Finite Group Databasebeta

← #46845627

Subgroups — #46845627

Order 34688.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576205C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3C2×D8×D271C211yesmaximal, characteristic, complemented
4#31774463C211yesmaximal, characteristic, complemented
5#31774044C211yesmaximal, characteristic, complemented
6#31774043C211yesmaximal, characteristic, complemented
7#31774040C211yesmaximal, characteristic, complemented
8#31774039C211yesmaximal, characteristic, complemented
9#31774032C211yesmaximal, characteristic, complemented
10#31774031C211yesmaximal, characteristic, complemented
11#31774029C211yesmaximal, characteristic, complemented
12#31773900C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#447×C271C211yesmaximal, characteristic, complemented
16#22576205—24no—
17#22576205—24no—
18C2×#16864654C2211yescharacteristic
19C2×#16864654C2211yescharacteristic, complemented
20C2×#16864653—21no—
21C2×#16864653C2211yescharacteristic
22C2×#16864653C2211yescharacteristic, complemented
23C22×C22⋊D271—21no—
24C22×C22⋊D271—21no—
25C2×D4×D271—21no—
26C2×D4×D271C2211yescharacteristic
27C2×D4×D271C2211yescharacteristic, complemented
28C2×D4×D271C2211yescharacteristic, complemented
29#22576184—24no—
30C2×#16864643C2211yescharacteristic, complemented
31C2×#16864642C2211yescharacteristic, complemented
32C2×D2168C2211yescharacteristic, complemented
33C2×#16864639C2211yescharacteristic, complemented
34C2×#16864639C2211yescharacteristic, complemented
35C2×#16864639C2211yescharacteristic, complemented
36D8×D271—24no—
37C2×#16864637C2211yescharacteristic, complemented
38C2×C8×D271C2211yescharacteristic, complemented
39#22576156—21no—
40#22576146—21no—
41#22576135C2211yescharacteristic, complemented
42#22576134C2211yescharacteristic, complemented
43#22576131C2211yescharacteristic, complemented
44#22576130—21no—
45#22576128—21no—
46#22576118C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52#22576080C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C22×D4×C271C2211yescharacteristic, complemented
57C2×D8×C271C2211yescharacteristic, complemented
58C2×D8×C271C2211yescharacteristic, complemented
59#121×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271—21no—
65C23×D271—21no—
66#16864654—22no—
67#16864654—22no—
68#16864654—22no—
69#16864654—22no—
70#16864653—28no—
71#16864653—22no—
72#16864653—22no—
73#16864653—22no—
74#16864653D414yescomplemented
75C2×Dic542D411yescharacteristic
76C2×Dic542C2311yescharacteristic
77C2×D1084—21no—
78C2×D1084D411yescharacteristic
79C2×D1084C2311yescharacteristic
80C2×D1084C2311yescharacteristic
81C2×C22⋊D271—28no—
82C2×C22⋊D271—28no—
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86C2×C22⋊D271—21no—
87C2×C22⋊D271—21no—
88C2×C22⋊D271—21no—
89C2×C22⋊D271—21no—
90C2×C22⋊D271—21no—
91C2×C22⋊D271D411yescharacteristic
92C2×C22⋊D271D411yescharacteristic
93D4×D271—28no—
94D4×D271—22no—
95D4×D271—22no—
96D4×D271—22no—
97D4×D271—22no—
98D4×D271—22no—
99D4×D271D414yescomplemented
100C22×Dic271—21no—
101C22×Dic271—21no—
102C2×C4×D271—21no—
103C2×C4×D271C2311yescharacteristic
