Subgroups — C₉×#3038 #11767759

Order 1152.

876 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#460	C₂	1	4	yes	maximal, complemented
5	C₉×#459	C₂	1	4	yes	maximal, complemented
6	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#418	C₂	1	2	yes	maximal, complemented
8	C₃×#3038	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	—	2	2	no	—
11	C₂²×D₄×C₉	C₂²	1	2	yes	—
12	C₉×AGL(1,ℤ/8)	—	2	8	no	—
13	C₂×C₉×SD₁₆	C₂²	1	4	yes	complemented
14	C₂×C₉×D₈	C₂²	1	4	yes	complemented
15	C₂×C₉×M₁₆	C₂²	1	2	yes	—
16	C₉×#127	—	2	1	no	—
17	C₉×#127	—	2	1	no	—
18	C₉×#127	C₂²	1	4	yes	—
19	C₄×Q₈×C₉	C₂²	1	2	yes	complemented
20	C₄×D₄×C₉	—	2	2	no	—
21	C₄×D₄×C₉	C₂²	1	2	yes	complemented
22	C₉×#117	—	2	1	no	—
23	C₉×#117	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₉	—	2	1	no	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
27	C₉×#102	C₂²	1	8	yes	complemented
28	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	—
29	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
30	C₃×C₄×#41	C₂×C₃	1	1	yes	characteristic
31	C₃×#460	C₂×C₃	1	4	yes	—
32	C₃×#459	C₂×C₃	1	4	yes	—
33	C₃×#423	C₂×C₃	1	1	yes	characteristic
34	C₃×#418	C₂×C₃	1	2	yes	—
35	C₂⁴×C₉	—	2	2	no	—
36	C₉×#41	—	2	2	no	—
37	C₉×#41	D₄	1	4	yes	complemented
38	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	—	2	4	no	—
41	C₂×D₄×C₉	—	2	2	no	—
42	C₂×D₄×C₉	—	2	4	no	—
43	C₂×D₄×C₉	—	2	4	no	—
44	C₂×D₄×C₉	—	2	4	no	—
45	C₂×D₄×C₉	—	4	8	no	—
46	C₂×D₄×C₉	C₂³	1	4	yes	—
47	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	D₄	1	2	yes	—
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₉×SD₁₆	—	2	8	no	—
53	C₉×D₈	—	2	8	no	—
54	C₉×M₁₆	—	2	4	no	—
55	C₂×C₈×C₉	C₂³	1	4	yes	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	C₂³	1	2	yes	—
58	C₉×C₂²⋊C₄	—	2	1	no	—
59	C₉×C₂²⋊C₄	—	2	2	no	—
60	C₄²×C₉	—	2	1	no	—
61	C₄²×C₉	—	2	2	no	—
62	C₄²×C₉	D₄	1	1	yes	characteristic
63	C₄²×C₉	D₄	1	1	yes	characteristic
64	C₄²×C₉	C₂³	1	2	yes	—
65	#3038	C₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₃×D₄	—	2	2	no	—
68	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
69	C₃×AGL(1,ℤ/8)	—	2	8	no	—
70	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
71	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
72	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
73	C₃×#127	—	2	1	no	—
74	C₃×#127	—	2	1	no	—
75	C₃×#127	C₂²×C₃	1	4	yes	—
76	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
77	C₃×C₄×D₄	—	2	2	no	—
78	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
79	C₃×#117	—	2	1	no	—
80	C₃×#117	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₃×C₄²	—	2	1	no	—
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
84	C₃×#102	C₂²×C₃	1	8	yes	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	—	2	4	no	—
87	C₂³×C₉	—	4	8	no	—
88	C₂³×C₉	—	4	4	no	—
89	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₉	—	2	1	no	—
91	Q₈×C₉	C₂×D₄	1	2	yes	complemented
92	D₄×C₉	—	2	4	no	—
93	D₄×C₉	—	2	4	no	—
94	D₄×C₉	—	2	1	no	—
95	D₄×C₉	—	2	4	no	—
96	D₄×C₉	—	4	8	no	—
97	D₄×C₉	—	4	4	no	—
98	D₄×C₉	—	4	8	no	—
99	D₄×C₉	C₂×D₄	1	2	yes	complemented
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	4	no	—
102	C₂×C₄×C₉	—	2	4	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	4	4	no	—
105	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₉	#41	1	2	yes	—
110	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
