Finite Group Databasebeta

← #19146610

Subgroups — #19146610

Order 5504.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D172C211yesmaximal, characteristic
2#14583265C211yesmaximal, characteristic, complemented
3#14583223C211yesmaximal, characteristic, complemented
4#14583212C211yesmaximal, characteristic, complemented
5#14583196C211yesmaximal, characteristic, complemented
6#14583147C211yesmaximal, characteristic, complemented
7#14583130C211yesmaximal, characteristic, complemented
8#14583099C211yesmaximal, characteristic, complemented
9#14583049C211yesmaximal, characteristic, complemented
10#14583049C211yesmaximal, characteristic, complemented
11#14582997C211yesmaximal, characteristic, complemented
12#14582971C211yesmaximal, characteristic, complemented
13#14582903C211yesmaximal, characteristic, complemented
14C2×#13063245C211yesmaximal, characteristic, complemented
15C43×#552C211yesmaximal, characteristic, complemented
16C22×D172C2211yescharacteristic
17C22×D172C2211yescharacteristic
18C2×D4×D43—21no—
19C2×#10621884C2211yescharacteristic
20C2×#10621884C2211yescharacteristic, complemented
21C2×#10621884C2211yescharacteristic, complemented
22C2×#10621877C2211yescharacteristic, complemented
23C22×C43⋊C8C2211yescharacteristic, complemented
24#13063322C2211yescharacteristic
25#13063300—21no—
26#13063299C2211yescharacteristic
27#13063299C2211yescharacteristic
28#13063295—21no—
29#13063295—21no—
30C4×D172C2211yescharacteristic
31#13063258C2211yescharacteristic, complemented
32#13063258C2211yescharacteristic, complemented
33#13063256C2211yescharacteristic, complemented
34C4⋊(C43⋊C8)C2211yescharacteristic, complemented
35#13063249C2211yescharacteristic, complemented
36#13063245C2211yescharacteristic
37#13063245C2211yescharacteristic
38#13063245C2211yescharacteristic
39#13063245C2211yescharacteristic, complemented
40#13063245C2211yescharacteristic, complemented
41#13063245C2211yescharacteristic, complemented
42#13063245C2211yescharacteristic, complemented
43#13063245C2211yescharacteristic, complemented
44#13063245C2211yescharacteristic, complemented
45C22⋊(C43⋊C8)C2211yescharacteristic, complemented
46C22⋊(C43⋊C8)C2211yescharacteristic, complemented
47#13063236C2211yescharacteristic, complemented
48#126×C43—21no—
49#125×C43—21no—
50#125×C43C2211yescharacteristic, complemented
51#123×C43—21no—
52C22⋊Q8×C43C2211yescharacteristic, complemented
53C22⋊Q8×C43C2211yescharacteristic, complemented
54C4×Q8×C43C2211yescharacteristic, complemented
55C4×D4×C43C2211yescharacteristic
56#117×C43—21no—
57#117×C43C2211yescharacteristic, complemented
58C2×C4⋊C4×C43C2211yescharacteristic, complemented
59C23×D43—21no—
60C23×D43—21no—
61C2×Q8×C43C2311yescharacteristic
62C2×D4×C43D411yescharacteristic
63C22×C4×C43—21no—
64C22×C4×C43C2311yescharacteristic
65C22×C4×C43C2311yescharacteristic
66C2×C22⋊D43—21no—
67C2×C22⋊D43—21no—
68D4×D43—44no—
69C2×D172—21no—
70C2×D172—21no—
71C2×D172—21no—
72C2×D172—21no—
73C2×D172—22no—
74C2×D172—22no—
75C2×D172—41no—
76C2×D172—41no—
77C2×D172C2311yescharacteristic
78C2×D172C2311yescharacteristic
79C2×D172C2311yescharacteristic
80C2×C4×D43—21no—
81C4⋊C4×C43—21no—
82C4⋊C4×C43—21no—
83C4⋊C4×C43—21no—
84C4⋊C4×C43—21no—
85C4⋊C4×C43—21no—
86C4⋊C4×C43—21no—
87C4⋊C4×C43D411yescharacteristic
88C4⋊C4×C43C2311yescharacteristic
89C4⋊C4×C43C2311yescharacteristic
90C4⋊C4×C43C2311yescharacteristic
91C22⋊C4×C43—21no—
92C22⋊C4×C43—21no—
93C22⋊C4×C43—21no—
94C22⋊C4×C43D411yescharacteristic
