Subgroups — #13144042

Order 1728.

470 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6226	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11720538	C₂	1	1	yes	maximal, characteristic
4	#11720537	C₂	1	1	yes	maximal, characteristic
5	#11720537	C₂	1	1	yes	maximal, characteristic
6	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
7	#11720516	C₂	1	1	yes	maximal, characteristic
8	#10592527	—	3	1	no	maximal
9	#10588510	—	3	3	no	maximal
10	C₂×#6226	C₄	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊Dic₉)	C₄	1	1	yes	characteristic
15	#88534	C₄	1	1	yes	characteristic
16	#88534	C₄	1	1	yes	characteristic
17	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄×#634	—	3	1	no	—
22	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
23	#63642	—	3	1	no	—
24	#63641	—	3	1	no	—
25	#63641	—	3	1	no	—
26	C₈×C₃⋊Dic₃	—	3	1	no	—
27	#63630	—	3	1	no	—
28	C₄×Dic₁₈	—	3	3	no	—
29	C₉×C₄⋊C₈	D₃	1	3	yes	—
30	#63375	—	3	3	no	—
31	Dic₉⋊C₈	—	3	3	no	—
32	Dic₉⋊C₈	—	3	3	no	—
33	C₈×Dic₉	—	3	3	no	—
34	#63362	—	3	3	no	—
35	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	#6226	—	2	2	no	—
43	C₃×C₈×C₉	—	2	1	no	—
44	C₃×C₈×C₉	—	2	1	no	—
45	C₃⋊(C₉⋊C₈)	—	2	1	no	—
46	C₃⋊(C₉⋊C₈)	Q₈	1	2	yes	complemented
47	#4828	—	9	3	no	—
48	#4828	—	9	1	no	—
49	C₂×#634	—	3	1	no	—
50	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
52	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
54	#3750	—	3	1	no	—
55	#3750	—	3	1	no	—
56	C₄×C₃⋊Dic₃	—	3	1	no	—
57	C₄×C₃⋊Dic₃	—	3	1	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
59	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
60	C₂×Dic₁₈	—	3	3	no	—
61	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
62	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
63	C₄²×C₉	C₂×D₃	1	3	yes	—
64	C₄⋊Dic₉	—	3	3	no	—
65	Dic₉⋊C₄	—	3	3	no	—
66	Dic₉⋊C₄	—	3	3	no	—
67	C₄×Dic₉	—	3	3	no	—
68	C₄×Dic₉	—	3	3	no	—
69	C₂×C₉⋊C₈	—	3	3	no	—
70	C₂×C₉⋊C₈	—	3	3	no	—
71	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄×C₉	—	2	1	no	—
73	C₃×C₄×C₉	—	2	1	no	—
74	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
75	C₃×C₄×C₉	#41	1	1	yes	characteristic
76	C₃⋊Dic₉	—	2	1	no	—
77	C₃⋊Dic₉	—	2	1	no	—
78	C₃⋊Dic₉	—	2	1	no	—
79	C₃⋊Dic₉	M₁₆	1	2	yes	—
80	C₄×Dic₆	—	9	3	no	—
81	C₄×Dic₆	—	9	1	no	—
82	C₃×C₄⋊C₈	D₉	1	3	yes	—
83	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
84	#815	—	9	3	no	—
85	#815	—	9	1	no	—
86	Dic₃⋊C₈	—	9	3	no	—
87	Dic₃⋊C₈	—	9	3	no	—
88	Dic₃⋊C₈	—	9	1	no	—
89	Dic₃⋊C₈	—	9	1	no	—
90	C₈×Dic₃	—	9	3	no	—
91	C₈×Dic₃	—	9	1	no	—
92	#803	—	9	3	no	—
93	#803	—	9	1	no	—
94	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	—	3	1	no	—
98	C₂×C₃⋊Dic₃	—	3	1	no	—
99	C₂×C₃⋊Dic₃	—	3	1	no	—
100	C₂×C₃⋊Dic₃	—	3	1	no	—
101	#634	—	6	2	no	—
102	C₃²×C₈	—	2	1	no	—
103	C₃²×C₈	—	2	1	no	—
104	C₃⋊(C₃⋊C₈)	—	3	2	no	—
105	C₃⋊(C₃⋊C₈)	—	6	1	no	—
106	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
107	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
108	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
109	C₂×Dic₉	—	3	3	no	—
110	C₂×Dic₉	—	3	3	no	—
111	C₂×Dic₉	—	3	3	no	—
112	C₂×Dic₉	—	3	3	no	—
113	Dic₁₈	—	6	6	no	—
114	C₈×C₉	—	2	3	no	—
115	C₈×C₉	—	2	3	no	—
116	C₉⋊C₈	—	3	6	no	—
117	C₉⋊C₈	—	6	3	no	—
118	#446	—	27	1	no	Sylow
119	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₃×C₉	#131	1	1	yes	characteristic
122	C₂×Dic₆	—	9	3	no	—
123	C₂×Dic₆	—	9	1	no	—
124	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
125	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
126	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
128	C₃×C₄²	C₂×D₉	1	3	yes	—
129	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
130	C₄⋊Dic₃	—	9	3	no	—
131	C₄⋊Dic₃	—	9	1	no	—
132	Dic₃⋊C₄	—	9	3	no	—
133	Dic₃⋊C₄	—	9	3	no	—
134	Dic₃⋊C₄	—	9	1	no	—
135	Dic₃⋊C₄	—	9	1	no	—
136	C₄×Dic₃	—	9	3	no	—
137	C₄×Dic₃	—	9	3	no	—
138	C₄×Dic₃	—	9	1	no	—
139	C₄×Dic₃	—	9	1	no	—
140	C₂×C₃⋊C₈	—	9	3	no	—
141	C₂×C₃⋊C₈	—	9	1	no	—
