Subgroups — #72252904

Order 3456.

675 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#3663⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#3663⋊C₃	C₂	1	2	yes	maximal
3	3₋¹⁺²×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588131⋊C₃	C₂	1	2	yes	maximal, complemented
5	C₈×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#68075	—	3	1	no	maximal
7	#11773693	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×#3663⋊C₃	C₄	1	2	yes	—
9	C₂×#3663⋊C₃	C₄	1	4	yes	—
10	C₄²×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
11	C₄×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₄×C₈×3₋¹⁺²	C₂²	1	2	yes	—
13	#63375⋊C₃	C₄	1	2	yes	—
14	#63375⋊C₃	C₄	1	4	yes	—
15	C₄×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
16	C₄×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×ΓL(1,𝔽₂₅)	—	3	2	no	—
18	C₃×C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
19	C₃²×#322	D₃	1	1	yes	characteristic
20	C₃×#4454	—	3	2	no	—
21	C₃×C₈×C₃⋊C₈	—	3	1	no	—
22	C₄×#3663	C₂×C₃	1	2	yes	—
23	C₄×#3663	C₂×C₃	1	1	yes	characteristic
24	C₉×#322	—	2	1	no	—
25	C₉×#322	C₂×C₃	1	1	yes	characteristic, complemented
26	#10588131	C₂×C₃	1	2	yes	complemented
27	C₈×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₄×AGL(1,ℤ/9)	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×AGL(1,ℤ/9)	C₂×C₄	1	2	yes	—
30	C₂×C₈×3₋¹⁺²	C₂×C₄	1	4	yes	—
31	C₂×C₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
32	C₄²×3₋¹⁺²	C₂³	1	1	yes	characteristic
33	C₄×3₋¹⁺²⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₄×3₋¹⁺²⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×3₋¹⁺²⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×3₋¹⁺²⋊C₈	C₂×C₄	1	4	yes	—
37	#3663⋊C₃	—	2	4	no	—
38	#3663⋊C₃	—	2	8	no	—
39	C₃×#1455	—	9	1	no	—
40	#68075	—	3	1	no	—
41	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	4	no	—
42	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	2	no	—
43	C₃×C₄²×D₃	—	3	1	no	—
44	C₃²×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	C₃²×C₄×C₈	C₂×D₃	1	2	yes	—
46	C₃×#815	—	3	4	no	—
47	C₃×#815	—	3	2	no	—
48	C₃×C₄×C₃⋊C₈	—	3	2	no	—
49	C₃×C₄×C₃⋊C₈	—	3	1	no	—
50	C₂×#3663	C₃×C₄	1	2	yes	—
51	C₂×#3663	C₃×C₄	1	4	yes	—
52	C₄²×D₉	C₂²×C₃	1	1	yes	characteristic
53	C₄×C₈×C₉	—	2	1	no	—
54	C₄×C₈×C₉	—	2	2	no	—
55	C₄×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
56	C₄×C₈×C₉	C₂²×C₃	1	2	yes	—
57	#63375	C₃×C₄	1	2	yes	—
58	#63375	C₃×C₄	1	4	yes	—
59	C₄×C₉⋊C₈	C₂²×C₃	1	2	yes	—
60	C₄×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×AGL(1,ℤ/9)	C₄²	1	1	yes	characteristic
62	C₂×C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×3₋¹⁺²	C₂²×C₄	1	2	yes	—