95C22⋊C4×C43D411yescharacteristic
96C42×C43—21no—
97C42×C43—21no—
98C42×C43C2311yescharacteristic
99#10621884—22no—
100#10621884—41no—
101#10621884—41no—
102(C2×D43)⋊C4—21no—
103(C2×D43)⋊C4—21no—
104(C2×D43)⋊C4—21no—
105C4⋊Dic43C2311yescharacteristic
106#10621877—41no—
107C2×C43⋊C8—41no—
108C2×C43⋊C8C2311yescharacteristic
109C2×C43⋊C8C2311yescharacteristic
110C2×C43⋊C8C2311yescharacteristic
111C2×C43⋊C8C2311yescharacteristic
112C23×C43C2×D411yescharacteristic
113C23×C43C2×D411yescharacteristic
114C22×D43—21no—
115C22×D43—21no—
116C22×D43—21no—
117C22×D43—42no—
118C22×D43—42no—
119C22×D43—42no—
120C22×D43—42no—
121C22×D43—81no—
122C22×D43—81no—
123Q8×C43—21no—
124Q8×C43—21no—
125D4×C43—22no—
126C2×C4×C43—21no—
127C2×C4×C43—21no—
128C2×C4×C43—21no—
129C2×C4×C43—21no—
130C2×C4×C43—21no—
131C2×C4×C43—21no—
132C2×C4×C43—21no—
133C2×C4×C43—21no—
134C2×C4×C43—42no—
135C2×C4×C43C2×D411yescharacteristic
136C2×C4×C43C2×D411yescharacteristic
137C2×C4×C43C2×D411yescharacteristic
138C2×C4×C43C2×D411yescharacteristic
139C2×C4×C43C2411yescharacteristic
140C22⋊D43—42no—
141C22⋊D43—42no—
142C2×Dic43—21no—
143D172—21no—
144D172—21no—
145D172—21no—
146D172—21no—
147D172—21no—
148D172—42no—
149D172—42no—
150D172—42no—
151D172—81no—
152D172#4112yes—
153C4×D43—42no—
154C43⋊C8—21no—
155C43⋊C8—21no—
156C43⋊C8—21no—
157C43⋊C8—21no—
158C22×C43—21no—
159C22×C43—21no—
160C22×C43—21no—
161C22×C43—21no—
162C22×C43—21no—
163C22×C43AGL(1,ℤ/8)12yes—
164C22×C43C22×D411yescharacteristic
165C2×D43—22no—
166C2×D43—41no—
167C2×D43—41no—
168C2×D43—41no—
169C2×D43—41no—
170C2×D43—41no—
171C2×D43—41no—
172C2×D43—41no—
173C2×D43—41no—
174C2×D43—84no—
175C2×D43—84no—
176C4×C43—21no—
177C4×C43—21no—
178C4×C43—21no—
179C4×C43—21no—
180C4×C43—41no—
181C4×C43—41no—
182C4×C43—41no—
183C4×C43—41no—
184C4×C43C2×#4111yescharacteristic
185C4×C43H(5,𝔽2)11yescharacteristic
186Dic43—41no—
187#3310—431nomaximal, Sylow
188C2×C43—22no—
189C2×C43—41no—
190C2×C43#54611yescharacteristic
191C2×C43C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C43#57711yescharacteristic
193D43—42no—
194D43—81no—
195D43—81no—
196#552D4311yescharacteristic, complemented
197D42—431no—
198#490—431no—
199#482—431no—
200#480—431no—
201#478—431no—
202#468—431no—
203#465—431no—
204#460—431no—
205#450—431no—
206#450—431no—
207#440—431no—
208#435—431no—
209#418—431no—
210C2×#102—431no—
211C43#331011yescharacteristic, complemented, Sylow
212C22×D4—431no—
213C22×D4—431no—
214C22×D4—861no—
215C2×SD16—431no—
216C2×SD16—431no—
217C2×SD16—431no—
218C2×M16—431no—
219C22×C8—431no—
220#127—431no—
221#126—21no—
222#125—21no—
223#125C2×D4311yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D4311yescharacteristic, complemented
226C22⋊Q8C2×D4311yescharacteristic, complemented
227#121—431no—
228#121—431no—
229#121—861no—
230#120—861no—
231#120—861no—
232C4×Q8C2×D4311yescharacteristic, complemented
233C4×D4—431no—
234C4×D4C2×D4311yescharacteristic
235#117—21no—
236#117C2×D4311yescharacteristic, complemented
237C2×C4⋊C4C2×D4311yescharacteristic, complemented
238#107—431no—
239#107—431no—
240#106—431no—
241C4⋊C8—431no—
242#103—431no—
243#102—431no—
244#102—431no—
245#102—431no—
246#102—431no—
247#102—431no—
248#102—431no—
249#102—431no—
250#102—431no—
251#102—431no—
252C22⋊C8—431no—
253C22⋊C8—431no—
254#97—431no—
255C24—861no—
256C24—861no—
257C2×Q8C22×D4311yescharacteristic
258C2×D4—431no—
259C2×D4—431no—
260C2×D4—431no—
261C2×D4—862no—
262C2×D4—862no—
263C2×D4—861no—
264C2×D4—861no—
