Subgroups — C₂×C₉×#532 #11768163

Order 1152.

2592 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×#41	C₂	1	2	yes	maximal, complemented
2	C₉×#532	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉×#124	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×#121	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×D₄×C₉	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#532	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	—	2	16	no	—
11	C₂×C₉×#41	C₂²	1	16	yes	complemented
12	C₂²×Q₈×C₉	C₂²	1	2	yes	—
13	C₂²×D₄×C₉	C₂²	1	4	yes	—
14	C₂²×D₄×C₉	C₂²	1	4	yes	—
15	C₂²×D₄×C₉	C₂²	1	2	yes	—
16	C₂³×C₄×C₉	C₂²	1	2	yes	—
17	C₂³×C₄×C₉	C₂²	1	4	yes	—
18	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
19	C₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₉×#127	C₂²	1	8	yes	complemented
21	C₉×#124	C₂²	1	16	yes	complemented
22	C₉×#121	C₂²	1	32	yes	complemented
23	C₄×D₄×C₉	C₂²	1	32	yes	complemented
24	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₉	C₂²	1	8	yes	complemented
30	C₂²×C₃×#41	C₂×C₃	1	2	yes	—
31	C₃×#532	C₂×C₃	1	16	yes	—
32	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
34	C₂×C₃×#124	C₂×C₃	1	2	yes	—
35	C₂×C₃×#121	C₂×C₃	1	4	yes	—
36	C₂×C₃×C₄×D₄	C₂×C₃	1	4	yes	—
37	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
38	C₂⁴×C₉	C₂³	1	4	yes	—
39	C₂⁴×C₉	C₂³	1	1	yes	characteristic
40	C₉×#41	—	2	64	no	—
41	C₂×Q₈×C₉	—	2	8	no	—
42	C₂×Q₈×C₉	C₂³	1	8	yes	—
43	C₂×D₄×C₉	—	2	16	no	—
44	C₂×D₄×C₉	—	2	8	no	—
45	C₂×D₄×C₉	—	2	16	no	—
46	C₂×D₄×C₉	C₂³	1	16	yes	—
47	C₂×D₄×C₉	C₂³	1	16	yes	—
48	C₂×D₄×C₉	C₂³	1	8	yes	—
49	C₂²×C₄×C₉	—	2	16	no	—
50	C₂²×C₄×C₉	—	2	8	no	—
51	C₂²×C₄×C₉	D₄	1	8	yes	complemented
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	8	yes	—
54	C₂²×C₄×C₉	C₂³	1	4	yes	—
55	C₂²×C₄×C₉	C₂³	1	16	yes	—
56	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₉	C₂³	1	4	yes	—
58	C₂²×C₄×C₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
60	C₉×C₄⋊C₄	C₂³	1	16	yes	—
61	C₉×C₂²⋊C₄	C₂³	1	32	yes	—
62	C₄²×C₉	C₂³	1	8	yes	—
63	C₄²×C₉	C₂³	1	4	yes	—
64	C₄²×C₉	C₂³	1	4	yes	—
65	C₂×#532	C₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃×#41	—	2	16	no	—
67	C₂×C₃×#41	C₂²×C₃	1	16	yes	—
68	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
69	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
70	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
71	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
72	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
73	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
74	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
76	C₃×#127	C₂²×C₃	1	8	yes	—
77	C₃×#124	C₂²×C₃	1	16	yes	—
78	C₃×#121	C₂²×C₃	1	32	yes	—
79	C₃×C₄×D₄	C₂²×C₃	1	32	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
82	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
83	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₃×C₄²	C₂²×C₃	1	8	yes	—
