Finite Group Databasebeta

← #20286385

Subgroups — #20286385

Order 6784.

620 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719922C211yesmaximal, characteristic, complemented
2C2×D8×D53C211yesmaximal, characteristic, complemented
3#16811257C211yesmaximal, characteristic, complemented
4D4×C22⋊D53C211yesmaximal, characteristic, complemented
5#16810998C211yesmaximal, characteristic, complemented
6#16810858C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810816C211yesmaximal, characteristic, complemented
10#16810759C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810703C211yesmaximal, characteristic, complemented
14#16810686C211yesmaximal, characteristic, complemented
15C53×#447C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic, complemented
17C22×C22⋊D53—21no—
18C2×D4×D53—21no—
19C2×D4×D53C2211yescharacteristic, complemented
20C2×D4×D53C2211yescharacteristic, complemented
21C2×D424C2211yescharacteristic, complemented
22C2×D424C2211yescharacteristic, complemented
23C2×#11719922—24no—
24C2×#11719922C2211yescharacteristic
25C2×#11719922C2211yescharacteristic, complemented
26C2×#11719922C2211yescharacteristic, complemented
27C2×#11719922C2211yescharacteristic, complemented
28C2×#11719922C2211yescharacteristic, complemented
29D8×D53—24no—
30C22×#88322C2211yescharacteristic, complemented
31C2×C8×D53C2211yescharacteristic, complemented
32#13133213C2211yescharacteristic, complemented
33#13133213C2211yescharacteristic, complemented
34#13133192C2211yescharacteristic, complemented
35#13133191C2211yescharacteristic, complemented
36#13133190—21no—
37#13133189—21no—
38#13133188C2211yescharacteristic, complemented
39#13133187—21no—
40#13133186—21no—
41#13133185—21no—
42#13133185—21no—
43D4×Dic53C2211yescharacteristic, complemented
44#13133179C2211yescharacteristic, complemented
45C4×C22⋊D53C2211yescharacteristic
46#13133150C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133138C2211yescharacteristic, complemented
51#13133137C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53C8×Dic53C2211yescharacteristic, complemented
54C22×D4×C53C2211yescharacteristic, complemented
55C2×D8×C53C2211yescharacteristic, complemented
56C2×D8×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic, complemented
60C22⋊C8×C53C2211yescharacteristic, complemented
61C24×C53—21no—
62C23×D53—21no—
63C23×D53—21no—
64C23×D53—21no—
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—41no—
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
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—44no—
82C2×C22⋊D53D411yescharacteristic
83C22×Dic53—21no—
84D4×D53—22no—
85D4×D53—22no—
86D4×D53—41no—
87D4×D53—41no—
88D4×D53—44no—
89C2×D212—22no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—41no—
93C2×D212C2311yescharacteristic
94C2×D212C2311yescharacteristic
95C2×D212C2311yescharacteristic
96C2×C4×D53—21no—
97C2×C4×D53C2311yescharacteristic
98D8×C53—22no—
99D8×C53—41no—
100C2×C8×C53C2311yescharacteristic
101C2×C8×C53C2311yescharacteristic
102C4⋊C4×C53C2311yescharacteristic
103C22⋊C4×C53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53—21no—
106C22⋊Dic53D411yescharacteristic
107#11719922—22no—
108#11719922—22no—
109#11719922—22no—
110#11719922—22no—
111#11719922D414yescomplemented
112#11719921—21no—
113#11719921—21no—
114#11719921—21no—
115#11719921D411yescharacteristic
116C4⋊Dic53C2311yescharacteristic
117#11719919D411yescharacteristic
118C4×Dic53C2311yescharacteristic
119C2×#88322—22no—
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122D424—22no—
123D424—41no—
124C8×D53—22no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—81no—
131C23×C53C2×D411yescharacteristic
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—42no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—81no—
