Subgroups — C₂²×C₉×#215 #13153640

Order 1728.

1050 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#215	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃×#215	C₃	1	1	yes	maximal, characteristic
6	C₂²×C₉×SD₁₆	—	3	1	no	maximal
7	C₂³×D₃×C₉	—	2	1	no	—
8	C₂×C₃×Q₈×C₉	—	2	3	no	—
9	C₂×C₃×Q₈×C₉	C₂²	1	6	yes	complemented
10	C₂²×C₃×C₄×C₉	—	2	1	no	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×D₁₂	—	2	3	no	—
13	C₂×C₉×D₁₂	C₂²	1	6	yes	—
14	C₉×#215	C₂²	1	16	yes	direct product factor
15	C₂×C₉×C₃⋊C₈	C₂²	1	6	yes	complemented
16	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×#215	C₂×C₃	1	12	yes	—
19	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂²×Q₈×C₉	—	2	1	no	—
21	C₂²×Q₈×C₉	D₃	1	1	yes	characteristic, complemented
22	C₂²×D₄×C₉	—	3	1	no	—
23	C₂×C₉×SD₁₆	—	3	12	no	—
24	C₂²×C₈×C₉	—	3	1	no	—
25	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
26	C₂²×D₃×C₉	—	2	6	no	—
27	C₂²×D₃×C₉	—	4	4	no	—
28	C₃×Q₈×C₉	—	2	6	no	—
29	C₃×Q₈×C₉	C₂³	1	4	yes	complemented
30	C₂×C₃×C₄×C₉	—	2	6	no	—
31	C₂×C₃×C₄×C₉	D₄	1	3	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
33	C₉×D₁₂	—	2	6	no	—
34	C₉×D₁₂	C₂³	1	4	yes	—
35	C₉×C₃⋊C₈	C₂³	1	4	yes	complemented
36	C₂²×C₃×SD₁₆	—	3	1	no	—
37	C₂²×#215	C₉	1	1	yes	characteristic, direct product factor
38	C₂³×C₃×D₃	—	2	1	no	—
39	C₂×C₃²×Q₈	—	2	3	no	—
40	C₂×C₃²×Q₈	C₂²×C₃	1	6	yes	—
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×D₁₂	—	2	3	no	—
44	C₂×C₃×D₁₂	C₂²×C₃	1	6	yes	—
45	C₂⁴×C₉	—	6	1	no	—
46	C₃×#215	C₂²×C₃	1	16	yes	—
47	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	6	yes	—
48	C₂×Q₈×C₉	—	2	6	no	—
49	C₂×Q₈×C₉	—	2	6	no	—
50	C₂×Q₈×C₉	—	2	3	no	—
51	C₂×Q₈×C₉	C₂×D₃	1	6	yes	complemented
52	C₂×D₄×C₉	—	3	6	no	—
53	C₂×D₄×C₉	—	6	3	no	—
54	C₂²×C₄×C₉	—	2	2	no	—
55	C₂²×C₄×C₉	—	2	1	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×SD₁₆	—	3	16	no	—
59	C₂×C₈×C₉	—	3	6	no	—
60	C₂²×C₃×C₉	SD₁₆	1	4	yes	complemented
61	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
62	C₂×D₃×C₉	—	2	4	no	—
63	C₂×D₃×C₉	—	4	12	no	—
64	C₃×C₄×C₉	—	2	4	no	—
65	C₃×C₄×C₉	C₂×D₄	1	3	yes	—
66	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
67	C₂²×C₃×Q₈	—	2	1	no	—
68	C₂²×C₃×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
69	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
70	C₂²×C₃×D₄	—	3	1	no	—
71	C₂²×D₁₂	C₂×C₉	1	1	yes	characteristic
72	C₂×C₃×SD₁₆	—	3	12	no	—
73	C₂²×C₃×C₈	—	3	1	no	—
74	C₂×#215	C₂×C₉	1	12	yes	direct product factor
75	C₂²×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
76	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
77	C₂²×C₃×D₃	—	2	6	no	—
78	C₂²×C₃×D₃	—	4	4	no	—
79	C₃²×Q₈	—	2	6	no	—
80	C₃²×Q₈	C₂³×C₃	1	4	yes	—
81	C₂×C₃²×C₄	—	2	6	no	—
82	C₂×C₃²×C₄	C₃×D₄	1	3	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
84	C₃×D₁₂	—	2	6	no	—
85	C₃×D₁₂	C₂³×C₃	1	4	yes	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	—	6	6	no	—
88	C₂³×C₉	—	12	4	no	—
89	C₂³×C₉	#67	1	1	yes	characteristic
90	C₃×C₃⋊C₈	C₂³×C₃	1	4	yes	—
91	Q₈×C₉	—	2	12	no	—
92	Q₈×C₉	—	2	6	no	—
93	Q₈×C₉	—	2	4	no	—
94	Q₈×C₉	C₂²×D₃	1	4	yes	complemented
95	D₄×C₉	—	3	4	no	—
96	D₄×C₉	—	6	6	no	—
97	C₂×C₄×C₉	—	2	12	no	—
