Subgroups — #13139676

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 280 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×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊Dic₉	C₂	1	4	yes	maximal
5	#10591349	—	3	1	no	maximal
6	#10588615	—	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)	—	2	2	no	—
12	C₄×C₉×Dic₃	C₄	1	4	yes	complemented
13	C₃×C₄⋊Dic₉	—	2	2	no	—
14	Dic₃⋊Dic₉	—	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×Dic₃⋊Dic₃	—	3	4	no	—
19	C₂×C₄²×C₉	—	3	1	no	—
20	C₂×C₄⋊Dic₉	—	3	1	no	—
21	C₂×C₄⋊Dic₉	—	3	4	no	—
22	C₂×C₄⋊Dic₉	—	6	1	no	—
23	C₂×C₄⋊Dic₉	D₃	1	1	yes	characteristic
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	—	2	2	no	—
29	C₂×C₃⋊Dic₉	—	2	4	no	—
30	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
31	C₂×C₉×Dic₃	D₄	1	4	yes	—
32	C₂×C₉×Dic₃	Q₈	1	4	yes	—
33	C₂×C₃×Dic₉	—	2	4	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	#4975	—	9	1	no	—
36	#4935	—	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)	—	6	2	no	—
42	C₃×C₄⋊Dic₃	—	6	2	no	—
43	C₃×C₄×Dic₃	Dic₃	1	4	yes	—
44	Dic₃⋊Dic₃	—	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	C₄²×C₉	—	3	4	no	—
52	C₄⋊Dic₉	—	2	2	no	—
53	C₄⋊Dic₉	—	6	8	no	—
54	C₄⋊Dic₉	—	6	4	no	—
55	C₄⋊Dic₉	—	6	2	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	—	4	2	no	—
63	C₉×Dic₃	C₄⋊C₄	1	8	yes	complemented
64	C₃×Dic₉	—	4	2	no	—
65	C₂×C₃×C₄⋊C₄	—	9	1	no	—
66	C₂×C₃×C₄²	—	3	1	no	—
67	C₂×C₄⋊Dic₃	—	3	1	no	—
68	C₂×C₄⋊Dic₃	—	9	4	no	—
69	C₂×C₄⋊Dic₃	—	9	1	no	—
70	C₂×C₄⋊Dic₃	—	9	1	no	—
71	C₂×C₄⋊Dic₃	—	18	1	no	—
72	C₂×Dic₃⋊C₄	—	9	4	no	—
73	C₂×C₄×Dic₃	D₉	1	1	yes	characteristic
74	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	Dic₆	1	2	yes	—
76	C₂×C₃²×C₄	D₁₂	1	2	yes	—
77	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
78	C₂×C₃⋊Dic₃	—	6	2	no	—
79	C₂×C₃⋊Dic₃	—	6	4	no	—
80	C₂×C₃×Dic₃	—	6	4	no	—
81	C₂×C₃×Dic₃	—	6	2	no	—
82	C₂×C₃×Dic₃	Dic₆	1	4	yes	—
83	C₂×C₃×Dic₃	D₁₂	1	4	yes	—
84	C₂×C₃×Dic₃	C₂×Dic₃	1	4	yes	—
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	—	3	4	no	—
91	C₂×C₄×C₉	—	3	4	no	—
92	C₂×C₄×C₉	—	3	4	no	—
93	C₂×C₄×C₉	Dic₆	1	2	yes	—
94	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
95	C₂×C₄×C₉	#67	1	2	yes	—
96	C₂×Dic₉	—	2	4	no	—
97	C₂×Dic₉	—	2	2	no	—
98	C₂×Dic₉	—	6	4	no	—
99	C₂×Dic₉	—	6	4	no	—
100	C₂×Dic₉	—	6	2	no	—
101	C₂×Dic₉	—	12	4	no	—
102	#384	—	27	1	no	Sylow
103	C₂×C₃×C₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₉	#127	1	1	yes	characteristic
106	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
107	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	3	2	no	—
110	C₂²×C₃×C₄	—	9	2	no	—
111	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
112	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
