Subgroups — C₉×#5510 #13152421

Order 1728.

974 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#235	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#930	C₂	1	4	yes	maximal, complemented
4	C₉×#897	C₂	1	4	yes	maximal, complemented
5	C₉×#895	C₂	1	4	yes	maximal, complemented
6	C₃×#5510	C₃	1	1	yes	maximal, characteristic
7	C₉×#537	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×#67	C₂²	1	4	yes	—
11	C₉×#235	—	2	8	no	—
12	C₂×C₉×D₁₂	C₂²	1	2	yes	—
13	C₂×C₉×D₁₂	C₂²	1	4	yes	—
14	C₂×C₄×D₃×C₉	C₂²	1	4	yes	—
15	C₂×C₉×Dic₆	C₂²	1	2	yes	—
16	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
17	C₉×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
18	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
19	C₂×C₃×#235	C₂×C₃	1	2	yes	—
20	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₃×#930	C₂×C₃	1	4	yes	—
22	C₃×#897	C₂×C₃	1	4	yes	—
23	C₃×#895	C₂×C₃	1	4	yes	—
24	C₂×C₉×#41	—	3	2	no	—
25	C₉×C₂²⋊Q₈	—	3	4	no	—
26	C₉×#121	—	3	4	no	—
27	C₉×#121	—	3	4	no	—
28	C₂×C₉×C₄⋊C₄	—	2	1	no	—
29	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
31	C₂²×D₃×C₉	C₂³	1	4	yes	—
32	C₂×C₃×C₄×C₉	—	4	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	4	yes	complemented
34	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
36	C₉×#67	—	2	8	no	—
37	C₂×C₉×Dic₃	C₂³	1	4	yes	—
38	C₉×D₁₂	—	2	4	no	—
39	C₉×D₁₂	—	4	4	no	—
40	C₄×D₃×C₉	—	2	8	no	—
41	C₉×Dic₆	—	2	4	no	—
42	C₃×#537	—	3	1	no	—
43	#5510	C₉	1	1	yes	characteristic, direct product factor
44	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
45	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₃×#67	C₂²×C₃	1	4	yes	—
47	C₃×#235	—	2	8	no	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
49	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
50	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
51	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
52	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
53	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
54	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
55	C₉×#41	—	6	8	no	—
56	C₂×Q₈×C₉	—	3	2	no	—
57	C₂×D₄×C₉	—	3	4	no	—
58	C₂×D₄×C₉	—	3	2	no	—
59	C₂×D₄×C₉	—	3	4	no	—
60	C₂²×C₄×C₉	—	2	1	no	—
61	C₂²×C₄×C₉	—	2	2	no	—
62	C₂²×C₄×C₉	—	3	4	no	—
63	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
64	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₉×C₄⋊C₄	—	2	4	no	—
66	C₉×C₄⋊C₄	—	3	4	no	—
67	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
68	C₉×C₂²⋊C₄	—	3	8	no	—
69	C₂²×C₃×C₉	—	2	1	no	—
70	C₂²×C₃×C₉	—	2	1	no	—
71	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
72	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
73	C₂×D₃×C₉	—	2	4	no	—
74	C₂×D₃×C₉	—	2	4	no	—
75	C₂×D₃×C₉	—	2	4	no	—
76	C₃×C₄×C₉	—	2	4	no	—
77	C₃×C₄×C₉	C₂×D₄	1	4	yes	—
78	C₉×Dic₃	—	2	4	no	—
79	C₂×C₃×#41	—	3	2	no	—
80	C₂×#235	C₂×C₉	1	2	yes	complemented
81	C₃×C₂²⋊Q₈	—	3	4	no	—
82	C₃×#121	—	3	4	no	—
83	C₃×#121	—	3	4	no	—
84	C₂×C₃×C₄⋊C₄	—	2	1	no	—
85	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
86	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
87	#930	C₂×C₉	1	4	yes	complemented
88	#897	C₂×C₉	1	4	yes	complemented
89	#895	C₂×C₉	1	4	yes	complemented
90	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
91	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
92	C₂×C₃²×C₄	—	4	2	no	—
93	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
94	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
95	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
96	C₃×#67	—	2	8	no	—
