Subgroups — #13139363

Order 1728.

486 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#11720273	C₂	1	4	yes	maximal, complemented
5	#10591120	—	3	1	no	maximal
6	#10588802	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
9	C₄×C₃⋊Dic₉	C₄	1	1	yes	characteristic
10	C₄×C₃⋊Dic₉	C₄	1	1	yes	characteristic
11	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
12	C₉×#208	—	2	2	no	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×#3667	—	2	2	no	—
15	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
17	C₂×C₃×#208	—	3	1	no	—
18	C₂×C₃×#208	D₃	1	1	yes	characteristic
19	#63572	—	3	4	no	—
20	C₂×C₉×M₁₆	—	3	1	no	—
21	C₂×C₄×Dic₉	—	3	1	no	—
22	C₂×C₄×Dic₉	—	6	1	no	—
23	C₂×#3667	D₃	1	1	yes	characteristic
24	Dic₉⋊C₈	—	3	4	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊Dic₉	—	2	4	no	—
31	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
32	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
33	C₉×C₃⋊C₈	—	2	2	no	—
34	C₃×C₉⋊C₈	—	2	2	no	—
35	#5118	—	9	1	no	—
36	#5118	—	9	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	3	2	no	—
39	C₄×C₃⋊Dic₃	—	3	1	no	—
40	C₄×C₃⋊Dic₃	—	3	1	no	—
41	C₄×C₃⋊Dic₃	—	3	2	no	—
42	C₃×#208	—	2	2	no	—
43	C₃×#208	—	6	2	no	—
44	C₂×C₃×C₃⋊C₈	—	3	2	no	—
45	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
48	C₂²×Dic₉	—	3	2	no	—
49	C₂²×Dic₉	—	6	2	no	—
50	C₉×M₁₆	—	6	2	no	—
51	C₂×C₈×C₉	—	3	2	no	—
52	C₄×Dic₉	—	3	1	no	—
53	C₄×Dic₉	—	3	1	no	—
54	C₄×Dic₉	—	3	2	no	—
55	C₄×Dic₉	—	6	1	no	—
56	C₄×Dic₉	—	6	1	no	—
57	C₄×Dic₉	—	6	2	no	—
58	#3667	—	2	2	no	—
59	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
64	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₃⋊Dic₉	—	2	2	no	—
67	C₃⋊Dic₉	M₁₆	1	4	yes	complemented
68	C₂×C₃×M₁₆	—	3	1	no	—
69	C₂×C₃×M₁₆	—	9	1	no	—
70	C₂×C₄×Dic₃	—	9	1	no	—
71	C₂×C₄×Dic₃	—	9	1	no	—
72	C₂×C₄×Dic₃	—	18	1	no	—
73	C₂×#208	—	3	1	no	—
74	C₂×#208	D₉	1	1	yes	characteristic
75	Dic₃⋊C₈	—	9	4	no	—
76	Dic₃⋊C₈	—	9	4	no	—
77	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	Dic₆	1	2	yes	—
79	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	#67	1	2	yes	—
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	—	3	2	no	—
83	C₂×C₃⋊Dic₃	—	3	2	no	—
84	C₂×C₃⋊Dic₃	—	6	4	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
87	C₃×C₃⋊C₈	—	2	2	no	—
88	C₃×C₃⋊C₈	—	6	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	Dic₆	1	2	yes	—
94	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
95	C₂×C₄×C₉	#67	1	2	yes	—
96	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
97	C₂×Dic₉	—	3	2	no	—
98	C₂×Dic₉	—	3	2	no	—
99	C₂×Dic₉	—	6	2	no	—
100	C₂×Dic₉	—	6	2	no	—
101	C₂×Dic₉	—	6	4	no	—
102	C₂×Dic₉	—	6	8	no	—
103	C₈×C₉	—	6	2	no	—
104	C₉⋊C₈	—	2	2	no	—
105	#423	—	27	1	no	Sylow
106	C₂×C₃×C₉	—	2	2	no	—
107	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
111	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
112	C₂²×Dic₃	—	9	2	no	—
113	C₂²×Dic₃	—	9	2	no	—
114	C₂²×Dic₃	—	18	2	no	—
115	C₃×M₁₆	—	6	2	no	—
116	C₃×M₁₆	—	18	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	—	9	2	no	—
119	C₄×Dic₃	—	9	1	no	—
120	C₄×Dic₃	—	9	1	no	—
121	C₄×Dic₃	—	9	2	no	—
122	C₄×Dic₃	—	9	1	no	—
123	C₄×Dic₃	—	9	1	no	—
124	C₄×Dic₃	—	9	2	no	—
125	C₄×Dic₃	—	18	1	no	—
126	C₄×Dic₃	—	18	1	no	—
127	C₄×Dic₃	—	18	2	no	—
128	#208	—	2	2	no	—
129	#208	—	6	2	no	—
130	C₂×C₃⋊C₈	—	3	2	no	—
131	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
132	C₂²×C₃²	—	2	2	no	—
133	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
134	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
136	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
137	C₃²×C₄	C₂×#67	1	1	yes	characteristic
138	C₃⋊Dic₃	—	3	4	no	—
139	C₃⋊Dic₃	—	6	2	no	—
140	C₂²×C₉	—	2	4	no	—
