Subgroups — C₉×#5727 #13153290

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₈×C₉	C₂	1	2	yes	maximal, complemented
2	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#946	C₂	1	2	yes	maximal, complemented
4	C₉×#896	C₂	1	4	yes	maximal, complemented
5	C₉×#893	C₂	1	2	yes	maximal
6	C₉×#888	C₂	1	2	yes	maximal, complemented
7	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
8	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#5727	C₃	1	1	yes	maximal, characteristic
10	D₄×Q₈×C₉	—	3	1	no	maximal
11	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
12	D₃×Q₈×C₉	—	2	8	no	—
13	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
15	C₂×C₄×D₃×C₉	C₂²	1	4	yes	—
16	C₂×C₉×Dic₆	C₂²	1	4	yes	—
17	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₉×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
23	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
27	C₄×C₉×Dic₃	C₂²	1	2	yes	—
28	C₂×C₃×D₃×Q₈	C₂×C₃	1	2	yes	—
29	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
30	C₃×#946	C₂×C₃	1	2	yes	—
31	C₃×#896	C₂×C₃	1	4	yes	—
32	C₃×#893	C₂×C₃	1	2	yes	—
33	C₃×#888	C₂×C₃	1	2	yes	—
34	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
35	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
36	C₂²×Q₈×C₉	—	3	2	no	—
37	C₉×C₄⋊Q₈	—	2	1	no	—
38	C₉×C₄⋊Q₈	—	3	2	no	—
39	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
40	C₉×C₂²⋊Q₈	—	3	2	no	—
41	C₉×C₂²⋊Q₈	—	3	4	no	—
42	C₄×Q₈×C₉	—	3	1	no	—
43	C₄×D₄×C₉	—	3	2	no	—
44	C₄×D₄×C₉	—	3	1	no	—
45	C₂²×D₃×C₉	C₂³	1	2	yes	—
46	C₃×Q₈×C₉	—	2	4	no	—
47	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
48	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₂×C₉×Dic₃	C₂³	1	2	yes	—
52	C₂×C₉×Dic₃	C₂³	1	2	yes	—
53	C₂×C₉×Dic₃	C₂³	1	2	yes	—
54	C₉×D₁₂	Q₈	1	4	yes	complemented
55	C₄×D₃×C₉	—	2	4	no	—
56	C₄×D₃×C₉	—	2	8	no	—
57	C₉×Dic₆	—	2	8	no	—
58	C₉×Dic₆	D₄	1	4	yes	complemented
59	C₃×D₄×Q₈	—	3	1	no	—
60	#5727	C₉	1	1	yes	characteristic, direct product factor
61	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
62	C₃×D₃×Q₈	—	2	8	no	—
63	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
65	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
66	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
67	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
68	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
71	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
72	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
73	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
75	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
76	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
77	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
78	C₂×Q₈×C₉	—	2	2	no	—
79	C₂×Q₈×C₉	—	3	1	no	—
80	C₂×Q₈×C₉	—	3	4	no	—
81	C₂×Q₈×C₉	—	6	8	no	—
82	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
83	C₂×D₄×C₉	—	3	1	no	—
84	C₂²×C₄×C₉	—	3	4	no	—
85	C₂²×C₄×C₉	—	3	2	no	—
86	C₉×C₄⋊C₄	—	2	2	no	—
87	C₉×C₄⋊C₄	—	2	2	no	—
88	C₉×C₄⋊C₄	—	3	1	no	—
89	C₉×C₄⋊C₄	—	3	2	no	—
90	C₉×C₄⋊C₄	—	3	1	no	—
91	C₉×C₄⋊C₄	—	3	4	no	—
92	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
93	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
94	C₉×C₂²⋊C₄	—	3	4	no	—
95	C₉×C₂²⋊C₄	—	3	2	no	—
96	C₄²×C₉	—	2	1	no	—
97	C₄²×C₉	—	3	2	no	—
98	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
