Finite Group Databasebeta

← #46845613

Subgroups — #46845613

Order 34688.

806 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D271C211yesmaximal, characteristic, complemented
2C2×#22576187C211yesmaximal, characteristic, complemented
3C2×#22576184C211yesmaximal, characteristic, complemented
4#121×D271C211yesmaximal, characteristic, complemented
5#31774044C211yesmaximal, characteristic, complemented
6#31774042C211yesmaximal, characteristic, complemented
7#31774040C211yesmaximal, characteristic, complemented
8#31774037C211yesmaximal, characteristic, complemented
9#31774034C211yesmaximal, characteristic, complemented
10#31774032C211yesmaximal, characteristic, complemented
11#31774029C211yesmaximal, characteristic, complemented
12#31773897C211yesmaximal, characteristic, complemented
13#31773895C211yesmaximal, characteristic, complemented
14#31773859C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864654C2211yescharacteristic, complemented
19C2×#16864653—21no—
20C2×#16864653C2211yescharacteristic
21#41×D271—24no—
22#41×D271—24no—
23C2×Q8×D271C2211yescharacteristic, complemented
24C2×D4×D271—21no—
25C2×D4×D271—21no—
26C2×D4×D271C2211yescharacteristic, complemented
27C2×D4×D271C2211yescharacteristic, complemented
28C22×C4×D271—21no—
29C22×C4×D271C2211yescharacteristic
30#22576187—24no—
31#22576184—24no—
32C2×#16864643C2211yescharacteristic, complemented
33C2×#16864642C2211yescharacteristic, complemented
34C2×#16864641C2211yescharacteristic, complemented
35C2×D2168C2211yescharacteristic, complemented
36C2×#16864639C2211yescharacteristic, complemented
37C2×#16864639C2211yescharacteristic, complemented
38C2×#16864637C2211yescharacteristic, complemented
39C2×#16864637C2211yescharacteristic, complemented
40#22576135C2211yescharacteristic
41#22576134C2211yescharacteristic, complemented
42#22576133—21no—
43#22576132—21no—
44#22576131C2211yescharacteristic
45C4⋊C4×D271C2211yescharacteristic, complemented
46C22⋊C4×D271—21no—
47#22576085C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52#22576080C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C2×#41×C271C2211yescharacteristic, complemented
57C2×SD16×C271C2211yescharacteristic, complemented
58C2×D8×C271C2211yescharacteristic, complemented
59#121×C271C2211yescharacteristic, complemented
60#103×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271—21no—
65C23×D271D411yescharacteristic
66#16864655—22no—
67#16864655D414yescomplemented
68#16864654—28no—
69#16864654—22no—
70#16864654—22no—
71#16864653—28no—
72#16864653—22no—
73#16864653D414yescomplemented
74C2×Dic542—21no—
75C2×Dic542C2311yescharacteristic
76Q8×D271—22no—
77Q8×D271—22no—
78C2×D1084—21no—
79C2×D1084—21no—
80C2×D1084C2311yescharacteristic
81C2×D1084C2311yescharacteristic
82C2×C22⋊D271—21no—
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86C2×C22⋊D271—21no—
87C2×C22⋊D271—21no—
88D4×D271—28no—
89D4×D271—22no—
90D4×D271—22no—
91D4×D271—22no—
92D4×D271—22no—
93D4×D271—44no—
94C22×Dic271—21no—
95C22×Dic271D411yescharacteristic
96C2×C4×D271—28no—
97C2×C4×D271—21no—
98C2×C4×D271—21no—
99C2×C4×D271—22no—
100C2×C4×D271—22no—
101C2×C4×D271—21no—
102C2×C4×D271—21no—