113	C₂²×Dic₃	—	3	2	no	—
114	C₂²×Dic₃	—	9	2	no	—
115	C₂²×Dic₃	—	9	2	no	—
116	C₂²×Dic₃	—	18	2	no	—
117	C₂²×Dic₃	C₂×D₉	1	2	yes	—
118	C₃×C₄⋊C₄	—	18	2	no	—
119	C₃×C₄²	—	3	4	no	—
120	C₄⋊Dic₃	—	6	2	no	—
121	C₄⋊Dic₃	—	18	8	no	—
122	C₄⋊Dic₃	—	18	4	no	—
123	C₄⋊Dic₃	—	18	2	no	—
124	C₄⋊Dic₃	—	18	2	no	—
125	Dic₃⋊C₄	—	18	8	no	—
126	C₄×Dic₃	Dic₉	1	4	yes	complemented
127	C₂²×C₃²	C₂×Dic₆	1	2	yes	—
128	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
129	C₂²×C₃²	C₂×D₁₂	1	2	yes	—
130	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
131	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
132	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
133	C₃⋊Dic₃	—	12	2	no	—
134	C₃×Dic₃	—	12	2	no	—
135	C₃×Dic₃	C₄⋊Dic₃	1	8	yes	—
136	C₂²×C₉	—	2	2	no	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	2	1	no	—
139	C₂²×C₉	—	2	1	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
142	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂²×C₉	D₃×D₄	1	2	yes	—
144	C₂²×C₉	D₃×Q₈	1	2	yes	—
145	C₂²×C₉	C₂×#67	1	1	yes	characteristic
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	—	3	8	no	—
148	C₄×C₉	Dic₃⋊C₄	1	4	yes	complemented
149	Dic₉	—	4	2	no	—
150	Dic₉	—	12	4	no	—
151	Dic₉	—	12	2	no	—
152	C₂×C₄⋊C₄	—	9	1	no	—
153	C₂×C₄⋊C₄	—	27	4	no	—
154	C₂×C₄⋊C₄	—	27	1	no	—
155	C₂×C₄²	—	3	1	no	—
156	C₃×C₉	#384	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
159	C₂³×C₃	C₂×D₃²	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	3	4	no	—
164	C₂×C₃×C₄	—	3	4	no	—
165	C₂×C₃×C₄	—	3	4	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	D₃₆	1	2	yes	—
170	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
171	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
172	C₂×C₃×C₄	#626	1	2	yes	—
173	C₂×C₃×C₄	#627	1	2	yes	—
174	C₂×Dic₃	—	6	4	no	—
175	C₂×Dic₃	—	6	2	no	—
176	C₂×Dic₃	—	18	4	no	—
177	C₂×Dic₃	—	18	4	no	—
178	C₂×Dic₃	—	18	2	no	—
179	C₂×Dic₃	—	18	4	no	—
180	C₂×Dic₃	—	18	2	no	—
181	C₂×Dic₃	—	36	4	no	—
182	C₂×Dic₃	Dic₁₈	1	4	yes	—
183	C₂×Dic₃	D₃₆	1	4	yes	—
184	C₂×Dic₃	C₂×Dic₉	1	4	yes	—
185	C₂×C₃²	#869	1	1	yes	characteristic
186	C₂×C₃²	#869	1	2	yes	—
187	C₂×C₃²	#874	1	1	yes	characteristic
188	C₂×C₃²	C₂×C₄⋊Dic₃	1	2	yes	—
189	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
190	C₂×C₉	—	2	1	no	—
191	C₂×C₉	—	2	2	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	2	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	#888	1	2	yes	—
196	C₂×C₉	D₃×C₄⋊C₄	1	2	yes	—
197	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
198	C₂×C₉	#940	1	1	yes	characteristic
199	C₂×C₉	#944	1	1	yes	characteristic
200	C₂²×C₄	—	3	2	no	—
201	C₂²×C₄	—	9	2	no	—
202	C₂²×C₄	—	27	2	no	—
203	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
204	C₄⋊C₄	—	18	2	no	—
