Subgroups — #11878820

Order 1152.

732 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	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#63436	C₂	1	1	yes	maximal, characteristic
3	#10588773	C₂	1	2	yes	maximal, complemented
4	#10588770	C₂	1	2	yes	maximal, complemented
5	#10588729	C₂	1	4	yes	maximal, complemented
6	#10588710	C₂	1	4	yes	maximal, complemented
7	#10588605	C₂	1	1	yes	maximal, characteristic, complemented
8	#83355	—	3	1	no	maximal
9	C₂²×D₃₆	—	2	1	no	—
10	C₂²×D₃₆	—	2	1	no	—
11	C₂²×D₃₆	C₂²	1	2	yes	—
12	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₉×#125	—	2	1	no	—
15	C₉×C₂²⋊Q₈	—	2	1	no	—
16	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₄×Q₈×C₉	—	2	1	no	—
18	C₄×Q₈×C₉	C₂²	1	2	yes	complemented
19	C₉×#117	—	2	1	no	—
20	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₂×#3675	C₂²	1	8	yes	complemented
22	#63436	—	2	1	no	—
23	#63436	—	2	1	no	—
24	#63436	C₂²	1	2	yes	—
25	#63436	C₂²	1	2	yes	—
26	#63389	C₂²	1	2	yes	—
27	#63369	C₂²	1	8	yes	complemented
28	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
29	C₄×C₉⋊C₈	C₂²	1	2	yes	complemented
30	C₃×#533	D₃	1	1	yes	characteristic
31	C₂×#874	—	3	1	no	—
32	#5088	—	3	2	no	—
33	#5085	—	3	2	no	—
34	#5046	—	3	4	no	—
35	#5027	—	3	4	no	—
36	#4925	—	3	1	no	—
37	C₂³×D₉	—	2	2	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×D₃₆	—	2	2	no	—
44	C₂×D₃₆	—	2	2	no	—
45	C₂×D₃₆	—	2	2	no	—
46	C₂×D₃₆	—	2	2	no	—
47	C₂×D₃₆	—	2	2	no	—
48	C₂×D₃₆	—	4	4	no	—
49	C₂×D₃₆	—	4	4	no	—
50	C₂×D₃₆	—	4	4	no	—
51	C₂×D₃₆	C₂³	1	4	yes	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	1	no	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	C₂³	1	2	yes	—
58	C₉×C₂²⋊C₄	—	2	2	no	—
59	C₄²×C₉	—	2	2	no	—
60	C₄²×C₉	D₄	1	1	yes	characteristic
61	C₄²×C₉	D₄	1	1	yes	characteristic
62	C₄²×C₉	C₂³	1	1	yes	characteristic
63	C₄²×C₉	C₂³	1	1	yes	characteristic
64	#3675	—	4	8	no	—
65	C₂×C₉⋊C₈	C₂³	1	4	yes	—
66	#3208	—	9	1	no	Sylow
67	C₂²×D₁₂	—	3	2	no	—
68	C₂²×D₁₂	—	6	1	no	—
69	C₂²×D₁₂	—	6	1	no	—
70	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
72	C₃×#125	—	2	1	no	—
73	C₃×C₂²⋊Q₈	—	2	1	no	—
74	C₃×C₂²⋊Q₈	C₂×D₃	1	2	yes	—
75	C₃×C₄×Q₈	—	2	1	no	—
76	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
77	C₃×#117	—	2	1	no	—
78	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
79	C₂×#215	—	3	8	no	—
80	#874	—	3	2	no	—
81	#874	—	3	2	no	—
82	#874	—	6	1	no	—
83	#874	—	6	1	no	—
84	#830	—	3	2	no	—
85	#809	—	3	8	no	—
86	C₄⋊(C₃⋊C₈)	—	3	2	no	—
87	C₄×C₃⋊C₈	—	3	2	no	—
88	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₉	—	2	4	no	—
90	C₂²×D₉	—	4	4	no	—
91	C₂²×D₉	—	4	4	no	—
92	C₂²×D₉	—	4	4	no	—
93	Q₈×C₉	—	2	4	no	—
94	Q₈×C₉	—	4	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	SD₁₆	1	4	yes	complemented
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
106	D₃₆	—	2	8	no	—
107	D₃₆	—	4	4	no	—
108	D₃₆	—	4	4	no	—
109	D₃₆	—	4	4	no	—
110	D₃₆	—	4	4	no	—
111	D₃₆	—	4	4	no	—
112	C₉⋊C₈	—	2	4	no	—
113	#533	D₉	1	1	yes	characteristic, complemented
114	C₂×#127	—	9	1	no	—