103C2×C4×D271D411yescharacteristic
104C2×C4×D271D411yescharacteristic
105C2×C4×D271C2311yescharacteristic
106C2×C4×D271C2311yescharacteristic
107#16864643—22no—
108#16864642—22no—
109#16864641—22no—
110D2168—22no—
111#16864639—22no—
112#16864639—22no—
113#16864637—22no—
114#16864637—22no—
115C2×C271⋊C8C2311yescharacteristic
116C2×C271⋊C8C2311yescharacteristic
117C4⋊Dic271C2311yescharacteristic
118Dic271⋊C4—21no—
119C22⋊Dic271—21no—
120(C2×D271)⋊C4—21no—
121(C2×D271)⋊C4—21no—
122#41×C271—22no—
123#41×C271—22no—
124C2×Q8×C271C2311yescharacteristic
125C2×D4×C271—21no—
126C2×D4×C271—21no—
127C2×D4×C271C2311yescharacteristic
128C2×D4×C271C2311yescharacteristic
129C22×C4×C271—21no—
130C22×C4×C271C2311yescharacteristic
131SD16×C271—22no—
132D8×C271—22no—
133C2×C8×C271C2311yescharacteristic
134C2×C8×C271C2311yescharacteristic
135C4⋊C4×C271C2311yescharacteristic
136C22⋊C4×C271—21no—
137C22×D271—24no—
138C22×D271—21no—
139C22×D271—22no—
140C22×D271—22no—
141C22×D271—21no—
142C22×D271—41no—
143C22×D271—42no—
144C22×D271—42no—
145C22×D271—42no—
146C22×D271—42no—
147C22×D271—42no—
148C22×D271C2×D411yescharacteristic
149C22×D271C2×D411yescharacteristic
150Dic542—24no—
151Dic542—21no—
152Dic542C2×D412yescomplemented
153D1084—24no—
154D1084—21no—
155D1084—21no—
156D1084—21no—
157D1084—42no—
158D1084C2×D412yescomplemented
159C22⋊D271—24no—
160C22⋊D271—24no—
161C22⋊D271—24no—
162C22⋊D271—42no—
163C22⋊D271—42no—
164C22⋊D271—42no—
165C2×Dic271—21no—
166C2×Dic271—24no—
167C2×Dic271—22no—
168C2×Dic271—21no—
169C2×Dic271C2×D411yescharacteristic
170C2×Dic271C2×D411yescharacteristic
171C4×D271—24no—
172C4×D271—24no—
173C4×D271—22no—
174C4×D271—21no—
175C4×D271—21no—
176C4×D271—22no—
177C4×D271—24no—
178C4×D271—42no—
179C4×D271C2×D412yes—
180C4×D271C2×D412yes—
181C271⋊C8—21no—
182C271⋊C8—21no—
183C23×C271—21no—
184C23×C271—21no—
185C23×C271C2×D411yescharacteristic
186Q8×C271—21no—
187Q8×C271C2×D412yescomplemented
188D4×C271—24no—
189D4×C271—21no—
190D4×C271—21no—
191D4×C271—21no—
192D4×C271—42no—
193D4×C271C2×D412yescomplemented
194C2×C4×C271—24no—
195C2×C4×C271—21no—
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—21no—
206C22×C271—21no—
207C22×C271—41no—
208C22×C271—41no—
209C22×C271—41no—
210C22×C271—41no—
211C22×C271—41no—
212C22×C271C22×D411yescharacteristic
213C2×D271—21no—
214C2×D271—21no—
215C2×D271—21no—
216C2×D271—21no—
217C2×D271—22no—
218C2×D271—44no—
219C2×D271—48no—
220C2×D271—41no—
221C2×D271—41no—
222C2×D271—41no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—84no—
226C2×D271#12012yes—
227C2×D271AGL(1,ℤ/8)12yescomplemented
228C4×C271—21no—
229C4×C271—22no—
230C4×C271—41no—
231C4×C271C22×D411yescharacteristic
232C4×C271C22×D411yescharacteristic
233Dic271—22no—
234Dic271—21no—
235Dic271—41no—
236Dic271#12012yes—
237C2×C271—41no—
238C2×C271—42no—
239C2×C271—81no—
240C2×C271C2×#12011yescharacteristic
241C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
242C2×C271#57511yescharacteristic
243D271—22no—
244D271—22no—
245D271—42no—
246D271—81no—
247C271#300611yescharacteristic, complemented, Sylow
248#3006—2711nomaximal, Sylow
249C22×#41—2711no—
250C2×AGL(1,ℤ/8)—2711no—
251C2×AGL(1,ℤ/8)—2711no—
