Subgroups — C₉×#2548 #11767269

Order 1152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#124	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#394	C₂	1	4	yes	maximal, complemented
5	C₉×#390	C₂	1	2	yes	maximal, complemented
6	C₉×#390	C₂	1	4	yes	maximal, complemented
7	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2548	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×D₄×C₉	C₂²	1	2	yes	—
12	C₂²×D₄×C₉	C₂²	1	2	yes	—
13	C₉×#127	—	2	4	no	—
14	C₉×#124	—	2	8	no	—
15	C₄×Q₈×C₉	—	2	4	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
21	C₂×C₄²×C₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
24	C₉×#95	C₂²	1	2	yes	—
25	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×#124	C₂×C₃	1	2	yes	—
27	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	C₃×#394	C₂×C₃	1	4	yes	—
29	C₃×#390	C₂×C₃	1	2	yes	—
30	C₃×#390	C₂×C₃	1	4	yes	—
31	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₂⁴×C₉	C₂³	1	4	yes	—
33	C₂×Q₈×C₉	—	2	2	no	—
34	C₂×Q₈×C₉	—	2	2	no	—
35	C₂×Q₈×C₉	D₄	1	4	yes	complemented
36	C₂×D₄×C₉	—	2	4	no	—
37	C₂×D₄×C₉	—	2	4	no	—
38	C₂×D₄×C₉	—	2	4	no	—
39	C₂×D₄×C₉	—	2	4	no	—
40	C₂×D₄×C₉	—	2	4	no	—
41	C₂×D₄×C₉	—	2	8	no	—
42	C₂×D₄×C₉	—	4	4	no	—
43	C₂²×C₄×C₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₉×C₄⋊C₄	—	2	4	no	—
52	C₉×C₂²⋊C₄	—	2	16	no	—
53	C₉×C₂²⋊C₄	—	4	8	no	—
54	C₄²×C₉	—	2	4	no	—
55	C₄²×C₉	—	2	4	no	—
56	C₄²×C₉	D₄	1	4	yes	—
57	#2548	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
60	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
61	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
62	C₃×#127	—	2	4	no	—
63	C₃×#124	—	2	8	no	—
64	C₃×C₄×Q₈	—	2	4	no	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
70	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
71	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
73	C₃×#95	C₂²×C₃	1	2	yes	—
74	C₂³×C₉	—	2	4	no	—
75	C₂³×C₉	—	2	4	no	—
76	C₂³×C₉	—	2	4	no	—
77	C₂³×C₉	—	2	8	no	—
78	C₂³×C₉	—	2	8	no	—
79	C₂³×C₉	C₂⁴	1	1	yes	characteristic
80	Q₈×C₉	—	2	8	no	—
81	D₄×C₉	—	4	8	no	—
82	D₄×C₉	—	4	8	no	—
83	D₄×C₉	—	4	8	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	—	2	4	no	—
92	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₉	#41	1	2	yes	—
98	C₂×C₄×C₉	#41	1	4	yes	—
99	C₂×#127	C₂×C₉	1	1	yes	characteristic
100	C₂×#124	C₂×C₉	1	2	yes	complemented
101	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
102	#394	C₂×C₉	1	4	yes	complemented
103	#390	C₂×C₉	1	4	yes	complemented
104	#390	C₂×C₉	1	2	yes	complemented
105	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
106	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
107	C₂×C₃×Q₈	—	2	2	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
110	C₂×C₃×D₄	—	2	4	no	—
111	C₂×C₃×D₄	—	2	4	no	—
112	C₂×C₃×D₄	—	2	4	no	—
113	C₂×C₃×D₄	—	2	4	no	—
114	C₂×C₃×D₄	—	2	4	no	—
115	C₂×C₃×D₄	—	2	8	no	—
116	C₂×C₃×D₄	—	4	4	no	—
117	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
118	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
119	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₃×C₄⋊C₄	—	2	4	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	4	no	—
126	C₃×C₂²⋊C₄	—	2	16	no	—
127	C₃×C₂²⋊C₄	—	4	8	no	—
128	C₃×C₄²	—	2	4	no	—
129	C₃×C₄²	—	2	4	no	—
130	C₃×C₄²	C₃×D₄	1	4	yes	—
131	C₂²×C₉	—	4	4	no	—
132	C₂²×C₉	—	4	4	no	—
133	C₂²×C₉	—	4	4	no	—
134	C₂²×C₉	—	4	8	no	—
135	C₂²×C₉	—	4	8	no	—
136	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₉	C₂×#41	1	1	yes	characteristic
139	C₂²×C₉	C₂×#41	1	2	yes	—
140	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
141	C₄×C₉	—	2	4	no	—
142	C₄×C₉	—	2	2	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	—	4	4	no	—
145	C₄×C₉	#121	1	4	yes	—
146	C₄×C₉	#124	1	4	yes	complemented
147	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
148	C₂²×D₄	C₂²×C₉	1	2	yes	—
149	C₂²×D₄	C₂²×C₉	1	2	yes	—
150	C₂²×D₄	C₂²×C₉	1	2	yes	—
151	#127	—	2	4	no	—
152	#124	—	2	8	no	—
153	C₄×Q₈	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
159	C₂×C₄²	C₂²×C₉	1	2	yes	—