115	#492	—	9	2	no	—
116	#486	—	9	2	no	—
117	#460	—	9	4	no	—
118	#450	—	9	4	no	—
119	#431	—	9	1	no	—
120	C₂³×D₃	—	6	2	no	—
121	C₂×C₃×Q₈	—	2	1	no	—
122	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
123	C₂²×C₃×C₄	#67	1	1	yes	characteristic
124	C₂²×C₃×C₄	#67	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂×D₁₂	—	3	4	no	—
127	C₂×D₁₂	—	6	2	no	—
128	C₂×D₁₂	—	6	2	no	—
129	C₂×D₁₂	—	6	2	no	—
130	C₂×D₁₂	—	6	2	no	—
131	C₂×D₁₂	—	6	2	no	—
132	C₂×D₁₂	—	12	4	no	—
133	C₂×D₁₂	—	12	4	no	—
134	C₂×D₁₂	—	12	4	no	—
135	C₃×C₄⋊C₄	—	2	1	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
141	C₃×C₂²⋊C₄	—	2	2	no	—
142	C₃×C₄²	—	2	2	no	—
143	C₃×C₄²	#67	1	1	yes	characteristic
144	C₃×C₄²	#67	1	1	yes	characteristic
145	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
146	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
147	#215	—	12	8	no	—
148	C₂×C₃⋊C₈	—	3	4	no	—
149	C₂²×C₉	—	2	1	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
152	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₉	—	4	4	no	—
154	C₂×D₉	—	4	4	no	—
155	C₂×D₉	—	4	4	no	—
156	C₂×D₉	—	8	8	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	—	4	2	no	—
161	C₄×C₉	C₂×SD₁₆	1	2	yes	—
162	C₄×C₉	C₂×SD₁₆	1	2	yes	—
163	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
164	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
165	C₂²×D₄	—	9	2	no	—
166	C₂²×D₄	—	18	1	no	—
167	C₂²×D₄	—	18	1	no	—
168	C₂×SD₁₆	—	9	8	no	—
169	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic, complemented
170	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic, complemented
171	#127	—	9	2	no	—
172	#127	—	9	2	no	—
173	#127	—	18	1	no	—
174	#127	—	18	1	no	—
175	#125	—	2	1	no	—
176	C₂²⋊Q₈	—	2	1	no	—
177	C₂²⋊Q₈	C₂×D₉	1	2	yes	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	C₂×D₉	1	2	yes	complemented
180	#117	—	2	1	no	—
181	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
182	C₄⋊C₈	—	9	2	no	—
183	#102	—	9	8	no	—
184	C₂²⋊C₈	—	9	2	no	—
185	C₄×C₈	—	9	2	no	—
186	C₂³×C₃	C₂×#67	1	1	yes	characteristic
187	C₂²×D₃	—	6	4	no	—
188	C₂²×D₃	—	12	4	no	—
189	C₂²×D₃	—	12	4	no	—
190	C₂²×D₃	—	12	4	no	—
191	C₃×Q₈	—	2	4	no	—
192	C₃×Q₈	—	4	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	#215	1	4	yes	—
198	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
204	D₁₂	—	6	8	no	—
205	D₁₂	—	12	4	no	—
206	D₁₂	—	12	4	no	—
207	D₁₂	—	12	4	no	—
208	D₁₂	—	12	4	no	—
209	D₁₂	—	12	4	no	—
210	C₃⋊C₈	—	6	4	no	—
211	C₂×C₉	—	2	2	no	—
212	C₂×C₉	#535	1	1	yes	characteristic
213	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
214	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	D₉	—	8	4	no	—
216	C₂⁴	—	18	2	no	—
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	C₂²×D₉	1	2	yes	—
219	C₂×D₄	—	9	4	no	—
220	C₂×D₄	—	18	2	no	—
221	C₂×D₄	—	18	2	no	—
222	C₂×D₄	—	18	2	no	—
223	C₂×D₄	—	18	2	no	—
224	C₂×D₄	—	18	2	no	—
225	C₂×D₄	—	36	4	no	—
226	C₂×D₄	—	36	4	no	—
227	C₂×D₄	—	36	4	no	—
228	C₂²×C₄	#611	1	1	yes	characteristic
