Finite Group Databasebeta

← #20286199

Subgroups — #20286199

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×#13133244C211yesmaximal, characteristic, complemented
2C22×#11719922C211yesmaximal, characteristic, complemented
3#16811264C211yesmaximal, characteristic, complemented
4D4×C22⋊D53C211yesmaximal, characteristic, complemented
5#16811009C211yesmaximal, characteristic, complemented
6#16810875C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810817C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11#16810759C211yesmaximal, characteristic, complemented
12#16810758C211yesmaximal, characteristic, complemented
13#16810732C211yesmaximal, characteristic, complemented
14#16810691C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C22×D212C2211yescharacteristic, complemented
18C22×C22⋊D53—21no—
19C2×D4×D53C2211yescharacteristic, complemented
20#13133244—24no—
21C2×#11719924C2211yescharacteristic, complemented
22C2×D424C2211yescharacteristic, complemented
23C2×D424C2211yescharacteristic, complemented
24C2×#11719922—24no—
25C2×#11719922C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27C2×#11719922C2211yescharacteristic, complemented
28C2×#11719922C2211yescharacteristic, complemented
29C2×#11719912C2211yescharacteristic, complemented
30C22×#88322C2211yescharacteristic, complemented
31#13133213C2211yescharacteristic, complemented
32#13133206C2211yescharacteristic, complemented
33#13133202—21no—
34#13133192—21no—
35#13133192C2211yescharacteristic, complemented
36#13133191C2211yescharacteristic
37#13133189—21no—
38#13133188C2211yescharacteristic, complemented
39#13133187—21no—
40#13133186—21no—
41#13133185—21no—
42D4×Dic53C2211yescharacteristic, complemented
43#13133179C2211yescharacteristic, complemented
44C4×C22⋊D53C2211yescharacteristic
45C22⋊C4×D53—21no—
46#13133148C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133141C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic, complemented
51#13133137C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53#13133127C2211yescharacteristic, complemented
54C22×D4×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic, complemented
56C2×SD16×C53C2211yescharacteristic, complemented
57C22⋊Q8×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic, complemented
60C22⋊C8×C53C2211yescharacteristic, complemented
61C24×C53—21no—
62C23×D53—21no—
63C23×D53—21no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—41no—
68C2×D4×C53C2311yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—44no—
77C2×C22⋊D53D411yescharacteristic
78C22×Dic53—21no—
79#11719949—22no—
80#11719949—41no—
81D4×D53—22no—
82D4×D53—41no—
83C2×D212—22no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—41no—
88C2×D212C2311yescharacteristic
89C2×D212C2311yescharacteristic
90C2×D212C2311yescharacteristic
91C2×C4×D53—21no—
92C2×C4×D53—21no—
93C2×C4×D53C2311yescharacteristic
94SD16×C53—22no—
95SD16×C53—41no—
96C2×C8×C53C2311yescharacteristic
97C2×C8×C53C2311yescharacteristic
98C4⋊C4×C53—21no—
99C4⋊C4×C53C2311yescharacteristic
100C22⋊C4×C53—21no—
101C22⋊Dic53—21no—
102C22⋊Dic53—21no—
103C22⋊Dic53D411yescharacteristic
104#11719924—22no—
105#11719922—22no—
106#11719922—22no—
107#11719922—22no—
108#11719922D414yescomplemented
109#11719921—21no—
110#11719921—21no—
111#11719921—21no—
112#11719921—21no—
113#11719921—21no—
114#11719921D411yescharacteristic
115C4⋊Dic53C2311yescharacteristic
116#11719919D411yescharacteristic
117C4×Dic53C2311yescharacteristic
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121D424—22no—
122D424—41no—
123#11719912—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—81no—
129C23×C53C2×D411yescharacteristic
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—42no—
136C22×D53—42no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53C2×D411yescharacteristic
140Q8×C53—21no—
141Q8×C53—21no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53—42no—
146D4×C53C2×D412yes—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2411yescharacteristic
153C22⋊D53—22no—
154C22⋊D53—42no—
155C22⋊D53—42no—
156C22⋊D53—42no—
157C22⋊D53—42no—
158C22⋊D53—81no—
159C2×Dic53—21no—
160C2×Dic53—42no—
161C2×Dic53C2×D411yescharacteristic
162C2×Dic53C2×D411yescharacteristic
163C2×Dic53C2×D411yescharacteristic
164D212—21no—
165D212—21no—
166D212—21no—
167D212—21no—
168D212—21no—
169D212—42no—
170D212—42no—
171D212—81no—
172D212C2×D412yescomplemented
173C4×D53—21no—
174C4×D53—21no—
175C4×D53—42no—
176C4×D53—81no—
177C8×C53—21no—
178C8×C53—21no—
179#88322—21no—
180#88322C2×D412yescomplemented
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—22no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—84no—
190C22×C53AGL(1,ℤ/8)12yescomplemented
191C22×C53C22×D411yescharacteristic
192C2×D53—21no—
193C2×D53—22no—
194C2×D53—21no—
195C2×D53—21no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—41no—
203C2×D53—41no—
204C2×D53—84no—
205C2×D53—84no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53C22×D411yescharacteristic
210C4×C53H(5,𝔽2)11yescharacteristic
211Dic53—21no—
212Dic53—21no—
213Dic53—21no—
214Dic53—41no—
215#3264—531nomaximal, Sylow
