Subgroups — C₉×#5689 #13153164

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#897	C₂	1	2	yes	maximal
3	C₉×#896	C₂	1	2	yes	maximal
4	C₉×#892	C₂	1	1	yes	maximal, characteristic
5	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₉×#890	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#889	C₂	1	2	yes	maximal, complemented
8	C₉×#888	C₂	1	2	yes	maximal, complemented
9	C₉×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₉×#872	C₂	1	1	yes	maximal, characteristic
11	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#5689	C₃	1	1	yes	maximal, characteristic
13	C₉×#544	—	3	1	no	maximal
14	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
15	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
16	C₂×C₉×Dic₆	C₂²	1	2	yes	—
17	C₂×C₉×Dic₆	C₂²	1	2	yes	—
18	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₉×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
30	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
36	C₄×C₉×Dic₃	C₂²	1	2	yes	—
37	C₃²×#125	C₂×C₃	1	1	yes	characteristic
38	C₃×#897	C₂×C₃	1	2	yes	—
39	C₃×#896	C₂×C₃	1	2	yes	—
40	C₃×#892	C₂×C₃	1	1	yes	characteristic
41	C₃×D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
42	C₃×#890	C₂×C₃	1	1	yes	characteristic
43	C₃×#889	C₂×C₃	1	2	yes	—
44	C₃×#888	C₂×C₃	1	2	yes	—
45	C₃×#888	C₂×C₃	1	1	yes	characteristic
46	C₃×#872	C₂×C₃	1	1	yes	characteristic
47	C₃×#869	C₂×C₃	1	1	yes	characteristic
48	C₉×C₄⋊Q₈	—	3	2	no	—
49	C₉×C₄⋊Q₈	—	3	1	no	—
50	C₉×C₄⋊Q₈	—	3	1	no	—
51	C₉×#125	—	2	1	no	—
52	C₉×#125	—	3	2	no	—
53	C₉×#125	—	3	1	no	—
54	C₉×#125	D₃	1	1	yes	characteristic, complemented
55	C₉×C₂²⋊Q₈	—	3	2	no	—
56	C₉×C₂²⋊Q₈	—	3	2	no	—
57	C₉×#117	—	3	1	no	—
58	C₉×#117	—	3	1	no	—
59	C₂×C₉×C₄⋊C₄	—	3	1	no	—
60	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
63	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
64	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
66	C₂×C₉×Dic₃	C₂³	1	2	yes	—
67	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₉×Dic₃	C₂³	1	2	yes	—
69	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
71	C₄×D₃×C₉	—	4	2	no	—
72	C₄×D₃×C₉	Q₈	1	4	yes	complemented
73	C₉×Dic₆	—	4	2	no	—
74	C₉×Dic₆	—	4	2	no	—
75	C₃×#544	—	3	1	no	—
76	#5689	C₉	1	1	yes	characteristic, direct product factor
77	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
80	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
81	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
86	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
87	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
90	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
91	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
92	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
93	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
94	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
95	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
96	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
98	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
99	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
100	C₂×Q₈×C₉	—	3	2	no	—
101	C₂×Q₈×C₉	—	3	2	no	—
