Subgroups — C₉×#5729 #13153310

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#946	C₂	1	2	yes	maximal
3	C₉×#944	C₂	1	2	yes	maximal, complemented
4	C₉×#896	C₂	1	2	yes	maximal
5	C₉×#892	C₂	1	1	yes	maximal, characteristic
6	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₉×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#889	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×#870	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#5729	C₃	1	1	yes	maximal, characteristic
13	C₉×#544	—	3	1	no	maximal
14	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
15	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
16	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
21	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
23	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₉×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₉×Dic₃⋊C₄	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×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
38	C₃×#946	C₂×C₃	1	2	yes	—
39	C₃×#944	C₂×C₃	1	2	yes	—
40	C₃×#896	C₂×C₃	1	2	yes	—
41	C₃×#892	C₂×C₃	1	1	yes	characteristic
42	C₃×D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
43	C₃×#890	C₂×C₃	1	1	yes	characteristic
44	C₃×#889	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×#870	C₂×C₃	1	1	yes	characteristic
48	C₉×C₄⋊Q₈	—	2	1	no	—
49	C₉×C₄⋊Q₈	—	3	2	no	—
50	C₉×C₄⋊Q₈	—	3	1	no	—
51	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
52	C₉×#125	—	3	2	no	—
53	C₉×#125	—	3	1	no	—
54	C₉×#125	—	3	1	no	—
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	C₃×Q₈×C₉	—	4	2	no	—
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×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
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	2	yes	—
71	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
72	C₄×D₃×C₉	—	4	2	no	—
73	C₄×D₃×C₉	Q₈	1	4	yes	complemented
74	C₉×Dic₆	—	4	2	no	—
75	C₃×#544	—	3	1	no	—
76	#5729	C₉	1	1	yes	characteristic, direct product factor
77	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
79	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
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	1	yes	characteristic
84	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
86	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₃×C₄⋊Dic₃	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×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
92	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
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	1	yes	characteristic
99	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
100	C₂×Q₈×C₉	—	2	2	no	—
101	C₂×Q₈×C₉	—	3	2	no	—
102	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
103	C₂²×C₄×C₉	—	3	1	no	—
104	C₂²×C₄×C₉	—	3	2	no	—
105	C₉×C₄⋊C₄	—	2	1	no	—
106	C₉×C₄⋊C₄	—	2	1	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	—	3	2	no	—
118	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
119	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
120	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
121	C₉×C₂²⋊C₄	—	3	2	no	—
122	C₉×C₂²⋊C₄	—	3	1	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	1	no	—
140	C₉×Dic₃	—	2	2	no	—
141	C₉×Dic₃	C₂×Q₈	1	2	yes	—
142	C₃×C₄⋊Q₈	—	2	1	no	—
143	C₃×C₄⋊Q₈	—	3	2	no	—
144	C₃×C₄⋊Q₈	—	3	1	no	—
145	C₃×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
146	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
147	C₃×#125	—	3	2	no	—
148	C₃×#125	—	3	1	no	—
149	C₃×#125	—	3	1	no	—