104C2×C4×D271C2311yescharacteristic
105#16864643—22no—
106#16864642—22no—
107D2168—22no—
108#16864639—22no—
109#16864639—22no—
110#16864639—22no—
111#16864637—22no—
112C2×C271⋊C8C2311yescharacteristic
113C2×C271⋊C8C2311yescharacteristic
114C8×D271—22no—
115C4⋊Dic271C2311yescharacteristic
116C22⋊Dic271—21no—
117(C2×D271)⋊C4—21no—
118(C2×D271)⋊C4—21no—
119(C2×D271)⋊C4—21no—
120C4×Dic271—21no—
121C24×C271—21no—
122C2×D4×C271—22no—
123C2×D4×C271—22no—
124C2×D4×C271—21no—
125C2×D4×C271—21no—
126C2×D4×C271C2311yescharacteristic
127C2×D4×C271C2311yescharacteristic
128C2×D4×C271C2311yescharacteristic
129C22×C4×C271C2311yescharacteristic
130D8×C271—22no—
131D8×C271—22no—
132C2×C8×C271C2311yescharacteristic
133C2×C8×C271C2311yescharacteristic
134C4⋊C4×C271C2311yescharacteristic
135C22⋊C4×C271—21no—
136C22×D271—21no—
137C22×D271—24no—
138C22×D271—24no—
139C22×D271—21no—
140C22×D271—42no—
141C22×D271—81no—
142C22×D271—81no—
143C22×D271—81no—
144C22×D271—81no—
145C22×D271—81no—
146C22×D271—81no—
147C22×D271C2×D411yescharacteristic
148C22×D271C2×D411yescharacteristic
149Dic542—21no—
150Dic542—22no—
151Dic542C2×D412yes—
152D1084—22no—
153D1084—21no—
154D1084—21no—
155D1084—21no—
156D1084—81no—
157D1084C2×D412yes—
158C22⋊D271—24no—
159C22⋊D271—24no—
160C22⋊D271—24no—
161C22⋊D271—24no—
162C22⋊D271—22no—
163C22⋊D271—24no—
164C22⋊D271—22no—
165C22⋊D271—42no—
166C22⋊D271—81no—
167C22⋊D271—81no—
168C2×Dic271—24no—
169C2×Dic271—21no—
170C2×Dic271—24no—
171C2×Dic271—21no—
172C2×Dic271C2×D411yescharacteristic
173C2×Dic271C2×D411yescharacteristic
174C4×D271—24no—
175C4×D271—21no—
176C4×D271—21no—
177C4×D271C2×D412yes—
178C4×D271C2×D412yes—
179C271⋊C8—21no—
180C271⋊C8—21no—
181C23×C271—24no—
182C23×C271—21no—
183C23×C271—21no—
184C23×C271—21no—
185C23×C271—81no—
186C23×C271—81no—
187C23×C271C2×D411yescharacteristic
188D4×C271—24no—
189D4×C271—21no—
190D4×C271—21no—
191D4×C271—21no—
192D4×C271—21no—
193D4×C271—81no—
194D4×C271C2×D412yes—
195D4×C271C2×D412yes—
196C2×C4×C271—21no—
197C2×C4×C271—21no—
198C2×C4×C271—21no—
199C2×C4×C271C2×D411yescharacteristic
200C2×C4×C271C2411yescharacteristic
201C8×C271—21no—
202C8×C271—21no—
203C22×C271—21no—
204C22×C271—22no—
205C22×C271—22no—
206C22×C271—21no—
207C22×C271—21no—
208C22×C271—41no—
209C22×C271—41no—
210C22×C271—41no—
211C22×C271—41no—
212C22×C271—41no—
213C22×C271—41no—
214C22×C271—41no—
215C22×C271—84no—
216C22×C271C22×D411yescharacteristic
217C2×D271—21no—
218C2×D271—21no—
219C2×D271—21no—
220C2×D271—22no—
221C2×D271—21no—
222C2×D271—21no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271—41no—
227C2×D271—41no—
228C2×D271—84no—
229C2×D271—84no—
230C2×D271—84no—
231C2×D271#12012yes—
232C4×C271—21no—
233C4×C271—41no—
234C4×C271C22×D411yescharacteristic
235C4×C271C22×D411yescharacteristic
236Dic271—22no—
237Dic271—21no—
238Dic271—41no—
239Dic271#12012yes—
240C2×C271—42no—
241C2×C271—42no—
242C2×C271—41no—
243C2×C271—81no—
244C2×C271C2×#12011yescharacteristic
245C2×C271#57611yescharacteristic
246C2×C271#57611yescharacteristic
247D271—22no—
248D271—41no—
249D271—42no—
250D271—81no—
251C271#300811yescharacteristic, complemented, Sylow