86	C₂³×C₉	—	2	4	no	—
87	C₂³×C₉	—	2	8	no	—
88	C₂³×C₉	—	2	16	no	—
89	C₂³×C₉	C₂×D₄	1	4	yes	—
90	C₂³×C₉	C₂⁴	1	8	yes	—
91	C₂³×C₉	C₂⁴	1	2	yes	—
92	C₂³×C₉	C₂⁴	1	1	yes	characteristic
93	Q₈×C₉	—	2	16	no	—
94	D₄×C₉	—	2	32	no	—
95	D₄×C₉	—	2	16	no	—
96	D₄×C₉	—	2	32	no	—
97	C₂×C₄×C₉	—	2	8	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	32	no	—
100	C₂×C₄×C₉	—	2	16	no	—
101	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
102	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₉	C₂×D₄	1	16	yes	complemented
104	C₂×C₄×C₉	#41	1	8	yes	complemented
105	C₂×C₄×C₉	C₂⁴	1	4	yes	—
106	C₂×C₄×C₉	C₂⁴	1	8	yes	—
107	C₂×C₄×C₉	C₂⁴	1	2	yes	—
108	C₂×C₄×C₉	C₂⁴	1	2	yes	—
109	C₂×C₄×C₉	C₂⁴	1	4	yes	—
110	C₂²×#41	C₂×C₉	1	2	yes	complemented
111	#532	C₂×C₉	1	16	yes	direct product factor
112	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
113	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
114	C₂×#124	C₂×C₉	1	2	yes	complemented
115	C₂×#121	C₂×C₉	1	4	yes	complemented
116	C₂×C₄×D₄	C₂×C₉	1	4	yes	complemented
117	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic, complemented
118	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
119	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
120	C₃×#41	—	2	64	no	—
121	C₂×C₃×Q₈	—	2	8	no	—
122	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
123	C₂×C₃×D₄	—	2	16	no	—
124	C₂×C₃×D₄	—	2	8	no	—
125	C₂×C₃×D₄	—	2	16	no	—
126	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
127	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
128	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
129	C₂²×C₃×C₄	—	2	16	no	—
130	C₂²×C₃×C₄	—	2	8	no	—
131	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
132	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
134	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
135	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
136	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
138	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
140	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
141	C₃×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
142	C₃×C₄²	C₂³×C₃	1	8	yes	—
143	C₃×C₄²	C₂³×C₃	1	4	yes	—
144	C₃×C₄²	C₂³×C₃	1	4	yes	—
145	C₂²×C₉	—	2	8	no	—
146	C₂²×C₉	—	2	4	no	—
147	C₂²×C₉	—	2	8	no	—
148	C₂²×C₉	—	2	16	no	—
149	C₂²×C₉	—	4	16	no	—
150	C₂²×C₉	C₂²×D₄	1	4	yes	—
151	C₂²×C₉	C₂²×D₄	1	2	yes	—
152	C₂²×C₉	C₂×#41	1	4	yes	—
153	C₂²×C₉	C₂⁵	1	1	yes	characteristic
154	C₄×C₉	—	2	8	no	—
155	C₄×C₉	—	2	4	no	—
156	C₄×C₉	C₂²×D₄	1	4	yes	—
157	C₄×C₉	C₂²×D₄	1	4	yes	—
158	C₄×C₉	C₂×#41	1	8	yes	complemented
159	C₂×#41	—	2	16	no	—
160	C₂×#41	C₂²×C₉	1	16	yes	complemented
161	C₂²×Q₈	C₂²×C₉	1	2	yes	—
162	C₂²×D₄	C₂²×C₉	1	4	yes	—
163	C₂²×D₄	C₂²×C₉	1	4	yes	—
164	C₂²×D₄	C₂²×C₉	1	2	yes	—
165	C₂³×C₄	C₂²×C₉	1	4	yes	—
166	C₂³×C₄	C₂²×C₉	1	2	yes	—
167	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
168	C₄⋊Q₈	C₂²×C₉	1	8	yes	complemented