64	C₂×3₋¹⁺²⋊C₄	C₄²	1	1	yes	characteristic
65	C₂×3₋¹⁺²⋊C₄	C₄²	1	2	yes	—
66	C₄×AGL(1,ℤ/9)	—	2	2	no	—
67	C₄×AGL(1,ℤ/9)	—	2	4	no	—
68	C₈×3₋¹⁺²	—	2	4	no	—
69	C₈×3₋¹⁺²	M₁₆	1	4	yes	complemented
70	3₋¹⁺²⋊C₈	—	2	2	no	—
71	3₋¹⁺²⋊C₈	M₁₆	1	8	yes	complemented
72	C₃×C₄×M₁₆	—	9	1	no	—
73	C₃×C₄×M₁₆	—	9	2	no	—
74	C₄×ΓL(1,𝔽₂₅)	—	3	2	no	—
75	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
76	C₃×#322	—	6	1	no	—
77	C₃×#322	—	9	2	no	—
78	C₃×#322	C₃×D₃	1	1	yes	characteristic
79	C₃×C₈²	—	9	1	no	—
80	#4454	—	3	2	no	—
81	C₈×C₃⋊C₈	—	3	1	no	—
82	C₂×C₃×C₄×D₃	—	3	1	no	—
83	C₂×C₃×C₄×D₃	—	3	2	no	—
84	C₂×C₃²×C₈	C₄×D₃	1	4	yes	—
85	C₂×C₃²×C₈	C₄×D₃	1	2	yes	—
86	C₃²×C₄²	C₂²×D₃	1	1	yes	characteristic
87	C₃×C₄×Dic₃	—	3	1	no	—
88	C₃×C₄×Dic₃	—	3	2	no	—
89	C₂×C₃×C₃⋊C₈	—	3	4	no	—
90	C₂×C₃×C₃⋊C₈	—	3	2	no	—
91	C₃×ΓL(1,𝔽₂₅)	—	6	8	no	—
92	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
93	C₂×C₄×D₉	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₄×D₉	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₈×C₉	—	2	2	no	—
96	C₂×C₈×C₉	—	2	4	no	—
97	C₂×C₈×C₉	C₂×C₃×C₄	1	4	yes	—
98	C₂×C₈×C₉	C₂×C₃×C₄	1	2	yes	—
99	C₄²×C₉	—	2	1	no	—
100	C₄²×C₉	C₂³×C₃	1	1	yes	characteristic
101	C₄×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
102	C₄×Dic₉	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₉⋊C₈	C₂×C₃×C₄	1	2	yes	—
104	C₂×C₉⋊C₈	C₂×C₃×C₄	1	4	yes	—
105	#3663	—	2	4	no	—
106	#3663	—	2	8	no	—
107	#1455	—	9	1	no	Sylow
108	C₂²×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
109	C₂×AGL(1,ℤ/9)	—	2	1	no	—
110	C₂×AGL(1,ℤ/9)	—	2	2	no	—
111	C₄×3₋¹⁺²	C₂×M₁₆	1	2	yes	—
112	C₄×3₋¹⁺²	C₂×M₁₆	1	4	yes	—
113	3₋¹⁺²⋊C₄	—	2	1	no	—
114	3₋¹⁺²⋊C₄	—	2	2	no	—
115	C₂×C₃×M₁₆	—	9	2	no	—
116	C₂×C₃×M₁₆	—	9	4	no	—
117	C₂×C₃×C₄²	—	9	1	no	—
118	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
119	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
120	C₄²×D₃	—	3	1	no	—
121	C₃×#97	—	9	2	no	—
122	C₃×#97	—	9	4	no	—
123	C₃×C₄×C₈	—	6	1	no	—
124	C₃×C₄×C₈	—	6	2	no	—
125	C₃×C₄×C₈	—	9	2	no	—
126	C₃×C₄×C₈	—	9	1	no	—
127	C₃×C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
128	C₃×C₄×C₈	C₂×C₃×D₃	1	2	yes	—
129	#815	—	3	4	no	—
130	#815	—	3	2	no	—
131	C₄×C₃⋊C₈	—	3	2	no	—
132	C₄×C₃⋊C₈	—	3	1	no	—
133	C₂²×C₃×D₃	—	3	1	no	—
134	C₂×C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
136	C₂×C₃×Dic₃	—	3	1	no	—
137	C₂×C₃×Dic₃	—	3	2	no	—
138	C₃×C₄×D₃	—	6	2	no	—
139	C₃×C₄×D₃	—	6	4	no	—
140	C₂²×D₉	C₃×C₄²	1	1	yes	characteristic
141	C₃²×C₈	—	2	4	no	—
142	C₃²×C₈	ΓL(1,𝔽₂₅)	1	4	yes	—
143	C₃×C₃⋊C₈	—	3	8	no	—
144	C₃×C₃⋊C₈	—	6	2	no	—
145	C₂×C₄×C₉	—	2	1	no	—
146	C₂×C₄×C₉	—	2	2	no	—
147	C₂×C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₂×C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
