Subgroups — #10589623⋊C₃ #13152535

Order 1728.

1052 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	#63488⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#63455⋊C₃	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×AGL(1,ℤ/9)	C₂	1	4	yes	maximal, complemented
6	C₄×C₄⋊AGL(1,ℤ/9)	C₂	1	4	yes	maximal, complemented
7	C₃×#5521	—	3	1	no	maximal
8	#10589623	C₃	1	1	yes	maximal, characteristic, complemented
9	C₂³×AGL(1,ℤ/9)	C₂²	1	2	yes	—
10	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊AGL(1,ℤ/9)	C₄	1	8	yes	complemented
12	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	2	yes	—
13	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	2	yes	—
14	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	8	yes	complemented
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	complemented
16	C₂²⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
17	C₄²×3₋¹⁺²	C₂²	1	2	yes	complemented
18	#3676⋊C₃	C₂²	1	2	yes	—
19	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	8	yes	complemented
20	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	complemented
21	C₄×3₋¹⁺²⋊C₄	C₂²	1	2	yes	complemented
22	C₂²×C₃×D₁₂	—	3	1	no	—
23	C₃²×#117	D₃	1	1	yes	characteristic
24	C₃×#926	—	3	1	no	—
25	C₃×#893	—	3	4	no	—
26	C₃×D₃×C₂²⋊C₄	—	3	4	no	—
27	C₃×C₄×D₁₂	—	3	4	no	—
28	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic
29	C₉×#117	—	2	1	no	—
30	C₉×#117	C₂×C₃	1	1	yes	characteristic, complemented
31	#63488	C₂×C₃	1	1	yes	characteristic, complemented
32	#63455	C₂×C₃	1	4	yes	complemented
33	C₂²⋊C₄×D₉	C₂×C₃	1	4	yes	complemented
34	C₄×D₃₆	C₂×C₃	1	4	yes	complemented
35	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
36	C₂²×AGL(1,ℤ/9)	—	4	4	no	—
37	C₂²×AGL(1,ℤ/9)	C₂×C₄	1	8	yes	complemented
38	C₂²×AGL(1,ℤ/9)	C₂³	1	4	yes	—
39	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
40	C₂×C₄×3₋¹⁺²	C₂³	1	4	yes	—
41	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
42	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
43	C₂×3₋¹⁺²⋊C₄	C₂³	1	4	yes	—
44	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	8	yes	complemented
45	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	8	yes	complemented
46	C₄×AGL(1,ℤ/9)	—	4	8	no	—
47	C₃×#518	—	9	1	no	—
48	#5521	—	3	1	no	—
49	C₂³×C₃×D₃	—	3	2	no	—
50	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×D₁₂	—	3	8	no	—
52	C₂×C₃×D₁₂	—	3	2	no	—
53	C₂×C₃×D₁₂	—	3	2	no	—
54	C₂×C₃×C₄×D₃	—	3	8	no	—
55	C₂³×D₉	C₂²×C₃	1	2	yes	—
56	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
57	C₃²×C₂²⋊C₄	C₂×D₃	1	2	yes	—
58	C₃²×C₄²	C₂×D₃	1	2	yes	—
59	C₃×#217	—	3	2	no	—
60	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
61	C₃×C₄⋊Dic₃	—	3	2	no	—
62	C₃×C₄×Dic₃	—	3	2	no	—
63	C₂²×C₄×C₉	—	2	1	no	—
64	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
65	C₂×D₃₆	C₃×C₄	1	8	yes	complemented
66	C₂×D₃₆	C₂²×C₃	1	2	yes	—
67	C₂×D₃₆	C₂²×C₃	1	2	yes	—
68	C₂×C₄×D₉	C₂²×C₃	1	8	yes	complemented
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
71	C₉×C₂²⋊C₄	—	2	2	no	—