265C2×D4—861no—
266C2×D4—861no—
267C2×D4—861no—
268C2×D4—861no—
269C2×D4—1721no—
270C2×D4—1721no—
271C2×D4—1724no—
272C2×D4C22⋊D4311yescharacteristic
273C22×C4—21no—
274C22×C4—861no—
275C22×C4C22×D4311yescharacteristic
276C22×C4C22×D4311yescharacteristic
277SD16—862no—
278SD16—1721no—
279SD16—1721no—
280M16—1721no—
281C2×C8—431no—
282C2×C8—431no—
283C2×C8—431no—
284C2×C8—431no—
285C2×C8—1721no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—431no—
293C4⋊C4C22⋊D4311yescharacteristic
294C4⋊C4C22×D4311yescharacteristic
295C4⋊C4C22×D4311yescharacteristic
296C4⋊C4C22×D4311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—861no—
301C22⋊C4—861no—
302C22⋊C4—861no—
303C22⋊C4C22⋊D4311yescharacteristic
304C22⋊C4C22⋊D4311yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D4311yescharacteristic
308C23—861no—
309C23—861no—
310C23—861no—
311C23—3441no—
312C23—3441no—
313C23—1722no—
314C23—1722no—
315C23—1722no—
316C23—1722no—
317C23C2×C22⋊D4311yescharacteristic
318C23C2×C22⋊D4311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—432no—
323D4—861no—
324D4—861no—
325D4—861no—
326D4—861no—
327D4—861no—
328D4—3441no—
329D4—1722no—
330D4—1722no—
331D4—1722no—
332D4—1722no—
333D4—1722no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—861no—
344C2×C4—1722no—
345C2×C4C2×C22⋊D4311yescharacteristic
346C2×C4C2×C22⋊D4311yescharacteristic
347C2×C4C2×C22⋊D4311yescharacteristic
348C2×C4C2×C22⋊D4311yescharacteristic
349C2×C4C23×D4311yescharacteristic
350C8—861no—
351C8—861no—
352C8—861no—
353C8—861no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—862no—
360C22—3444no—
361C22—3444no—
362C22—1721no—
363C22—1721no—
364C22—1721no—
365C22—1721no—
366C22—1721no—
367C22—1721no—
368C22—1721no—
369C22—1721no—
370C22#1306335012yes—
371C22C22×C22⋊D4311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1721no—
381C4#41×D4311yescharacteristic
382C4#1306337111yescharacteristic
383C2—22no—
384C2—41no—
385C2—3441no—
386C2—3441no—
387C2—1722no—
388C2#1458353711yescharacteristic
389C2C2×#1306335011yescharacteristic
390C2#1458370711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#552C2
socleC22×C43C22×D4
hypercenter#552D43
hypocenterC43#3310
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #14583265C2semidirect
3. #14583223C2semidirect
4. #14583212C2semidirect
5. #14583196C2semidirect
6. #14583147C2semidirect
7. #14583130C2semidirect
8. #14583099C2semidirect
9. #14583049C2semidirect
10. #14583049C2semidirect
11. #14582997C2semidirect
12. #14582971C2semidirect
13. #14582903C2semidirect
14. C2×#13063245C2semidirect
15. C43×#552C2semidirect
20. C2×#10621884C22semidirect
21. C2×#10621884C22semidirect
22. C2×#10621877C22semidirect
23. C22×C43⋊C8C22semidirect
31. #13063258C22semidirect
32. #13063258C22semidirect
33. #13063256C22semidirect
34. C4⋊(C43⋊C8)C22semidirect
35. #13063249C22semidirect
39. #13063245C22semidirect
40. #13063245C22semidirect
41. #13063245C22semidirect
42. #13063245C22semidirect
43. #13063245C22semidirect
44. #13063245C22semidirect
45. C22⋊(C43⋊C8)C22semidirect
46. C22⋊(C43⋊C8)C22semidirect
47. #13063236C22semidirect
50. #125×C43C22semidirect
52. C22⋊Q8×C43C22semidirect
53. C22⋊Q8×C43C22semidirect
54. C4×Q8×C43C22semidirect
57. #117×C43C22semidirect
58. C2×C4⋊C4×C43C22semidirect
196. #552D43semidirect
211. C43#3310semidirect
223. #125C2×D43semidirect
225. C22⋊Q8C2×D43semidirect
226. C22⋊Q8C2×D43semidirect
232. C4×Q8C2×D43semidirect
236. #117C2×D43semidirect
237. C2×C4⋊C4C2×D43semidirect