Subgroups — #11886903

Order 1152.

714 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589871	C₂	1	2	yes	maximal, complemented
2	#10589854	C₂	1	2	yes	maximal, complemented
3	C₉×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588823	C₂	1	2	yes	maximal, complemented
5	#10588577	C₂	1	2	yes	maximal, complemented
6	#10588502	C₂	1	4	yes	maximal, complemented
7	#10588380	C₂	1	1	yes	maximal, characteristic, complemented
8	#10588377	C₂	1	1	yes	maximal, characteristic, complemented
9	#79894	—	3	1	no	maximal
10	#63715	C₂²	1	4	yes	complemented
11	C₉×#135	C₂²	1	1	yes	characteristic
12	C₉×#131	C₂²	1	2	yes	complemented
13	#63512	C₂²	1	4	yes	complemented
14	#63481	—	2	2	no	—
15	#63476	—	2	4	no	—
16	D₈×D₉	—	2	2	no	—
17	#63470	—	2	4	no	—
18	#63470	C₂²	1	4	yes	complemented
19	#63467	—	2	2	no	—
20	#63467	C₂²	1	1	yes	characteristic
21	C₂×D₇₂	—	2	2	no	—
22	C₂×D₇₂	C₂²	1	2	yes	complemented
23	C₂×D₇₂	C₂²	1	2	yes	complemented
24	#63464	C₂²	1	2	yes	complemented
25	#63436	C₂²	1	2	yes	complemented
26	C₉×#108	C₂²	1	1	yes	characteristic, complemented
27	C₉×#104	C₂²	1	2	yes	complemented
28	C₄×C₈×C₉	C₂²	1	1	yes	characteristic, complemented
29	#63383	C₂²	1	4	yes	complemented
30	#63372	C₂²	1	1	yes	characteristic, complemented
31	#63364	C₂²	1	2	yes	complemented
32	#5769	—	3	2	no	—
33	#5752	—	3	2	no	—
34	C₃×#443	D₃	1	1	yes	characteristic
35	#5142	—	3	2	no	—
36	#4897	—	3	2	no	—
37	#4822	—	3	4	no	—
38	#4700	—	3	1	no	—
39	#4695	—	3	1	no	—
40	C₉×#41	C₂³	1	2	yes	—
41	#3701	—	2	4	no	—
42	D₄×D₉	—	2	4	no	—
43	D₄×D₉	—	4	4	no	—
44	#3697	—	2	4	no	—
45	#3697	C₂³	1	2	yes	—
46	C₂×D₃₆	—	2	4	no	—
47	C₂×D₃₆	—	4	2	no	—
48	C₂×D₃₆	C₂³	1	4	yes	—
49	C₉×Q₁₆	D₄	1	1	yes	characteristic, complemented
50	C₉×SD₁₆	D₄	1	2	yes	complemented
51	C₉×D₈	D₄	1	1	yes	characteristic, complemented
52	C₉×M₁₆	—	2	2	no	—
53	C₉×M₁₆	C₂³	1	2	yes	—
54	C₂×C₈×C₉	—	2	2	no	—
55	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
57	C₄²×C₉	C₂³	1	1	yes	characteristic
58	#3675	—	2	4	no	—
59	#3673	—	2	4	no	—
60	#3667	C₂³	1	2	yes	—
61	D₇₂	—	2	4	no	—
62	D₇₂	—	2	2	no	—
63	D₇₂	—	2	2	no	—
64	D₇₂	—	2	2	no	—
65	D₇₂	—	2	2	no	—
66	D₇₂	D₄	1	1	yes	characteristic, complemented
67	#3664	—	2	4	no	—
68	#3664	D₄	1	2	yes	complemented
69	#3663	—	2	2	no	—
70	C₈×D₉	—	2	2	no	—
71	Dic₃₆	D₄	1	1	yes	characteristic, complemented
72	#3292	—	9	1	no	Sylow
73	#1009	—	3	4	no	—
74	C₃×#135	C₂×D₃	1	1	yes	characteristic
