#410	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₃	C₂×Dic₉	1	6	yes	—
149	C₂³×C₃	C₂²×D₉	1	3	yes	—
150	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
151	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
152	C₃×Q₈	—	2	6	no	—
153	C₃×Q₈	—	2	2	no	—
154	C₃×D₄	—	2	12	no	—
155	C₃×D₄	—	2	4	no	—
156	C₃×D₄	—	2	6	no	—
157	C₃×D₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	6	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
163	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
164	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
165	C₂×C₃×C₄	#611	1	6	yes	—
166	C₂×C₃×C₄	#611	1	6	yes	—
167	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
170	C₂×C₃×C₄	#638	1	2	yes	—
171	C₂×C₃×C₄	#638	1	2	yes	—
172	C₂×Dic₃	—	18	12	no	—
173	C₂×Dic₃	—	18	4	no	—
174	C₂×C₃²	—	2	2	no	—
175	C₂×C₃²	—	2	1	no	—
176	C₂×C₃²	—	4	2	no	—
177	C₂×C₃²	C₂×#217	1	1	yes	characteristic
178	C₂×C₉	—	2	6	no	—
179	C₂×C₉	—	2	3	no	—
180	C₂×C₉	—	4	6	no	—
181	C₂×C₉	C₂×#217	1	3	yes	—
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	C₂×Q₈	C₃⋊Dic₉	1	1	yes	characteristic, complemented
185	C₂×D₄	C₃⋊Dic₉	1	1	yes	characteristic, complemented
186	C₂×D₄	C₂×C₃⋊D₉	1	2	yes	—
187	C₂²×C₄	C₃⋊Dic₉	1	2	yes	complemented
188	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
189	C₄⋊C₄	—	54	2	no	—
190	C₂²⋊C₄	—	27	4	no	—
191	C₄²	—	54	2	no	—
192	C₂²×C₃	—	2	6	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	4	3	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	6	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
199	C₂²×C₃	C₂×#611	1	6	yes	—
200	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
201	C₂²×C₃	C₂×#638	1	2	yes	—
202	C₃×C₄	—	2	6	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	#3676	1	3	yes	—
205	C₃×C₄	#3676	1	6	yes	—
206	C₃×C₄	#3676	1	3	yes	—
207	C₃×C₄	#3757	1	1	yes	characteristic
208	C₃×C₄	#3757	1	2	yes	—
209	C₃×C₄	#3757	1	1	yes	characteristic
210	Dic₃	—	36	4	no	—
211	Dic₃	—	36	12	no	—
212	C₃²	#5240	1	1	yes	characteristic
213	C₉	#5240	1	3	yes	complemented
214	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
215	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	54	4	no	—
222	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
223	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
225	C₂×C₄	#6233	1	2	yes	—
226	C₂×C₄	#6233	1	2	yes	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	6	no	—
230	C₂×C₃	—	2	3	no	—
231	C₂×C₃	—	4	2	no	—
232	C₂×C₃	—	4	6	no	—
233	C₂×C₃	C₂×#3676	1	3	yes	—
234	C₂×C₃	C₂×#3757	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
239	C₂²	C₂×#6233	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	108	4	no	—
242	C₄	#88551	1	1	yes	characteristic
243	C₄	#88551	1	2	yes	—
244	C₄	#88551	1	1	yes	characteristic
245	C₃	#10588920	1	3	yes	complemented
246	C₃	#10592939	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	2	1	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88551
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂×C₃²	C₂×#217
prime radical	C₂×C₃²×#41	D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂×#41	C₃⋊D₉
hypocenter	C₃×C₉	#410
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ 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. #11720577	C₂	semidirect
8. C₂×C₃×Q₈×C₉	C₄	semidirect
9. C₂×C₃×D₄×C₉	C₄	semidirect
11. C₂²×C₃×C₄×C₉	C₄	semidirect
49. C₂×Q₈×C₉	Dic₃	semidirect
50. C₂×D₄×C₉	Dic₃	semidirect
52. C₂²×C₄×C₉	Dic₃	semidirect
108. C₂×C₃×Q₈	Dic₉	semidirect
110. C₂×C₃×D₄	Dic₉	semidirect
114. C₂²×C₃×C₄	Dic₉	semidirect
147. C₃×C₉	#410	semidirect
184. C₂×Q₈	C₃⋊Dic₉	semidirect
185. C₂×D₄	C₃⋊Dic₉	semidirect
187. C₂²×C₄	C₃⋊Dic₉	semidirect
213. C₉	#5240	semidirect
245. C₃	#10588920	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
#11720577	2
#10592939	3
#10588920	3