141	C₂²×C₉	—	2	2	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	Dic₃⋊C₄	1	2	yes	—
145	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
146	C₄×C₉	—	2	2	no	—
147	C₄×C₉	—	2	1	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
150	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
151	C₄×C₉	C₂×#67	1	1	yes	characteristic
152	Dic₉	—	3	4	no	—
153	Dic₉	—	6	4	no	—
154	Dic₉	—	6	4	no	—
155	Dic₉	—	6	2	no	—
156	C₂×M₁₆	—	3	1	no	—
157	C₂×M₁₆	—	9	1	no	—
158	C₂×C₄²	—	27	1	no	—
159	C₄⋊C₈	—	27	4	no	—
160	C₃×C₉	#423	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
163	C₂³×C₃	#624	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
169	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
170	C₂×C₃×C₄	#611	1	2	yes	—
171	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
172	C₂×C₃×C₄	#624	1	1	yes	characteristic
173	C₂×C₃×C₄	#625	1	2	yes	—
174	C₂×C₃×C₄	#627	1	2	yes	—
175	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
176	C₂×Dic₃	—	9	2	no	—
177	C₂×Dic₃	—	9	2	no	—
178	C₂×Dic₃	—	9	2	no	—
179	C₂×Dic₃	—	9	2	no	—
180	C₂×Dic₃	—	18	2	no	—
181	C₂×Dic₃	—	18	2	no	—
182	C₂×Dic₃	—	18	4	no	—
183	C₂×Dic₃	—	18	4	no	—
184	C₂×Dic₃	—	18	8	no	—
185	C₃×C₈	—	6	2	no	—
186	C₃×C₈	—	18	2	no	—
187	C₃⋊C₈	—	2	2	no	—
188	C₃⋊C₈	—	6	2	no	—
189	C₂×C₃²	—	2	2	no	—
190	C₂×C₃²	D₃×M₁₆	1	2	yes	—
191	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
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	D₃×M₁₆	1	2	yes	—
197	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	—	27	2	no	—
199	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
200	M₁₆	—	6	2	no	—
201	M₁₆	—	18	2	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	9	2	no	—
204	C₄²	—	27	1	no	—
205	C₄²	—	27	1	no	—
206	C₄²	—	27	2	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
213	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
214	C₂²×C₃	#3724	1	2	yes	—
215	C₂²×C₃	C₂×#624	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
220	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
221	C₃×C₄	C₂×#611	1	1	yes	characteristic
222	C₃×C₄	#3724	1	2	yes	—
223	C₃×C₄	C₂×#625	1	1	yes	characteristic
224	C₃×C₄	C₂×#627	1	1	yes	characteristic
225	Dic₃	—	9	4	no	—
226	Dic₃	—	9	4	no	—
227	Dic₃	—	18	4	no	—
228	Dic₃	—	18	4	no	—
229	Dic₃	—	18	2	no	—
230	Dic₃	—	18	2	no	—
231	C₃²	#5118	1	1	yes	characteristic
232	C₉	—	2	1	no	—
233	C₉	#5118	1	1	yes	characteristic, complemented
234	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	—	27	2	no	—
236	C₂×C₄	—	27	2	no	—
237	C₂×C₄	—	54	4	no	—
238	C₂×C₄	#6189	1	2	yes	—
239	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
240	C₂×C₄	#6194	1	2	yes	—
241	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
242	C₈	—	6	2	no	—
243	C₈	—	18	2	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	2	no	—
249	C₂×C₃	M₁₆×D₉	1	2	yes	—
250	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
251	C₂×C₃	#63818	1	2	yes	—
252	C₂×C₃	C₂×#3724	1	1	yes	characteristic
253	C₂²	—	2	2	no	—
254	C₂²	#88442	1	2	yes	—
255	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
256	C₄	—	27	4	no	—
257	C₄	—	54	2	no	—
258	C₄	#88442	1	2	yes	—
259	C₄	C₂×#6189	1	1	yes	characteristic
260	C₄	C₂×#6194	1	1	yes	characteristic
261	C₃	—	2	1	no	—
262	C₃	#10588802	1	1	yes	characteristic, complemented
263	C₃	#10591120	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	#11720863	1	2	yes	—
266	C₂	C₂×#88442	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#423
LCS	G ▷ C₂²×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. #11720273	C₂	semidirect
67. C₃⋊Dic₉	M₁₆	semidirect
160. C₃×C₉	#423	semidirect
233. C₉	#5118	semidirect
262. C₃	#10588802	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×#208	2
C₂×C₃×#3667	2
#11720273	2
#10591120	3
#10588802	3