75	C₃×#131	C₂×D₃	1	2	yes	—
76	#949	—	3	4	no	—
77	#919	—	6	2	no	—
78	#914	—	6	4	no	—
79	D₃×D₈	—	6	2	no	—
80	#908	—	3	4	no	—
81	#908	—	6	4	no	—
82	#904	—	3	1	no	—
83	#904	—	6	2	no	—
84	C₂×D₂₄	—	3	2	no	—
85	C₂×D₂₄	—	3	2	no	—
86	C₂×D₂₄	—	6	2	no	—
87	#901	—	3	2	no	—
88	#874	—	3	2	no	—
89	C₃×#108	C₂×D₃	1	1	yes	characteristic
90	C₃×#104	C₂×D₃	1	2	yes	—
91	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
92	#823	—	3	4	no	—
93	#819	—	3	1	no	—
94	#805	—	3	2	no	—
95	C₂²×D₉	—	2	4	no	—
96	C₂²×D₉	—	4	4	no	—
97	Q₈×C₉	C₂×D₄	1	2	yes	—
98	D₄×C₉	—	2	2	no	—
99	D₄×C₉	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
103	#611	—	2	2	no	—
104	#611	—	4	4	no	—
105	D₃₆	—	2	4	no	—
106	D₃₆	—	2	4	no	—
107	D₃₆	—	4	4	no	—
108	D₃₆	—	8	4	no	—
109	D₃₆	C₂×D₄	1	2	yes	—
110	C₄×D₉	—	2	2	no	—
111	C₄×D₉	—	4	4	no	—
112	Dic₁₈	C₂×D₄	1	2	yes	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	C₂×D₄	1	1	yes	characteristic
115	C₈×C₉	C₂×D₄	1	1	yes	characteristic
116	C₈×C₉	C₂×D₄	1	1	yes	characteristic
117	C₈×C₉	C₂×D₄	1	1	yes	characteristic
118	C₉⋊C₈	—	2	2	no	—
119	#576	—	9	2	no	—
120	#576	—	9	2	no	—
121	#493	—	9	1	no	—
122	#472	—	9	2	no	—
123	#472	—	9	2	no	—
124	#453	—	9	4	no	—
125	#443	—	9	1	no	—
126	#443	D₉	1	1	yes	characteristic, complemented
127	C₃×#41	C₂²×D₃	1	2	yes	—
128	#239	—	6	4	no	—
129	D₃×D₄	—	6	4	no	—
130	D₃×D₄	—	12	4	no	—
131	#235	—	3	2	no	—
132	#235	—	6	4	no	—
133	C₂×D₁₂	—	3	4	no	—
134	C₂×D₁₂	—	6	4	no	—
135	C₂×D₁₂	—	12	2	no	—
136	C₃×Q₁₆	D₁₂	1	1	yes	characteristic
137	C₃×SD₁₆	D₁₂	1	2	yes	—
138	C₃×D₈	D₁₂	1	1	yes	characteristic
139	C₃×M₁₆	—	2	2	no	—
140	C₃×M₁₆	C₂²×D₃	1	2	yes	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
145	#215	—	6	4	no	—
146	#213	—	6	4	no	—
147	#208	—	3	2	no	—
148	Dic₁₂	—	3	1	no	—
149	D₂₄	—	3	1	no	—
150	D₂₄	—	6	4	no	—
151	D₂₄	—	6	2	no	—
152	D₂₄	—	6	2	no	—
153	D₂₄	—	6	2	no	—
154	D₂₄	—	6	2	no	—
155	#204	—	3	2	no	—
156	#204	—	6	4	no	—
157	ΓL(1,𝔽₂₅)	—	6	2	no	—
158	D₃×C₈	—	6	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×D₉	—	2	2	no	—
162	C₂×D₉	—	4	4	no	—
163	C₂×D₉	—	8	4	no	—
164	C₂×D₉	—	16	4	no	—
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
168	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
169	Dic₉	—	2	2	no	—
170	H(5,𝔽₂)	—	9	4	no	—
171	AGL(1,ℤ/8)	—	9	4	no	—
172	AGL(1,ℤ/8)	—	9	4	no	—
173	AGL(1,ℤ/8)	—	18	4	no	—
174	AGL(1,ℤ/8)	—	18	4	no	—
175	#135	—	9	1	no	—
176	#135	—	18	2	no	—