216C2×C53—22no—
217C2×C53—42no—
218C2×C53—81no—
219C2×C53D4211yescharacteristic
220C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C53#57611yescharacteristic
222D53—41no—
223D53—42no—
224D53—81no—
225D53—81no—
226C2×AGL(1,ℤ/8)—531no—
227C22×D8—531no—
228#546—531no—
229D42—531no—
230#496—531no—
231#469—531no—
232#465—531no—
233#463—531no—
234#459—531no—
235#450D5311yescharacteristic, complemented
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—531no—
245C22×D4—1061no—
246C22×D4C2×D5311yescharacteristic
247AGL(1,ℤ/8)—1064no—
248C2×SD16—531no—
249C2×SD16C2×D5311yescharacteristic, complemented
250C2×SD16C2×D5311yescharacteristic, complemented
251C2×D8—531no—
252C2×D8—531no—
253C2×D8—531no—
254C2×D8—531no—
255C2×D8—531no—
256C2×D8—531no—
257C2×D8—1064no—
258C2×M16—531no—
259C22×C8—531no—
260#127—531no—
261#124—531no—
262#123—1061no—
263C22⋊Q8C2×D5311yescharacteristic, complemented
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#102C2×D5311yescharacteristic, complemented
283#102C2×D5311yescharacteristic, complemented
284C22⋊C8—531no—
285C22⋊C8C2×D5311yescharacteristic, complemented
286#97—531no—
287C24—21no—
288C24—1061no—
289C24—1061no—
290#41—1062no—
291#41—2121no—
292C2×Q8C22×D5311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—531no—
297C2×D4—531no—
298C2×D4—531no—
299C2×D4—531no—
300C2×D4—1062no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1062no—
307C2×D4—1061no—
308C2×D4—1061no—
309C2×D4—1061no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2124no—
313C2×D4C22×D5311yescharacteristic
314C2×D4C22×D5311yescharacteristic
315C22×C4—531no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4C22×D5311yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—1062no—
323D8—534no—
324D8—1062no—
325D8—1062no—
326D8—1062no—
327D8—1062no—
328D8—2121no—
329M16—1062no—
330C2×C8—531no—
331C2×C8—531no—
332C2×C8—1062no—
333C2×C8C22×D5311yescharacteristic
334C2×C8C22×D5311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—531no—
337C4⋊C4—531no—
338C4⋊C4C22×D5311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—531no—
341C22⋊C4—531no—
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—42no—
353C23—42no—
354C23—81no—
355C23—531no—
356C23—1061no—
357C23—1061no—
358C23—1061no—
359C23—4241no—
360C23—4241no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23—2122no—
365C23D4×D5311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—532no—
373D4—1062no—
374D4—1061no—
375D4—1061no—
376D4—1061no—
377D4—1061no—
378D4—1061no—
379D4—4241no—
380D4—4241no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4—2122no—
387D4D4×D5312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—531no—
393C2×C4—531no—
394C2×C4—531no—
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—21no—
405C8—532no—
406C8—1061no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—1061no—
417C22—1061no—
418C22—1061no—
419C22—1062no—
420C22—4244no—
421C22—4244no—
422C22—2121no—
423C22—2121no—
424C22—2121no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22—2121no—
429C22—2121no—
430C22#1313324412yescomplemented
431C22C2×D4×D5311yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—1061no—
436C4—1061no—
437C4—1061no—
438C4—2121no—
439C4C2×D4×D5311yescharacteristic
440C4#1313326211yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—4241no—
445C2—4241no—
446C2—2121no—
447C2—2122no—
448C2#1681125511yescharacteristic
449C2C2×#1313324411yescharacteristic
450C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3264
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. C22×#11719922C2semidirect
3. #16811264C2semidirect
4. D4×C22⋊D53C2semidirect
5. #16811009C2semidirect
6. #16810875C2semidirect
7. #16810854C2semidirect
8. #16810840C2semidirect
9. #16810817C2semidirect
10. #16810767C2semidirect
11. #16810759C2semidirect
12. #16810758C2semidirect
13. #16810732C2semidirect
14. #16810691C2semidirect
15. C53×#450C2semidirect
16. C2×#11719949C22semidirect
17. C22×D212C22semidirect
19. C2×D4×D53C22semidirect
21. C2×#11719924C22semidirect
22. C2×D424C22semidirect
23. C2×D424C22semidirect
27. C2×#11719922C22semidirect
28. C2×#11719922C22semidirect
29. C2×#11719912C22semidirect
30. C22×#88322C22semidirect
31. #13133213C22semidirect
32. #13133206C22semidirect
35. #13133192C22semidirect
38. #13133188C22semidirect
42. D4×Dic53C22semidirect
43. #13133179C22semidirect
46. #13133148C22semidirect
47. #13133146C22semidirect
48. #13133141C22semidirect
49. #13133139C22semidirect
50. #13133139C22semidirect
51. #13133137C22semidirect
52. #13133129C22semidirect
53. #13133127C22semidirect
55. C2×SD16×C53C22semidirect
56. C2×SD16×C53C22semidirect
57. C22⋊Q8×C53C22semidirect
58. #102×C53C22semidirect
59. #102×C53C22semidirect
60. C22⋊C8×C53C22semidirect
108. #11719922D4semidirect
172. D212C2×D4semidirect
180. #88322C2×D4semidirect
190. C22×C53AGL(1,ℤ/8)semidirect
235. #450D53semidirect
241. C53#3264semidirect
249. C2×SD16C2×D53semidirect
250. C2×SD16C2×D53semidirect
263. C22⋊Q8C2×D53semidirect
282. #102C2×D53semidirect
283. #102C2×D53semidirect
285. C22⋊C8C2×D53semidirect
430. C22#13133244semidirect