Subgroups — #13139318

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₉×#926	C₂	1	1	yes	maximal, characteristic
3	C₃×#63488	C₂	1	1	yes	maximal, characteristic
4	#11720265	C₂	1	2	yes	maximal, complemented
5	#11720263	C₂	1	2	yes	maximal, complemented
6	#10591138	—	3	1	no	maximal
7	#10588676	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₉⋊C₈)	C₄	1	4	yes	complemented
10	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₉×#217	—	2	1	no	—
13	C₃×#3676	—	2	1	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₄×C₉×Dic₃	—	2	1	no	—
16	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
17	C₃×C₄×Dic₉	—	2	1	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	C₃×#926	—	3	1	no	—
20	C₃×#926	D₃	1	1	yes	characteristic
21	#63576	—	3	2	no	—
22	#63574	—	3	2	no	—
23	C₉×#117	—	3	1	no	—
24	#63488	D₃	1	1	yes	characteristic
25	C₂²×C₉⋊C₈	—	3	1	no	—
26	C₂²×C₉⋊C₈	—	6	1	no	—
27	#63367	—	3	2	no	—
28	#63366	—	3	2	no	—
29	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
37	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
38	#4998	—	9	1	no	—
39	#4998	—	9	1	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
44	C₃×#217	—	2	1	no	—
45	C₃×#217	—	6	1	no	—
46	C₃×C₄⋊Dic₃	—	3	2	no	—
47	C₃×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
48	C₃×C₄×Dic₃	—	2	1	no	—
49	C₃×C₄×Dic₃	—	6	1	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
52	C₉×C₄⋊C₄	—	3	2	no	—
53	C₉×C₂²⋊C₄	—	6	1	no	—
54	C₄²×C₉	—	6	1	no	—
55	#3676	—	2	1	no	—
56	C₄⋊Dic₉	C₂×D₃	1	2	yes	—
57	C₄×Dic₉	—	2	1	no	—
58	C₂×C₉⋊C₈	—	3	4	no	—
59	C₂×C₉⋊C₈	—	3	1	no	—
60	C₂×C₉⋊C₈	—	3	1	no	—
61	C₂×C₉⋊C₈	—	6	4	no	—
62	C₂×C₉⋊C₈	—	6	1	no	—
63	C₂×C₉⋊C₈	—	6	1	no	—
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
68	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
70	C₉×Dic₃	—	4	2	no	—
71	C₃×Dic₉	—	4	2	no	—
72	C₃×#117	—	3	1	no	—
73	C₃×#117	—	9	1	no	—
74	#926	—	3	1	no	—
75	#926	D₉	1	1	yes	characteristic
76	C₂²×C₃⋊C₈	—	9	1	no	—
77	C₂²×C₃⋊C₈	—	9	1	no	—
78	C₂²×C₃⋊C₈	—	18	1	no	—
79	#808	—	9	2	no	—
80	#808	—	9	2	no	—
81	#807	—	9	2	no	—
82	#807	—	9	2	no	—
83	C₂³×C₃²	#67	1	1	yes	characteristic
84	C₂×C₃²×C₄	Dic₆	1	2	yes	—
85	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
86	C₂×C₃²×C₄	#67	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₂×C₃×Dic₃	—	6	2	no	—
90	C₂³×C₉	—	2	1	no	—
91	C₂³×C₉	#67	1	1	yes	characteristic
92	C₃⋊(C₃⋊C₈)	—	3	2	no	—
93	C₃⋊(C₃⋊C₈)	—	3	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	6	2	no	—
99	C₂×C₄×C₉	Dic₆	1	2	yes	—
100	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
101	C₂×C₄×C₉	#67	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
103	C₂×Dic₉	—	2	2	no	—
104	C₉⋊C₈	—	3	2	no	—
105	C₉⋊C₈	—	3	2	no	—
106	C₉⋊C₈	—	6	2	no	—
107	C₉⋊C₈	—	6	2	no	—
108	#427	—	27	1	no	Sylow
109	C₂×C₃×C₉	—	2	2	no	—
110	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₃×C₉	#135	1	2	yes	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
114	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
115	C₃×C₄⋊C₄	—	3	2	no	—
116	C₃×C₄⋊C₄	—	9	2	no	—
117	C₃×C₂²⋊C₄	—	6	1	no	—
118	C₃×C₂²⋊C₄	—	18	1	no	—
119	C₃×C₄²	—	6	1	no	—
120	C₃×C₄²	—	18	1	no	—
121	#217	—	2	1	no	—
122	#217	—	6	1	no	—
123	C₄⋊Dic₃	—	3	2	no	—
124	C₄⋊Dic₃	C₂×D₉	1	2	yes	—
125	C₄×Dic₃	—	2	1	no	—
126	C₄×Dic₃	—	6	1	no	—
127	C₂×C₃⋊C₈	—	9	4	no	—
128	C₂×C₃⋊C₈	—	9	1	no	—
129	C₂×C₃⋊C₈	—	9	1	no	—
130	C₂×C₃⋊C₈	—	9	4	no	—
131	C₂×C₃⋊C₈	—	9	1	no	—
132	C₂×C₃⋊C₈	—	9	1	no	—
133	C₂×C₃⋊C₈	—	18	4	no	—
134	C₂×C₃⋊C₈	—	18	1	no	—
135	C₂×C₃⋊C₈	—	18	1	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
138	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
139	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
140	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
141	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
142	C₃×Dic₃	—	4	2	no	—
143	C₃×Dic₃	—	12	2	no	—
144	C₂²×C₉	—	2	4	no	—
145	C₂²×C₉	—	2	2	no	—