100	C₂×D₃×C₉	—	2	2	no	—
101	C₂×D₃×C₉	—	2	2	no	—
102	C₂×D₃×C₉	C₂×Q₈	1	4	yes	complemented
103	C₃×C₄×C₉	—	2	4	no	—
104	C₃×C₄×C₉	—	2	1	no	—
105	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
106	C₃×C₄×C₉	C₂×Q₈	1	2	yes	complemented
107	C₉×Dic₃	—	2	2	no	—
108	C₉×Dic₃	—	2	2	no	—
109	C₉×Dic₃	C₂×D₄	1	4	yes	—
110	C₂²×C₃×Q₈	—	3	2	no	—
111	C₂×D₃×Q₈	C₂×C₉	1	2	yes	complemented
112	C₃×C₄⋊Q₈	—	2	1	no	—
113	C₃×C₄⋊Q₈	—	3	2	no	—
114	C₃×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
115	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
116	C₃×C₂²⋊Q₈	—	3	2	no	—
117	C₃×C₂²⋊Q₈	—	3	4	no	—
118	C₃×C₄×Q₈	—	3	1	no	—
119	C₃×C₄×D₄	—	3	2	no	—
120	C₃×C₄×D₄	—	3	1	no	—
121	#946	C₂×C₉	1	2	yes	complemented
122	#896	C₂×C₉	1	4	yes	complemented
123	#893	C₂×C₉	1	2	yes	—
124	#888	C₂×C₉	1	2	yes	complemented
125	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic
126	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
127	C₂²×C₃×D₃	C₂³×C₃	1	2	yes	—
128	C₃²×Q₈	—	2	4	no	—
129	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
131	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
134	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
135	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
136	C₃×D₁₂	C₃×Q₈	1	4	yes	—
137	C₃×C₄×D₃	—	2	4	no	—
138	C₃×C₄×D₃	—	2	8	no	—
139	C₃×Dic₆	—	2	8	no	—
140	C₃×Dic₆	C₃×D₄	1	4	yes	—
141	C₂³×C₉	—	3	2	no	—
142	Q₈×C₉	—	2	4	no	—
143	Q₈×C₉	—	3	4	no	—
144	Q₈×C₉	—	4	4	no	—
145	Q₈×C₉	—	6	8	no	—
146	D₄×C₉	—	3	4	no	—
147	C₂×C₄×C₉	—	2	1	no	—
148	C₂×C₄×C₉	—	2	1	no	—
149	C₂×C₄×C₉	—	2	1	no	—
150	C₂×C₄×C₉	—	2	4	no	—
151	C₂×C₄×C₉	—	3	2	no	—
152	C₂×C₄×C₉	—	3	2	no	—
153	C₂×C₄×C₉	—	3	2	no	—
154	C₂×C₄×C₉	—	6	8	no	—
155	C₂×C₄×C₉	—	6	4	no	—
156	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
158	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
159	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
160	D₄×Q₈	—	3	1	no	Sylow
161	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
164	D₃×C₉	—	2	4	no	—
165	C₂×C₃×Q₈	—	2	2	no	—
166	C₂×C₃×Q₈	—	3	1	no	—
167	C₂×C₃×Q₈	—	3	4	no	—
168	C₂×C₃×Q₈	—	6	8	no	—
169	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
170	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
171	C₂×C₃×D₄	—	3	1	no	—
172	C₂²×C₃×C₄	—	3	4	no	—
173	C₂²×C₃×C₄	—	3	2	no	—
174	D₃×Q₈	—	2	8	no	—
175	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
176	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
177	C₂×C₄×D₃	C₂²×C₉	1	4	yes	—
178	C₂×Dic₆	C₂²×C₉	1	4	yes	—
179	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
180	C₃×C₄⋊C₄	—	2	2	no	—
181	C₃×C₄⋊C₄	—	2	2	no	—
182	C₃×C₄⋊C₄	—	3	2	no	—
183	C₃×C₄⋊C₄	—	3	4	no	—
184	C₃×C₄⋊C₄	—	3	1	no	—
185	C₃×C₄⋊C₄	—	3	1	no	—
186	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
187	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
188	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
189	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
190	C₃×C₂²⋊C₄	—	3	4	no	—
191	C₃×C₂²⋊C₄	—	3	2	no	—
192	C₃×C₄²	—	2	1	no	—
193	C₃×C₄²	—	3	2	no	—
194	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
195	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
196	(C₂×D₃)⋊C₄	C₂²×C₉	1	2	yes	—
197	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
198	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