149	C₂×Dic₉	C₃×C₄²	1	1	yes	characteristic
150	C₂×Dic₉	C₃×C₄²	1	2	yes	—
151	C₄×D₉	—	2	2	no	—
152	C₄×D₉	—	2	4	no	—
153	C₈×C₉	—	2	4	no	—
154	C₈×C₉	—	2	4	no	—
155	C₈×C₉	—	4	4	no	—
156	C₈×C₉	C₃×M₁₆	1	4	yes	complemented
157	C₉⋊C₈	—	2	2	no	—
158	C₉⋊C₈	C₃×M₁₆	1	8	yes	complemented
159	C₄×M₁₆	—	9	2	no	—
160	C₄×M₁₆	—	9	1	no	—
161	#322	—	9	2	no	—
162	#322	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
163	C₈²	—	9	1	no	—
164	C₂×3₋¹⁺²	C₄×M₁₆	1	1	yes	characteristic
165	C₂×3₋¹⁺²	C₄×M₁₆	1	2	yes	—
166	AGL(1,ℤ/9)	—	4	1	no	—
167	C₂²×C₃×C₄	—	9	1	no	—
168	C₂²×C₃×C₄	—	9	2	no	—
169	C₂×C₄×D₃	—	3	1	no	—
170	C₂×C₄×D₃	—	3	2	no	—
171	C₃×M₁₆	—	18	4	no	—
172	C₃×M₁₆	—	18	8	no	—
173	C₂×C₃×C₈	—	6	2	no	—
174	C₂×C₃×C₈	—	6	4	no	—
175	C₂×C₃×C₈	—	9	4	no	—
176	C₂×C₃×C₈	—	9	2	no	—
177	C₂×C₃×C₈	C₃×C₄×D₃	1	4	yes	—
178	C₂×C₃×C₈	C₃×C₄×D₃	1	2	yes	—
179	C₃×C₄²	—	6	1	no	—
180	C₃×C₄²	—	9	1	no	—
181	C₃×C₄²	—	9	2	no	—
182	C₃×C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
183	C₄×Dic₃	—	3	1	no	—
184	C₄×Dic₃	—	3	2	no	—
185	C₂×C₃⋊C₈	—	3	4	no	—
186	C₂×C₃⋊C₈	—	3	2	no	—
187	ΓL(1,𝔽₂₅)	—	6	8	no	—
188	ΓL(1,𝔽₂₅)	—	6	4	no	—
189	C₂²×C₃²	C₄²×D₃	1	1	yes	characteristic
190	C₂×C₃×D₃	—	6	1	no	—
191	C₂×C₃×D₃	—	6	2	no	—
192	C₃²×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
193	C₃²×C₄	D₃×M₁₆	1	4	yes	—
194	C₃×Dic₃	—	6	1	no	—
195	C₃×Dic₃	—	6	2	no	—
196	C₂²×C₉	—	2	1	no	—
197	C₂²×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
198	C₂×D₉	—	2	1	no	—
199	C₂×D₉	—	2	2	no	—
200	C₄×C₉	—	2	2	no	—
201	C₄×C₉	—	2	4	no	—
202	C₄×C₉	C₂×C₃×M₁₆	1	2	yes	—
203	C₄×C₉	C₂×C₃×M₁₆	1	4	yes	—
204	Dic₉	—	2	1	no	—
205	Dic₉	—	2	2	no	—
206	C₂×M₁₆	—	9	4	no	—
207	C₂×M₁₆	—	9	2	no	—
208	C₂×C₄²	—	9	1	no	—
209	#97	—	9	4	no	—
210	#97	—	9	2	no	—
211	C₄×C₈	—	9	2	no	—
212	C₄×C₈	—	9	1	no	—
213	C₄×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
214	C₄×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
215	3₋¹⁺²	#1455	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₃	—	9	1	no	—
217	C₂²×D₃	—	3	1	no	—
218	C₂×C₃×C₄	—	6	1	no	—
219	C₂×C₃×C₄	—	6	2	no	—
220	C₂×C₃×C₄	—	9	2	no	—
221	C₂×C₃×C₄	—	9	1	no	—
222	C₂×C₃×C₄	—	18	2	no	—
223	C₂×C₃×C₄	—	18	4	no	—
224	C₂×C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
226	C₂×Dic₃	—	3	1	no	—
227	C₂×Dic₃	—	3	2	no	—
228	C₄×D₃	—	6	2	no	—
229	C₄×D₃	—	6	4	no	—
230	C₃×C₈	—	2	4	no	—
231	C₃×C₈	—	6	4	no	—
232	C₃×C₈	—	9	8	no	—
233	C₃×C₈	—	12	4	no	—
234	C₃×C₈	—	18	2	no	—
235	C₃×C₈	C₃×ΓL(1,𝔽₂₅)	1	4	yes	—
236	C₃⋊C₈	—	3	8	no	—
237	C₃⋊C₈	—	6	2	no	—
238	C₂×C₃²	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
239	C₂×C₃²	#4707	1	2	yes	—