72	C₉×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₄²×C₉	—	2	2	no	—
74	C₄²×C₉	C₂²×C₃	1	2	yes	complemented
75	#3676	C₂²×C₃	1	2	yes	—
76	(C₂×D₉)⋊C₄	C₂²×C₃	1	8	yes	complemented
77	C₄⋊Dic₉	C₂²×C₃	1	2	yes	complemented
78	C₄×Dic₉	C₂²×C₃	1	2	yes	complemented
79	C₂²×3₋¹⁺²	—	2	2	no	—
80	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
81	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
82	C₂×AGL(1,ℤ/9)	—	2	8	no	—
83	C₂×AGL(1,ℤ/9)	—	4	8	no	—
84	C₂×AGL(1,ℤ/9)	C₂²×C₄	1	8	yes	—
85	C₄×3₋¹⁺²	—	2	4	no	—
86	C₄×3₋¹⁺²	C₂²×C₄	1	2	yes	—
87	C₄×3₋¹⁺²	C₂²×C₄	1	2	yes	—
88	3₋¹⁺²⋊C₄	—	2	4	no	—
89	C₂²×C₃×D₄	—	9	1	no	—
90	C₂²×D₁₂	—	3	1	no	—
91	C₃×C₄×D₄	—	9	4	no	—
92	C₃×C₄×D₄	—	9	4	no	—
93	C₃×#117	—	3	1	no	—
94	C₃×#117	—	9	1	no	—
95	C₃×#117	C₃×D₃	1	1	yes	characteristic
96	C₂×C₃×C₂²⋊C₄	—	9	4	no	—
97	#926	—	3	1	no	—
98	#893	—	3	4	no	—
99	D₃×C₂²⋊C₄	—	3	4	no	—
100	C₄×D₁₂	—	3	4	no	—
101	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₃×D₃	—	3	4	no	—
103	C₂²×C₃×D₃	—	3	8	no	—
104	C₂²×C₃×D₃	—	12	4	no	—
105	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
106	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
107	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	—	3	4	no	—
110	C₃×D₁₂	—	3	8	no	—
111	C₃×D₁₂	—	3	8	no	—
112	C₃×C₄×D₃	—	12	8	no	—
113	C₂³×C₉	—	2	1	no	—
114	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
115	C₂²×D₉	—	4	4	no	—
116	C₂²×D₉	C₂×C₃×C₄	1	8	yes	complemented
117	C₂²×D₉	C₂³×C₃	1	4	yes	—
118	C₂×C₄×C₉	—	2	4	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	4	no	—
122	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	complemented
123	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
124	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
126	C₂×Dic₉	C₂³×C₃	1	4	yes	—
127	D₃₆	C₂×C₃×C₄	1	8	yes	complemented
128	D₃₆	C₂×C₃×C₄	1	8	yes	complemented
129	C₄×D₉	—	4	8	no	—
130	#518	—	9	1	no	Sylow
131	C₂×3₋¹⁺²	—	2	2	no	—
132	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
133	C₂×3₋¹⁺²	H(5,𝔽₂)	1	2	yes	—
134	AGL(1,ℤ/9)	—	2	8	no	—
135	C₂⁴×C₃	—	9	2	no	—
136	C₂³×D₃	—	3	2	no	—
137	C₂×C₃×D₄	—	9	2	no	—
138	C₂×C₃×D₄	—	9	2	no	—
139	C₂×C₃×D₄	—	9	8	no	—
140	C₂²×C₃×C₄	—	3	1	no	—
141	C₂²×C₃×C₄	—	9	8	no	—
142	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
143	C₂×D₁₂	—	3	8	no	—
144	C₂×D₁₂	—	3	2	no	—
145	C₂×D₁₂	—	3	2	no	—
146	C₂×C₄×D₃	—	3	8	no	—
147	C₃×C₄⋊C₄	—	3	2	no	—
148	C₃×C₄⋊C₄	—	9	2	no	—
149	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
150	C₃×C₂²⋊C₄	—	3	2	no	—
151	C₃×C₂²⋊C₄	—	9	8	no	—
152	C₃×C₂²⋊C₄	—	9	2	no	—
153	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
154	C₃×C₄²	—	3	2	no	—
155	C₃×C₄²	—	9	2	no	—
156	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
157	#217	—	3	2	no	—
158	(C₂×D₃)⋊C₄	—	3	8	no	—
159	C₄⋊Dic₃	—	3	2	no	—
160	C₄×Dic₃	—	3	2	no	—
161	C₂²×C₃²	—	2	2	no	—
162	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
163	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
164	C₂×C₃×D₃	—	3	8	no	—
165	C₂×C₃×D₃	—	6	8	no	—