177	#135	—	18	2	no	—
178	#135	C₂×D₉	1	1	yes	characteristic
179	C₂×D₈	—	9	2	no	—
180	C₂×D₈	—	9	2	no	—
181	C₂×D₈	—	18	2	no	—
182	C₂×D₈	—	18	2	no	—
183	#131	—	9	2	no	—
184	#131	C₂×D₉	1	2	yes	complemented
185	#127	—	9	2	no	—
186	#108	—	9	1	no	—
187	#108	C₂×D₉	1	1	yes	characteristic, complemented
188	#104	—	9	2	no	—
189	#104	C₂×D₉	1	2	yes	complemented
190	#100	—	9	4	no	—
191	C₄×C₈	C₂×D₉	1	1	yes	characteristic, complemented
192	C₂²×D₃	—	6	4	no	—
193	C₂²×D₃	—	12	4	no	—
194	C₃×Q₈	C₂×D₁₂	1	2	yes	—
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	C₂×D₁₂	1	2	yes	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
200	#67	—	6	2	no	—
201	#67	—	12	4	no	—
202	D₁₂	—	3	2	no	—
203	D₁₂	—	6	4	no	—
204	D₁₂	—	6	4	no	—
205	D₁₂	—	12	4	no	—
206	D₁₂	—	24	4	no	—
207	C₄×D₃	—	6	2	no	—
208	C₄×D₃	—	12	4	no	—
209	Dic₆	—	3	2	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
212	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
213	C₃×C₈	D₃×D₄	1	1	yes	characteristic
214	C₃×C₈	D₃×D₄	1	1	yes	characteristic
215	C₃⋊C₈	—	6	2	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	4	2	no	—
218	C₂×C₉	D₄²	1	1	yes	characteristic
219	D₉	—	4	2	no	—
220	D₉	—	8	4	no	—
221	#41	—	9	2	no	—
222	#41	—	18	4	no	—
223	#41	—	18	4	no	—
224	#41	C₂²×D₉	1	2	yes	—
225	C₂×D₄	—	9	4	no	—
226	C₂×D₄	—	18	4	no	—
227	C₂×D₄	—	18	4	no	—
228	C₂×D₄	—	36	2	no	—
229	C₂×D₄	—	36	4	no	—
230	Q₁₆	—	9	1	no	—
231	Q₁₆	D₃₆	1	1	yes	characteristic, complemented
232	SD₁₆	—	9	2	no	—
233	SD₁₆	—	18	4	no	—
234	SD₁₆	—	18	4	no	—
235	SD₁₆	D₃₆	1	2	yes	complemented
236	D₈	—	9	1	no	—
237	D₈	—	18	4	no	—
238	D₈	—	18	4	no	—
239	D₈	—	18	2	no	—
240	D₈	—	18	2	no	—
241	D₈	—	18	2	no	—
242	D₈	—	18	2	no	—
243	D₈	D₃₆	1	1	yes	characteristic, complemented
244	M₁₆	—	2	2	no	—
245	M₁₆	—	9	2	no	—
246	M₁₆	—	18	2	no	—
247	M₁₆	C₂²×D₉	1	2	yes	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	18	2	no	—
250	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
252	C₄²	C₂²×D₉	1	1	yes	characteristic
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	#1009	1	1	yes	characteristic
255	C₂×D₃	—	6	2	no	—
256	C₂×D₃	—	12	4	no	—
257	C₂×D₃	—	24	4	no	—
258	C₂×D₃	—	48	4	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
262	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
263	Dic₃	—	6	2	no	—
264	C₉	#3292	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	18	4	no	—
266	C₂³	—	36	4	no	—
267	Q₈	—	9	2	no	—