252#3008—2711nomaximal, Sylow
253C2×H(5,𝔽2)—2711no—
254C2×AGL(1,ℤ/8)—2711no—
255C22×D8—2711no—
256#535—2711no—
257#449—2711no—
258#449—2711no—
259#449—2711no—
260#449—2711no—
261#447—2711no—
262#447—2711no—
263#447—2711no—
264#447D27111yescharacteristic, complemented
265#418—2711no—
266#416—2711no—
267#408—2711no—
268H(5,𝔽2)—5424no—
269H(5,𝔽2)—5424no—
270C2×#41—2711no—
271C2×#41—2711no—
272C2×#41—2711no—
273C2×#41—2711no—
274C2×#41—5421no—
275C22×D4—2711no—
276C22×D4—2711no—
277C22×D4—2711no—
278C22×D4—5421no—
279C22×D4—5421no—
280C22×D4—5421no—
281C22×D4C2×D27111yescharacteristic, complemented
282AGL(1,ℤ/8)—5424no—
283C2×SD16—2711no—
284C2×SD16—2711no—
285C2×D8—2711no—
286C2×D8—2711no—
287C2×D8—2711no—
288C2×D8—2711no—
289C2×D8—5424no—
290C2×D8C2×D27111yescharacteristic, complemented
291C2×D8C2×D27111yescharacteristic, complemented
292C2×M16—2711no—
293C22×C8—2711no—
294#127—5421no—
295#124—5421no—
296#121—2711no—
297#121—2711no—
298#121—2711no—
299#121C2×D27111yescharacteristic, complemented
300#120—5421no—
301#120—5421no—
302#117—2711no—
303#103—2711no—
304#103—2711no—
305#102—2711no—
306#102—2711no—
307#102—2711no—
308#102—2711no—
309#102C2×D27111yescharacteristic, complemented
310#102C2×D27111yescharacteristic, complemented
311C22⋊C8—2711no—
312C22⋊C8—2711no—
313C22⋊C8—2711no—
314C22⋊C8C2×D27111yescharacteristic, complemented
315C24—21no—
316C24—5421no—
317C24—5421no—
318C24—5421no—
319#41—2714no—
320#41—5422no—
321#41—5422no—
322#41—5422no—
323#41—5422no—
324#41—5422no—
325#41—5422no—
326#41—5428no—
327#41—5422no—
328C2×Q8—2711no—
329C2×Q8—2711no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—2714no—
335C2×D4—2711no—
336C2×D4—2711no—
337C2×D4—2711no—
338C2×D4—2711no—
339C2×D4—2711no—
340C2×D4—5421no—
341C2×D4—5422no—
342C2×D4—5422no—
343C2×D4—5422no—
344C2×D4—5421no—
345C2×D4—5421no—
346C2×D4—5421no—
347C2×D4—5422no—
348C2×D4—5428no—
349C2×D4—5421no—
350C2×D4—5421no—
351C2×D4—5428no—
352C2×D4—5428no—
353C2×D4—5421no—
354C2×D4—5422no—
355C2×D4—5421no—
356C2×D4—5421no—
357C2×D4C22×D27111yescharacteristic
358C2×D4C22×D27111yescharacteristic
359C2×D4C22×D27111yescharacteristic
360C22×C4—2711no—
361C22×C4—2711no—
362C22×C4—5421no—
363C22×C4—5421no—
364C22×C4—5421no—
365C22×C4C22×D27111yescharacteristic
366SD16—5422no—
367SD16—5422no—
368D8—22no—
369D8—22no—
370D8—5422no—
371D8—5422no—
372D8—5422no—
373D8—5422no—
374M16—5422no—
375C2×C8—2711no—
376C2×C8—2711no—
377C2×C8—5422no—
378C2×C8C22×D27111yescharacteristic
379C2×C8C22×D27111yescharacteristic
380C4⋊C4—2711no—
381C4⋊C4C22×D27111yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—5421no—
384C22⋊C4—5421no—
385C22⋊C4—5421no—
386C22⋊C4—5421no—
387C42—5421no—
388C23—24no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—81no—
393C23—81no—
394C23—2711no—
395C23—2711no—
396C23—5424no—
397C23—5424no—
398C23—5421no—
399C23—5421no—
400C23—10842no—
401C23—21681no—
402C23—21681no—
403C23—21681no—
404C23—21681no—
405C23—21681no—
406C23—21681no—
407C23D4×D27111yescharacteristic
408Q8—2712no—
409Q8—5422no—
410Q8—5421no—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—21no—
416D4—81no—
417D4—2712no—
