Subgroups — C₉×#2584 #11767305

Order 1152.

702 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#398	C₂	1	2	yes	maximal
4	C₉×#395	C₂	1	2	yes	maximal
5	C₉×#389	C₂	1	2	yes	maximal
6	C₉×#389	C₂	1	2	yes	maximal
7	C₉×#389	C₂	1	4	yes	maximal
8	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₃×#2584	C₃	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#125	—	2	4	no	—
12	C₄×Q₈×C₉	—	2	4	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₉×#95	C₂²	1	2	yes	—
26	C₉×#95	C₂²	1	4	yes	—
27	C₉×#95	C₂²	1	4	yes	—
28	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
30	C₃×#398	C₂×C₃	1	2	yes	—
31	C₃×#395	C₂×C₃	1	2	yes	—
32	C₃×#389	C₂×C₃	1	2	yes	—
33	C₃×#389	C₂×C₃	1	4	yes	—
34	C₃×#389	C₂×C₃	1	2	yes	—
35	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
36	C₂×Q₈×C₉	—	2	2	no	—
37	C₂×Q₈×C₉	—	2	4	no	—
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₉×C₄⋊C₄	—	2	4	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	2	4	no	—
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₉×C₄⋊C₄	—	2	2	no	—
52	C₉×C₄⋊C₄	—	2	8	no	—
53	C₉×C₄⋊C₄	—	4	4	no	—
54	C₉×C₄⋊C₄	Q₈	1	8	yes	complemented
55	C₄²×C₉	—	2	4	no	—
56	C₄²×C₉	—	4	2	no	—
57	C₄²×C₉	D₄	1	4	yes	—
58	#2584	C₉	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×#125	—	2	4	no	—
61	C₃×C₄×Q₈	—	2	4	no	—
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
73	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
74	C₃×#95	C₂²×C₃	1	2	yes	—
75	C₃×#95	C₂²×C₃	1	4	yes	—
76	C₃×#95	C₂²×C₃	1	4	yes	—
77	C₂³×C₉	C₂⁴	1	1	yes	characteristic
78	Q₈×C₉	—	2	8	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
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	C₂×Q₈	1	8	yes	—
98	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
99	C₂×C₄×C₉	#41	1	4	yes	—
100	C₂×#125	C₂×C₉	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
102	#398	C₂×C₉	1	2	yes	—
103	#395	C₂×C₉	1	2	yes	—
104	#389	C₂×C₉	1	2	yes	—
105	#389	C₂×C₉	1	4	yes	—
106	#389	C₂×C₉	1	2	yes	—
107	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
111	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
112	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
116	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	8	no	—
125	C₃×C₄⋊C₄	—	4	4	no	—
126	C₃×C₄⋊C₄	C₃×Q₈	1	8	yes	—
127	C₃×C₄²	—	2	4	no	—
128	C₃×C₄²	—	4	2	no	—
129	C₃×C₄²	C₃×D₄	1	4	yes	—
130	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₉	C₂²×Q₈	1	2	yes	—
132	C₂²×C₉	C₂×#41	1	2	yes	—
133	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂²×C₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₄×C₉	—	2	2	no	—
136	C₄×C₉	—	2	4	no	—
137	C₄×C₉	—	2	2	no	—
138	C₄×C₉	—	2	8	no	—
139	C₄×C₉	—	4	2	no	—
140	C₄×C₉	—	4	2	no	—
141	C₄×C₉	—	4	2	no	—
142	C₄×C₉	C₄⋊Q₈	1	4	yes	—
143	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
144	#125	—	2	4	no	—
145	C₄×Q₈	—	2	4	no	—
146	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
153	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
154	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
155	C₂×C₄²	C₂²×C₉	1	2	yes	—
156	C₂×C₄²	C₂²×C₉	1	2	yes	—
157	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
158	#95	C₂²×C₉	1	2	yes	—
159	#95	C₂²×C₉	1	4	yes	—
160	#95	C₂²×C₉	1	4	yes	—
161	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
162	C₃×Q₈	—	2	8	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
182	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
183	C₂×C₃×C₄	C₃×#41	1	4	yes	—
184	C₂×C₉	#525	1	1	yes	characteristic
185	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₉	#537	1	1	yes	characteristic
187	C₂×C₉	#554	1	2	yes	—
188	C₂×C₉	#557	1	2	yes	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂²×C₄	C₂³×C₉	1	2	yes	—
192	C₂²×C₄	C₂³×C₉	1	2	yes	—
193	C₂²×C₄	C₂³×C₉	1	2	yes	—
194	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₉	1	4	yes	—
197	C₂²×C₄	C₂³×C₉	1	2	yes	—
198	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	8	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	Q₈×C₉	1	8	yes	complemented
208	C₄²	—	2	4	no	—
209	C₄²	—	4	2	no	—
210	C₄²	D₄×C₉	1	4	yes	—
211	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
212	C₂²×C₃	C₂²×C₃×Q₈	1	2	yes	—
213	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
214	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
215	C₂²×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	8	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	—	4	2	no	—
223	C₃×C₄	C₃×C₄⋊Q₈	1	4	yes	—
224	C₉	#2584	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	C₂⁴×C₉	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
243	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
245	C₂×C₄	C₂×Q₈×C₉	1	8	yes	—
246	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
247	C₂×C₄	C₉×#41	1	4	yes	—
248	C₂×C₃	C₃×#525	1	1	yes	characteristic
249	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
250	C₂×C₃	C₃×#537	1	1	yes	characteristic
251	C₂×C₃	C₃×#554	1	2	yes	—
252	C₂×C₃	C₃×#557	1	2	yes	—
253	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
254	C₂²	C₂²×Q₈×C₉	1	2	yes	—
255	C₂²	C₂×C₉×#41	1	2	yes	—
256	C₂²	C₉×H(5,𝔽₂)	1	1	yes	characteristic
257	C₂²	C₉×2₋¹⁺⁴	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	8	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	C₉×C₄⋊Q₈	1	4	yes	—
266	C₃	C₃×#2584	1	1	yes	characteristic
267	C₂	C₉×#525	1	1	yes	characteristic
268	C₂	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic
269	C₂	C₉×#537	1	1	yes	characteristic
270	C₂	C₉×#554	1	2	yes	—
271	C₂	C₉×#557	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂⁴×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
54. C₉×C₄⋊C₄	Q₈	semidirect
58. #2584	C₉	direct (N × H)
207. C₄⋊C₄	Q₈×C₉	semidirect
224. C₉	#2584	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×#125	2
C₂×C₄×Q₈×C₉	2
C₉×#398	2
C₉×#395	2
C₉×#389	2
C₄×C₉×C₄⋊C₄	2
C₃×#2584	3

Subgroups — C9×#2584 #11767305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2584 #11767305