166	C₂×C₃×D₃	—	12	8	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
169	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
170	C₃×Dic₃	—	6	4	no	—
171	C₂²×C₉	—	2	2	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	4	2	no	—
175	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
176	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×D₉	—	2	8	no	—
178	C₂×D₉	—	4	8	no	—
179	C₂×D₉	C₂²×C₃×C₄	1	8	yes	—
180	C₄×C₉	—	2	4	no	—
181	C₄×C₉	—	2	2	no	—
182	C₄×C₉	—	2	2	no	—
183	C₄×C₉	—	4	4	no	—
184	C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
185	C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
186	Dic₉	—	2	4	no	—
187	C₂²×D₄	—	9	1	no	—
188	C₄×D₄	—	9	4	no	—
189	C₄×D₄	—	9	4	no	—
190	#117	—	9	1	no	—
191	#117	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
192	C₂×C₂²⋊C₄	—	9	4	no	—
193	3₋¹⁺²	#518	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₃	—	3	1	no	—
195	C₂³×C₃	—	9	4	no	—
196	C₂³×C₃	—	9	8	no	—
197	C₂³×C₃	—	36	4	no	—
198	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
199	C₂²×D₃	—	3	4	no	—
200	C₂²×D₃	—	3	8	no	—
201	C₂²×D₃	—	12	4	no	—
202	C₃×D₄	—	9	8	no	—
203	C₃×D₄	—	9	8	no	—
204	C₂×C₃×C₄	—	3	4	no	—
205	C₂×C₃×C₄	—	3	1	no	—
206	C₂×C₃×C₄	—	3	1	no	—
207	C₂×C₃×C₄	—	3	4	no	—
208	C₂×C₃×C₄	—	9	4	no	—
209	C₂×C₃×C₄	—	36	8	no	—
210	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
214	C₂×Dic₃	—	3	4	no	—
215	D₁₂	—	3	8	no	—
216	D₁₂	—	3	8	no	—
217	C₄×D₃	—	12	8	no	—
218	C₂×C₃²	—	2	2	no	—
219	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
220	C₂×C₃²	#1009	1	2	yes	—
221	C₃×D₃	—	6	8	no	—
222	C₂×C₉	—	2	2	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	2	no	—
225	C₂×C₉	—	4	2	no	—
226	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
227	C₂×C₉	C₃×H(5,𝔽₂)	1	2	yes	—
228	D₉	—	2	8	no	—
229	C₂⁴	—	9	2	no	—
230	C₂×D₄	—	9	8	no	—
231	C₂×D₄	—	9	2	no	—
232	C₂×D₄	—	9	2	no	—
233	C₂²×C₄	—	9	8	no	—
234	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
235	C₄⋊C₄	—	9	2	no	—
236	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	2	yes	complemented
237	C₂²⋊C₄	—	9	2	no	—
238	C₂²⋊C₄	—	9	8	no	—
239	C₂²⋊C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
240	C₄²	—	9	2	no	—
241	C₄²	C₂×AGL(1,ℤ/9)	1	2	yes	complemented
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	3	1	no	—
244	C₂²×C₃	—	3	2	no	—
245	C₂²×C₃	—	6	2	no	—
246	C₂²×C₃	—	9	8	no	—
247	C₂²×C₃	—	18	8	no	—
248	C₂²×C₃	—	36	8	no	—
249	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
250	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
251	C₂×D₃	—	3	8	no	—
252	C₂×D₃	—	6	8	no	—
253	C₂×D₃	—	12	8	no	—
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	3	2	no	—
256	C₃×C₄	—	3	2	no	—
257	C₃×C₄	—	6	4	no	—
258	C₃×C₄	—	18	4	no	—
259	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
260	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
261	Dic₃	—	6	4	no	—
262	C₃²	#5521	1	1	yes	characteristic
263	C₉	—	2	1	no	—
264	C₉	C₃×#518	1	1	yes	characteristic, complemented
265	C₂³	—	9	4	no	—
266	C₂³	—	9	8	no	—
