Subgroups — C₉×#2434 #11767155

Order 1152.

1200 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₉×#395	C₂	1	4	yes	maximal, complemented
5	C₉×#391	C₂	1	2	yes	maximal, complemented
6	C₉×#385	C₂	1	4	yes	maximal
7	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×#2434	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	—	2	8	no	—
10	C₂²×Q₈×C₉	—	2	4	no	—
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Q₈	—	2	8	no	—
17	C₉×C₂²⋊Q₈	—	2	4	no	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²×C₉	C₂²	1	2	yes	—
24	C₉×#95	C₂²	1	2	yes	—
25	C₉×#95	C₂²	1	8	yes	—
26	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
28	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
29	C₃×#395	C₂×C₃	1	4	yes	—
30	C₃×#391	C₂×C₃	1	2	yes	—
31	C₃×#385	C₂×C₃	1	4	yes	—
32	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
33	C₂⁴×C₉	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₉	—	2	32	no	—
35	C₂×Q₈×C₉	—	2	8	no	—
36	C₂×Q₈×C₉	—	2	4	no	—
37	C₂×Q₈×C₉	—	2	4	no	—
38	C₂×Q₈×C₉	—	2	4	no	—
39	C₂×Q₈×C₉	—	2	2	no	—
40	C₂×Q₈×C₉	—	2	4	no	—
41	C₂×Q₈×C₉	D₄	1	8	yes	—
42	C₂²×C₄×C₉	—	2	4	no	—
43	C₂²×C₄×C₉	—	2	8	no	—
44	C₂²×C₄×C₉	—	2	4	no	—
45	C₂²×C₄×C₉	D₄	1	4	yes	—
46	C₂²×C₄×C₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂³	1	4	yes	—
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₉×C₄⋊C₄	—	2	2	no	—
53	C₉×C₄⋊C₄	—	2	4	no	—
54	C₉×C₄⋊C₄	—	2	16	no	—
55	C₉×C₂²⋊C₄	—	2	4	no	—
56	C₉×C₂²⋊C₄	Q₈	1	8	yes	complemented
57	C₄²×C₉	—	2	4	no	—
58	#2434	C₉	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₃×Q₈	—	2	8	no	—
60	C₂²×C₃×Q₈	—	2	4	no	—
61	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
62	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
64	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
65	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
66	C₃×C₄⋊Q₈	—	2	8	no	—
67	C₃×C₂²⋊Q₈	—	2	4	no	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
69	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
74	C₃×#95	C₂²×C₃	1	2	yes	—
75	C₃×#95	C₂²×C₃	1	8	yes	—
76	C₂³×C₉	—	2	2	no	—
77	C₂³×C₉	—	2	2	no	—
78	C₂³×C₉	C₂×D₄	1	2	yes	—
79	C₂³×C₉	C₂×Q₈	1	4	yes	—
80	C₂³×C₉	C₂⁴	1	1	yes	characteristic
81	Q₈×C₉	—	2	16	no	—
82	Q₈×C₉	—	2	16	no	—
83	Q₈×C₉	—	4	4	no	—
84	C₂×C₄×C₉	—	2	16	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	4	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	8	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	C₂×D₄	1	8	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
98	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₉	C₂×Q₈	1	8	yes	complemented
100	C₂³×Q₈	C₂×C₉	1	1	yes	characteristic
101	C₂×C₄⋊Q₈	C₂×C₉	1	2	yes	complemented
102	C₂×C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
103	#395	C₂×C₉	1	4	yes	complemented
104	#391	C₂×C₉	1	2	yes	complemented
105	#385	C₂×C₉	1	4	yes	—
106	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
107	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	2	32	no	—
109	C₂×C₃×Q₈	—	2	8	no	—
110	C₂×C₃×Q₈	—	2	4	no	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	2	4	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	2	4	no	—
115	C₂×C₃×Q₈	C₃×D₄	1	8	yes	—
116	C₂²×C₃×C₄	—	2	4	no	—
117	C₂²×C₃×C₄	—	2	8	no	—
118	C₂²×C₃×C₄	—	2	4	no	—
119	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
122	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
123	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
125	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	2	4	no	—
128	C₃×C₄⋊C₄	—	2	16	no	—
129	C₃×C₂²⋊C₄	—	2	4	no	—
130	C₃×C₂²⋊C₄	C₃×Q₈	1	8	yes	—
131	C₃×C₄²	—	2	4	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	2	no	—
134	C₂²×C₉	—	2	8	no	—
135	C₂²×C₉	C₄⋊Q₈	1	4	yes	complemented
136	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₉	C₂²×D₄	1	2	yes	—
138	C₂²×C₉	C₂²×Q₈	1	2	yes	—
139	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
140	C₄×C₉	—	2	8	no	—
141	C₄×C₉	—	2	2	no	—
142	C₄×C₉	—	2	8	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	—	4	2	no	—
145	C₄×C₉	#120	1	4	yes	—
146	C₂²×Q₈	—	2	8	no	—
147	C₂²×Q₈	—	2	4	no	—
148	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
149	C₂²×Q₈	C₂²×C₉	1	2	yes	—