142	C₂×C₃⋊C₈	—	9	3	no	—
143	C₂×C₃⋊C₈	—	9	1	no	—
144	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
145	C₃²×C₄	—	2	1	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	#237	1	1	yes	characteristic
148	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
149	C₃⋊Dic₃	—	3	2	no	—
150	C₃⋊Dic₃	—	6	1	no	—
151	C₃⋊Dic₃	—	6	1	no	—
152	C₃⋊Dic₃	—	6	1	no	—
153	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
154	C₄×C₉	—	2	3	no	—
155	C₄×C₉	—	2	3	no	—
156	C₄×C₉	#237	1	3	yes	—
157	C₄×C₉	D₃×Q₈	1	3	yes	—
158	Dic₉	—	3	6	no	—
159	Dic₉	—	6	3	no	—
160	Dic₉	—	6	3	no	—
161	Dic₉	—	6	3	no	—
162	C₄×Q₈	—	27	1	no	—
163	C₄⋊C₈	—	27	1	no	—
164	C₄⋊C₈	—	27	1	no	—
165	C₄⋊C₈	C₃⋊D₉	1	1	yes	characteristic
166	#97	—	27	1	no	—
167	#97	—	27	1	no	—
168	C₄×C₈	—	27	1	no	—
169	C₃×C₉	#446	1	1	yes	characteristic, complemented, Sylow
170	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
171	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
172	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
173	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
176	C₂×Dic₃	—	9	3	no	—
177	C₂×Dic₃	—	9	1	no	—
178	C₂×Dic₃	—	9	3	no	—
179	C₂×Dic₃	—	9	3	no	—
180	C₂×Dic₃	—	9	3	no	—
181	C₂×Dic₃	—	9	1	no	—
182	C₂×Dic₃	—	9	1	no	—
183	C₂×Dic₃	—	9	1	no	—
184	Dic₆	—	18	2	no	—
185	Dic₆	—	18	6	no	—
186	C₃×C₈	—	2	3	no	—
187	C₃×C₈	—	2	3	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃⋊C₈	—	9	6	no	—
191	C₃⋊C₈	—	9	2	no	—
192	C₃⋊C₈	—	18	3	no	—
193	C₃⋊C₈	—	18	1	no	—
194	C₂×C₃²	#887	1	1	yes	characteristic
195	C₂×C₃²	#901	1	1	yes	characteristic
196	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
197	C₂×C₉	#887	1	3	yes	—
198	C₂×C₉	#901	1	3	yes	—
199	C₂×C₉	D₃×M₁₆	1	3	yes	—
200	C₂×Q₈	—	27	1	no	—
201	C₂×C₈	—	27	1	no	—
202	C₂×C₈	—	27	1	no	—
203	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
204	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
205	C₄⋊C₄	—	27	1	no	—
206	C₄⋊C₄	—	27	1	no	—
207	C₄⋊C₄	—	27	1	no	—
208	C₄²	—	27	1	no	—
209	C₄²	—	27	1	no	—
210	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
211	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
212	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	3	no	—
216	C₃×C₄	—	2	3	no	—
217	C₃×C₄	#3699	1	3	yes	—
218	C₃×C₄	Q₈×D₉	1	3	yes	—
219	C₃×C₄	#3830	1	1	yes	characteristic
220	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
221	Dic₃	—	9	2	no	—
222	Dic₃	—	9	6	no	—
223	Dic₃	—	18	1	no	—
224	Dic₃	—	18	1	no	—
225	Dic₃	—	18	1	no	—
226	Dic₃	—	18	3	no	—
227	Dic₃	—	18	3	no	—
228	Dic₃	—	18	3	no	—
229	C₃²	#4828	1	1	yes	characteristic
230	C₉	#4828	1	3	yes	complemented
231	Q₈	—	54	2	no	—
232	C₂×C₄	—	27	1	no	—
233	C₂×C₄	—	27	1	no	—
234	C₂×C₄	—	27	1	no	—
235	C₂×C₄	—	27	1	no	—
236	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
237	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	27	2	no	—
242	C₈	—	54	1	no	—
243	C₂×C₃	#63449	1	3	yes	—
244	C₂×C₃	#63464	1	3	yes	—
245	C₂×C₃	M₁₆×D₉	1	3	yes	—
246	C₂×C₃	#64096	1	1	yes	characteristic
247	C₂×C₃	#64110	1	1	yes	characteristic
248	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
249	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	27	2	no	—
253	C₄	—	54	1	no	—
254	C₄	—	54	1	no	—
255	C₄	—	54	1	no	—
256	C₄	#88742	1	1	yes	characteristic
257	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
258	C₃	#10588510	1	3	yes	complemented
259	C₃	#10592527	1	1	yes	characteristic
260	C₂	#11721743	1	1	yes	characteristic
261	C₂	#11721770	1	1	yes	characteristic
262	C₂	M₁₆×C₃⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃²	C₂×C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄⋊C₈	C₃⋊D₉
hypocenter	C₃×C₉	#446
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
46. C₃⋊(C₉⋊C₈)	Q₈	semidirect
169. C₃×C₉	#446	semidirect
230. C₉	#4828	semidirect
258. C₃	#10588510	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6226	2
C₃×C₉×C₄⋊C₈	2
#11720538	2
#11720537	2
C₈×C₃⋊Dic₉	2
#11720516	2
#10592527	3
#10588510	3