267	C₂³	—	36	4	no	—
268	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
269	D₄	—	9	8	no	—
270	D₄	—	9	8	no	—
271	C₂×C₄	—	9	4	no	—
272	C₂×C₄	—	36	8	no	—
273	C₂×C₄	C₄×AGL(1,ℤ/9)	1	4	yes	complemented
274	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	4	yes	—
275	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
276	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
277	C₂×C₃	—	2	2	no	—
278	C₂×C₃	—	3	1	no	—
279	C₂×C₃	—	3	2	no	—
280	C₂×C₃	—	6	2	no	—
281	C₂×C₃	—	18	8	no	—
282	C₂×C₃	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
283	C₂×C₃	C₃×#1009	1	2	yes	—
284	D₃	—	6	8	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	9	8	no	—
287	C₂²	—	18	8	no	—
288	C₂²	—	36	8	no	—
289	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
290	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
291	C₄	—	2	4	no	—
292	C₄	—	18	4	no	—
293	C₄	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
294	C₄	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
295	C₃	—	3	1	no	—
296	C₃	C₃×#5521	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	18	8	no	—
299	C₂	C₂²×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
300	C₂	#11722641	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	3₋¹⁺²×#117	C₂
prime radical	C₂²×C₄⋊AGL(1,ℤ/9)	C₂
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	#117	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#518
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. 3₋¹⁺²×#117	C₂	semidirect
3. #63488⋊C₃	C₂	semidirect
4. #63455⋊C₃	C₂	semidirect
5. C₂²⋊C₄×AGL(1,ℤ/9)	C₂	semidirect
6. C₄×C₄⋊AGL(1,ℤ/9)	C₂	semidirect
8. #10589623	C₃	semidirect
11. C₂×C₄⋊AGL(1,ℤ/9)	C₄	semidirect
14. C₂×C₄×AGL(1,ℤ/9)	C₂²	semidirect
15. C₄⋊C₄×3₋¹⁺²	C₂²	semidirect
17. C₄²×3₋¹⁺²	C₂²	semidirect
19. (C₂×AGL(1,ℤ/9))⋊C₄	C₂²	semidirect
20. C₄⋊(3₋¹⁺²⋊C₄)	C₂²	semidirect
21. C₄×3₋¹⁺²⋊C₄	C₂²	semidirect
30. C₉×#117	C₂×C₃	semidirect
31. #63488	C₂×C₃	semidirect
32. #63455	C₂×C₃	semidirect
33. C₂²⋊C₄×D₉	C₂×C₃	semidirect
34. C₄×D₃₆	C₂×C₃	semidirect
37. C₂²×AGL(1,ℤ/9)	C₂×C₄	semidirect
39. C₂×C₄×3₋¹⁺²	C₂×C₄	semidirect
44. C₄⋊AGL(1,ℤ/9)	C₂×C₄	semidirect
45. C₄⋊AGL(1,ℤ/9)	C₂×C₄	semidirect
65. C₂×D₃₆	C₃×C₄	semidirect
68. C₂×C₄×D₉	C₂²×C₃	semidirect
70. C₉×C₄⋊C₄	C₂²×C₃	semidirect
74. C₄²×C₉	C₂²×C₃	semidirect
76. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
77. C₄⋊Dic₉	C₂²×C₃	semidirect
78. C₄×Dic₉	C₂²×C₃	semidirect
116. C₂²×D₉	C₂×C₃×C₄	semidirect
122. C₂×C₄×C₉	C₂×C₃×C₄	semidirect
127. D₃₆	C₂×C₃×C₄	semidirect
128. D₃₆	C₂×C₃×C₄	semidirect
191. #117	AGL(1,ℤ/9)	semidirect
193. 3₋¹⁺²	#518	semidirect
236. C₄⋊C₄	C₂×AGL(1,ℤ/9)	semidirect
241. C₄²	C₂×AGL(1,ℤ/9)	semidirect
264. C₉	C₃×#518	semidirect
273. C₂×C₄	C₄×AGL(1,ℤ/9)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊AGL(1,ℤ/9)	2
3₋¹⁺²×#117	2
#63488⋊C₃	2
#63455⋊C₃	2
C₂²⋊C₄×AGL(1,ℤ/9)	2
C₄×C₄⋊AGL(1,ℤ/9)	2
C₃×#5521	3
#10589623	3

Subgroups — #10589623⋊C3 #13152535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10589623⋊C₃ #13152535