97	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
98	C₃×D₁₂	—	2	4	no	—
99	C₃×D₁₂	—	4	4	no	—
100	C₃×C₄×D₃	—	2	8	no	—
101	C₃×Dic₆	—	2	4	no	—
102	C₂³×C₉	—	2	1	no	—
103	C₂³×C₉	—	3	4	no	—
104	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
105	Q₈×C₉	—	6	4	no	—
106	D₄×C₉	—	6	8	no	—
107	D₄×C₉	—	6	4	no	—
108	D₄×C₉	—	12	4	no	—
109	C₂×C₄×C₉	—	2	4	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	4	no	—
112	C₂×C₄×C₉	—	3	4	no	—
113	C₂×C₄×C₉	—	4	2	no	—
114	C₂×C₄×C₉	—	4	4	no	—
115	C₂×C₄×C₉	—	6	8	no	—
116	C₂×C₄×C₉	D₁₂	1	4	yes	complemented
117	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
118	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
119	#537	—	3	1	no	Sylow
120	C₂×C₃×C₉	—	2	2	no	—
121	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
124	D₃×C₉	—	4	4	no	—
125	C₃×#41	—	6	8	no	—
126	C₂×C₃×Q₈	—	3	2	no	—
127	C₂×C₃×D₄	—	3	4	no	—
128	C₂×C₃×D₄	—	3	2	no	—
129	C₂×C₃×D₄	—	3	4	no	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	—	3	4	no	—
133	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
134	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂×#67	C₂²×C₉	1	4	yes	—
138	#235	—	2	8	no	—
139	C₂×D₁₂	C₂²×C₉	1	2	yes	—
140	C₂×D₁₂	C₂²×C₉	1	4	yes	—
141	C₂×C₄×D₃	C₂²×C₉	1	4	yes	—
142	C₂×Dic₆	C₂²×C₉	1	2	yes	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	3	4	no	—
145	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
146	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
147	C₃×C₂²⋊C₄	—	3	8	no	—
148	(C₂×D₃)⋊C₄	C₂²×C₉	1	8	yes	—
149	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
150	C₂²×C₃²	—	2	1	no	—
151	C₂²×C₃²	—	2	1	no	—
152	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
153	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
154	C₂×C₃×D₃	—	2	4	no	—
155	C₂×C₃×D₃	—	2	4	no	—
156	C₂×C₃×D₃	—	2	4	no	—
157	C₃²×C₄	—	2	4	no	—
158	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
159	C₃×Dic₃	—	2	4	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	2	1	no	—
165	C₂²×C₉	—	2	2	no	—
166	C₂²×C₉	—	6	4	no	—
167	C₂²×C₉	—	6	4	no	—
168	C₂²×C₉	—	6	4	no	—
169	C₂²×C₉	C₂×D₁₂	1	2	yes	—
170	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
171	C₄×C₉	—	2	4	no	—
172	C₄×C₉	—	2	4	no	—
173	C₄×C₉	—	4	4	no	—
174	C₄×C₉	—	6	4	no	—
175	C₄×C₉	C₂×D₁₂	1	4	yes	—
176	C₂×#41	—	3	2	no	—
177	C₂²⋊Q₈	—	3	4	no	—
178	#121	—	3	4	no	—
179	#121	—	3	4	no	—
180	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic, complemented
181	C₃×C₉	#537	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₃	—	2	1	no	—
183	C₂³×C₃	—	3	4	no	—
184	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
185	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
186	C₂²×D₃	C₂³×C₉	1	4	yes	—
187	C₃×Q₈	—	6	4	no	—
188	C₃×D₄	—	6	8	no	—
189	C₃×D₄	—	6	4	no	—
190	C₃×D₄	—	12	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	3	4	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	4	4	no	—
198	C₂×C₃×C₄	—	6	8	no	—
199	C₂×C₃×C₄	D₄×C₉	1	4	yes	complemented
200	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
201	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
202	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
205	#67	—	2	8	no	—
206	C₂×Dic₃	C₂³×C₉	1	4	yes	—
207	D₁₂	—	2	4	no	—
208	D₁₂	—	4	4	no	—
209	C₄×D₃	—	2	8	no	—
210	Dic₆	—	2	4	no	—
211	C₂×C₃²	—	2	2	no	—
212	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
214	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
215	C₃×D₃	—	4	4	no	—
216	C₂×C₉	—	2	4	no	—
217	C₂×C₉	—	2	2	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	—	12	4	no	—