146	C₂²×C₉	—	2	2	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	Dic₃⋊C₄	1	2	yes	—
149	C₂²×C₉	C₂×#67	1	1	yes	characteristic
150	C₄×C₉	—	2	2	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	12	2	no	—
154	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
155	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
156	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
157	Dic₉	—	4	2	no	—
158	C₂²×C₈	—	27	1	no	—
159	#117	—	3	1	no	—
160	#117	—	9	1	no	—
161	#107	—	27	2	no	—
162	#106	—	27	2	no	—
163	C₃×C₉	#427	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	#611	1	1	yes	characteristic
166	C₂³×C₃	#625	1	1	yes	characteristic
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	6	2	no	—
172	C₂×C₃×C₄	—	18	2	no	—
173	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
174	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
175	C₂×C₃×C₄	#611	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
177	C₂×C₃×C₄	#624	1	2	yes	—
178	C₂×C₃×C₄	#625	1	1	yes	characteristic
179	C₂×C₃×C₄	#627	1	2	yes	—
180	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
181	C₂×Dic₃	—	2	2	no	—
182	C₂×Dic₃	—	6	2	no	—
183	C₃⋊C₈	—	9	2	no	—
184	C₃⋊C₈	—	9	2	no	—
185	C₃⋊C₈	—	9	2	no	—
186	C₃⋊C₈	—	9	2	no	—
187	C₃⋊C₈	—	18	2	no	—
188	C₃⋊C₈	—	18	2	no	—
189	C₂×C₃²	—	2	2	no	—
190	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
191	C₂×C₃²	#950	1	2	yes	—
192	C₂×C₉	—	2	4	no	—
193	C₂×C₉	—	2	2	no	—
194	C₂×C₉	—	2	2	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
197	C₂×C₉	#950	1	2	yes	—
198	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
199	C₂×C₈	—	27	4	no	—
200	C₂×C₈	—	27	1	no	—
201	C₂×C₈	—	27	1	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	—	9	2	no	—
204	C₂²⋊C₄	—	6	1	no	—
205	C₂²⋊C₄	—	18	1	no	—
206	C₄²	—	6	1	no	—
207	C₄²	—	18	1	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
214	C₂²×C₃	C₂×#611	1	1	yes	characteristic
215	C₂²×C₃	#3724	1	2	yes	—
216	C₂²×C₃	C₂×#625	1	1	yes	characteristic
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	12	2	no	—
221	C₃×C₄	—	36	2	no	—
222	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
223	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
224	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
225	C₃×C₄	#3724	1	2	yes	—
226	C₃×C₄	C₂×#624	1	1	yes	characteristic
227	C₃×C₄	C₂×#627	1	1	yes	characteristic
228	Dic₃	—	4	2	no	—
229	Dic₃	—	12	2	no	—
230	C₃²	#4998	1	1	yes	characteristic
231	C₉	—	2	1	no	—
232	C₉	#4998	1	1	yes	characteristic, complemented
233	C₂³	#6194	1	1	yes	characteristic
234	C₂×C₄	—	6	2	no	—
235	C₂×C₄	—	18	2	no	—
236	C₂×C₄	#6189	1	2	yes	—
237	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
238	C₂×C₄	#6194	1	1	yes	characteristic
239	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
240	C₈	—	27	2	no	—
241	C₈	—	27	2	no	—
242	C₂×C₃	—	2	4	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
248	C₂×C₃	#63513	1	2	yes	—
249	C₂×C₃	#63822	1	2	yes	—
250	C₂×C₃	C₂×#3724	1	1	yes	characteristic
251	C₂²	—	2	2	no	—
252	C₂²	#88442	1	2	yes	—
253	C₂²	C₂×#6194	1	1	yes	characteristic
254	C₄	—	12	2	no	—
255	C₄	—	36	2	no	—
256	C₄	#88442	1	2	yes	—
257	C₄	C₂×#6189	1	1	yes	characteristic
258	C₄	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
259	C₃	—	2	1	no	—
260	C₃	#10588676	1	1	yes	characteristic, complemented
261	C₃	#10591138	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	#11720881	1	2	yes	—
264	C₂	C₂×#88442	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6194
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#427
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720265	C₂	semidirect
5. #11720263	C₂	semidirect
9. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
163. C₃×C₉	#427	semidirect
232. C₉	#4998	semidirect
260. C₃	#10588676	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₉⋊C₈)	2
C₉×#926	2
C₃×#63488	2
#11720265	2
#11720263	2
#10591138	3
#10588676	3