98	C₂×C₄×C₉	—	2	6	no	—
99	C₂×C₄×C₉	—	2	3	no	—
100	C₂×C₄×C₉	—	2	3	no	—
101	C₂×C₄×C₉	#67	1	3	yes	—
102	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
103	C₈×C₉	—	3	4	no	—
104	C₂²×SD₁₆	—	3	1	no	Sylow
105	C₂×C₃×C₉	C₂×SD₁₆	1	6	yes	complemented
106	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
107	D₃×C₉	—	4	4	no	—
108	C₂⁴×C₃	—	6	1	no	—
109	C₂³×D₃	—	2	1	no	—
110	C₂×C₃×Q₈	—	2	6	no	—
111	C₂×C₃×Q₈	—	2	3	no	—
112	C₂×C₃×Q₈	—	2	3	no	—
113	C₂×C₃×Q₈	—	2	6	no	—
114	C₂×C₃×Q₈	C₂²×C₉	1	6	yes	complemented
115	C₂×C₃×Q₈	C₂×C₃×D₃	1	6	yes	—
116	C₂×C₃×D₄	—	3	6	no	—
117	C₂×C₃×D₄	—	6	3	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
124	C₂×D₁₂	—	2	3	no	—
125	C₂×D₁₂	C₂²×C₉	1	6	yes	—
126	C₃×SD₁₆	—	3	16	no	—
127	C₂×C₃×C₈	—	3	6	no	—
128	#215	C₂²×C₉	1	16	yes	direct product factor
129	C₂×C₃⋊C₈	C₂²×C₉	1	6	yes	complemented
130	C₂²×C₃²	C₃×SD₁₆	1	4	yes	—
131	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
132	C₂×C₃×D₃	—	2	4	no	—
133	C₂×C₃×D₃	—	4	12	no	—
134	C₃²×C₄	—	2	4	no	—
135	C₃²×C₄	C₂×C₃×D₄	1	3	yes	—
136	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	2	3	no	—
139	C₂²×C₉	—	6	4	no	—
140	C₂²×C₉	—	12	12	no	—
141	C₂²×C₉	#215	1	4	yes	direct product factor
142	C₂²×C₉	C₂×#67	1	3	yes	—
143	C₄×C₉	—	2	8	no	—
144	C₄×C₉	—	2	4	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	2	3	no	—
147	C₄×C₉	C₂×#67	1	3	yes	—
148	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
149	C₂²×Q₈	D₃×C₉	1	1	yes	characteristic, complemented
150	C₂²×D₄	—	3	1	no	—
151	C₂×SD₁₆	—	3	12	no	—
152	C₂²×C₈	—	3	1	no	—
153	C₃×C₉	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₃	—	2	1	no	—
155	C₂³×C₃	—	6	6	no	—
156	C₂³×C₃	—	12	4	no	—
157	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
158	C₂³×C₃	C₃×#67	1	1	yes	characteristic
159	C₂²×D₃	—	2	6	no	—
160	C₂²×D₃	—	4	4	no	—
161	C₃×Q₈	—	2	12	no	—
162	C₃×Q₈	—	2	6	no	—
163	C₃×Q₈	—	2	6	no	—
164	C₃×Q₈	—	2	4	no	—
165	C₃×Q₈	C₂³×C₉	1	4	yes	complemented
166	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
167	C₃×D₄	—	3	4	no	—
168	C₃×D₄	—	6	6	no	—
169	C₂×C₃×C₄	—	2	12	no	—
170	C₂×C₃×C₄	—	2	6	no	—
171	C₂×C₃×C₄	—	2	6	no	—
172	C₂×C₃×C₄	—	2	3	no	—
173	C₂×C₃×C₄	—	2	3	no	—
174	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
175	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
176	C₂×C₃×C₄	C₃×#67	1	3	yes	—
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
178	D₁₂	—	2	6	no	—
179	D₁₂	C₂³×C₉	1	4	yes	—
180	C₃×C₈	—	3	4	no	—
181	C₃⋊C₈	C₂³×C₉	1	4	yes	complemented
182	C₂×C₃²	C₂×C₃×SD₁₆	1	6	yes	—
183	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₃×D₃	—	4	4	no	—
185	C₂×C₉	—	2	6	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	12	4	no	—
188	C₂×C₉	C₂×#215	1	6	yes	direct product factor
189	C₂×C₉	C₂²×#67	1	1	yes	characteristic
190	C₂⁴	—	6	1	no	—
191	C₂×Q₈	—	2	3	no	—
192	C₂×Q₈	C₂×D₃×C₉	1	6	yes	complemented
193	C₂×D₄	—	3	6	no	—
194	C₂×D₄	—	6	3	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
197	SD₁₆	—	3	16	no	—
198	C₂×C₈	—	3	6	no	—
199	C₂²×C₃	—	2	3	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	6	4	no	—
202	C₂²×C₃	—	12	12	no	—
203	C₂²×C₃	C₉×SD₁₆	1	4	yes	complemented
204	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