102	C₂²×C₄×C₉	—	3	1	no	—
103	C₂²×C₄×C₉	—	3	2	no	—
104	C₉×C₄⋊C₄	—	2	1	no	—
105	C₉×C₄⋊C₄	—	2	1	no	—
106	C₉×C₄⋊C₄	—	2	2	no	—
107	C₉×C₄⋊C₄	—	2	2	no	—
108	C₉×C₄⋊C₄	—	3	2	no	—
109	C₉×C₄⋊C₄	—	3	1	no	—
110	C₉×C₄⋊C₄	—	3	1	no	—
111	C₉×C₄⋊C₄	—	3	1	no	—
112	C₉×C₄⋊C₄	—	3	1	no	—
113	C₉×C₄⋊C₄	—	3	2	no	—
114	C₉×C₄⋊C₄	—	3	2	no	—
115	C₉×C₄⋊C₄	—	3	2	no	—
116	C₉×C₄⋊C₄	—	3	2	no	—
117	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
118	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
119	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
120	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
121	C₉×C₂²⋊C₄	—	3	1	no	—
122	C₉×C₂²⋊C₄	—	3	2	no	—
123	C₉×C₂²⋊C₄	—	3	1	no	—
124	C₄²×C₉	—	2	1	no	—
125	C₄²×C₉	—	3	2	no	—
126	C₄²×C₉	—	3	1	no	—
127	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
128	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
129	C₂×D₃×C₉	—	2	1	no	—
130	C₂×D₃×C₉	—	2	1	no	—
131	C₂×D₃×C₉	C₂×Q₈	1	2	yes	—
132	C₃×C₄×C₉	—	2	1	no	—
133	C₃×C₄×C₉	—	2	2	no	—
134	C₃×C₄×C₉	—	2	2	no	—
135	C₃×C₄×C₉	—	2	1	no	—
136	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
137	C₉×Dic₃	—	2	1	no	—
138	C₉×Dic₃	—	2	2	no	—
139	C₉×Dic₃	—	2	2	no	—
140	C₉×Dic₃	—	2	1	no	—
141	C₉×Dic₃	C₂×Q₈	1	2	yes	—
142	C₃×C₄⋊Q₈	—	3	2	no	—
143	C₃×C₄⋊Q₈	—	3	1	no	—
144	C₃×C₄⋊Q₈	—	3	1	no	—
145	C₃×#125	—	2	1	no	—
146	C₃×#125	—	3	2	no	—
147	C₃×#125	—	3	1	no	—
148	C₃×#125	C₂×C₉	1	1	yes	characteristic, complemented
149	C₃×#125	C₃×D₃	1	1	yes	characteristic
150	C₃×C₂²⋊Q₈	—	3	2	no	—
151	C₃×C₂²⋊Q₈	—	3	2	no	—
152	C₃×#117	—	3	1	no	—
153	C₃×#117	—	3	1	no	—
154	C₂×C₃×C₄⋊C₄	—	3	1	no	—
155	#897	C₂×C₉	1	2	yes	—
156	#896	C₂×C₉	1	2	yes	—
157	#892	C₂×C₉	1	1	yes	characteristic
158	D₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
159	#890	C₂×C₉	1	1	yes	characteristic, complemented
160	#889	C₂×C₉	1	2	yes	complemented
161	#888	C₂×C₉	1	1	yes	characteristic, complemented
162	#888	C₂×C₉	1	2	yes	complemented
163	#872	C₂×C₉	1	1	yes	characteristic
164	#869	C₂×C₉	1	1	yes	characteristic, complemented
165	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
166	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
167	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
168	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
169	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
172	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
173	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
174	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
175	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
176	C₃×C₄×D₃	—	4	2	no	—
177	C₃×C₄×D₃	C₃×Q₈	1	4	yes	—
178	C₃×Dic₆	—	4	2	no	—
179	C₃×Dic₆	—	4	2	no	—
180	C₂³×C₉	—	3	1	no	—
181	Q₈×C₉	—	12	2	no	—
182	Q₈×C₉	—	12	2	no	—
183	C₂×C₄×C₉	—	2	1	no	—
184	C₂×C₄×C₉	—	2	1	no	—
185	C₂×C₄×C₉	—	2	1	no	—
186	C₂×C₄×C₉	—	2	2	no	—
187	C₂×C₄×C₉	—	2	2	no	—
188	C₂×C₄×C₉	—	3	1	no	—
189	C₂×C₄×C₉	—	3	4	no	—
190	C₂×C₄×C₉	—	3	1	no	—
191	C₂×C₄×C₉	—	3	1	no	—
192	C₂×C₄×C₉	—	3	2	no	—
193	C₂×C₄×C₉	—	3	2	no	—
194	C₂×C₄×C₉	—	12	2	no	—
195	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
