Subgroups — #13139637

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#88442	C₂	1	4	yes	maximal
5	#10591359	—	3	1	no	maximal
6	#10588655	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
9	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₄×C₃⋊Dic₉	C₄	1	4	yes	complemented
12	C₉×C₄⋊Dic₃	—	2	2	no	—
13	C₃×C₄⋊Dic₉	—	2	2	no	—
14	#88442	—	2	8	no	—
15	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
16	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
17	C₂×C₃×C₄⋊Dic₃	D₃	1	1	yes	characteristic
18	C₂×#3724	—	3	4	no	—
19	C₂×C₉×C₄⋊C₄	—	3	1	no	—
20	C₂×C₄⋊Dic₉	D₃	1	1	yes	characteristic
21	C₂×Dic₉⋊C₄	—	3	4	no	—
22	C₂×C₄×Dic₉	—	3	1	no	—
23	C₂×C₄×Dic₉	—	6	1	no	—
24	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
28	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
29	C₂×C₃⋊Dic₉	D₄	1	4	yes	—
30	C₂×C₃⋊Dic₉	Q₈	1	4	yes	—
31	C₂×C₉×Dic₃	—	2	4	no	—
32	C₂×C₉×Dic₃	—	2	2	no	—
33	C₂×C₃×Dic₉	—	2	4	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	#4975	—	9	1	no	—
36	#4975	—	9	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	3	2	no	—
39	C₂²×C₃×Dic₃	—	3	2	no	—
40	C₂²×C₃×Dic₃	C₂×D₃	1	2	yes	—
41	C₄×C₃⋊Dic₃	—	3	4	no	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	6	2	no	—
44	#3724	—	6	8	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	—	3	2	no	—
47	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
48	C₂²×Dic₉	—	3	2	no	—
49	C₂²×Dic₉	—	6	2	no	—
50	C₂²×Dic₉	C₂×D₃	1	2	yes	—
51	C₉×C₄⋊C₄	—	6	2	no	—
52	C₄⋊Dic₉	—	2	2	no	—
53	Dic₉⋊C₄	—	6	8	no	—
54	C₄×Dic₉	—	3	4	no	—
55	C₄×Dic₉	—	6	4	no	—
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
61	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
62	C₃⋊Dic₉	C₄⋊C₄	1	8	yes	complemented
63	C₉×Dic₃	—	4	2	no	—
64	C₃×Dic₉	—	4	2	no	—
65	C₂×C₃×C₄⋊C₄	—	3	1	no	—
66	C₂×C₃×C₄⋊C₄	—	9	1	no	—
67	C₂×C₄⋊Dic₃	—	3	1	no	—
68	C₂×C₄⋊Dic₃	D₉	1	1	yes	characteristic
69	C₂×Dic₃⋊C₄	—	9	4	no	—
70	C₂×Dic₃⋊C₄	—	9	4	no	—
71	C₂×C₄×Dic₃	—	9	1	no	—
72	C₂×C₄×Dic₃	—	9	1	no	—
73	C₂×C₄×Dic₃	—	18	1	no	—
74	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	Dic₆	1	2	yes	—
76	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
77	C₂×C₃²×C₄	#67	1	2	yes	—
78	C₂×C₃⋊Dic₃	—	3	4	no	—
79	C₂×C₃⋊Dic₃	—	3	4	no	—
80	C₂×C₃⋊Dic₃	—	3	4	no	—
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	—	2	2	no	—
83	C₂×C₃×Dic₃	—	6	4	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	6	4	no	—
91	C₂×C₄×C₉	—	6	2	no	—
92	C₂×C₄×C₉	Dic₆	1	2	yes	—
93	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
94	C₂×C₄×C₉	#67	1	2	yes	—
95	C₂×Dic₉	—	2	4	no	—
96	C₂×Dic₉	—	2	2	no	—
97	C₂×Dic₉	—	3	4	no	—
98	C₂×Dic₉	—	3	4	no	—
99	C₂×Dic₉	—	3	4	no	—
100	C₂×Dic₉	—	6	4	no	—
101	C₂×Dic₉	—	6	4	no	—
102	C₂×Dic₉	—	6	4	no	—
103	#384	—	27	1	no	Sylow
104	C₂×C₃×C₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₉	#127	1	1	yes	characteristic