169	#127	C₂²×C₉	1	8	yes	complemented
170	#124	C₂²×C₉	1	16	yes	complemented
171	#121	C₂²×C₉	1	32	yes	complemented
172	C₄×D₄	C₂²×C₉	1	32	yes	complemented
173	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
175	C₂×C₄²	C₂²×C₉	1	2	yes	—
176	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₉	1	8	yes	complemented
179	C₂³×C₃	—	2	4	no	—
180	C₂³×C₃	—	2	8	no	—
181	C₂³×C₃	—	2	16	no	—
182	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
183	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
184	C₂³×C₃	C₂⁴×C₃	1	2	yes	—
185	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
186	C₃×Q₈	—	2	16	no	—
187	C₃×D₄	—	2	32	no	—
188	C₃×D₄	—	2	16	no	—
189	C₃×D₄	—	2	32	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	32	no	—
193	C₂×C₃×C₄	—	2	16	no	—
194	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
196	C₂×C₃×C₄	C₂×C₃×D₄	1	16	yes	—
197	C₂×C₃×C₄	C₃×#41	1	8	yes	—
198	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
199	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
200	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
201	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
202	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
203	C₂×C₉	—	2	4	no	—
204	C₂×C₉	—	4	8	no	—
205	C₂×C₉	#532	1	4	yes	direct product factor
206	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
207	C₂×C₉	C₂²×#41	1	2	yes	—
208	C₂⁴	C₂³×C₉	1	4	yes	—
209	C₂⁴	C₂³×C₉	1	1	yes	characteristic
210	#41	—	2	64	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	C₂³×C₉	1	8	yes	—
213	C₂×D₄	—	2	16	no	—
214	C₂×D₄	—	2	8	no	—
215	C₂×D₄	—	2	16	no	—
216	C₂×D₄	C₂³×C₉	1	16	yes	—
217	C₂×D₄	C₂³×C₉	1	16	yes	—
218	C₂×D₄	C₂³×C₉	1	8	yes	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	D₄×C₉	1	8	yes	complemented
222	C₂²×C₄	C₂³×C₉	1	2	yes	—
223	C₂²×C₄	C₂³×C₉	1	8	yes	—
224	C₂²×C₄	C₂³×C₉	1	4	yes	—
225	C₂²×C₄	C₂³×C₉	1	16	yes	—
226	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
228	C₂²×C₄	C₂³×C₉	1	4	yes	—
229	C₂²×C₄	C₂³×C₉	1	2	yes	—
230	C₄⋊C₄	C₂³×C₉	1	16	yes	—
231	C₂²⋊C₄	C₂³×C₉	1	32	yes	—
232	C₄²	C₂³×C₉	1	8	yes	—
233	C₄²	C₂³×C₉	1	4	yes	—
234	C₄²	C₂³×C₉	1	4	yes	—
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	2	8	no	—
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	16	no	—
239	C₂²×C₃	—	4	16	no	—
240	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
241	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
242	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
243	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
244	C₃×C₄	—	2	8	no	—
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
247	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
248	C₃×C₄	C₂×C₃×#41	1	8	yes	—
249	C₉	C₂×#532	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	4	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	16	no	—
253	C₂³	C₂×D₄×C₉	1	4	yes	—
254	C₂³	C₂⁴×C₉	1	8	yes	—
255	C₂³	C₂⁴×C₉	1	2	yes	—