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	#946	C₂×C₉	1	2	yes	—
156	#944	C₂×C₉	1	2	yes	complemented
157	#896	C₂×C₉	1	2	yes	—
158	#892	C₂×C₉	1	1	yes	characteristic
159	D₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
160	#890	C₂×C₉	1	1	yes	characteristic, complemented
161	#889	C₂×C₉	1	2	yes	complemented
162	#888	C₂×C₉	1	1	yes	characteristic, complemented
163	#872	C₂×C₉	1	1	yes	characteristic
164	#870	C₂×C₉	1	1	yes	characteristic, complemented
165	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
166	C₃²×Q₈	—	4	2	no	—
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	2	yes	—
171	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
173	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
174	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
175	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
176	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
177	C₃×C₄×D₃	—	4	2	no	—
178	C₃×C₄×D₃	C₃×Q₈	1	4	yes	—
179	C₃×Dic₆	—	4	2	no	—
180	C₂³×C₉	—	3	1	no	—
181	Q₈×C₉	—	4	2	no	—
182	Q₈×C₉	—	4	4	no	—
183	Q₈×C₉	—	12	2	no	—
184	C₂×C₄×C₉	—	2	1	no	—
185	C₂×C₄×C₉	—	2	1	no	—
186	C₂×C₄×C₉	—	2	1	no	—
187	C₂×C₄×C₉	—	2	2	no	—
188	C₂×C₄×C₉	—	2	2	no	—
189	C₂×C₄×C₉	—	3	1	no	—
190	C₂×C₄×C₉	—	3	4	no	—
191	C₂×C₄×C₉	—	3	1	no	—
192	C₂×C₄×C₉	—	3	2	no	—
193	C₂×C₄×C₉	—	3	1	no	—
194	C₂×C₄×C₉	—	3	2	no	—
195	C₂×C₄×C₉	—	12	2	no	—
196	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
197	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
198	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
199	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
200	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
201	#544	—	3	1	no	Sylow
202	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
203	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
205	D₃×C₉	—	2	2	no	—
206	C₂×C₃×Q₈	—	2	2	no	—
207	C₂×C₃×Q₈	—	3	2	no	—
208	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
209	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
210	C₂²×C₃×C₄	—	3	1	no	—
211	C₂²×C₃×C₄	—	3	2	no	—
212	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
213	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
214	C₂×Dic₆	C₂²×C₉	1	2	yes	—
215	C₃×C₄⋊C₄	—	2	1	no	—
216	C₃×C₄⋊C₄	—	2	1	no	—
217	C₃×C₄⋊C₄	—	2	2	no	—
218	C₃×C₄⋊C₄	—	3	2	no	—
219	C₃×C₄⋊C₄	—	3	1	no	—
220	C₃×C₄⋊C₄	—	3	1	no	—
221	C₃×C₄⋊C₄	—	3	2	no	—
222	C₃×C₄⋊C₄	—	3	2	no	—
223	C₃×C₄⋊C₄	—	3	2	no	—
224	C₃×C₄⋊C₄	—	3	2	no	—
225	C₃×C₄⋊C₄	—	3	1	no	—
226	C₃×C₄⋊C₄	—	3	1	no	—
227	C₃×C₄⋊C₄	—	3	2	no	—
228	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
229	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
230	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
231	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
232	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
233	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
234	C₃×C₂²⋊C₄	—	3	2	no	—
235	C₃×C₂²⋊C₄	—	3	1	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	1	yes	characteristic
247	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
248	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
249	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
250	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
251	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
252	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
253	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
254	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