107	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
108	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	3	2	no	—
111	C₂²×C₃×C₄	—	9	2	no	—
112	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
113	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
114	C₂²×Dic₃	—	3	2	no	—
115	C₂²×Dic₃	—	9	2	no	—
116	C₂²×Dic₃	—	9	2	no	—
117	C₂²×Dic₃	—	18	2	no	—
118	C₂²×Dic₃	C₂×D₉	1	2	yes	—
119	C₃×C₄⋊C₄	—	6	2	no	—
120	C₃×C₄⋊C₄	—	18	2	no	—
121	C₄⋊Dic₃	—	2	2	no	—
122	C₄⋊Dic₃	—	6	2	no	—
123	Dic₃⋊C₄	—	18	8	no	—
124	Dic₃⋊C₄	—	18	8	no	—
125	C₄×Dic₃	—	9	4	no	—
126	C₄×Dic₃	—	9	4	no	—
127	C₄×Dic₃	—	18	4	no	—
128	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
129	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
130	C₂²×C₃²	D₃×D₄	1	2	yes	—
131	C₂²×C₃²	D₃×Q₈	1	2	yes	—
132	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
133	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
134	C₃⋊Dic₃	—	3	8	no	—
135	C₃×Dic₃	—	4	2	no	—
136	C₃×Dic₃	—	12	2	no	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	1	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
143	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂²×C₉	D₃×D₄	1	2	yes	—
145	C₂²×C₉	D₃×Q₈	1	2	yes	—
146	C₂²×C₉	C₂×#67	1	1	yes	characteristic
147	C₄×C₉	—	2	4	no	—
148	C₄×C₉	—	12	2	no	—
149	C₄×C₉	Dic₃⋊C₄	1	4	yes	complemented
150	Dic₉	—	3	8	no	—
151	Dic₉	—	4	2	no	—
152	Dic₉	—	6	8	no	—
153	C₂×C₄⋊C₄	—	3	1	no	—
154	C₂×C₄⋊C₄	—	9	1	no	—
155	C₂×C₄⋊C₄	—	27	4	no	—
156	C₂×C₄²	—	27	1	no	—
157	C₃×C₉	#384	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₃	—	2	1	no	—
159	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
160	C₂³×C₃	C₂×D₃²	1	1	yes	characteristic
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	6	4	no	—
165	C₂×C₃×C₄	—	6	2	no	—
166	C₂×C₃×C₄	—	18	4	no	—
167	C₂×C₃×C₄	—	18	2	no	—
168	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
169	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
170	C₂×C₃×C₄	#611	1	2	yes	—
171	C₂×C₃×C₄	#624	1	2	yes	—
172	C₂×C₃×C₄	#625	1	2	yes	—
173	C₂×C₃×C₄	#627	1	2	yes	—
174	C₂×Dic₃	—	2	4	no	—
175	C₂×Dic₃	—	2	2	no	—
176	C₂×Dic₃	—	6	4	no	—
177	C₂×Dic₃	—	6	2	no	—
178	C₂×Dic₃	—	9	4	no	—
179	C₂×Dic₃	—	9	4	no	—
180	C₂×Dic₃	—	9	4	no	—
181	C₂×Dic₃	—	9	4	no	—
182	C₂×Dic₃	—	9	4	no	—
183	C₂×Dic₃	—	9	4	no	—
184	C₂×Dic₃	—	18	4	no	—
185	C₂×Dic₃	—	18	4	no	—
186	C₂×Dic₃	—	18	4	no	—
187	C₂×C₃²	#888	1	2	yes	—
188	C₂×C₃²	D₃×C₄⋊C₄	1	2	yes	—
189	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
190	C₂×C₃²	#940	1	1	yes	characteristic
191	C₂×C₃²	#944	1	1	yes	characteristic
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	2	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	#888	1	2	yes	—
198	C₂×C₉	D₃×C₄⋊C₄	1	2	yes	—
199	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
200	C₂×C₉	#940	1	1	yes	characteristic
