Subgroups — C₉×#5728 #13153309

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 386 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	Q₈×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₉×#897	C₂	1	2	yes	maximal
5	C₉×#892	C₂	1	2	yes	maximal
6	C₉×#890	C₂	1	2	yes	maximal, complemented
7	C₉×#887	C₂	1	2	yes	maximal, complemented
8	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic
9	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#5728	C₃	1	1	yes	maximal, characteristic
11	C₉×#533	—	3	1	no	maximal
12	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
13	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
16	C₂×C₉×Dic₆	C₂²	1	2	yes	—
17	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
21	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
24	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
25	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
26	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
27	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
28	C₄×C₉×Dic₃	C₂²	1	4	yes	—
29	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
30	C₃×#946	C₂×C₃	1	2	yes	—
31	C₃×Q₈×Dic₃	C₂×C₃	1	2	yes	—
32	C₃×#897	C₂×C₃	1	2	yes	—
33	C₃×#892	C₂×C₃	1	2	yes	—
34	C₃×#890	C₂×C₃	1	2	yes	—
35	C₃×#887	C₂×C₃	1	2	yes	—
36	C₃×C₄²×D₃	C₂×C₃	1	1	yes	characteristic
37	C₃×#869	C₂×C₃	1	1	yes	characteristic
38	C₉×C₄⋊Q₈	—	2	1	no	—
39	C₉×C₄⋊Q₈	—	3	1	no	—
40	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
41	C₉×#125	—	3	2	no	—
42	C₉×C₂²⋊Q₈	—	3	2	no	—
43	C₉×C₂²⋊Q₈	—	3	2	no	—
44	C₄×Q₈×C₉	—	3	2	no	—
45	C₄×Q₈×C₉	—	3	2	no	—
46	C₉×#117	—	3	2	no	—
47	C₂×C₄²×C₉	—	3	1	no	—
48	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
49	C₃×Q₈×C₉	—	2	4	no	—
50	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
51	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₂×C₉×Dic₃	C₂³	1	4	yes	—
55	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
58	C₄×D₃×C₉	—	2	2	no	—
59	C₄×D₃×C₉	—	2	2	no	—
60	C₄×D₃×C₉	Q₈	1	4	yes	complemented
61	C₉×Dic₆	—	2	4	no	—
62	C₃×#533	—	3	1	no	—
63	#5728	C₉	1	1	yes	characteristic, direct product factor
64	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
69	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
72	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
73	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
74	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
75	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
76	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
77	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
78	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
79	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
80	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
81	C₂×Q₈×C₉	—	2	2	no	—
82	C₂×Q₈×C₉	—	3	2	no	—
83	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
84	C₂²×C₄×C₉	—	3	1	no	—
85	C₂²×C₄×C₉	—	3	1	no	—
86	C₂²×C₄×C₉	—	3	1	no	—
87	C₉×C₄⋊C₄	—	2	2	no	—
88	C₉×C₄⋊C₄	—	2	2	no	—
89	C₉×C₄⋊C₄	—	3	2	no	—
90	C₉×C₄⋊C₄	—	3	4	no	—
91	C₉×C₄⋊C₄	—	3	4	no	—
92	C₉×C₄⋊C₄	—	3	2	no	—
93	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
94	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
95	C₉×C₂²⋊C₄	—	3	4	no	—
96	C₄²×C₉	—	2	1	no	—
97	C₄²×C₉	—	3	1	no	—
98	C₄²×C₉	—	3	1	no	—
99	C₄²×C₉	—	3	1	no	—
100	C₄²×C₉	—	3	4	no	—