252C2×#121—2711no—
253#449—2711no—
254#449—2711no—
255#449—2711no—
256#449D27111yescharacteristic, complemented
257#448—2711no—
258#448—2711no—
259#447—2711no—
260#447—2711no—
261#417—2711no—
262#417—2711no—
263#407—2711no—
264C2×#41—2711no—
265C2×#41—2711no—
266C2×#41—2711no—
267C2×#41—5421no—
268C2×#41—5421no—
269C2×#41—5424no—
270C2×#41—5424no—
271C2×#41C2×D27111yescharacteristic, complemented
272C22×Q8—2711no—
273C22×D4—2711no—
274C22×D4—2711no—
275C22×D4—5421no—
276C22×D4—5421no—
277C23×C4—2711no—
278C23×C4—5421no—
279AGL(1,ℤ/8)—5424no—
280AGL(1,ℤ/8)—5424no—
281C2×SD16—2711no—
282C2×SD16—2711no—
283C2×SD16—2711no—
284C2×SD16C2×D27111yescharacteristic, complemented
285C2×D8—2711no—
286C2×D8—2711no—
287C2×D8—2711no—
288C2×D8C2×D27111yescharacteristic, complemented
289C2×M16—2711no—
290C2×M16—2711no—
291#121—2711no—
292#121—2711no—
293#121—2711no—
294#121—5421no—
295#121—5421no—
296#121C2×D27111yescharacteristic, complemented
297C2×C4⋊C4—2711no—
298C2×C22⋊C4—5421no—
299#103—2711no—
300#103—2711no—
301#103—2711no—
302#103C2×D27111yescharacteristic, complemented
303#102—2711no—
304#102—2711no—
305#102—2711no—
306#102C2×D27111yescharacteristic, complemented
307C22⋊C8—2711no—
308C22⋊C8—2711no—
309C22⋊C8—2711no—
310C22⋊C8C2×D27111yescharacteristic, complemented
311C24—2711no—
312C24—5421no—
313C24—5421no—
314#41—22no—
315#41—22no—
316#41—2714no—
317#41—2714no—
318#41—5422no—
319#41—5422no—
320#41—5428no—
321#41—5422no—
322#41—5422no—
323#41—5428no—
324C2×Q8—2711no—
325C2×Q8—5421no—
326C2×Q8—5422no—
327C2×Q8—5422no—
328C2×Q8C22×D27111yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—2711no—
332C2×D4—2711no—
333C2×D4—5421no—
334C2×D4—5421no—
335C2×D4—5422no—
336C2×D4—5422no—
337C2×D4—5421no—
338C2×D4—5421no—
339C2×D4—5422no—
340C2×D4—5428no—
341C2×D4—5421no—
342C2×D4—5421no—
343C2×D4—5422no—
344C2×D4—5421no—
345C2×D4—5421no—
346C2×D4—10844no—
347C2×D4C22×D27111yescharacteristic
348C2×D4C22×D27111yescharacteristic
349C22×C4—21no—
350C22×C4—2711no—
351C22×C4—2711no—
352C22×C4—2711no—
353C22×C4—2711no—
354C22×C4—2711no—
355C22×C4—5422no—
356C22×C4—5422no—
357C22×C4—5421no—
358C22×C4—5421no—
359C22×C4—5428no—
360C22×C4—5421no—
361C22×C4—5421no—
362C22×C4—5421no—
363C22×C4C22×D27111yescharacteristic
364SD16—22no—
365SD16—5422no—
366SD16—5422no—
367SD16—5422no—
368D8—22no—
369D8—5422no—
370D8—5422no—
371D8—5422no—
372M16—5422no—
373M16—5422no—
374C2×C8—2711no—
375C2×C8—2711no—
376C2×C8C22×D27111yescharacteristic
377C2×C8C22×D27111yescharacteristic
378C4⋊C4—2711no—
379C4⋊C4—5421no—
380C4⋊C4C22×D27111yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—5421no—
383C22⋊C4—5421no—
384C22⋊C4—5421no—
385C23—21no—
386C23—21no—
387C23—2711no—
388C23—2711no—
389C23—5422no—
390C23—5422no—
391C23—5424no—
392C23—5421no—
393C23—5421no—
394C23—10841no—
395C23—10842no—
396C23—10842no—
397C23—10842no—
398C23—10842no—
399C23—10842no—
400C23D4×D27111yescharacteristic
401Q8—21no—
402Q8—2712no—
403Q8—5424no—
404Q8—5421no—
405Q8D4×D27112yescomplemented
406D4—24no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—42no—
411D4—2712no—
412D4—5424no—
413D4—5424no—
414D4—5424no—
415D4—5424no—
