Finite Group Databasebeta

← #20286195

Subgroups — #20286195

Order 6784.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133241C211yesmaximal, characteristic, complemented
2#16811267C211yesmaximal, characteristic, complemented
3D4×C22⋊D53C211yesmaximal, characteristic, complemented
4#16811011C211yesmaximal, characteristic, complemented
5#16810873C211yesmaximal, characteristic, complemented
6#16810852C211yesmaximal, characteristic, complemented
7#16810836C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810774C211yesmaximal, characteristic, complemented
10#16810765C211yesmaximal, characteristic, complemented
11#16810757C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810735C211yesmaximal, characteristic, complemented
14#16810694C211yesmaximal, characteristic, complemented
15C22×D8×C53C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic, complemented
17C22×C22⋊D53—21no—
18C2×D4×D53C2211yescharacteristic, complemented
19#13133241—24no—
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic, complemented
22C2×#11719923C2211yescharacteristic, complemented
23C2×#11719922C2211yescharacteristic, complemented
24C2×#11719922C2211yescharacteristic, complemented
25C2×#11719912C2211yescharacteristic, complemented
26#13133213C2211yescharacteristic, complemented
27#13133204C2211yescharacteristic, complemented
28#13133198—21no—
29#13133196C2211yescharacteristic, complemented
30#13133191C2211yescharacteristic
31#13133189—21no—
32#13133189—21no—
33#13133188C2211yescharacteristic, complemented
34#13133188C2211yescharacteristic, complemented
35#13133186—21no—
36#13133186—21no—
37#13133185—21no—
38D4×Dic53C2211yescharacteristic, complemented
39D4×Dic53C2211yescharacteristic, complemented
40C4×C22⋊D53C2211yescharacteristic
41C2×C22⋊Dic53—21no—
42#13133149C2211yescharacteristic, complemented
43#13133146C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49#13133138C2211yescharacteristic, complemented
50C22⋊#88322C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#13133127C2211yescharacteristic, complemented
53C22×D4×C53C2211yescharacteristic
54C22×D4×C53C2211yescharacteristic, complemented
55C2×D8×C53—24no—
56C2×D8×C53C2211yescharacteristic
57C2×D8×C53C2211yescharacteristic
58C2×D8×C53C2211yescharacteristic, complemented
59C2×D8×C53C2211yescharacteristic, complemented
60C22×C8×C53C2211yescharacteristic, complemented
61C24×C53—21no—
62C24×C53—21no—
63C23×D53—21no—
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53C2311yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C2×C22⋊D53—21no—
82C2×C22⋊D53—44no—
83C2×C22⋊D53D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87#11719947—22no—
88#11719947—41no—
89D4×D53—22no—
90D4×D53—41no—
91C2×D212C2311yescharacteristic
92C2×C4×D53C2311yescharacteristic
93C2×Dic106C2311yescharacteristic
94D8×C53—22no—
95D8×C53—22no—
96D8×C53—22no—
97D8×C53D414yescomplemented
98C2×C8×C53—22no—
99C2×C8×C53C2311yescharacteristic
100C2×C8×C53C2311yescharacteristic
101C22⋊Dic53—21no—
102C22⋊Dic53—21no—
103C22⋊Dic53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53—21no—
106C22⋊Dic53—21no—
107C22⋊Dic53—21no—
108C22⋊Dic53D411yescharacteristic
109#11719923—22no—
110#11719923—41no—
111#11719922—22no—
112#11719922—41no—
113#11719921—21no—
114#11719921D411yescharacteristic
115C4⋊Dic53C2311yescharacteristic
116C4⋊Dic53C2311yescharacteristic
117#11719919—21no—
118#11719919D411yescharacteristic
119C4×Dic53C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122#11719913—22no—