101	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
103	C₂×D₃×C₉	—	2	1	no	—
104	C₂×D₃×C₉	—	2	1	no	—
105	C₂×D₃×C₉	C₂×Q₈	1	2	yes	—
106	C₃×C₄×C₉	—	2	4	no	—
107	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₉	#41	1	2	yes	—
109	C₃×C₄×C₉	#41	1	2	yes	—
110	C₉×Dic₃	—	2	4	no	—
111	C₉×Dic₃	—	2	1	no	—
112	C₉×Dic₃	—	2	1	no	—
113	C₉×Dic₃	C₂×Q₈	1	2	yes	—
114	C₃×C₄⋊Q₈	—	2	1	no	—
115	C₃×C₄⋊Q₈	—	3	1	no	—
116	C₃×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
117	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
118	C₃×#125	—	3	2	no	—
119	C₃×C₂²⋊Q₈	—	3	2	no	—
120	C₃×C₂²⋊Q₈	—	3	2	no	—
121	C₃×C₄×Q₈	—	3	2	no	—
122	C₃×C₄×Q₈	—	3	2	no	—
123	C₃×#117	—	3	2	no	—
124	C₂×C₃×C₄²	—	3	1	no	—
125	#946	C₂×C₉	1	2	yes	—
126	Q₈×Dic₃	C₂×C₉	1	2	yes	complemented
127	#897	C₂×C₉	1	2	yes	—
128	#892	C₂×C₉	1	2	yes	—
129	#890	C₂×C₉	1	2	yes	complemented
130	#887	C₂×C₉	1	2	yes	complemented
131	C₄²×D₃	C₂×C₉	1	1	yes	characteristic
132	#869	C₂×C₉	1	1	yes	characteristic, complemented
133	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
134	C₃²×Q₈	—	2	4	no	—
135	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
136	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
142	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
143	C₃×C₄×D₃	—	2	2	no	—
144	C₃×C₄×D₃	—	2	2	no	—
145	C₃×C₄×D₃	C₃×Q₈	1	4	yes	—
146	C₃×Dic₆	—	2	4	no	—
147	C₂³×C₉	—	3	1	no	—
148	Q₈×C₉	—	2	4	no	—
149	Q₈×C₉	—	4	4	no	—
150	Q₈×C₉	—	6	4	no	—
151	C₂×C₄×C₉	—	2	1	no	—
152	C₂×C₄×C₉	—	2	1	no	—
153	C₂×C₄×C₉	—	2	1	no	—
154	C₂×C₄×C₉	—	2	4	no	—
155	C₂×C₄×C₉	—	3	1	no	—
156	C₂×C₄×C₉	—	3	4	no	—
157	C₂×C₄×C₉	—	3	1	no	—
158	C₂×C₄×C₉	—	3	1	no	—
159	C₂×C₄×C₉	—	3	4	no	—
160	C₂×C₄×C₉	—	6	2	no	—
161	C₂×C₄×C₉	—	6	2	no	—
162	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
163	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
164	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
165	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
166	#533	—	3	1	no	Sylow
167	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
169	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
170	D₃×C₉	—	2	2	no	—
171	C₂×C₃×Q₈	—	2	2	no	—
172	C₂×C₃×Q₈	—	3	2	no	—
173	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
174	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
175	C₂²×C₃×C₄	—	3	1	no	—
176	C₂²×C₃×C₄	—	3	1	no	—
177	C₂²×C₃×C₄	—	3	1	no	—
178	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
179	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
180	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
181	C₂×Dic₆	C₂²×C₉	1	2	yes	—
182	C₃×C₄⋊C₄	—	2	2	no	—
183	C₃×C₄⋊C₄	—	2	2	no	—
184	C₃×C₄⋊C₄	—	3	4	no	—
185	C₃×C₄⋊C₄	—	3	2	no	—
186	C₃×C₄⋊C₄	—	3	4	no	—
187	C₃×C₄⋊C₄	—	3	2	no	—
188	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
189	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
190	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
191	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
192	C₃×C₂²⋊C₄	—	3	4	no	—
193	C₃×C₄²	—	2	1	no	—
194	C₃×C₄²	—	3	1	no	—
195	C₃×C₄²	—	3	1	no	—
196	C₃×C₄²	—	3	1	no	—
197	C₃×C₄²	—	3	4	no	—
198	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
199	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
200	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
201	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