150	C₂²×Q₈	C₂²×C₉	1	2	yes	—
151	C₂³×C₄	C₂²×C₉	1	2	yes	—
152	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	8	no	—
154	C₂²⋊Q₈	—	2	4	no	—
155	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
157	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
160	C₂×C₄²	C₂²×C₉	1	2	yes	—
161	#95	C₂²×C₉	1	2	yes	—
162	#95	C₂²×C₉	1	8	yes	—
163	C₂³×C₃	—	2	2	no	—
164	C₂³×C₃	—	2	2	no	—
165	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
166	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
167	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
168	C₃×Q₈	—	2	16	no	—
169	C₃×Q₈	—	2	16	no	—
170	C₃×Q₈	—	4	4	no	—
171	C₂×C₃×C₄	—	2	16	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	8	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
185	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
187	C₂×C₉	—	2	4	no	—
188	C₂×C₉	C₂×#120	1	1	yes	characteristic
189	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
190	C₂×C₉	#536	1	1	yes	characteristic
191	C₂×C₉	D₄×Q₈	1	4	yes	—
192	C₂⁴	C₂³×C₉	1	1	yes	characteristic
193	C₂×Q₈	—	2	32	no	—
194	C₂×Q₈	—	2	8	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	D₄×C₉	1	8	yes	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	D₄×C₉	1	4	yes	—
205	C₂²×C₄	C₂³×C₉	1	2	yes	—
206	C₂²×C₄	C₂³×C₉	1	2	yes	—
207	C₂²×C₄	C₂³×C₉	1	4	yes	—
208	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₉	1	4	yes	—
210	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	16	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	Q₈×C₉	1	8	yes	complemented
216	C₄²	—	2	4	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	8	no	—
220	C₂²×C₃	C₃×C₄⋊Q₈	1	4	yes	—
221	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
222	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
223	C₂²×C₃	C₂²×C₃×Q₈	1	2	yes	—
224	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	8	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	—	4	2	no	—
230	C₃×C₄	C₃×#120	1	4	yes	—
231	C₉	#2434	1	1	yes	characteristic, direct product factor, Sylow
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×D₄×C₉	1	2	yes	—
235	C₂³	C₂×Q₈×C₉	1	4	yes	—
236	C₂³	C₂⁴×C₉	1	1	yes	characteristic
237	Q₈	—	2	16	no	—
238	Q₈	—	2	16	no	—
239	Q₈	—	4	4	no	—
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	C₂×D₄×C₉	1	8	yes	—
252	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
253	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
254	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
255	C₂×C₄	C₂×Q₈×C₉	1	8	yes	complemented
256	C₂×C₃	—	2	4	no	—
257	C₂×C₃	C₂×C₃×#120	1	1	yes	characteristic
258	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
259	C₂×C₃	C₃×#536	1	1	yes	characteristic
260	C₂×C₃	C₃×D₄×Q₈	1	4	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	C₉×C₄⋊Q₈	1	4	yes	complemented
265	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
266	C₂²	C₂²×D₄×C₉	1	2	yes	—
267	C₂²	C₂²×Q₈×C₉	1	2	yes	—
268	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
269	C₄	—	2	8	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	8	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	C₉×#120	1	4	yes	—
275	C₃	C₃×#2434	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	C₂×C₉×#120	1	1	yes	characteristic
278	C₂	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic
279	C₂	C₉×#536	1	1	yes	characteristic
280	C₂	D₄×Q₈×C₉	1	4	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³×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×C₄⋊Q₈	C₂	semidirect
4. C₉×#395	C₂	semidirect
5. C₉×#391	C₂	semidirect
56. C₉×C₂²⋊C₄	Q₈	semidirect
58. #2434	C₉	direct (N × H)
99. C₂×C₄×C₉	C₂×Q₈	semidirect
101. C₂×C₄⋊Q₈	C₂×C₉	semidirect
103. #395	C₂×C₉	semidirect
104. #391	C₂×C₉	semidirect
135. C₂²×C₉	C₄⋊Q₈	semidirect
215. C₂²⋊C₄	Q₈×C₉	semidirect
231. C₉	#2434	direct (N × H)
255. C₂×C₄	C₂×Q₈×C₉	semidirect
264. C₂²	C₉×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₉	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×C₂²⋊Q₈	2
C₉×#395	2
C₉×#391	2
C₉×#385	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2434	3

Subgroups — C9×#2434 #11767155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2434 #11767155