229	C₂²×C₄	#611	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
231	SD₁₆	—	36	8	no	—
232	C₂×C₈	—	9	4	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂²×D₉	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	#611	1	1	yes	characteristic
242	C₄²	#611	1	1	yes	characteristic
243	C₄²	C₂²×D₉	1	1	yes	characteristic
244	C₄²	C₂²×D₉	1	1	yes	characteristic
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	C₂×#215	1	2	yes	—
248	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
249	C₂×D₃	—	12	4	no	—
250	C₂×D₃	—	12	4	no	—
251	C₂×D₃	—	12	4	no	—
252	C₂×D₃	—	24	8	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	C₂×#215	1	2	yes	—
258	C₃×C₄	C₂×#215	1	2	yes	—
259	C₃×C₄	#949	1	2	yes	—
260	C₃×C₄	#1009	1	2	yes	—
261	C₉	#3208	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	18	4	no	—
263	C₂³	—	36	4	no	—
264	C₂³	—	36	4	no	—
265	C₂³	—	36	4	no	—
266	C₂³	C₂×#611	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	4	2	no	—
269	D₄	—	18	8	no	—
270	D₄	—	36	4	no	—
271	D₄	—	36	4	no	—
272	D₄	—	36	4	no	—
273	D₄	—	36	4	no	—
274	D₄	—	36	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	#3675	1	4	yes	complemented
280	C₂×C₄	C₂×#611	1	1	yes	characteristic
281	C₂×C₄	C₂×#611	1	1	yes	characteristic
282	C₂×C₄	C₂×#611	1	1	yes	characteristic
283	C₂×C₄	C₂×#611	1	1	yes	characteristic
284	C₂×C₄	C₂×#611	1	1	yes	characteristic
285	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
286	C₈	—	18	4	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	C₂²×#215	1	1	yes	characteristic
289	C₂×C₃	C₂×#949	1	1	yes	characteristic
290	C₂×C₃	#5830	1	1	yes	characteristic
291	D₃	—	24	4	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	36	4	no	—
295	C₂²	—	36	4	no	—
296	C₂²	—	36	4	no	—
297	C₂²	—	72	8	no	—
298	C₂²	C₂×#3675	1	2	yes	—
299	C₂²	C₂²×#611	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	C₂×#3675	1	2	yes	—
305	C₄	C₂×#3675	1	2	yes	—
306	C₄	#63512	1	2	yes	—
307	C₄	#63715	1	2	yes	—
308	C₃	#83355	1	1	yes	characteristic
309	C₂	—	2	2	no	—
310	C₂	—	72	4	no	—
311	C₂	C₂²×#3675	1	1	yes	characteristic
312	C₂	C₂×#63512	1	1	yes	characteristic
313	C₂	#10589932	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#533	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂×#63436	C₂
hypercenter	#533	D₉
hypocenter	C₉	#3208
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₉×#533	C₂	semidirect
3. #10588773	C₂	semidirect
4. #10588770	C₂	semidirect
5. #10588729	C₂	semidirect
6. #10588710	C₂	semidirect
7. #10588605	C₂	semidirect
12. C₉×C₄⋊Q₈	C₂²	semidirect
13. C₉×C₄⋊Q₈	C₂²	semidirect
18. C₄×Q₈×C₉	C₂²	semidirect
21. C₂×#3675	C₂²	semidirect
27. #63369	C₂²	semidirect
28. C₄⋊(C₉⋊C₈)	C₂²	semidirect
29. C₄×C₉⋊C₈	C₂²	semidirect
99. C₂×C₄×C₉	SD₁₆	semidirect
113. #533	D₉	semidirect
169. C₄⋊Q₈	C₂×D₉	semidirect
170. C₄⋊Q₈	C₂×D₉	semidirect
179. C₄×Q₈	C₂×D₉	semidirect
261. C₉	#3208	semidirect
279. C₂×C₄	#3675	semidirect

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₂×#63436	2
#10588773	2
#10588770	2
#10588729	2
#10588710	2
#10588605	2
#83355	3