255	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
256	C₄×Dic₃	C₂²×C₉	1	2	yes	—
257	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
258	C₂×C₃×D₃	—	2	1	no	—
259	C₂×C₃×D₃	—	2	1	no	—
260	C₂×C₃×D₃	C₂×C₃×Q₈	1	2	yes	—
261	C₃²×C₄	—	2	1	no	—
262	C₃²×C₄	—	2	1	no	—
263	C₃²×C₄	—	2	2	no	—
264	C₃²×C₄	—	2	2	no	—
265	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
266	C₃×Dic₃	—	2	1	no	—
267	C₃×Dic₃	—	2	2	no	—
268	C₃×Dic₃	—	2	1	no	—
269	C₃×Dic₃	—	2	2	no	—
270	C₃×Dic₃	C₂×C₃×Q₈	1	2	yes	—
271	C₂²×C₉	—	2	1	no	—
272	C₂²×C₉	—	3	2	no	—
273	C₂²×C₉	—	6	1	no	—
274	C₂²×C₉	—	6	1	no	—
275	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
276	C₄×C₉	—	2	2	no	—
277	C₄×C₉	—	2	2	no	—
278	C₄×C₉	—	2	2	no	—
279	C₄×C₉	—	2	1	no	—
280	C₄×C₉	—	2	2	no	—
281	C₄×C₉	—	2	1	no	—
282	C₄×C₉	—	2	2	no	—
283	C₄×C₉	—	3	2	no	—
284	C₄×C₉	—	4	2	no	—
285	C₄×C₉	—	4	2	no	—
286	C₄×C₉	—	6	2	no	—
287	C₄×C₉	—	6	1	no	—
288	C₄×C₉	—	6	1	no	—
289	C₄×C₉	—	6	2	no	—
290	C₄×C₉	D₃×Q₈	1	2	yes	—
291	C₄⋊Q₈	—	3	2	no	—
292	C₄⋊Q₈	—	3	1	no	—
293	C₄⋊Q₈	D₃×C₉	1	1	yes	characteristic, complemented
294	#125	—	3	2	no	—
295	#125	—	3	1	no	—
296	#125	—	3	1	no	—
297	C₂²⋊Q₈	—	3	2	no	—
298	C₂²⋊Q₈	—	3	2	no	—
299	#117	—	3	1	no	—
300	#117	—	3	1	no	—
301	C₂×C₄⋊C₄	—	3	1	no	—
302	C₃×C₉	#544	1	1	yes	characteristic, complemented, Sylow
303	C₂³×C₃	—	3	1	no	—
304	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
305	C₃×Q₈	—	4	2	no	—
306	C₃×Q₈	—	4	2	no	—
307	C₃×Q₈	—	4	4	no	—
308	C₃×Q₈	—	12	2	no	—
309	C₂×C₃×C₄	—	2	1	no	—
310	C₂×C₃×C₄	—	2	1	no	—
311	C₂×C₃×C₄	—	2	1	no	—
312	C₂×C₃×C₄	—	2	2	no	—
313	C₂×C₃×C₄	—	2	2	no	—
314	C₂×C₃×C₄	—	3	1	no	—
315	C₂×C₃×C₄	—	3	4	no	—
316	C₂×C₃×C₄	—	3	1	no	—
317	C₂×C₃×C₄	—	3	2	no	—
318	C₂×C₃×C₄	—	3	1	no	—
319	C₂×C₃×C₄	—	3	2	no	—
320	C₂×C₃×C₄	—	12	2	no	—
321	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
322	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
323	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
324	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
325	C₂×C₃×C₄	C₂³×C₉	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×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
329	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
330	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
331	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
332	C₂×Dic₃	C₂³×C₉	1	2	yes	—
333	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
334	C₂×Dic₃	C₂³×C₉	1	2	yes	—
335	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
336	C₄×D₃	—	4	2	no	—
337	C₄×D₃	Q₈×C₉	1	4	yes	complemented
338	Dic₆	—	4	2	no	—
339	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
340	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
341	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
342	C₃×D₃	—	2	2	no	—
343	C₂×C₉	—	2	1	no	—
344	C₂×C₉	—	2	1	no	—
345	C₂×C₉	—	2	1	no	—
346	C₂×C₉	—	6	2	no	—
347	C₂×C₉	#1004	1	1	yes	characteristic
348	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
349	C₂×C₉	#1007	1	1	yes	characteristic
350	C₂×Q₈	—	3	2	no	—
351	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
352	C₂²×C₄	—	3	1	no	—
353	C₂²×C₄	—	3	2	no	—
354	C₄⋊C₄	—	3	2	no	—
355	C₄⋊C₄	—	3	1	no	—
356	C₄⋊C₄	—	3	1	no	—
357	C₄⋊C₄	—	3	1	no	—
358	C₄⋊C₄	—	3	1	no	—
359	C₄⋊C₄	—	3	2	no	—
360	C₄⋊C₄	—	3	2	no	—
361	C₄⋊C₄	—	3	2	no	—
362	C₄⋊C₄	—	3	2	no	—
363	C₄⋊C₄	—	3	2	no	—
364	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
365	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