416D4—5421no—
417D4—5421no—
418D4—5421no—
419D4—10842no—
420D4—10842no—
421D4—10842no—
422D4—10842no—
423D4D4×D27112yescomplemented
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—2711no—
430C2×C4—2712no—
431C2×C4—2712no—
432C2×C4—2711no—
433C2×C4—5422no—
434C2×C4—5424no—
435C2×C4—5422no—
436C2×C4—5424no—
437C2×C4—5424no—
438C2×C4—5422no—
439C2×C4—5424no—
440C2×C4—5421no—
441C2×C4—5421no—
442C2×C4—5421no—
443C2×C4—5421no—
444C2×C4—10842no—
445C2×C4D4×D27111yescharacteristic
446C2×C4C23×D27111yescharacteristic
447C8—21no—
448C8—21no—
449C8—5421no—
450C8—5421no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—2712no—
461C22—2712no—
462C22—5421no—
463C22—5421no—
464C22—5421no—
465C22—5421no—
466C22—5422no—
467C22—10844no—
468C22—10848no—
469C22—10841no—
470C22—10841no—
471C22—10841no—
472C22—10841no—
473C22—10841no—
474C22—21684no—
475C22C2×D4×D27111yescharacteristic
476C4—21no—
477C4—22no—
478C4—41no—
479C4—2712no—
480C4—5421no—
481C4—5422no—
482C4—10841no—
483C4C2×D4×D27111yescharacteristic
484C4C2×D4×D27111yescharacteristic
485C2—41no—
486C2—42no—
487C2—81no—
488C2—5422no—
489C2—5422no—
490C2—10842no—
491C2—21681no—
492C2#120×D27111yescharacteristic
493C2AGL(1,ℤ/8)×D27111yescharacteristic
494C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3006
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D271C2semidirect
2. C2×#22576187C2semidirect
3. C2×#22576184C2semidirect
4. #121×D271C2semidirect
5. #31774044C2semidirect
6. #31774042C2semidirect
7. #31774040C2semidirect
8. #31774037C2semidirect
9. #31774034C2semidirect
10. #31774032C2semidirect
11. #31774029C2semidirect
12. #31773897C2semidirect
13. #31773895C2semidirect
14. #31773859C2semidirect
15. #449×C271C2semidirect
18. C2×#16864654C22semidirect
23. C2×Q8×D271C22semidirect
26. C2×D4×D271C22semidirect
27. C2×D4×D271C22semidirect
32. C2×#16864643C22semidirect
33. C2×#16864642C22semidirect
34. C2×#16864641C22semidirect
35. C2×D2168C22semidirect
36. C2×#16864639C22semidirect
37. C2×#16864639C22semidirect
38. C2×#16864637C22semidirect
39. C2×#16864637C22semidirect
41. #22576134C22semidirect
45. C4⋊C4×D271C22semidirect
47. #22576085C22semidirect
48. #22576084C22semidirect
49. #22576083C22semidirect
50. #22576082C22semidirect
51. #22576081C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
56. C2×#41×C271C22semidirect
57. C2×SD16×C271C22semidirect
58. C2×D8×C271C22semidirect
59. #121×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
67. #16864655D4semidirect
73. #16864653D4semidirect
152. Dic542C2×D4semidirect
158. D1084C2×D4semidirect
187. Q8×C271C2×D4semidirect
193. D4×C271C2×D4semidirect
227. C2×D271AGL(1,ℤ/8)semidirect
247. C271#3006semidirect
256. #449D271semidirect
271. C2×#41C2×D271semidirect
284. C2×SD16C2×D271semidirect
288. C2×D8C2×D271semidirect
296. #121C2×D271semidirect
302. #103C2×D271semidirect
306. #102C2×D271semidirect
310. C22⋊C8C2×D271semidirect
405. Q8D4×D271semidirect
423. D4D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2712
C2×#225761872
C2×#225761842
#121×D2712
#317740442
#317740422
#317740402
#317740372
#317740342
#317740322
#317740292
#317738972
#317738952
#317738592
#449×C2712
#3006271