144C22×D53C2×D411yescharacteristic
145D4×C53—21no—
146D4×C53—21no—
147D4×C53—21no—
148D4×C53—21no—
149D4×C53—21no—
150D4×C53—42no—
151D4×C53—42no—
152D4×C53C2×D412yescomplemented
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53C2×D411yescharacteristic
157C2×C4×C53C2411yescharacteristic
158C22⋊D53—42no—
159C22⋊D53—42no—
160C22⋊D53—42no—
161C22⋊D53—42no—
162C22⋊D53—42no—
163C22⋊D53—42no—
164C22⋊D53—81no—
165C22⋊D53—81no—
166C22⋊D53D814yescomplemented
167C2×Dic53—21no—
168C2×Dic53—42no—
169C2×Dic53C2×D411yescharacteristic
170C2×Dic53C2×D411yescharacteristic
171C2×Dic53C2×D411yescharacteristic
172D212—21no—
173D212—21no—
174D212—21no—
175D212—21no—
176D212—21no—
177D212—42no—
178D212—42no—
179D212C2×D412yescomplemented
180C4×D53—21no—
181C4×D53—21no—
182C4×D53—42no—
183C8×C53—21no—
184C8×C53—21no—
185#88322—21no—
186#88322C2×D412yescomplemented
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—22no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—84no—
199C22×C53C2×D812yescomplemented
200C22×C53C22×D411yescharacteristic
201C2×D53—21no—
202C2×D53—21no—
203C2×D53—22no—
204C2×D53—41no—
205C2×D53—41no—
206C2×D53—41no—
207C2×D53—41no—
208C2×D53—41no—
209C2×D53—41no—
210C2×D53—41no—
211C2×D53—41no—
212C2×D53—84no—
213C2×D53—84no—
214C2×D53—84no—
215C2×D53C2×D812yescomplemented
216C4×C53—21no—
217C4×C53—41no—
218C4×C53C22×D411yescharacteristic
219C4×C53H(5,𝔽2)11yescharacteristic
220Dic53—21no—
221Dic53—21no—
222Dic53—41no—
223Dic53C2×D812yescomplemented
224D4×D8—531nomaximal, Sylow
225C2×C53—22no—
226C2×C53—42no—
227C2×C53—81no—
228C2×C53—81no—
229C2×C53D4211yescharacteristic
230C2×C53C22×D811yescharacteristic
231C2×C53#57611yescharacteristic
232D53—22no—
233D53—42no—
234D53—81no—
235D53—81no—
236C22×D8—531no—
237C22×D8—531no—
238D42—531no—
239D42—531no—
240#493—531no—
241#466—531no—
242#466—531no—
243#459—531no—
244#459—531no—
245#447—531no—
246#447—531no—
247#447—531no—
248#447D5311yescharacteristic, complemented
249C4×D8—531no—
250C8×D4—531no—
251C53D4×D811yescharacteristic, complemented, Sylow
252C22×D4—531no—
253C22×D4—531no—
254C22×D4—531no—
255C22×D4—1061no—
256C22×D4—1061no—
257C22×D4C2×D5311yescharacteristic, complemented
258C2×D8—531no—
259C2×D8—531no—
260C2×D8—531no—
261C2×D8—531no—
262C2×D8—531no—
263C2×D8—531no—
264C2×D8—531no—
265C2×D8—1064no—
266C2×D8—1064no—
267C2×D8C2×D5311yescharacteristic, complemented
268C2×D8C2×D5311yescharacteristic, complemented
269C22×C8—531no—
270C22×C8—531no—
271#127—531no—
272#127—531no—
273#121—531no—
274#121—531no—
275#121—531no—
276#121—1061no—
277#121—1061no—
278#121C2×D5311yescharacteristic, complemented
279#120—1061no—
280#120—1061no—
281#120—1061no—
282#120—1061no—
283C4×D4—531no—
284C4×D4—531no—
285C4×D4—531no—
286#107—531no—
287C4⋊C8—531no—
288#102—531no—
289#102—531no—
290#102—531no—
291#102—531no—
292#102C2×D5311yescharacteristic, complemented
293#102C2×D5311yescharacteristic, complemented
294C22⋊C8—531no—
295C22⋊C8C2×D5311yescharacteristic, complemented
296C4×C8—531no—
297C24—21no—
298C24—1061no—
299C24—1061no—
300C24—1061no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—531no—
306C2×D4—531no—
307C2×D4—531no—
308C2×D4—531no—
309C2×D4—1062no—
310C2×D4—1061no—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4—1061no—
315C2×D4—1061no—
316C2×D4—1061no—
317C2×D4—1062no—
318C2×D4—1062no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2124no—
326C2×D4—2124no—
327C2×D4C22×D5311yescharacteristic
328C2×D4C22×D5311yescharacteristic
329C2×D4C22×D5311yescharacteristic
330C22×C4—531no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4C22×D5311yescharacteristic
334D8—22no—
335D8—41no—
336D8—534no—
337D8—1062no—
338D8—1062no—
339D8—1062no—
340D8—1062no—
341D8—1062no—
342D8—2121no—
343C2×C8—531no—
344C2×C8—531no—
345C2×C8—1062no—