240	C₃×D₃	—	12	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	2	no	—
243	C₂×C₉	C₃×C₄×M₁₆	1	1	yes	characteristic
244	C₂×C₉	C₃×C₄×M₁₆	1	2	yes	—
245	D₉	—	4	1	no	—
246	C₂²×C₄	—	9	1	no	—
247	C₂²×C₄	—	9	2	no	—
248	M₁₆	—	18	8	no	—
249	M₁₆	—	18	4	no	—
250	C₂×C₈	—	9	4	no	—
251	C₂×C₈	—	9	2	no	—
252	C₂×C₈	C₄×AGL(1,ℤ/9)	1	4	yes	—
253	C₂×C₈	C₄×AGL(1,ℤ/9)	1	2	yes	—
254	C₄²	—	9	1	no	—
255	C₄²	—	9	2	no	—
256	C₄²	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
257	C₂²×C₃	—	6	1	no	—
258	C₂²×C₃	—	18	2	no	—
259	C₂²×C₃	—	18	1	no	—
260	C₂²×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
261	C₂×D₃	—	6	1	no	—
262	C₂×D₃	—	6	2	no	—
263	C₃×C₄	—	6	2	no	—
264	C₃×C₄	—	6	4	no	—
265	C₃×C₄	—	18	2	no	—
266	C₃×C₄	—	18	1	no	—
267	C₃×C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
268	C₃×C₄	C₃×D₃×M₁₆	1	4	yes	—
269	Dic₃	—	6	1	no	—
270	Dic₃	—	6	2	no	—
271	C₃²	#68075	1	1	yes	characteristic
272	C₉	—	2	1	no	—
273	C₉	C₃×#1455	1	1	yes	characteristic, complemented
274	C₂³	—	9	1	no	—
275	C₂×C₄	—	9	2	no	—
276	C₂×C₄	—	9	1	no	—
277	C₂×C₄	—	18	2	no	—
278	C₂×C₄	—	18	4	no	—
279	C₂×C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	9	8	no	—
283	C₈	—	18	2	no	—
284	C₈	#3663⋊C₃	1	4	yes	complemented
285	C₂×C₃	—	6	2	no	—
286	C₂×C₃	—	6	1	no	—
287	C₂×C₃	—	36	1	no	—
288	C₂×C₃	C₃×C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
289	C₂×C₃	C₃×#4707	1	2	yes	—
290	D₃	—	12	1	no	—
291	C₂²	—	18	2	no	—
292	C₂²	—	18	1	no	—
293	C₂²	C₄²×AGL(1,ℤ/9)	1	1	yes	characteristic
294	C₄	—	18	2	no	—
295	C₄	—	18	1	no	—
296	C₄	C₂×#3663⋊C₃	1	2	yes	—
297	C₄	M₁₆×AGL(1,ℤ/9)	1	4	yes	—
298	C₃	—	6	1	no	—
299	C₃	C₃×#68075	1	1	yes	characteristic
300	C₂	—	36	1	no	—
301	C₂	C₄×#3663⋊C₃	1	1	yes	characteristic
302	C₂	#10588388⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂×C₃×C₄²
Frattini	C₃×C₄²	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#322	C₂
socle	C₂²×C₃	C₃×C₄²×D₃
prime radical	C₂²×AGL(1,ℤ/9)	C₄²
cosocle	C₄²×C₉	C₂³×C₃
hypercenter	#322	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#1455
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×#322	C₂	semidirect
4. #10588131⋊C₃	C₂	semidirect
5. C₈×3₋¹⁺²⋊C₈	C₂	semidirect
7. #11773693	C₃	semidirect
25. C₉×#322	C₂×C₃	semidirect
26. #10588131	C₂×C₃	semidirect
27. C₈×C₉⋊C₈	C₂×C₃	semidirect
69. C₈×3₋¹⁺²	M₁₆	semidirect
71. 3₋¹⁺²⋊C₈	M₁₆	semidirect
156. C₈×C₉	C₃×M₁₆	semidirect
158. C₉⋊C₈	C₃×M₁₆	semidirect
162. #322	AGL(1,ℤ/9)	semidirect
215. 3₋¹⁺²	#1455	semidirect
273. C₉	C₃×#1455	semidirect
284. C₈	#3663⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3663⋊C₃	2
3₋¹⁺²×#322	2
#10588131⋊C₃	2
C₈×3₋¹⁺²⋊C₈	2
C₃×#68075	3
#11773693	3