196	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
197	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
198	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
199	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
200	#544	—	3	1	no	Sylow
201	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
204	D₃×C₉	—	2	2	no	—
205	C₂×C₃×Q₈	—	3	2	no	—
206	C₂×C₃×Q₈	—	3	2	no	—
207	C₂²×C₃×C₄	—	3	1	no	—
208	C₂²×C₃×C₄	—	3	2	no	—
209	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
210	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
211	C₂×Dic₆	C₂²×C₉	1	2	yes	—
212	C₂×Dic₆	C₂²×C₉	1	2	yes	—
213	C₃×C₄⋊C₄	—	2	1	no	—
214	C₃×C₄⋊C₄	—	2	1	no	—
215	C₃×C₄⋊C₄	—	2	2	no	—
216	C₃×C₄⋊C₄	—	2	2	no	—
217	C₃×C₄⋊C₄	—	3	2	no	—
218	C₃×C₄⋊C₄	—	3	1	no	—
219	C₃×C₄⋊C₄	—	3	1	no	—
220	C₃×C₄⋊C₄	—	3	2	no	—
221	C₃×C₄⋊C₄	—	3	2	no	—
222	C₃×C₄⋊C₄	—	3	1	no	—
223	C₃×C₄⋊C₄	—	3	1	no	—
224	C₃×C₄⋊C₄	—	3	2	no	—
225	C₃×C₄⋊C₄	—	3	2	no	—
226	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
227	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
228	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
229	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
230	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
231	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
232	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
233	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
234	C₃×C₂²⋊C₄	—	3	1	no	—
235	C₃×C₂²⋊C₄	—	3	2	no	—
236	C₃×C₂²⋊C₄	—	3	1	no	—
237	C₃×C₄²	—	2	1	no	—
238	C₃×C₄²	—	3	2	no	—
239	C₃×C₄²	—	3	1	no	—
240	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
241	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
242	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
243	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
244	(C₂×D₃)⋊C₄	C₂²×C₉	1	2	yes	—
245	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
246	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
247	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
248	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
249	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
250	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
251	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
252	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
253	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
254	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
255	C₄×Dic₃	C₂²×C₉	1	2	yes	—
256	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
257	C₂×C₃×D₃	—	2	1	no	—
258	C₂×C₃×D₃	—	2	1	no	—
259	C₂×C₃×D₃	C₂×C₃×Q₈	1	2	yes	—
260	C₃²×C₄	—	2	1	no	—
261	C₃²×C₄	—	2	2	no	—
262	C₃²×C₄	—	2	2	no	—
263	C₃²×C₄	—	2	1	no	—
264	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
265	C₃×Dic₃	—	2	1	no	—
266	C₃×Dic₃	—	2	2	no	—
267	C₃×Dic₃	—	2	2	no	—
268	C₃×Dic₃	—	2	1	no	—
269	C₃×Dic₃	C₂×C₃×Q₈	1	2	yes	—
270	C₂²×C₉	—	2	1	no	—
271	C₂²×C₉	—	3	2	no	—
272	C₂²×C₉	—	6	1	no	—
273	C₂²×C₉	—	6	1	no	—
274	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
275	C₄×C₉	—	2	2	no	—
276	C₄×C₉	—	2	1	no	—
277	C₄×C₉	—	2	2	no	—
278	C₄×C₉	—	2	1	no	—
279	C₄×C₉	—	2	2	no	—
280	C₄×C₉	—	3	2	no	—