346C2×C8—1062no—
347C2×C8C22×D5311yescharacteristic
348C2×C8C22×D5311yescharacteristic
349C4⋊C4—531no—
350C4⋊C4—531no—
351C4⋊C4C22×D5311yescharacteristic
352C22⋊C4—21no—
353C22⋊C4—531no—
354C22⋊C4—531no—
355C22⋊C4—1061no—
356C22⋊C4—1061no—
357C22⋊C4—1061no—
358C22⋊C4—1061no—
359C22⋊C4—1061no—
360C42—531no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—531no—
368C23—1061no—
369C23—1061no—
370C23—1061no—
371C23—4241no—
372C23—4241no—
373C23—4241no—
374C23—2122no—
375C23—2122no—
376C23—2122no—
377C23—2122no—
378C23—2122no—
379C23—2122no—
380C23D4×D5311yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—21no—
386D4—42no—
387D4—42no—
388D4—534no—
389D4—532no—
390D4—1061no—
391D4—1061no—
392D4—1061no—
393D4—1061no—
394D4—1061no—
395D4—4241no—
396D4—4241no—
397D4—2122no—
398D4—2122no—
399D4—2122no—
400D4—2122no—
401D4—2122no—
402D4—2122no—
403D4—2122no—
404D4—2122no—
405D4D4×D5312yescomplemented
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—531no—
410C2×C4—531no—
411C2×C4—531no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—2122no—
416C2×C4—2122no—
417C2×C4D4×D5311yescharacteristic
418C2×C4C23×D5311yescharacteristic
419C8—21no—
420C8—21no—
421C8—532no—
422C8—1061no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—84no—
435C22—532no—
436C22—1061no—
437C22—1061no—
438C22—1062no—
439C22—4244no—
440C22—4244no—
441C22—4244no—
442C22—2121no—
443C22—2121no—
444C22—2121no—
445C22—2121no—
446C22—2121no—
447C22—2121no—
448C22—2121no—
449C22—2121no—
450C22D8×D5312yescomplemented
451C22C2×D4×D5311yescharacteristic
452C4—21no—
453C4—41no—
454C4—532no—
455C4—1061no—
456C4—1061no—
457C4—2121no—
458C4C2×D4×D5311yescharacteristic
459C4#1313326211yescharacteristic
460C2—22no—
461C2—42no—
462C2—81no—
463C2—81no—
464C2—1062no—
465C2—4241no—
466C2—4241no—
467C2—2122no—
468C2#1681125511yescharacteristic
469C2C2×D8×D5311yescharacteristic
470C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53D4×D8
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#11719922C2semidirect
2. C2×D8×D53C2semidirect
3. #16811257C2semidirect
4. D4×C22⋊D53C2semidirect
5. #16810998C2semidirect
6. #16810858C2semidirect
7. #16810857C2semidirect
8. #16810817C2semidirect
9. #16810816C2semidirect
10. #16810759C2semidirect
11. #16810758C2semidirect
12. #16810755C2semidirect
13. #16810703C2semidirect
14. #16810686C2semidirect
15. C53×#447C2semidirect
16. C22×D212C22semidirect
19. C2×D4×D53C22semidirect
20. C2×D4×D53C22semidirect
21. C2×D424C22semidirect
22. C2×D424C22semidirect
25. C2×#11719922C22semidirect
26. C2×#11719922C22semidirect
27. C2×#11719922C22semidirect
28. C2×#11719922C22semidirect
30. C22×#88322C22semidirect
31. C2×C8×D53C22semidirect
32. #13133213C22semidirect
33. #13133213C22semidirect
34. #13133192C22semidirect
35. #13133191C22semidirect
38. #13133188C22semidirect
43. D4×Dic53C22semidirect
44. #13133179C22semidirect
46. #13133150C22semidirect
47. #13133146C22semidirect
48. #13133139C22semidirect
49. #13133139C22semidirect
50. #13133138C22semidirect
51. #13133137C22semidirect
52. #13133129C22semidirect
53. C8×Dic53C22semidirect
54. C22×D4×C53C22semidirect
55. C2×D8×C53C22semidirect
56. C2×D8×C53C22semidirect
57. #121×C53C22semidirect
58. #102×C53C22semidirect
59. #102×C53C22semidirect
60. C22⋊C8×C53C22semidirect
111. #11719922D4semidirect
152. D4×C53C2×D4semidirect
166. C22⋊D53D8semidirect
179. D212C2×D4semidirect
186. #88322C2×D4semidirect
199. C22×C53C2×D8semidirect
215. C2×D53C2×D8semidirect
223. Dic53C2×D8semidirect
248. #447D53semidirect
251. C53D4×D8semidirect
257. C22×D4C2×D53semidirect
267. C2×D8C2×D53semidirect
268. C2×D8C2×D53semidirect
278. #121C2×D53semidirect
292. #102C2×D53semidirect
293. #102C2×D53semidirect
295. C22⋊C8C2×D53semidirect
405. D4D4×D53semidirect
450. C22D8×D53semidirect