111	C₈×C₉	—	2	4	no	—
112	C₂×AGL(1,ℤ/8)	C₂×C₉	1	2	yes	complemented
113	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
114	C₄×#41	C₂×C₉	1	1	yes	characteristic, complemented
115	#460	C₂×C₉	1	4	yes	complemented
116	#459	C₂×C₉	1	4	yes	complemented
117	#423	C₂×C₉	1	1	yes	characteristic, complemented
118	#418	C₂×C₉	1	2	yes	complemented
119	C₂⁴×C₃	—	2	2	no	—
120	C₃×#41	—	2	2	no	—
121	C₃×#41	C₃×D₄	1	4	yes	—
122	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	—	2	4	no	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	—	2	4	no	—
127	C₂×C₃×D₄	—	2	4	no	—
128	C₂×C₃×D₄	—	2	4	no	—
129	C₂×C₃×D₄	—	4	8	no	—
130	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
131	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₃×C₄	—	2	1	no	—
133	C₂²×C₃×C₄	—	2	2	no	—
134	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₃×SD₁₆	—	2	8	no	—
137	C₃×D₈	—	2	8	no	—
138	C₃×M₁₆	—	2	4	no	—
139	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
142	C₃×C₂²⋊C₄	—	2	1	no	—
143	C₃×C₂²⋊C₄	—	2	2	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
147	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
148	C₃×C₄²	C₂³×C₃	1	2	yes	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	4	2	no	—
152	C₂²×C₉	—	4	4	no	—
153	C₂²×C₉	—	4	8	no	—
154	C₂²×C₉	—	8	8	no	—
155	C₂²×C₉	#121	1	2	yes	—
156	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₉	—	2	2	no	—
158	C₄×C₉	—	2	2	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	#121	1	2	yes	—
161	C₄×C₉	AGL(1,ℤ/8)	1	4	yes	complemented
162	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
163	C₄×C₉	C₂×#41	1	1	yes	characteristic
164	C₂×#41	C₂²×C₉	1	1	yes	characteristic
165	C₂²×D₄	—	2	2	no	—
166	C₂²×D₄	C₂²×C₉	1	2	yes	—
167	AGL(1,ℤ/8)	—	2	8	no	—
168	C₂×SD₁₆	C₂²×C₉	1	4	yes	complemented
169	C₂×D₈	C₂²×C₉	1	4	yes	complemented
170	C₂×M₁₆	C₂²×C₉	1	2	yes	—
171	#127	—	2	1	no	—
172	#127	—	2	1	no	—
173	#127	C₂²×C₉	1	4	yes	—
174	C₄×Q₈	C₂²×C₉	1	2	yes	complemented
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	C₂²×C₉	1	2	yes	complemented
177	#117	—	2	1	no	—
178	#117	C₂²×C₉	1	1	yes	characteristic
179	C₂×C₄²	—	2	1	no	—
180	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
181	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
182	#102	C₂²×C₉	1	8	yes	complemented
183	C₂³×C₃	—	2	1	no	—
184	C₂³×C₃	—	2	4	no	—
185	C₂³×C₃	—	4	8	no	—
186	C₂³×C₃	—	4	4	no	—
187	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
188	C₃×Q₈	—	2	1	no	—
189	C₃×Q₈	C₂×C₃×D₄	1	2	yes	—
190	C₃×D₄	—	2	4	no	—
191	C₃×D₄	—	2	4	no	—
192	C₃×D₄	—	2	1	no	—
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	—	4	8	no	—
195	C₃×D₄	—	4	4	no	—
196	C₃×D₄	—	4	8	no	—
197	C₃×D₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	4	no	—
203	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
207	C₂×C₃×C₄	C₃×#41	1	2	yes	—
208	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
209	C₃×C₈	—	2	4	no	—
210	C₂×C₉	—	2	2	no	—
211	C₂×C₉	—	4	2	no	—
212	C₂×C₉	—	8	4	no	—
213	C₂×C₉	C₂×#121	1	1	yes	characteristic
214	C₂×C₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
215	C₂⁴	—	2	2	no	—
216	#41	—	2	2	no	—
217	#41	D₄×C₉	1	4	yes	complemented
218	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	8	no	—
226	C₂×D₄	C₂³×C₉	1	4	yes	—
227	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	D₄×C₉	1	2	yes	—
231	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