202	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
203	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
204	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
205	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
206	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
207	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
208	C₄×Dic₃	C₂²×C₉	1	4	yes	—
209	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
210	C₂×C₃×D₃	—	2	1	no	—
211	C₂×C₃×D₃	—	2	1	no	—
212	C₂×C₃×D₃	C₂×C₃×Q₈	1	2	yes	—
213	C₃²×C₄	—	2	4	no	—
214	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
215	C₃²×C₄	C₃×#41	1	2	yes	—
216	C₃²×C₄	C₃×#41	1	2	yes	—
217	C₃×Dic₃	—	2	4	no	—
218	C₃×Dic₃	—	2	1	no	—
219	C₃×Dic₃	—	2	1	no	—
220	C₃×Dic₃	C₂×C₃×Q₈	1	2	yes	—
221	C₂²×C₉	—	2	1	no	—
222	C₂²×C₉	—	3	2	no	—
223	C₂²×C₉	—	6	1	no	—
224	C₂²×C₉	—	6	1	no	—
225	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
226	C₄×C₉	—	2	4	no	—
227	C₄×C₉	—	2	2	no	—
228	C₄×C₉	—	2	2	no	—
229	C₄×C₉	—	2	2	no	—
230	C₄×C₉	—	3	2	no	—
231	C₄×C₉	—	4	4	no	—
232	C₄×C₉	—	6	1	no	—
233	C₄×C₉	—	6	1	no	—
234	C₄×C₉	—	6	4	no	—
235	C₄×C₉	#237	1	2	yes	—
236	C₄×C₉	D₃×Q₈	1	2	yes	—
237	C₄×C₉	#239	1	2	yes	—
238	C₄⋊Q₈	—	3	1	no	—
239	C₄⋊Q₈	D₃×C₉	1	1	yes	characteristic, complemented
240	#125	—	3	2	no	—
241	C₂²⋊Q₈	—	3	2	no	—
242	C₂²⋊Q₈	—	3	2	no	—
243	C₄×Q₈	—	3	2	no	—
244	C₄×Q₈	—	3	2	no	—
245	#117	—	3	2	no	—
246	C₂×C₄²	—	3	1	no	—
247	C₃×C₉	#533	1	1	yes	characteristic, complemented, Sylow
248	C₂³×C₃	—	3	1	no	—
249	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
250	C₃×Q₈	—	2	4	no	—
251	C₃×Q₈	—	2	4	no	—
252	C₃×Q₈	—	4	4	no	—
253	C₃×Q₈	—	6	4	no	—
254	C₂×C₃×C₄	—	2	1	no	—
255	C₂×C₃×C₄	—	2	1	no	—
256	C₂×C₃×C₄	—	2	1	no	—
257	C₂×C₃×C₄	—	2	4	no	—
258	C₂×C₃×C₄	—	3	1	no	—
259	C₂×C₃×C₄	—	3	4	no	—
260	C₂×C₃×C₄	—	3	1	no	—
261	C₂×C₃×C₄	—	3	1	no	—
262	C₂×C₃×C₄	—	3	4	no	—
263	C₂×C₃×C₄	—	6	2	no	—
264	C₂×C₃×C₄	—	6	2	no	—
265	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
266	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
267	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
268	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
269	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
270	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
271	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
272	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
273	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
274	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
275	C₂×Dic₃	C₂³×C₉	1	4	yes	—
276	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
277	C₄×D₃	—	2	2	no	—
278	C₄×D₃	—	2	2	no	—
279	C₄×D₃	Q₈×C₉	1	4	yes	complemented
280	Dic₆	—	2	4	no	—
281	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
282	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
283	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
284	C₃×D₃	—	2	2	no	—
285	C₂×C₉	—	2	1	no	—
286	C₂×C₉	—	2	1	no	—
287	C₂×C₉	—	2	1	no	—
288	C₂×C₉	—	6	2	no	—
289	C₂×C₉	C₂×#237	1	1	yes	characteristic
290	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
291	C₂×C₉	C₂×#239	1	1	yes	characteristic
292	C₂×Q₈	—	3	2	no	—
293	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
294	C₂²×C₄	—	3	1	no	—
295	C₂²×C₄	—	3	1	no	—
296	C₂²×C₄	—	3	1	no	—
297	C₄⋊C₄	—	3	2	no	—
298	C₄⋊C₄	—	3	4	no	—
299	C₄⋊C₄	—	3	4	no	—