366	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
367	C₂²⋊C₄	—	3	2	no	—
368	C₂²⋊C₄	—	3	1	no	—
369	C₂²⋊C₄	—	3	1	no	—
370	C₄²	—	3	2	no	—
371	C₄²	—	3	1	no	—
372	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
373	C₂²×C₃	—	2	1	no	—
374	C₂²×C₃	—	3	2	no	—
375	C₂²×C₃	—	6	1	no	—
376	C₂²×C₃	—	6	1	no	—
377	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
378	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
379	C₂×D₃	—	2	1	no	—
380	C₂×D₃	—	2	1	no	—
381	C₂×D₃	C₂×Q₈×C₉	1	2	yes	—
382	C₃×C₄	—	2	2	no	—
383	C₃×C₄	—	2	2	no	—
384	C₃×C₄	—	2	2	no	—
385	C₃×C₄	—	2	1	no	—
386	C₃×C₄	—	2	2	no	—
387	C₃×C₄	—	2	1	no	—
388	C₃×C₄	—	2	2	no	—
389	C₃×C₄	—	2	2	no	—
390	C₃×C₄	—	2	1	no	—
391	C₃×C₄	—	2	1	no	—
392	C₃×C₄	—	2	2	no	—
393	C₃×C₄	—	3	2	no	—
394	C₃×C₄	—	4	2	no	—
395	C₃×C₄	—	4	2	no	—
396	C₃×C₄	—	6	2	no	—
397	C₃×C₄	—	6	1	no	—
398	C₃×C₄	—	6	1	no	—
399	C₃×C₄	—	6	2	no	—
400	C₃×C₄	C₂×Q₈×C₉	1	2	yes	—
401	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
402	Dic₃	—	2	1	no	—
403	Dic₃	—	2	2	no	—
404	Dic₃	—	2	1	no	—
405	Dic₃	—	2	2	no	—
406	Dic₃	C₂×Q₈×C₉	1	2	yes	—
407	C₃²	C₃×#544	1	1	yes	characteristic
408	C₉	—	2	1	no	—
409	C₉	#5729	1	1	yes	characteristic, direct product factor
410	C₂³	—	3	1	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	12	2	no	—
413	C₂×C₄	—	3	1	no	—
414	C₂×C₄	—	3	4	no	—
415	C₂×C₄	—	3	1	no	—
416	C₂×C₄	—	3	2	no	—
417	C₂×C₄	—	3	1	no	—
418	C₂×C₄	—	3	2	no	—
419	C₂×C₄	—	12	2	no	—
420	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
421	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
424	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
425	C₂×C₃	—	2	1	no	—
426	C₂×C₃	—	2	1	no	—
427	C₂×C₃	—	2	1	no	—
428	C₂×C₃	—	6	2	no	—
429	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
430	C₂×C₃	C₉×H(5,𝔽₂)	1	1	yes	characteristic
431	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
432	C₂×C₃	C₃×#1004	1	1	yes	characteristic
433	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
434	C₂×C₃	C₃×#1007	1	1	yes	characteristic
435	D₃	—	2	2	no	—
436	C₂²	—	3	2	no	—
437	C₂²	—	6	1	no	—
438	C₂²	—	6	1	no	—
439	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	1	no	—
444	C₄	—	3	2	no	—
445	C₄	—	6	2	no	—
446	C₄	—	6	1	no	—
447	C₄	—	6	1	no	—
448	C₄	—	6	2	no	—
449	C₄	D₃×Q₈×C₉	1	2	yes	—
450	C₃	—	2	1	no	—
451	C₃	C₉×#544	1	1	yes	characteristic, complemented
452	C₃	C₃×#5729	1	1	yes	characteristic
453	C₂	—	6	2	no	—
454	C₂	C₉×#1004	1	1	yes	characteristic
455	C₂	C₂×D₃×Q₈×C₉	1	1	yes	characteristic
456	C₂	C₉×#1007	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×C₄⋊Q₈	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂²×C₃×D₃	C₂³×C₃
hypercenter	C₉×C₄⋊Q₈	D₃
hypocenter	C₃	C₉×#544
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. C₉×#944	C₂	semidirect
7. C₉×#890	C₂	semidirect
8. C₉×#889	C₂	semidirect
9. C₉×#888	C₂	semidirect
11. C₉×#870	C₂	semidirect
51. C₉×C₄⋊Q₈	D₃	semidirect
73. C₄×D₃×C₉	Q₈	semidirect
76. #5729	C₉	direct (N × H)
145. C₃×C₄⋊Q₈	C₂×C₉	semidirect
156. #944	C₂×C₉	semidirect
160. #890	C₂×C₉	semidirect
161. #889	C₂×C₉	semidirect
162. #888	C₂×C₉	semidirect
164. #870	C₂×C₉	semidirect
293. C₄⋊Q₈	D₃×C₉	semidirect
302. C₃×C₉	#544	semidirect
337. C₄×D₃	Q₈×C₉	semidirect
409. C₉	#5729	direct (N × H)
451. 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×C₄⋊Q₈	2
C₉×#946	2
C₉×#944	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×#870	2
C₃×#5729	3
C₉×#544	3

Subgroups — C9×#5729 #13153310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5729 #13153310