Finite Group Databasebeta

← #46845663

Subgroups — #46845663

Order 34688.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 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×#22576179C211yesmaximal, characteristic, complemented
4#31774476C211yesmaximal, characteristic, complemented
5#31774061C211yesmaximal, characteristic, complemented
6#31774060C211yesmaximal, characteristic, complemented
7#31774049C211yesmaximal, characteristic, complemented
8#31774048C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic, complemented
10#31774039C211yesmaximal, characteristic, complemented
11#31774029C211yesmaximal, characteristic, complemented
12#31773895C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#447×C271C211yesmaximal, characteristic, complemented
16#22576205—24no—
17#22576205—24no—
18C2×#16864654C2211yescharacteristic
19C2×#16864654C2211yescharacteristic
20C2×#16864653—21no—
21C2×#16864653C2211yescharacteristic
22C2×#16864653C2211yescharacteristic
23C2×#16864653C2211yescharacteristic, complemented
24C22×C22⋊D271—21no—
25C22×C22⋊D271—21no—
26C2×D4×D271—21no—
27C2×D4×D271C2211yescharacteristic
28C2×D4×D271C2211yescharacteristic
29#22576184—24no—
30#22576179—24no—
31C2×Dic1084C2211yescharacteristic, complemented
32C2×#16864643C2211yescharacteristic, complemented
33C2×#16864642C2211yescharacteristic, complemented
34C2×#16864642C2211yescharacteristic, complemented
35C2×#16864642C2211yescharacteristic, complemented
36C2×#16864639C2211yescharacteristic, complemented
37C2×#16864637C2211yescharacteristic, complemented
38C2×C8×D271C2211yescharacteristic, complemented
39#22576140C2211yescharacteristic, complemented
40#22576139C2211yescharacteristic, complemented
41#22576137—21no—
42#22576133—21no—
43#22576132—21no—
44#22576131—21no—
45#22576131C2211yescharacteristic, complemented
46C4⋊C4×D271C2211yescharacteristic, 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
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—
65#16864654—22no—
66#16864654—22no—
67#16864654—22no—
68#16864654—22no—
69#16864653—28no—
70#16864653—22no—
71#16864653—22no—
72#16864653—22no—
73#16864653—22no—
74#16864653—22no—
75#16864653D414yescomplemented
76C2×Dic542D411yescharacteristic
77C2×Dic542C2311yescharacteristic
78C2×Dic542C2311yescharacteristic
79C2×D1084D411yescharacteristic
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⋊D271D411yescharacteristic
91C2×C22⋊D271D411yescharacteristic
92D4×D271—28no—
93D4×D271—22no—
94D4×D271—22no—
95D4×D271—22no—
96D4×D271D414yescomplemented
97C22×Dic271—21no—
98C22×Dic271—21no—
99C22×Dic271—21no—
100C2×C4×D271—21no—
101C2×C4×D271—21no—
102C2×C4×D271C2311yescharacteristic
103C2×C4×D271C2311yescharacteristic
104Dic1084—22no—
105#16864643—22no—
106#16864642—22no—
107#16864642—22no—
108#16864642—22no—
109#16864639—22no—
110#16864637—22no—
111C2×C271⋊C8C2311yescharacteristic
112C2×C271⋊C8C2311yescharacteristic
113C8×D271—22no—
114C4⋊Dic271—21no—
115C4⋊Dic271C2311yescharacteristic
116Dic271⋊C4—21no—
117Dic271⋊C4—21no—
118C22⋊Dic271—21no—
119C22⋊Dic271—21no—
120(C2×D271)⋊C4—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—81no—
140C22×D271—81no—
141C22×D271—81no—
142C22×D271—81no—
143C22×D271C2×D411yescharacteristic
144C22×D271C2×D411yescharacteristic
145Dic542—21no—
146Dic542—22no—
147Dic542—21no—
148Dic542—21no—
149Dic542C2×D412yes—
150D1084—22no—
151D1084—21no—
152D1084C2×D412yes—
153C22⋊D271—24no—
154C22⋊D271—24no—
155C22⋊D271—24no—
156C22⋊D271—24no—
157C22⋊D271—22no—
158C22⋊D271—24no—
159C22⋊D271—22no—
160C22⋊D271—42no—
161C22⋊D271—81no—
162C2×Dic271—24no—
163C2×Dic271—21no—
164C2×Dic271—24no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271—42no—
168C2×Dic271C2×D411yescharacteristic
169C2×Dic271C2×D411yescharacteristic
170C4×D271—24no—
171C4×D271—21no—
172C4×D271—21no—
173C4×D271—81no—
174C4×D271C2×D412yes—
175C4×D271C2×D412yes—
176C271⋊C8—21no—
177C271⋊C8—21no—
178C23×C271—24no—
179C23×C271—21no—
180C23×C271—21no—
181C23×C271—21no—
182C23×C271—81no—
183C23×C271—81no—
184C23×C271C2×D411yescharacteristic
185D4×C271—24no—