160	C₂×C₄²	C₂²×C₉	1	2	yes	—
161	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
162	#95	C₂²×C₉	1	2	yes	—
163	C₂³×C₃	—	2	4	no	—
164	C₂³×C₃	—	2	4	no	—
165	C₂³×C₃	—	2	4	no	—
166	C₂³×C₃	—	2	8	no	—
167	C₂³×C₃	—	2	8	no	—
168	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
169	C₃×Q₈	—	2	8	no	—
170	C₃×D₄	—	4	8	no	—
171	C₃×D₄	—	4	8	no	—
172	C₃×D₄	—	4	8	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
185	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₃×#41	1	2	yes	—
187	C₂×C₃×C₄	C₃×#41	1	4	yes	—
188	C₂×C₉	—	8	4	no	—
189	C₂×C₉	C₂×#121	1	1	yes	characteristic
190	C₂×C₉	C₂×#124	1	1	yes	characteristic
191	C₂×C₉	#535	1	1	yes	characteristic
192	C₂×C₉	#550	1	2	yes	—
193	C₂×C₉	#559	1	2	yes	—
194	C₂⁴	C₂³×C₉	1	4	yes	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	D₄×C₉	1	4	yes	complemented
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	8	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂²×C₄	C₂³×C₉	1	4	yes	—
206	C₂²×C₄	C₂³×C₉	1	2	yes	—
207	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
208	C₂²×C₄	C₂³×C₉	1	2	yes	—
209	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	2	16	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	D₄×C₉	1	4	yes	—
219	C₂²×C₃	—	4	8	no	—
220	C₂²×C₃	—	4	4	no	—
221	C₂²×C₃	—	4	4	no	—
222	C₂²×C₃	—	4	4	no	—
223	C₂²×C₃	—	4	8	no	—
224	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
225	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
227	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
228	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
229	C₃×C₄	—	2	4	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	4	2	no	—
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	C₃×#121	1	4	yes	—
234	C₃×C₄	C₃×#124	1	4	yes	—
235	C₉	#2548	1	1	yes	characteristic, direct product factor, Sylow
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	2	8	no	—
241	C₂³	C₂⁴×C₉	1	1	yes	characteristic
242	Q₈	—	2	8	no	—
243	D₄	—	4	8	no	—
244	D₄	—	4	8	no	—
245	D₄	—	4	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
256	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
258	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
259	C₂×C₄	C₉×#41	1	2	yes	—
260	C₂×C₄	C₉×#41	1	4	yes	—
261	C₂×C₃	—	8	4	no	—
262	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
263	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
264	C₂×C₃	C₃×#535	1	1	yes	characteristic
265	C₂×C₃	C₃×#550	1	2	yes	—
266	C₂×C₃	C₃×#559	1	2	yes	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	8	no	—
271	C₂²	—	4	8	no	—
272	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
273	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
274	C₂²	C₂×C₉×#41	1	1	yes	characteristic
275	C₂²	C₂×C₉×#41	1	2	yes	—
276	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
277	C₄	—	2	4	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	4	no	—
281	C₄	C₉×#121	1	4	yes	—
282	C₄	C₉×#124	1	4	yes	complemented
283	C₃	C₃×#2548	1	1	yes	characteristic
284	C₂	—	8	4	no	—
285	C₂	C₂×C₉×#121	1	1	yes	characteristic
286	C₂	C₂×C₉×#124	1	1	yes	characteristic
287	C₂	C₉×#535	1	1	yes	characteristic
288	C₂	C₉×#550	1	2	yes	—
289	C₂	C₉×#559	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#127	C₂×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#124	C₂	semidirect
3. C₂×C₄×Q₈×C₉	C₂	semidirect
4. C₉×#394	C₂	semidirect
5. C₉×#390	C₂	semidirect
6. C₉×#390	C₂	semidirect
7. C₄×C₉×C₄⋊C₄	C₂	semidirect
35. C₂×Q₈×C₉	D₄	semidirect
57. #2548	C₉	direct (N × H)
100. C₂×#124	C₂×C₉	semidirect
101. C₂×C₄×Q₈	C₂×C₉	semidirect
102. #394	C₂×C₉	semidirect
103. #390	C₂×C₉	semidirect
104. #390	C₂×C₉	semidirect
105. C₄×C₄⋊C₄	C₂×C₉	semidirect
146. C₄×C₉	#124	semidirect
197. C₂×Q₈	D₄×C₉	semidirect
235. C₉	#2548	direct (N × H)
282. C₄	C₉×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#127	2
C₂×C₉×#124	2
C₂×C₄×Q₈×C₉	2
C₉×#394	2
C₉×#390	2
C₄×C₉×C₄⋊C₄	2
C₃×#2548	3

Subgroups — C9×#2548 #11767269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2548 #11767269