201	C₂×C₉	#944	1	1	yes	characteristic
202	C₂²×C₄	—	3	2	no	—
203	C₂²×C₄	—	9	2	no	—
204	C₂²×C₄	—	27	2	no	—
205	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
206	C₄⋊C₄	—	6	2	no	—
207	C₄⋊C₄	—	18	2	no	—
208	C₄⋊C₄	—	54	8	no	—
209	C₄²	—	27	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
216	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
217	C₂²×C₃	D₄×D₉	1	2	yes	—
218	C₂²×C₃	Q₈×D₉	1	2	yes	—
219	C₂²×C₃	C₂×#611	1	1	yes	characteristic
220	C₂²×C₃	#3797	1	2	yes	—
221	C₂²×C₃	#3802	1	2	yes	—
222	C₂²×C₃	C₂×#624	1	1	yes	characteristic
223	C₂²×C₃	C₂×#625	1	1	yes	characteristic
224	C₂²×C₃	C₂×#627	1	1	yes	characteristic
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	12	2	no	—
227	C₃×C₄	—	36	2	no	—
228	C₃×C₄	Dic₉⋊C₄	1	4	yes	complemented
229	C₃×C₄	#3724	1	4	yes	—
230	Dic₃	—	4	2	no	—
231	Dic₃	—	9	8	no	—
232	Dic₃	—	9	8	no	—
233	Dic₃	—	12	2	no	—
234	Dic₃	—	18	8	no	—
235	C₃²	#4975	1	1	yes	characteristic
236	C₉	—	2	1	no	—
237	C₉	#4975	1	1	yes	characteristic, complemented
238	C₂³	C₂×D₃×D₉	1	1	yes	characteristic
239	C₂×C₄	—	6	4	no	—
240	C₂×C₄	—	6	2	no	—
241	C₂×C₄	—	18	4	no	—
242	C₂×C₄	—	18	2	no	—
243	C₂×C₄	—	27	4	no	—
244	C₂×C₄	—	27	4	no	—
245	C₂×C₄	—	27	4	no	—
246	C₂×C₄	#6189	1	2	yes	—
247	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
248	C₂×C₄	#6194	1	2	yes	—
249	C₂×C₃	—	2	2	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	#63450	1	2	yes	—
255	C₂×C₃	C₄⋊C₄×D₉	1	2	yes	—
256	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
257	C₂×C₃	#63502	1	1	yes	characteristic
258	C₂×C₃	#63506	1	1	yes	characteristic
259	C₂×C₃	#63873	1	1	yes	characteristic
260	C₂×C₃	#63900	1	2	yes	—
261	C₂×C₃	#63914	1	1	yes	characteristic
262	C₂×C₃	#63919	1	2	yes	—
263	C₂×C₃	C₂×#3724	1	1	yes	characteristic
264	C₂²	(C₃⋊D₉)⋊Q₈	1	2	yes	—
265	C₂²	#88646	1	2	yes	—
266	C₂²	C₂×#6189	1	1	yes	characteristic
267	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
268	C₂²	C₂×#6194	1	1	yes	characteristic
269	C₄	—	12	2	no	—
270	C₄	—	27	8	no	—
271	C₄	—	36	2	no	—
272	C₄	#88442	1	4	yes	complemented
273	C₃	—	2	1	no	—
274	C₃	#10588655	1	1	yes	characteristic, complemented
275	C₃	#10591359	1	1	yes	characteristic
276	C₂	#11720930	1	1	yes	characteristic
277	C₂	#11720971	1	2	yes	—
278	C₂	#11721000	1	2	yes	—
279	C₂	#11721043	1	1	yes	characteristic
280	C₂	C₂×#88442	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#384
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
11. C₄×C₃⋊Dic₉	C₄	semidirect
61. C₃×C₄×C₉	C₄⋊C₄	semidirect
62. C₃⋊Dic₉	C₄⋊C₄	semidirect
149. C₄×C₉	Dic₃⋊C₄	semidirect
157. C₃×C₉	#384	semidirect
228. C₃×C₄	Dic₉⋊C₄	semidirect
237. C₉	#4975	semidirect
272. C₄	#88442	semidirect
274. C₃	#10588655	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊Dic₉	2
C₂×C₉×C₄⋊Dic₃	2
C₂×C₃×C₄⋊Dic₉	2
C₂×#88442	2
#10591359	3
#10588655	3