281	C₄×C₉	—	4	2	no	—
282	C₄×C₉	—	4	1	no	—
283	C₄×C₉	—	4	1	no	—
284	C₄×C₉	—	4	2	no	—
285	C₄×C₉	—	6	2	no	—
286	C₄×C₉	—	6	1	no	—
287	C₄×C₉	—	6	2	no	—
288	C₄×C₉	—	6	1	no	—
289	C₄×C₉	D₃×Q₈	1	2	yes	—
290	C₄⋊Q₈	—	3	2	no	—
291	C₄⋊Q₈	—	3	1	no	—
292	C₄⋊Q₈	—	3	1	no	—
293	#125	—	3	2	no	—
294	#125	—	3	1	no	—
295	#125	D₃×C₉	1	1	yes	characteristic, complemented
296	C₂²⋊Q₈	—	3	2	no	—
297	C₂²⋊Q₈	—	3	2	no	—
298	#117	—	3	1	no	—
299	#117	—	3	1	no	—
300	C₂×C₄⋊C₄	—	3	1	no	—
301	C₃×C₉	#544	1	1	yes	characteristic, complemented, Sylow
302	C₂³×C₃	—	3	1	no	—
303	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
304	C₃×Q₈	—	12	2	no	—
305	C₃×Q₈	—	12	2	no	—
306	C₂×C₃×C₄	—	2	1	no	—
307	C₂×C₃×C₄	—	2	1	no	—
308	C₂×C₃×C₄	—	2	1	no	—
309	C₂×C₃×C₄	—	2	2	no	—
310	C₂×C₃×C₄	—	2	2	no	—
311	C₂×C₃×C₄	—	3	1	no	—
312	C₂×C₃×C₄	—	3	4	no	—
313	C₂×C₃×C₄	—	3	1	no	—
314	C₂×C₃×C₄	—	3	1	no	—
315	C₂×C₃×C₄	—	3	2	no	—
316	C₂×C₃×C₄	—	3	2	no	—
317	C₂×C₃×C₄	—	12	2	no	—
318	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
319	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
320	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
321	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
322	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
323	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
324	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
325	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
326	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
327	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
328	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
329	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
330	C₂×Dic₃	C₂³×C₉	1	2	yes	—
331	C₂×Dic₃	C₂³×C₉	1	2	yes	—
332	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
333	C₄×D₃	—	4	2	no	—
334	C₄×D₃	Q₈×C₉	1	4	yes	complemented
335	Dic₆	—	4	2	no	—
336	Dic₆	—	4	2	no	—
337	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
338	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
340	C₃×D₃	—	2	2	no	—
341	C₂×C₉	—	2	1	no	—
342	C₂×C₉	—	2	1	no	—
343	C₂×C₉	—	2	1	no	—
344	C₂×C₉	—	6	2	no	—
345	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
346	C₂×C₉	#1009	1	1	yes	characteristic
347	C₂×C₉	#1010	1	1	yes	characteristic
348	C₂×Q₈	—	3	2	no	—
349	C₂×Q₈	—	3	2	no	—
350	C₂²×C₄	—	3	1	no	—
351	C₂²×C₄	—	3	2	no	—
352	C₄⋊C₄	—	3	2	no	—
353	C₄⋊C₄	—	3	1	no	—
354	C₄⋊C₄	—	3	1	no	—
355	C₄⋊C₄	—	3	1	no	—
356	C₄⋊C₄	—	3	1	no	—
357	C₄⋊C₄	—	3	2	no	—
358	C₄⋊C₄	—	3	2	no	—
359	C₄⋊C₄	—	3	2	no	—
360	C₄⋊C₄	—	3	2	no	—
361	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
362	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
363	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
364	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
365	C₂²⋊C₄	—	3	1	no	—
366	C₂²⋊C₄	—	3	2	no	—
367	C₂²⋊C₄	—	3	1	no	—
368	C₄²	—	3	2	no	—
369	C₄²	—	3	1	no	—
370	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
371	C₂²×C₃	—	2	1	no	—
372	C₂²×C₃	—	3	2	no	—
373	C₂²×C₃	—	6	1	no	—
374	C₂²×C₃	—	6	1	no	—
375	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