268	Q₈	C₂×D₃₆	1	2	yes	—
269	D₄	—	2	2	no	—
270	D₄	—	9	2	no	—
271	D₄	—	18	2	no	—
272	D₄	—	18	4	no	—
273	D₄	—	18	4	no	—
274	D₄	—	36	4	no	—
275	D₄	—	36	4	no	—
276	D₄	—	72	4	no	—
277	D₄	C₂×D₃₆	1	2	yes	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	18	2	no	—
281	C₂×C₄	—	36	4	no	—
282	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	18	2	no	—
285	C₈	C₂×D₃₆	1	1	yes	characteristic
286	C₈	C₂×D₃₆	1	1	yes	characteristic
287	C₈	D₄×D₉	1	1	yes	characteristic
288	C₈	D₄×D₉	1	1	yes	characteristic
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	4	2	no	—
291	C₂×C₃	D₄×D₁₂	1	1	yes	characteristic
292	D₃	—	12	2	no	—
293	D₃	—	24	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	18	2	no	—
296	C₂²	—	36	4	no	—
297	C₂²	—	72	4	no	—
298	C₂²	—	144	4	no	—
299	C₂²	#63715	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	18	2	no	—
303	C₄	C₂²×D₃₆	1	1	yes	characteristic
304	C₄	C₂×D₄×D₉	1	1	yes	characteristic
305	C₃	#79894	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	4	2	no	—
308	C₂	—	36	2	no	—
309	C₂	—	72	4	no	—
310	C₂	D₄×D₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#443	C₂
socle	C₂×C₃	D₄×D₁₂
hypercenter	#443	D₉
hypocenter	C₉	#3292
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #10589871	C₂	semidirect
2. #10589854	C₂	semidirect
3. C₉×#443	C₂	semidirect
4. #10588823	C₂	semidirect
5. #10588577	C₂	semidirect
6. #10588502	C₂	semidirect
7. #10588380	C₂	semidirect
8. #10588377	C₂	semidirect
10. #63715	C₂²	semidirect
12. C₉×#131	C₂²	semidirect
13. #63512	C₂²	semidirect
18. #63470	C₂²	semidirect
22. C₂×D₇₂	C₂²	semidirect
23. C₂×D₇₂	C₂²	semidirect
24. #63464	C₂²	semidirect
25. #63436	C₂²	semidirect
26. C₉×#108	C₂²	semidirect
27. C₉×#104	C₂²	semidirect
28. C₄×C₈×C₉	C₂²	semidirect
29. #63383	C₂²	semidirect
30. #63372	C₂²	semidirect
31. #63364	C₂²	semidirect
49. C₉×Q₁₆	D₄	semidirect
50. C₉×SD₁₆	D₄	semidirect
51. C₉×D₈	D₄	semidirect
66. D₇₂	D₄	semidirect
68. #3664	D₄	semidirect
71. Dic₃₆	D₄	semidirect
126. #443	D₉	semidirect
184. #131	C₂×D₉	semidirect
187. #108	C₂×D₉	semidirect
189. #104	C₂×D₉	semidirect
191. C₄×C₈	C₂×D₉	semidirect
231. Q₁₆	D₃₆	semidirect
235. SD₁₆	D₃₆	semidirect
243. D₈	D₃₆	semidirect
264. C₉	#3292	semidirect

Maximal subgroups

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

subgroup	index
#10589871	2
#10589854	2
C₉×#443	2
#10588823	2
#10588577	2
#10588502	2
#10588380	2
#10588377	2
#79894	3