232	SD₁₆	—	2	8	no	—
233	D₈	—	2	8	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	C₂³×C₉	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	C₂³×C₉	1	2	yes	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	D₄×C₉	1	1	yes	characteristic
243	C₄²	D₄×C₉	1	1	yes	characteristic
244	C₄²	C₂³×C₉	1	2	yes	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	4	8	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	—	8	8	no	—
251	C₂²×C₃	C₃×#121	1	2	yes	—
252	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	C₃×#121	1	2	yes	—
257	C₃×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
258	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
259	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
260	C₉	#3038	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	2	1	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	8	no	—
264	C₂³	—	4	4	no	—
265	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
266	Q₈	—	2	1	no	—
267	Q₈	C₂×D₄×C₉	1	2	yes	complemented
268	D₄	—	2	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	1	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	8	no	—
275	D₄	C₂×D₄×C₉	1	2	yes	complemented
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
282	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
284	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
285	C₂×C₄	C₉×#41	1	2	yes	—
286	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	—	4	2	no	—
290	C₂×C₃	—	8	4	no	—
291	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
292	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	8	8	no	—
299	C₂²	C₉×#121	1	2	yes	—
300	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	2	no	—
304	C₄	C₉×#121	1	2	yes	—
305	C₄	C₉×AGL(1,ℤ/8)	1	4	yes	complemented
306	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
307	C₄	C₂×C₉×#41	1	1	yes	characteristic
308	C₃	C₃×#3038	1	1	yes	characteristic
309	C₂	—	2	2	no	—
310	C₂	—	4	2	no	—
311	C₂	—	8	4	no	—
312	C₂	C₂×C₉×#121	1	1	yes	characteristic
313	C₂	C₂×C₉×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3038	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₉×#127	C₂	semidirect
3. C₄×C₉×#41	C₂	semidirect
4. C₉×#460	C₂	semidirect
5. C₉×#459	C₂	semidirect
6. C₉×#423	C₂	semidirect
7. C₉×#418	C₂	semidirect
13. C₂×C₉×SD₁₆	C₂²	semidirect
14. C₂×C₉×D₈	C₂²	semidirect
19. C₄×Q₈×C₉	C₂²	semidirect
21. C₄×D₄×C₉	C₂²	semidirect
26. C₉×C₄⋊C₈	C₂²	semidirect
27. C₉×#102	C₂²	semidirect
37. C₉×#41	D₄	semidirect
65. #3038	C₉	direct (N × H)
91. Q₈×C₉	C₂×D₄	semidirect
99. D₄×C₉	C₂×D₄	semidirect
112. C₂×AGL(1,ℤ/8)	C₂×C₉	semidirect
113. C₂×#127	C₂×C₉	semidirect
114. C₄×#41	C₂×C₉	semidirect
115. #460	C₂×C₉	semidirect
116. #459	C₂×C₉	semidirect
117. #423	C₂×C₉	semidirect
118. #418	C₂×C₉	semidirect
161. C₄×C₉	AGL(1,ℤ/8)	semidirect
168. C₂×SD₁₆	C₂²×C₉	semidirect
169. C₂×D₈	C₂²×C₉	semidirect
174. C₄×Q₈	C₂²×C₉	semidirect
176. C₄×D₄	C₂²×C₉	semidirect
181. C₄⋊C₈	C₂²×C₉	semidirect
182. #102	C₂²×C₉	semidirect
217. #41	D₄×C₉	semidirect
260. C₉	#3038	direct (N × H)
267. Q₈	C₂×D₄×C₉	semidirect
275. D₄	C₂×D₄×C₉	semidirect
305. C₄	C₉×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×AGL(1,ℤ/8)	2
C₂×C₉×#127	2
C₄×C₉×#41	2
C₉×#460	2
C₉×#459	2
C₉×#423	2
C₉×#418	2
C₃×#3038	3

Subgroups — C9×#3038 #11767759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3038 #11767759