376	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
377	C₂×D₃	—	2	1	no	—
378	C₂×D₃	—	2	1	no	—
379	C₂×D₃	C₂×Q₈×C₉	1	2	yes	—
380	C₃×C₄	—	2	2	no	—
381	C₃×C₄	—	2	1	no	—
382	C₃×C₄	—	2	2	no	—
383	C₃×C₄	—	2	1	no	—
384	C₃×C₄	—	2	1	no	—
385	C₃×C₄	—	2	2	no	—
386	C₃×C₄	—	2	2	no	—
387	C₃×C₄	—	2	1	no	—
388	C₃×C₄	—	2	2	no	—
389	C₃×C₄	—	3	2	no	—
390	C₃×C₄	—	4	2	no	—
391	C₃×C₄	—	4	1	no	—
392	C₃×C₄	—	4	1	no	—
393	C₃×C₄	—	4	2	no	—
394	C₃×C₄	—	6	2	no	—
395	C₃×C₄	—	6	1	no	—
396	C₃×C₄	—	6	2	no	—
397	C₃×C₄	—	6	1	no	—
398	C₃×C₄	C₂×Q₈×C₉	1	2	yes	—
399	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
400	Dic₃	—	2	1	no	—
401	Dic₃	—	2	2	no	—
402	Dic₃	—	2	2	no	—
403	Dic₃	—	2	1	no	—
404	Dic₃	C₂×Q₈×C₉	1	2	yes	—
405	C₃²	C₃×#544	1	1	yes	characteristic
406	C₉	—	2	1	no	—
407	C₉	#5689	1	1	yes	characteristic, direct product factor
408	C₂³	—	3	1	no	—
409	Q₈	—	12	2	no	—
410	Q₈	—	12	2	no	—
411	C₂×C₄	—	3	1	no	—
412	C₂×C₄	—	3	4	no	—
413	C₂×C₄	—	3	1	no	—
414	C₂×C₄	—	3	1	no	—
415	C₂×C₄	—	3	2	no	—
416	C₂×C₄	—	3	2	no	—
417	C₂×C₄	—	12	2	no	—
418	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
419	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
420	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
422	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
423	C₂×C₃	—	2	1	no	—
424	C₂×C₃	—	2	1	no	—
425	C₂×C₃	—	2	1	no	—
426	C₂×C₃	—	6	2	no	—
427	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
428	C₂×C₃	C₉×H(5,𝔽₂)	1	1	yes	characteristic
429	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
430	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
431	C₂×C₃	C₃×#1009	1	1	yes	characteristic
432	C₂×C₃	C₃×#1010	1	1	yes	characteristic
433	D₃	—	2	2	no	—
434	C₂²	—	3	2	no	—
435	C₂²	—	6	1	no	—
436	C₂²	—	6	1	no	—
437	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	3	2	no	—
443	C₄	—	6	2	no	—
444	C₄	—	6	1	no	—
445	C₄	—	6	2	no	—
446	C₄	—	6	1	no	—
447	C₄	D₃×Q₈×C₉	1	2	yes	—
448	C₃	—	2	1	no	—
449	C₃	C₉×#544	1	1	yes	characteristic, complemented
450	C₃	C₃×#5689	1	1	yes	characteristic
451	C₂	—	6	2	no	—
452	C₂	C₂×D₃×Q₈×C₉	1	1	yes	characteristic
453	C₂	C₉×#1009	1	1	yes	characteristic
454	C₂	C₉×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂²×C₃×D₃	C₂³×C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#544
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#125	C₂	semidirect
6. C₉×#890	C₂	semidirect
7. C₉×#889	C₂	semidirect
8. C₉×#888	C₂	semidirect
9. C₉×#888	C₂	semidirect
11. C₉×#869	C₂	semidirect
54. C₉×#125	D₃	semidirect
72. C₄×D₃×C₉	Q₈	semidirect
76. #5689	C₉	direct (N × H)
148. C₃×#125	C₂×C₉	semidirect
159. #890	C₂×C₉	semidirect
160. #889	C₂×C₉	semidirect
161. #888	C₂×C₉	semidirect
162. #888	C₂×C₉	semidirect
164. #869	C₂×C₉	semidirect
295. #125	D₃×C₉	semidirect
301. C₃×C₉	#544	semidirect
334. C₄×D₃	Q₈×C₉	semidirect
407. C₉	#5689	direct (N × H)
449. C₃	C₉×#544	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#125	2
C₉×#897	2
C₉×#896	2
C₉×#892	2
D₃×C₉×C₄⋊C₄	2
C₉×#890	2
C₉×#889	2
C₉×#888	2
C₉×#872	2
C₉×#869	2
C₃×#5689	3
C₉×#544	3

Subgroups — C9×#5689 #13153164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5689 #13153164