222	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
223	C₂×C₉	#1004	1	1	yes	characteristic
224	C₂×C₉	#1007	1	1	yes	characteristic
225	#41	—	6	8	no	—
226	C₂×Q₈	—	3	2	no	—
227	C₂×D₄	—	3	4	no	—
228	C₂×D₄	—	3	2	no	—
229	C₂×D₄	—	3	4	no	—
230	C₂²×C₄	—	3	4	no	—
231	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
232	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
233	C₄⋊C₄	—	3	4	no	—
234	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
235	C₂²⋊C₄	—	3	8	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	6	4	no	—
245	C₂²×C₃	—	6	4	no	—
246	C₂²×C₃	—	6	4	no	—
247	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
248	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
249	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
250	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
251	C₂×D₃	—	2	4	no	—
252	C₂×D₃	—	2	4	no	—
253	C₂×D₃	—	2	4	no	—
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	—	6	4	no	—
259	C₃×C₄	C₂×D₄×C₉	1	4	yes	—
260	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
261	Dic₃	—	2	4	no	—
262	C₃²	C₃×#537	1	1	yes	characteristic
263	C₉	—	2	1	no	—
264	C₉	#5510	1	1	yes	characteristic, direct product factor
265	C₂³	—	3	4	no	—
266	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
267	Q₈	—	6	4	no	—
268	D₄	—	6	8	no	—
269	D₄	—	6	4	no	—
270	D₄	—	12	4	no	—
271	C₂×C₄	—	3	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	6	8	no	—
274	C₂×C₄	C₉×D₁₂	1	4	yes	complemented
275	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
276	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
277	C₂×C₃	—	2	4	no	—
278	C₂×C₃	—	2	2	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	12	4	no	—
284	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
285	C₂×C₃	C₉×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
287	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
288	C₂×C₃	C₃×#1004	1	1	yes	characteristic
289	C₂×C₃	C₃×#1007	1	1	yes	characteristic
290	D₃	—	4	4	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	6	4	no	—
294	C₂²	—	6	4	no	—
295	C₂²	—	6	4	no	—
296	C₂²	C₂×C₉×D₁₂	1	2	yes	—
297	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	6	4	no	—
300	C₄	C₂×C₉×D₁₂	1	4	yes	—
301	C₃	—	2	1	no	—
302	C₃	C₉×#537	1	1	yes	characteristic, complemented
303	C₃	C₃×#5510	1	1	yes	characteristic
304	C₂	—	2	2	no	—
305	C₂	—	12	4	no	—
306	C₂	C₂²×C₉×D₁₂	1	1	yes	characteristic
307	C₂	C₉×#1004	1	1	yes	characteristic
308	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	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5510	C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#537
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#235	C₂	semidirect
2. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
3. C₉×#930	C₂	semidirect
4. C₉×#897	C₂	semidirect
5. C₉×#895	C₂	semidirect
29. C₂×C₉×C₄⋊C₄	D₃	semidirect
33. C₂×C₃×C₄×C₉	D₄	semidirect
43. #5510	C₉	direct (N × H)
80. C₂×#235	C₂×C₉	semidirect
85. C₂×C₃×C₄⋊C₄	C₂×C₉	semidirect
87. #930	C₂×C₉	semidirect
88. #897	C₂×C₉	semidirect
89. #895	C₂×C₉	semidirect
116. C₂×C₄×C₉	D₁₂	semidirect
180. C₂×C₄⋊C₄	D₃×C₉	semidirect
181. C₃×C₉	#537	semidirect
199. C₂×C₃×C₄	D₄×C₉	semidirect
264. C₉	#5510	direct (N × H)
274. C₂×C₄	C₉×D₁₂	semidirect
302. C₃	C₉×#537	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#235	2
C₂×C₃×C₉×C₄⋊C₄	2
C₉×#930	2
C₉×#897	2
C₉×#895	2
C₃×#5510	3
C₉×#537	3

Subgroups — C9×#5510 #13152421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5510 #13152421