Subgroups — C₂×C₉×#444 #11767675

Order 1152.

696 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#131	C₂	1	2	yes	maximal, complemented
3	C₉×#444	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉×#104	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#444	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#41	C₂²	1	2	yes	—
9	C₉×#135	C₄	1	4	yes	complemented
10	C₉×#135	C₂²	1	4	yes	—
11	C₂×C₉×Q₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₉×SD₁₆	C₄	1	2	yes	complemented
13	C₂×C₉×D₈	C₄	1	1	yes	characteristic, complemented
14	C₉×#131	—	2	4	no	—
15	C₉×#131	C₂²	1	8	yes	complemented
16	C₂×C₉×M₁₆	—	2	2	no	—
17	C₂×C₉×M₁₆	C₂²	1	2	yes	—
18	C₂²×C₈×C₉	—	2	2	no	—
19	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₉×#108	C₂²	1	4	yes	complemented
23	C₉×#104	C₂²	1	8	yes	complemented
24	C₉×#97	C₂²	1	4	yes	complemented
25	C₂×C₃×#135	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×#131	C₂×C₃	1	2	yes	—
27	C₃×#444	C₂×C₃	1	8	yes	—
28	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×#104	C₂×C₃	1	2	yes	—
30	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
31	C₉×#41	—	2	2	no	—
32	C₉×#41	—	2	2	no	—
33	C₉×#41	C₂×C₄	1	4	yes	—
34	C₉×#41	C₂³	1	4	yes	—
35	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₉	—	2	2	no	—
37	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₉×Q₁₆	C₂×C₄	1	4	yes	complemented
42	C₉×SD₁₆	C₂×C₄	1	8	yes	complemented
43	C₉×D₈	C₂×C₄	1	4	yes	complemented
44	C₉×M₁₆	—	2	2	no	—
45	C₉×M₁₆	—	2	8	no	—
46	C₉×M₁₆	C₂³	1	4	yes	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	—	2	8	no	—
49	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
52	C₂×C₈×C₉	D₄	1	2	yes	—
53	C₂×C₈×C₉	D₄	1	2	yes	—
54	C₂×C₈×C₉	C₂³	1	2	yes	—
55	C₂×C₈×C₉	C₂³	1	2	yes	—
56	C₄²×C₉	—	2	1	no	—
57	C₄²×C₉	C₂³	1	2	yes	—
58	C₂×#444	C₉	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
60	C₃×#135	C₃×C₄	1	4	yes	—
61	C₃×#135	C₂²×C₃	1	4	yes	—
62	C₂×C₃×Q₁₆	C₃×C₄	1	1	yes	characteristic
63	C₂×C₃×SD₁₆	C₃×C₄	1	2	yes	—
64	C₂×C₃×D₈	C₃×C₄	1	1	yes	characteristic
65	C₃×#131	—	2	4	no	—
66	C₃×#131	C₂²×C₃	1	8	yes	—
67	C₂×C₃×M₁₆	—	2	2	no	—
68	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
69	C₂²×C₃×C₈	—	2	2	no	—
70	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
73	C₃×#108	C₂²×C₃	1	4	yes	—
74	C₃×#104	C₂²×C₃	1	8	yes	—
75	C₃×#97	C₂²×C₃	1	4	yes	—
76	C₂³×C₉	—	2	2	no	—
77	C₂³×C₉	#41	1	1	yes	characteristic
78	Q₈×C₉	—	2	2	no	—
79	Q₈×C₉	C₂²×C₄	1	4	yes	—
80	D₄×C₉	—	2	4	no	—
81	D₄×C₉	—	2	4	no	—
82	D₄×C₉	—	2	2	no	—
83	D₄×C₉	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	4	2	no	—
89	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	#41	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
95	C₈×C₉	—	2	4	no	—
96	C₈×C₉	C₂²×C₄	1	2	yes	—
97	C₈×C₉	C₂²×C₄	1	2	yes	—
98	C₈×C₉	C₂×D₄	1	4	yes	—
99	C₂×#135	C₂×C₉	1	1	yes	characteristic
100	C₂×#131	C₂×C₉	1	2	yes	complemented
101	#444	C₂×C₉	1	8	yes	direct product factor
102	C₂×#108	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂×#104	C₂×C₉	1	2	yes	complemented
104	C₂×#97	C₂×C₉	1	1	yes	characteristic, complemented
105	C₃×#41	—	2	2	no	—
106	C₃×#41	—	2	2	no	—
107	C₃×#41	C₂×C₃×C₄	1	4	yes	—
108	C₃×#41	C₂³×C₃	1	4	yes	—
109	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₃×D₄	—	2	2	no	—
111	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₃×Q₁₆	C₂×C₃×C₄	1	4	yes	—
116	C₃×SD₁₆	C₂×C₃×C₄	1	8	yes	—
117	C₃×D₈	C₂×C₃×C₄	1	4	yes	—
118	C₃×M₁₆	—	2	2	no	—
119	C₃×M₁₆	—	2	8	no	—
120	C₃×M₁₆	C₂³×C₃	1	4	yes	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	8	no	—
123	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
126	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
127	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
128	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
129	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	C₂³×C₃	1	2	yes	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	4	4	no	—
135	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
136	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
137	C₂²×C₉	C₂×#41	1	1	yes	characteristic
138	C₄×C₉	—	2	4	no	—
139	C₄×C₉	—	4	2	no	—
140	C₄×C₉	C₄×D₄	1	1	yes	characteristic
141	C₄×C₉	C₄×D₄	1	1	yes	characteristic
142	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×#41	C₂²×C₉	1	2	yes	—
145	#135	C₄×C₉	1	4	yes	complemented
146	#135	C₂²×C₉	1	4	yes	—
147	C₂×Q₁₆	C₄×C₉	1	1	yes	characteristic, complemented