123#11719912—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—81no—
133C23×C53—81no—
134C23×C53C2×D411yescharacteristic
135C22×D53—21no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—81no—
139C22×D53C2×D411yescharacteristic
140D4×C53—21no—
141D4×C53—21no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53—21no—
146D4×C53—42no—
147D4×C53—42no—
148D4×C53C2×D412yes—
149D4×C53C2×D412yescomplemented
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53C2×D411yescharacteristic
153C2×C4×C53C2411yescharacteristic
154C22⋊D53—22no—
155C22⋊D53—42no—
156C22⋊D53—42no—
157C22⋊D53—42no—
158C22⋊D53—42no—
159C22⋊D53—42no—
160C22⋊D53—81no—
161C22⋊D53—81no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—81no—
168C2×Dic53C2×D411yescharacteristic
169C2×Dic53C2×D411yescharacteristic
170C2×Dic53C2×D411yescharacteristic
171D212—21no—
172D212—21no—
173C4×D53—21no—
174C4×D53—21no—
175Dic106—21no—
176Dic106—21no—
177C8×C53—21no—
178C8×C53C2×D412yescomplemented
179#88322—21no—
180#88322—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—84no—
196C22×C53—84no—
197C22×C53AGL(1,ℤ/8)12yescomplemented
198C22×C53C22×D411yescharacteristic
199C2×D53—21no—
200C2×D53—21no—
201C2×D53—21no—
202C2×D53—41no—
203C2×D53—41no—
204C2×D53—41no—
205C2×D53—84no—
206C4×C53—21no—
207C4×C53C22×D411yescharacteristic
208C4×C53H(5,𝔽2)11yescharacteristic
209Dic53—21no—
210Dic53—21no—
211Dic53—21no—
212Dic53—41no—
213Dic53—41no—
214Dic53—41no—
215#3264—531nomaximal, Sylow
216C2×C53—22no—
217C2×C53—42no—
218C2×C53—42no—
219C2×C53—81no—
220C2×C53—81no—
221C2×C53D4211yescharacteristic
222C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
223C2×C53#57611yescharacteristic
224D53—41no—
225D53—81no—
226C2×AGL(1,ℤ/8)—531no—
227C22×D8D5311yescharacteristic, complemented
228#546—531no—
229D42—531no—
230#496—531no—
231#469—531no—
232#465—531no—
233#463—531no—
234#459—531no—
235#450—531no—
236#449—531no—
237#447—531no—
238#447—531no—
239#442—531no—
240#435—531no—
241C53#326411yescharacteristic, complemented, Sylow
242C2×#41—531no—
243C22×D4—531no—
244C22×D4—1061no—
245C22×D4C2×D5311yescharacteristic
246C22×D4C2×D5311yescharacteristic, complemented
247AGL(1,ℤ/8)—1064no—
248C2×SD16—531no—
249C2×SD16—531no—
250C2×SD16—531no—
251C2×D8—24no—
252C2×D8—531no—
253C2×D8—531no—
254C2×D8C2×D5311yescharacteristic
255C2×D8C2×D5311yescharacteristic
256C2×D8C2×D5311yescharacteristic, complemented
257C2×D8C2×D5311yescharacteristic, complemented
258C2×M16—531no—
259C22×C8C2×D5311yescharacteristic, complemented
260#127—531no—
261#124—531no—
262#123—1061no—
263C22⋊Q8—531no—
264#121—531no—
265#121—531no—
266#121—531no—
267#121—1061no—
268#121—1061no—
269#120—1061no—
270#120—1061no—
271#120—1061no—
272C4×D4—531no—
273C4×D4—531no—
274C4×D4—531no—
275C2×C22⋊C4—1061no—
276#106—531no—
277C4⋊C8—531no—
278#103—531no—
279#102—531no—
280#102—531no—
281#102—531no—
282#102—531no—
283#102—531no—
284C22⋊C8—531no—
285C22⋊C8—531no—
286#97—531no—
287C24—21no—
288C24—21no—
289C24—1061no—
290#41—1062no—
291#41—2121no—
292C2×Q8—531no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—531no—
300C2×D4—531no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—1062no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C2×D4—2121no—
310C2×D4—2124no—
311C2×D4C22×D5311yescharacteristic
312C2×D4C22×D5311yescharacteristic
313C2×D4C22×D5311yescharacteristic
314C2×D4C22×D5311yescharacteristic
315C22×C4—531no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4C22×D5311yescharacteristic
320SD16—1062no—