205	C₂²×C₃	C₃×#215	1	4	yes	—
206	C₂²×C₃	C₂×C₃×#67	1	3	yes	—
207	C₂×D₃	—	2	4	no	—
208	C₂×D₃	—	4	12	no	—
209	C₃×C₄	—	2	8	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	3	no	—
214	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
215	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
216	C₃×C₄	C₂×C₃×#67	1	3	yes	—
217	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
218	C₃²	C₂²×C₃×SD₁₆	1	1	yes	characteristic
219	C₉	—	2	1	no	—
220	C₉	C₂²×#215	1	1	yes	characteristic, direct product factor
221	C₂³	—	6	6	no	—
222	C₂³	—	12	4	no	—
223	C₂³	C₉×#67	1	1	yes	characteristic
224	Q₈	—	2	6	no	—
225	Q₈	C₂²×D₃×C₉	1	4	yes	complemented
226	D₄	—	3	4	no	—
227	D₄	—	6	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	C₉×#67	1	3	yes	—
230	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
231	C₈	—	3	4	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	6	no	—
234	C₂×C₃	—	12	4	no	—
235	C₂×C₃	C₂×C₉×SD₁₆	1	6	yes	complemented
236	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
237	C₂×C₃	C₂×C₃×#215	1	6	yes	—
238	C₂×C₃	C₂²×C₃×#67	1	1	yes	characteristic
239	D₃	—	4	4	no	—
240	C₂²	—	6	4	no	—
241	C₂²	—	12	12	no	—
242	C₂²	C₉×#215	1	4	yes	direct product factor
243	C₂²	C₂×C₉×#67	1	3	yes	—
244	C₄	—	2	4	no	—
245	C₄	C₂×C₉×#67	1	3	yes	—
246	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	C₂²×C₉×SD₁₆	1	1	yes	characteristic, complemented
249	C₃	C₂²×C₃×#215	1	1	yes	characteristic
250	C₂	—	12	4	no	—
251	C₂	C₂×C₉×#215	1	6	yes	direct product factor
252	C₂	C₂²×C₉×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	#67
commutator	C₃×C₄	C₂⁴×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle	C₂³×C₃²	C₃×D₄
prime radical	C₂²×C₃×D₁₂	C₂×C₃
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₂²×Q₈×C₉	D₃
hypocenter	C₃	C₂²×C₉×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈×C₉	C₂	semidirect
3. C₂×C₉×#215	C₂	direct (N × H)
4. C₂²×C₉×C₃⋊C₈	C₂	semidirect
9. C₂×C₃×Q₈×C₉	C₂²	semidirect
14. C₉×#215	C₂²	direct (N × H)
15. C₂×C₉×C₃⋊C₈	C₂²	semidirect
21. C₂²×Q₈×C₉	D₃	semidirect
29. C₃×Q₈×C₉	C₂³	semidirect
35. C₉×C₃⋊C₈	C₂³	semidirect
37. C₂²×#215	C₉	direct (N × H)
51. C₂×Q₈×C₉	C₂×D₃	semidirect
60. C₂²×C₃×C₉	SD₁₆	semidirect
68. C₂²×C₃×Q₈	C₂×C₉	semidirect
74. C₂×#215	C₂×C₉	direct (N × H)
75. C₂²×C₃⋊C₈	C₂×C₉	semidirect
94. Q₈×C₉	C₂²×D₃	semidirect
105. C₂×C₃×C₉	C₂×SD₁₆	semidirect
114. C₂×C₃×Q₈	C₂²×C₉	semidirect
128. #215	C₂²×C₉	direct (N × H)
129. C₂×C₃⋊C₈	C₂²×C₉	semidirect
141. C₂²×C₉	#215	direct (N × H)
149. C₂²×Q₈	D₃×C₉	semidirect
153. C₃×C₉	C₂²×SD₁₆	semidirect
165. C₃×Q₈	C₂³×C₉	semidirect
181. C₃⋊C₈	C₂³×C₉	semidirect
188. C₂×C₉	C₂×#215	direct (N × H)
192. C₂×Q₈	C₂×D₃×C₉	semidirect
203. C₂²×C₃	C₉×SD₁₆	semidirect
220. C₉	C₂²×#215	direct (N × H)
225. Q₈	C₂²×D₃×C₉	semidirect
235. C₂×C₃	C₂×C₉×SD₁₆	semidirect
242. C₂²	C₉×#215	direct (N × H)
248. C₃	C₂²×C₉×SD₁₆	semidirect
251. C₂	C₂×C₉×#215	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×C₃×Q₈×C₉	2
C₂²×C₉×D₁₂	2
C₂×C₉×#215	2
C₂²×C₉×C₃⋊C₈	2
C₂²×C₃×#215	3
C₂²×C₉×SD₁₆	3

Subgroups — C22×C9×#215 #13153640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×#215 #13153640