205	C₄⋊C₄	—	54	8	no	—
206	C₄⋊C₄	—	54	2	no	—
207	C₄²	—	3	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	C₂×Dic₁₈	1	2	yes	—
214	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
215	C₂²×C₃	C₂×D₃₆	1	2	yes	—
216	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
217	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
218	C₂²×C₃	D₃×Dic₆	1	2	yes	—
219	C₂²×C₃	D₃×D₁₂	1	2	yes	—
220	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
221	C₂²×C₃	C₂×#626	1	1	yes	characteristic
222	C₂²×C₃	C₂×#627	1	1	yes	characteristic
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	3	8	no	—
225	C₃×C₄	—	36	2	no	—
226	C₃×C₄	C₄⋊Dic₉	1	4	yes	complemented
227	C₃×C₄	Dic₃⋊Dic₃	1	4	yes	—
228	Dic₃	—	12	2	no	—
229	Dic₃	—	36	4	no	—
230	Dic₃	—	36	2	no	—
231	Dic₃	—	36	2	no	—
232	Dic₃	C₄⋊Dic₉	1	8	yes	complemented
233	C₃²	#4935	1	1	yes	characteristic
234	C₉	—	2	1	no	—
235	C₉	#4975	1	1	yes	characteristic, complemented
236	C₂³	C₂×D₃×D₉	1	1	yes	characteristic
237	C₂×C₄	—	3	4	no	—
238	C₂×C₄	—	3	4	no	—
239	C₂×C₄	—	3	4	no	—
240	C₂×C₄	—	18	4	no	—
241	C₂×C₄	—	18	2	no	—
242	C₂×C₄	—	54	4	no	—
243	C₂×C₄	—	54	2	no	—
244	C₂×C₄	#6189	1	2	yes	—
245	C₂×C₄	#6192	1	2	yes	—
246	C₂×C₄	D₃×Dic₉	1	2	yes	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	#63431	1	2	yes	—
253	C₂×C₃	#63431	1	1	yes	characteristic
254	C₂×C₃	#63436	1	1	yes	characteristic
255	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
256	C₂×C₃	C₂×C₄⋊Dic₉	1	2	yes	—
257	C₂×C₃	#63866	1	1	yes	characteristic
258	C₂×C₃	D₃×C₄⋊Dic₃	1	2	yes	—
259	C₂×C₃	#63911	1	1	yes	characteristic
260	C₂×C₃	#63918	1	2	yes	—
261	C₂×C₃	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
262	C₂²	D₃×Dic₁₈	1	2	yes	—
263	C₂²	D₃×D₃₆	1	2	yes	—
264	C₂²	C₂×#6189	1	1	yes	characteristic
265	C₂²	C₂×#6192	1	1	yes	characteristic
266	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
267	C₄	—	3	8	no	—
268	C₄	—	36	2	no	—
269	C₄	—	108	2	no	—
270	C₄	Dic₃⋊Dic₉	1	4	yes	complemented
271	C₃	—	2	1	no	—
272	C₃	#10588615	1	1	yes	characteristic, complemented
273	C₃	#10591349	1	1	yes	characteristic
274	C₂	#11720932	1	1	yes	characteristic
275	C₂	#11720970	1	2	yes	—
276	C₂	#11721006	1	1	yes	characteristic
277	C₂	D₃×C₄⋊Dic₉	1	2	yes	—
278	C₂	C₂×Dic₃⋊Dic₉	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
12. C₄×C₉×Dic₃	C₄	semidirect
61. C₃×C₄×C₉	C₄⋊C₄	semidirect
63. C₉×Dic₃	C₄⋊C₄	semidirect
126. C₄×Dic₃	Dic₉	semidirect
148. C₄×C₉	Dic₃⋊C₄	semidirect
156. C₃×C₉	#384	semidirect
226. C₃×C₄	C₄⋊Dic₉	semidirect
232. Dic₃	C₄⋊Dic₉	semidirect
235. C₉	#4975	semidirect
270. C₄	Dic₃⋊Dic₉	semidirect
272. C₃	#10588615	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×C₄×C₉×Dic₃	2
C₂×C₃×C₄⋊Dic₉	2
C₂×Dic₃⋊Dic₉	2
#10591349	3
#10588615	3