148	C₂×SD₁₆	C₄×C₉	1	2	yes	complemented
149	C₂×D₈	C₄×C₉	1	1	yes	characteristic, complemented
150	#131	—	2	4	no	—
151	#131	C₂²×C₉	1	8	yes	complemented
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	C₂²×C₉	1	2	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
156	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
157	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
158	#108	C₂²×C₉	1	4	yes	complemented
159	#104	C₂²×C₉	1	8	yes	complemented
160	#97	C₂²×C₉	1	4	yes	complemented
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	C₃×#41	1	1	yes	characteristic
163	C₃×Q₈	—	2	2	no	—
164	C₃×Q₈	C₂²×C₃×C₄	1	4	yes	—
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	C₂²×C₃×C₄	1	4	yes	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
180	C₃×C₈	—	2	4	no	—
181	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
182	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
183	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	4	4	no	—
187	C₂×C₉	#444	1	2	yes	direct product factor
188	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	C₂×C₄×C₉	1	4	yes	—
192	#41	C₂³×C₉	1	4	yes	—
193	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
199	Q₁₆	C₂×C₄×C₉	1	4	yes	complemented
200	SD₁₆	C₂×C₄×C₉	1	8	yes	complemented
201	D₈	C₂×C₄×C₉	1	4	yes	complemented
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	8	no	—
204	M₁₆	C₂³×C₉	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
210	C₂×C₈	D₄×C₉	1	2	yes	—
211	C₂×C₈	D₄×C₉	1	2	yes	—
212	C₂×C₈	C₂³×C₉	1	2	yes	—
213	C₂×C₈	C₂³×C₉	1	2	yes	—
214	C₄²	—	2	1	no	—
215	C₄²	C₂³×C₉	1	2	yes	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	4	4	no	—
219	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
225	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
226	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
227	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₉	C₂×#444	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	2	no	—
230	C₂³	C₉×#41	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	Q₈	C₂²×C₄×C₉	1	4	yes	—
233	D₄	—	2	4	no	—
234	D₄	—	2	4	no	—
235	D₄	—	2	2	no	—
236	D₄	C₂²×C₄×C₉	1	4	yes	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
246	C₂×C₄	C₉×#41	1	1	yes	characteristic
247	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	C₂²×C₄×C₉	1	2	yes	—
250	C₈	C₂²×C₄×C₉	1	2	yes	—
251	C₈	C₂×D₄×C₉	1	4	yes	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	4	4	no	—
255	C₂×C₃	C₃×#444	1	2	yes	—
256	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
261	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
262	C₂²	C₂×C₉×#41	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	4	2	no	—
265	C₄	C₄×D₄×C₉	1	1	yes	characteristic
266	C₄	C₄×D₄×C₉	1	1	yes	characteristic
267	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
268	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
269	C₃	C₂×C₃×#444	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	4	no	—
273	C₂	C₉×#444	1	2	yes	direct product factor
274	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₃×C₄×D₄
prime radical	C₂×C₃×#135	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#131	C₂	semidirect
3. C₉×#444	C₂	direct (N × H)
4. C₂×C₉×#108	C₂	semidirect
5. C₂×C₉×#104	C₂	semidirect
6. C₂×C₉×#97	C₂	semidirect
9. C₉×#135	C₄	semidirect
11. C₂×C₉×Q₁₆	C₄	semidirect
12. C₂×C₉×SD₁₆	C₄	semidirect
13. C₂×C₉×D₈	C₄	semidirect
15. C₉×#131	C₂²	semidirect
22. C₉×#108	C₂²	semidirect
23. C₉×#104	C₂²	semidirect
24. C₉×#97	C₂²	semidirect
41. C₉×Q₁₆	C₂×C₄	semidirect
42. C₉×SD₁₆	C₂×C₄	semidirect
43. C₉×D₈	C₂×C₄	semidirect
58. C₂×#444	C₉	direct (N × H)
100. C₂×#131	C₂×C₉	semidirect
101. #444	C₂×C₉	direct (N × H)
102. C₂×#108	C₂×C₉	semidirect
103. C₂×#104	C₂×C₉	semidirect
104. C₂×#97	C₂×C₉	semidirect
145. #135	C₄×C₉	semidirect
147. C₂×Q₁₆	C₄×C₉	semidirect
148. C₂×SD₁₆	C₄×C₉	semidirect
149. C₂×D₈	C₄×C₉	semidirect
151. #131	C₂²×C₉	semidirect
158. #108	C₂²×C₉	semidirect
159. #104	C₂²×C₉	semidirect
160. #97	C₂²×C₉	semidirect
187. C₂×C₉	#444	direct (N × H)
199. Q₁₆	C₂×C₄×C₉	semidirect
200. SD₁₆	C₂×C₄×C₉	semidirect
201. D₈	C₂×C₄×C₉	semidirect
228. C₉	C₂×#444	direct (N × H)
273. C₂	C₉×#444	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#135	2
C₂×C₉×#131	2
C₉×#444	2
C₂×C₉×#108	2
C₂×C₉×#104	2
C₂×C₉×#97	2
C₂×C₃×#444	3

Subgroups — C2×C9×#444 #11767675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#444 #11767675