199	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
200	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
201	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
202	C₄×Dic₃	C₂²×C₉	1	2	yes	—
203	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
204	C₂×C₃×D₃	—	2	2	no	—
205	C₂×C₃×D₃	—	2	2	no	—
206	C₂×C₃×D₃	C₂×C₃×Q₈	1	4	yes	—
207	C₃²×C₄	—	2	1	no	—
208	C₃²×C₄	—	2	4	no	—
209	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
210	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
211	C₃×Dic₃	—	2	2	no	—
212	C₃×Dic₃	—	2	2	no	—
213	C₃×Dic₃	C₂×C₃×D₄	1	4	yes	—
214	C₂²×C₉	—	2	1	no	—
215	C₂²×C₉	—	3	4	no	—
216	C₂²×C₉	—	6	2	no	—
217	C₂²×C₉	—	6	2	no	—
218	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
219	C₄×C₉	—	2	2	no	—
220	C₄×C₉	—	2	4	no	—
221	C₄×C₉	—	2	1	no	—
222	C₄×C₉	—	2	2	no	—
223	C₄×C₉	—	2	2	no	—
224	C₄×C₉	—	3	4	no	—
225	C₄×C₉	—	4	4	no	—
226	C₄×C₉	—	6	2	no	—
227	C₄×C₉	—	6	2	no	—
228	C₄×C₉	D₃×D₄	1	2	yes	—
229	C₄×C₉	D₃×Q₈	1	2	yes	complemented
230	C₂²×Q₈	—	3	2	no	—
231	C₄⋊Q₈	—	3	2	no	—
232	C₄⋊Q₈	D₃×C₉	1	1	yes	characteristic, complemented
233	C₂²⋊Q₈	—	3	2	no	—
234	C₂²⋊Q₈	—	3	4	no	—
235	C₄×Q₈	—	3	1	no	—
236	C₄×D₄	—	3	2	no	—
237	C₄×D₄	—	3	1	no	—
238	C₃×C₉	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
239	C₂³×C₃	—	3	2	no	—
240	C₂²×D₃	C₂³×C₉	1	2	yes	—
241	C₃×Q₈	—	2	4	no	—
242	C₃×Q₈	—	2	4	no	—
243	C₃×Q₈	—	3	4	no	—
244	C₃×Q₈	—	4	4	no	—
245	C₃×Q₈	—	6	8	no	—
246	C₃×D₄	—	3	4	no	—
247	C₂×C₃×C₄	—	2	1	no	—
248	C₂×C₃×C₄	—	2	1	no	—
249	C₂×C₃×C₄	—	2	1	no	—
250	C₂×C₃×C₄	—	2	4	no	—
251	C₂×C₃×C₄	—	3	2	no	—
252	C₂×C₃×C₄	—	3	2	no	—
253	C₂×C₃×C₄	—	3	2	no	—
254	C₂×C₃×C₄	—	6	8	no	—
255	C₂×C₃×C₄	—	6	4	no	—
256	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
258	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
259	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
260	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
261	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
262	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
264	C₂×Dic₃	C₂³×C₉	1	2	yes	—
265	C₂×Dic₃	C₂³×C₉	1	2	yes	—
266	C₂×Dic₃	C₂³×C₉	1	2	yes	—
267	D₁₂	Q₈×C₉	1	4	yes	complemented
268	C₄×D₃	—	2	4	no	—
269	C₄×D₃	—	2	8	no	—
270	Dic₆	—	2	8	no	—
271	Dic₆	D₄×C₉	1	4	yes	complemented
272	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
273	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
274	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
275	C₃×D₃	—	2	4	no	—
276	C₂×C₉	—	2	1	no	—
277	C₂×C₉	—	2	1	no	—
278	C₂×C₉	—	2	1	no	—
279	C₂×C₉	—	6	4	no	—
280	C₂×C₉	C₂×D₃×D₄	1	1	yes	characteristic
281	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
282	C₂×C₉	#1007	1	1	yes	characteristic
283	C₂×Q₈	—	3	1	no	—
284	C₂×Q₈	—	3	4	no	—
285	C₂×Q₈	—	6	8	no	—
286	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
287	C₂×D₄	—	3	1	no	—
288	C₂²×C₄	—	3	4	no	—
289	C₂²×C₄	—	3	2	no	—
290	C₄⋊C₄	—	3	1	no	—
291	C₄⋊C₄	—	3	2	no	—
292	C₄⋊C₄	—	3	1	no	—
293	C₄⋊C₄	—	3	4	no	—
294	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
295	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
296	C₂²⋊C₄	—	3	4	no	—
297	C₂²⋊C₄	—	3	2	no	—
298	C₄²	—	3	2	no	—
299	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
300	C₂²×C₃	—	2	1	no	—
301	C₂²×C₃	—	3	4	no	—
302	C₂²×C₃	—	6	2	no	—
303	C₂²×C₃	—	6	2	no	—