321SD16—1062no—
322SD16—2121no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—1062no—
327D8—2121no—
328D8C22⋊D5314yescomplemented
329M16—1062no—
330C2×C8—22no—
331C2×C8—531no—
332C2×C8—531no—
333C2×C8C22×D5311yescharacteristic
334C2×C8C22×D5311yescharacteristic
335C4⋊C4—531no—
336C4⋊C4—531no—
337C4⋊C4—531no—
338C4⋊C4—1061no—
339C22⋊C4—531no—
340C22⋊C4—531no—
341C22⋊C4—1061no—
342C22⋊C4—1061no—
343C22⋊C4—1061no—
344C22⋊C4—1061no—
345C22⋊C4—1061no—
346C22⋊C4—1061no—
347C22⋊C4—1061no—
348C22⋊C4—1061no—
349C42—531no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—531no—
361C23—1061no—
362C23—4241no—
363C23—2122no—
364C23—2122no—
365C23D4×D5311yescharacteristic
366Q8—1061no—
367Q8—1061no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—1062no—
377D4—1061no—
378D4—1061no—
379D4—4241no—
380D4—4241no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4C2×C22⋊D5312yes—
387D4C2×C22⋊D5312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—531no—
391C2×C4—531no—
392C2×C4—531no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—4241no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4D4×D5311yescharacteristic
402C2×C4C23×D5311yescharacteristic
403C8—21no—
404C8—1061no—
405C8—1061no—
406C8C2×C22⋊D5312yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—1061no—
424C22—1061no—
425C22—1061no—
426C22—4244no—
427C22—2121no—
428C22—2121no—
429C22—2121no—
430C22#1313324112yescomplemented
431C22C2×D4×D5311yescharacteristic
432C4—21no—
433C4—1061no—
434C4—1061no—
435C4—1061no—
436C4—2121no—
437C4—2121no—
438C4—2121no—
439C4C22×C22⋊D5311yescharacteristic
440C4#1313326211yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—4241no—
447C2—2121no—
448C2D4×C22⋊D5311yescharacteristic
449C2C2×#1313324111yescharacteristic
450C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC22×D8×C53C2
socleC22×C53C22×D4
hypercenterC22×D8D53
hypocenterC53#3264
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. #16811267C2semidirect
3. D4×C22⋊D53C2semidirect
4. #16811011C2semidirect
5. #16810873C2semidirect
6. #16810852C2semidirect
7. #16810836C2semidirect
8. #16810816C2semidirect
9. #16810774C2semidirect
10. #16810765C2semidirect
11. #16810757C2semidirect
12. #16810755C2semidirect
13. #16810735C2semidirect
14. #16810694C2semidirect
15. C22×D8×C53C2semidirect
16. C2×#11719947C22semidirect
18. C2×D4×D53C22semidirect
20. C2×#11719913C22semidirect
21. C2×#11719923C22semidirect
22. C2×#11719923C22semidirect
23. C2×#11719922C22semidirect
24. C2×#11719922C22semidirect
25. C2×#11719912C22semidirect
26. #13133213C22semidirect
27. #13133204C22semidirect
29. #13133196C22semidirect
33. #13133188C22semidirect
34. #13133188C22semidirect
38. D4×Dic53C22semidirect
39. D4×Dic53C22semidirect
42. #13133149C22semidirect
43. #13133146C22semidirect
44. #13133142C22semidirect
45. #13133139C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
48. #13133138C22semidirect
49. #13133138C22semidirect
50. C22⋊#88322C22semidirect
51. #13133129C22semidirect
52. #13133127C22semidirect
54. C22×D4×C53C22semidirect
58. C2×D8×C53C22semidirect
59. C2×D8×C53C22semidirect
60. C22×C8×C53C22semidirect
97. D8×C53D4semidirect
149. D4×C53C2×D4semidirect
178. C8×C53C2×D4semidirect
197. C22×C53AGL(1,ℤ/8)semidirect
227. C22×D8D53semidirect
241. C53#3264semidirect
246. C22×D4C2×D53semidirect
256. C2×D8C2×D53semidirect
257. C2×D8C2×D53semidirect
259. C22×C8C2×D53semidirect
328. D8C22⋊D53semidirect
387. D4C2×C22⋊D53semidirect
406. C8C2×C22⋊D53semidirect
430. C22#13133241semidirect