300	C₄⋊C₄	—	3	2	no	—
301	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
302	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
303	C₂²⋊C₄	—	3	4	no	—
304	C₄²	—	3	1	no	—
305	C₄²	—	3	1	no	—
306	C₄²	—	3	1	no	—
307	C₄²	—	3	4	no	—
308	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
309	C₂²×C₃	—	2	1	no	—
310	C₂²×C₃	—	3	2	no	—
311	C₂²×C₃	—	6	1	no	—
312	C₂²×C₃	—	6	1	no	—
313	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
314	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
315	C₂×D₃	—	2	1	no	—
316	C₂×D₃	—	2	1	no	—
317	C₂×D₃	C₂×Q₈×C₉	1	2	yes	—
318	C₃×C₄	—	2	4	no	—
319	C₃×C₄	—	2	4	no	—
320	C₃×C₄	—	2	2	no	—
321	C₃×C₄	—	2	2	no	—
322	C₃×C₄	—	2	2	no	—
323	C₃×C₄	—	3	2	no	—
324	C₃×C₄	—	4	4	no	—
325	C₃×C₄	—	6	1	no	—
326	C₃×C₄	—	6	1	no	—
327	C₃×C₄	—	6	4	no	—
328	C₃×C₄	C₂×Q₈×C₉	1	2	yes	—
329	C₃×C₄	C₉×#41	1	2	yes	—
330	C₃×C₄	C₉×#41	1	2	yes	—
331	C₃×C₄	C₃×#237	1	2	yes	—
332	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
333	C₃×C₄	C₃×#239	1	2	yes	—
334	Dic₃	—	2	4	no	—
335	Dic₃	—	2	1	no	—
336	Dic₃	—	2	1	no	—
337	Dic₃	C₂×Q₈×C₉	1	2	yes	—
338	C₃²	C₃×#533	1	1	yes	characteristic
339	C₉	—	2	1	no	—
340	C₉	#5728	1	1	yes	characteristic, direct product factor
341	C₂³	—	3	1	no	—
342	Q₈	—	2	4	no	—
343	Q₈	—	6	4	no	—
344	C₂×C₄	—	3	1	no	—
345	C₂×C₄	—	3	4	no	—
346	C₂×C₄	—	3	1	no	—
347	C₂×C₄	—	3	1	no	—
348	C₂×C₄	—	3	4	no	—
349	C₂×C₄	—	6	2	no	—
350	C₂×C₄	—	6	2	no	—
351	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
352	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
355	C₂×C₃	—	2	1	no	—
356	C₂×C₃	—	2	1	no	—
357	C₂×C₃	—	2	1	no	—
358	C₂×C₃	—	6	2	no	—
359	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
360	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
361	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
362	C₂×C₃	C₂×C₃×#237	1	1	yes	characteristic
363	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
364	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
365	D₃	—	2	2	no	—
366	C₂²	—	3	2	no	—
367	C₂²	—	6	1	no	—
368	C₂²	—	6	1	no	—
369	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
370	C₄	—	2	4	no	—
371	C₄	—	3	2	no	—
372	C₄	—	6	1	no	—
373	C₄	—	6	1	no	—
374	C₄	—	6	4	no	—
375	C₄	C₉×#237	1	2	yes	—
376	C₄	D₃×Q₈×C₉	1	2	yes	—
377	C₄	C₉×#239	1	2	yes	—
378	C₃	—	2	1	no	—
379	C₃	C₉×#533	1	1	yes	characteristic, complemented
380	C₃	C₃×#5728	1	1	yes	characteristic
381	C₂	—	6	2	no	—
382	C₂	C₂×C₉×#237	1	1	yes	characteristic
383	C₂	C₂×D₃×Q₈×C₉	1	1	yes	characteristic
384	C₂	C₂×C₉×#239	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×#533
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. Q₈×C₉×Dic₃	C₂	semidirect
6. C₉×#890	C₂	semidirect
7. C₉×#887	C₂	semidirect
9. C₉×#869	C₂	semidirect
40. C₉×C₄⋊Q₈	D₃	semidirect
60. C₄×D₃×C₉	Q₈	semidirect
63. #5728	C₉	direct (N × H)
116. C₃×C₄⋊Q₈	C₂×C₉	semidirect
126. Q₈×Dic₃	C₂×C₉	semidirect
129. #890	C₂×C₉	semidirect
130. #887	C₂×C₉	semidirect
132. #869	C₂×C₉	semidirect
239. C₄⋊Q₈	D₃×C₉	semidirect
247. C₃×C₉	#533	semidirect
279. C₄×D₃	Q₈×C₉	semidirect
340. C₉	#5728	direct (N × H)
379. C₃	C₉×#533	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
Q₈×C₉×Dic₃	2
C₉×#897	2
C₉×#892	2
C₉×#890	2
C₉×#887	2
C₄²×D₃×C₉	2
C₉×#869	2
C₃×#5728	3
C₉×#533	3

Subgroups — C9×#5728 #13153309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5728 #13153309