304	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
305	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
306	C₂×D₃	—	2	2	no	—
307	C₂×D₃	—	2	2	no	—
308	C₂×D₃	C₂×Q₈×C₉	1	4	yes	complemented
309	C₃×C₄	—	2	2	no	—
310	C₃×C₄	—	2	4	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	—	2	4	no	—
313	C₃×C₄	—	2	1	no	—
314	C₃×C₄	—	2	2	no	—
315	C₃×C₄	—	2	2	no	—
316	C₃×C₄	—	3	4	no	—
317	C₃×C₄	—	4	4	no	—
318	C₃×C₄	—	6	2	no	—
319	C₃×C₄	—	6	2	no	—
320	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
321	C₃×C₄	C₂×Q₈×C₉	1	2	yes	complemented
322	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
323	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
324	Dic₃	—	2	2	no	—
325	Dic₃	—	2	2	no	—
326	Dic₃	C₂×D₄×C₉	1	4	yes	—
327	C₃²	C₃×D₄×Q₈	1	1	yes	characteristic
328	C₉	—	2	1	no	—
329	C₉	#5727	1	1	yes	characteristic, direct product factor
330	C₂³	—	3	2	no	—
331	Q₈	—	2	4	no	—
332	Q₈	—	3	4	no	—
333	Q₈	—	6	8	no	—
334	D₄	—	3	4	no	—
335	C₂×C₄	—	3	2	no	—
336	C₂×C₄	—	3	2	no	—
337	C₂×C₄	—	3	2	no	—
338	C₂×C₄	—	6	8	no	—
339	C₂×C₄	—	6	4	no	—
340	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
342	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	6	4	no	—
348	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
349	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
350	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
351	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
352	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
353	C₂×C₃	C₃×#1007	1	1	yes	characteristic
354	D₃	—	2	4	no	—
355	C₂²	—	3	4	no	—
356	C₂²	—	6	2	no	—
357	C₂²	—	6	2	no	—
358	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
359	C₄	—	2	4	no	—
360	C₄	—	2	1	no	—
361	C₄	—	3	4	no	—
362	C₄	—	6	2	no	—
363	C₄	—	6	2	no	—
364	C₄	D₃×D₄×C₉	1	2	yes	—
365	C₄	D₃×Q₈×C₉	1	2	yes	complemented
366	C₃	—	2	1	no	—
367	C₃	D₄×Q₈×C₉	1	1	yes	characteristic, complemented
368	C₃	C₃×#5727	1	1	yes	characteristic
369	C₂	—	6	4	no	—
370	C₂	C₂×D₃×D₄×C₉	1	1	yes	characteristic
371	C₂	C₂×D₃×Q₈×C₉	1	1	yes	characteristic
372	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	C₂×C₃×D₁₂	C₂²×C₃
hypercenter	C₉×C₄⋊Q₈	D₃
hypocenter	C₃	D₄×Q₈×C₉
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. C₂×D₃×Q₈×C₉	C₂	semidirect
2. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. C₉×#946	C₂	semidirect
4. C₉×#896	C₂	semidirect
6. C₉×#888	C₂	semidirect
8. C₄×C₉×Dic₆	C₂	semidirect
39. C₉×C₄⋊Q₈	D₃	semidirect
54. C₉×D₁₂	Q₈	semidirect
58. C₉×Dic₆	D₄	semidirect
60. #5727	C₉	direct (N × H)
102. C₂×D₃×C₉	C₂×Q₈	semidirect
106. C₃×C₄×C₉	C₂×Q₈	semidirect
111. C₂×D₃×Q₈	C₂×C₉	semidirect
114. C₃×C₄⋊Q₈	C₂×C₉	semidirect
121. #946	C₂×C₉	semidirect
122. #896	C₂×C₉	semidirect
124. #888	C₂×C₉	semidirect
126. C₄×Dic₆	C₂×C₉	semidirect
229. C₄×C₉	D₃×Q₈	semidirect
232. C₄⋊Q₈	D₃×C₉	semidirect
238. C₃×C₉	D₄×Q₈	semidirect
267. D₁₂	Q₈×C₉	semidirect
271. Dic₆	D₄×C₉	semidirect
308. C₂×D₃	C₂×Q₈×C₉	semidirect
321. C₃×C₄	C₂×Q₈×C₉	semidirect
329. C₉	#5727	direct (N × H)
365. C₄	D₃×Q₈×C₉	semidirect
367. C₃	D₄×Q₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Q₈×C₉	2
C₃×C₉×C₄⋊Q₈	2
C₉×#946	2
C₉×#896	2
C₉×#893	2
C₉×#888	2
C₄×C₉×D₁₂	2
C₄×C₉×Dic₆	2
C₃×#5727	3
D₄×Q₈×C₉	3

Subgroups — C9×#5727 #13153290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5727 #13153290