256	C₂³	C₂⁴×C₉	1	1	yes	characteristic
257	Q₈	—	2	16	no	—
258	D₄	—	2	32	no	—
259	D₄	—	2	16	no	—
260	D₄	—	2	32	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	32	no	—
264	C₂×C₄	—	2	16	no	—
265	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
266	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
267	C₂×C₄	C₂×D₄×C₉	1	16	yes	complemented
268	C₂×C₄	C₉×#41	1	8	yes	complemented
269	C₂×C₄	C₂⁴×C₉	1	4	yes	—
270	C₂×C₄	C₂⁴×C₉	1	8	yes	—
271	C₂×C₄	C₂⁴×C₉	1	2	yes	—
272	C₂×C₄	C₂⁴×C₉	1	2	yes	—
273	C₂×C₄	C₂⁴×C₉	1	4	yes	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	4	8	no	—
276	C₂×C₃	C₃×#532	1	4	yes	—
277	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
278	C₂×C₃	C₂²×C₃×#41	1	2	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	16	no	—
283	C₂²	—	4	16	no	—
284	C₂²	C₂²×D₄×C₉	1	4	yes	—
285	C₂²	C₂²×D₄×C₉	1	2	yes	—
286	C₂²	C₂×C₉×#41	1	4	yes	—
287	C₂²	C₂⁵×C₉	1	1	yes	characteristic
288	C₄	—	2	8	no	—
289	C₄	—	2	4	no	—
290	C₄	C₂²×D₄×C₉	1	4	yes	—
291	C₄	C₂²×D₄×C₉	1	4	yes	—
292	C₄	C₂×C₉×#41	1	8	yes	complemented
293	C₃	C₂×C₃×#532	1	1	yes	characteristic
294	C₂	—	2	4	no	—
295	C₂	—	4	8	no	—
296	C₂	C₉×#532	1	4	yes	direct product factor
297	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
298	C₂	C₂²×C₉×#41	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#532	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×#41	C₂	semidirect
2. C₉×#532	C₂	direct (N × H)
3. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₂×C₉×#127	C₂	semidirect
5. C₂×C₉×#124	C₂	semidirect
6. C₂×C₉×#121	C₂	semidirect
7. C₂×C₄×D₄×C₉	C₂	semidirect
8. C₂²×C₄²×C₉	C₂	semidirect
11. C₂×C₉×#41	C₂²	semidirect
19. C₉×C₄⋊Q₈	C₂²	semidirect
20. C₉×#127	C₂²	semidirect
21. C₉×#124	C₂²	semidirect
22. C₉×#121	C₂²	semidirect
23. C₄×D₄×C₉	C₂²	semidirect
29. C₂×C₄²×C₉	C₂²	semidirect
51. C₂²×C₄×C₉	D₄	semidirect
65. C₂×#532	C₉	direct (N × H)
103. C₂×C₄×C₉	C₂×D₄	semidirect
104. C₂×C₄×C₉	#41	semidirect
110. C₂²×#41	C₂×C₉	semidirect
111. #532	C₂×C₉	direct (N × H)
112. C₂×C₄⋊Q₈	C₂×C₉	semidirect
113. C₂×#127	C₂×C₉	semidirect
114. C₂×#124	C₂×C₉	semidirect
115. C₂×#121	C₂×C₉	semidirect
116. C₂×C₄×D₄	C₂×C₉	semidirect
117. C₂²×C₄²	C₂×C₉	semidirect
158. C₄×C₉	C₂×#41	semidirect
160. C₂×#41	C₂²×C₉	semidirect
168. C₄⋊Q₈	C₂²×C₉	semidirect
169. #127	C₂²×C₉	semidirect
170. #124	C₂²×C₉	semidirect
171. #121	C₂²×C₉	semidirect
172. C₄×D₄	C₂²×C₉	semidirect
178. C₂×C₄²	C₂²×C₉	semidirect
205. C₂×C₉	#532	direct (N × H)
221. C₂²×C₄	D₄×C₉	semidirect
249. C₉	C₂×#532	direct (N × H)
267. C₂×C₄	C₂×D₄×C₉	semidirect
268. C₂×C₄	C₉×#41	semidirect
292. C₄	C₂×C₉×#41	semidirect
296. C₂	C₉×#532	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×#41	2
C₉×#532	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#127	2
C₂×C₉×#124	2
C₂×C₉×#121	2
C₂×C₄×D₄×C₉	2
C₂²×C₄²×C₉	2
C₂×C₃×#532	3

Subgroups — C2×C9×#532 #11768163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#532 #11768163