418D4—5422no—
419D4—5422no—
420D4—5424no—
421D4—5422no—
422D4—5424no—
423D4—5424no—
424D4—5424no—
425D4—5424no—
426D4—5421no—
427D4—5421no—
428D4—5421no—
429D4—10842no—
430D4—21681no—
431D4—21681no—
432D4—21681no—
433D4D4×D27112yes—
434D4D4×D27112yes—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—2711no—
439C2×C4—2712no—
440C2×C4—2712no—
441C2×C4—2711no—
442C2×C4—5424no—
443C2×C4—5424no—
444C2×C4—5424no—
445C2×C4—5421no—
446C2×C4—5421no—
447C2×C4—5421no—
448C2×C4—5421no—
449C2×C4D4×D27111yescharacteristic
450C2×C4C23×D27111yescharacteristic
451C8—21no—
452C8—21no—
453C8—5421no—
454C8—5421no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—84no—
468C22—2712no—
469C22—5421no—
470C22—5421no—
471C22—5422no—
472C22—5421no—
473C22—5421no—
474C22—5421no—
475C22—10841no—
476C22—10841no—
477C22—10841no—
478C22—10841no—
479C22—10841no—
480C22—21684no—
481C22—21684no—
482C22—21684no—
483C22C2×D4×D27111yescharacteristic
484C4—21no—
485C4—41no—
486C4—2712no—
487C4—5421no—
488C4—5422no—
489C4—10841no—
490C4C2×D4×D27111yescharacteristic
491C4C2×D4×D27111yescharacteristic
492C2—41no—
493C2—42no—
494C2—42no—
495C2—81no—
496C2—5422no—
497C2—10841no—
498C2—10842no—
499C2—21681no—
500C2#120×D27111yescharacteristic
501C2#3177469811yescharacteristic
502C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#447×C271C2
socleC22×C271C22×D4
hypercenter#447D271
hypocenterC271#3008
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#22576205C2semidirect
2. C2×#22576184C2semidirect
3. C2×D8×D271C2semidirect
4. #31774463C2semidirect
5. #31774044C2semidirect
6. #31774043C2semidirect
7. #31774040C2semidirect
8. #31774039C2semidirect
9. #31774032C2semidirect
10. #31774031C2semidirect
11. #31774029C2semidirect
12. #31773900C2semidirect
13. #31773890C2semidirect
14. #31773862C2semidirect
15. #447×C271C2semidirect
19. C2×#16864654C22semidirect
22. C2×#16864653C22semidirect
27. C2×D4×D271C22semidirect
28. C2×D4×D271C22semidirect
30. C2×#16864643C22semidirect
31. C2×#16864642C22semidirect
32. C2×D2168C22semidirect
33. C2×#16864639C22semidirect
34. C2×#16864639C22semidirect
35. C2×#16864639C22semidirect
37. C2×#16864637C22semidirect
38. C2×C8×D271C22semidirect
41. #22576135C22semidirect
42. #22576134C22semidirect
43. #22576131C22semidirect
46. #22576118C22semidirect
47. #22576085C22semidirect
48. #22576083C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576081C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
56. C22×D4×C271C22semidirect
57. C2×D8×C271C22semidirect
58. C2×D8×C271C22semidirect
59. #121×C271C22semidirect
60. #102×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
74. #16864653D4semidirect
99. D4×D271D4semidirect
251. C271#3008semidirect
264. #447D271semidirect
281. C22×D4C2×D271semidirect
290. C2×D8C2×D271semidirect
291. C2×D8C2×D271semidirect
299. #121C2×D271semidirect
309. #102C2×D271semidirect
310. #102C2×D271semidirect
314. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762052
C2×#225761842
C2×D8×D2712
#317744632
#317740442
#317740432
#317740402
#317740392
#317740322
#317740312
#317740292
#317739002
#317738902
#317738622
#447×C2712
#3008271