186D4×C271—21no—
187D4×C271—21no—
188D4×C271—21no—
189D4×C271—21no—
190D4×C271—81no—
191D4×C271C2×D412yes—
192D4×C271C2×D412yes—
193C2×C4×C271—21no—
194C2×C4×C271—21no—
195C2×C4×C271—21no—
196C2×C4×C271C2×D411yescharacteristic
197C2×C4×C271C2411yescharacteristic
198C8×C271—21no—
199C8×C271—21no—
200C22×C271—21no—
201C22×C271—22no—
202C22×C271—22no—
203C22×C271—21no—
204C22×C271—21no—
205C22×C271—41no—
206C22×C271—41no—
207C22×C271—41no—
208C22×C271—41no—
209C22×C271—41no—
210C22×C271—41no—
211C22×C271—41no—
212C22×C271—84no—
213C22×C271C22×D411yescharacteristic
214C2×D271—21no—
215C2×D271—21no—
216C2×D271—21no—
217C2×D271—22no—
218C2×D271—21no—
219C2×D271—21no—
220C2×D271—41no—
221C2×D271—41no—
222C2×D271—84no—
223C2×D271—84no—
224C2×D271#12012yes—
225C4×C271—21no—
226C4×C271—41no—
227C4×C271C22×D411yescharacteristic
228C4×C271C22×D411yescharacteristic
229Dic271—22no—
230Dic271—21no—
231Dic271—41no—
232Dic271—41no—
233Dic271#12012yes—
234C2×C271—42no—
235C2×C271—42no—
236C2×C271—41no—
237C2×C271—81no—
238C2×C271C2×#12011yescharacteristic
239C2×C271#57611yescharacteristic
240C2×C271#57711yescharacteristic
241D271—22no—
242D271—41no—
243D271—42no—
244C271#301211yescharacteristic, complemented, Sylow
245#3012—2711nomaximal, Sylow
246C2×H(5,𝔽2)—2711no—
247C2×AGL(1,ℤ/8)—2711no—
248C2×#135—2711no—
249#537—2711no—
250#452—2711no—
251#452—2711no—
252#450—2711no—
253#450—2711no—
254#449—2711no—
255#449—2711no—
256#447—2711no—
257#447D27111yescharacteristic, complemented
258#417—2711no—
259C2×#102—2711no—
260#408—2711no—
261H(5,𝔽2)—5424no—
262H(5,𝔽2)—5424no—
263C2×#41—2711no—
264C2×#41—2711no—
265C2×#41—2711no—
266C2×#41—2711no—
267C2×#41—2711no—
268C2×#41—5421no—
269C22×D4—2711no—
270C22×D4—2711no—
271C22×D4—5421no—
272C22×D4—5421no—
273C22×D4—5421no—
274C22×D4C2×D27111yescharacteristic
275AGL(1,ℤ/8)—5424no—
276#135—5424no—
277C2×Q16—2711no—
278C2×SD16—2711no—
279C2×SD16—2711no—
280C2×SD16—2711no—
281C2×SD16—2711no—
282C2×D8—2711no—
283C2×D8C2×D27111yescharacteristic, complemented
284C2×D8C2×D27111yescharacteristic, complemented
285C2×M16—2711no—
286C22×C8—2711no—
287C22⋊Q8—2711no—
288C22⋊Q8—2711no—
289C22⋊Q8—5421no—
290#121—2711no—
291#121—5421no—
292#121—5421no—
293#121—5421no—
294#121C2×D27111yescharacteristic, complemented
295C2×C4⋊C4—2711no—
296#103—2711no—
297#103—2711no—
298#102—2711no—
299#102—2711no—
300#102—2711no—
301#102—2711no—
302#102C2×D27111yescharacteristic, complemented
303#102C2×D27111yescharacteristic, complemented
304C22⋊C8—2711no—
305C22⋊C8—2711no—
306C22⋊C8—2711no—
307C22⋊C8C2×D27111yescharacteristic, complemented
308C24—21no—
309C24—5421no—
310C24—5421no—
311#41—2714no—
312#41—5422no—
313#41—5422no—
314#41—5422no—
315#41—5422no—
316#41—5422no—
317#41—5422no—
318#41—5422no—
319#41—5422no—
320#41—5428no—
321#41—5422no—
322C2×Q8—2711no—
323C2×Q8—2711no—
324C2×Q8—2711no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—2714no—
330C2×D4—2711no—
331C2×D4—2711no—
332C2×D4—2711no—
333C2×D4—2711no—
334C2×D4—5422no—
335C2×D4—5421no—
336C2×D4—5421no—
337C2×D4—5422no—
338C2×D4—5428no—
339C2×D4—5421no—
340C2×D4—5421no—
341C2×D4—5428no—
342C2×D4—5428no—
343C2×D4—5421no—
344C2×D4—5422no—
345C2×D4—5421no—
346C2×D4—5421no—
347C2×D4C22×D27111yescharacteristic
348C2×D4C22×D27111yescharacteristic
349C2×D4C22×D27111yescharacteristic
350C22×C4—2711no—
351C22×C4—2711no—
352C22×C4—5421no—
353C22×C4—5421no—
354C22×C4—5421no—
355C22×C4—5421no—
356C22×C4—5421no—
357C22×C4C22×D27111yescharacteristic
358Q16—5422no—
359SD16—5422no—
360SD16—5422no—
361SD16—5422no—
362SD16—5422no—
363D8—22no—
364D8—22no—
365D8—5422no—
366M16—5422no—
367C2×C8—2711no—
368C2×C8—2711no—
369C2×C8—5422no—
370C2×C8C22×D27111yescharacteristic
371C2×C8C22×D27111yescharacteristic
372C4⋊C4—2711no—
373C4⋊C4—5421no—
374C4⋊C4—5421no—
375C4⋊C4—5421no—
376C4⋊C4C22×D27111yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—5421no—
379C22⋊C4—5421no—
380C22⋊C4—5421no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—81no—
386C23—81no—
387C23—2711no—
388C23—2711no—
389C23—5424no—
390C23—5424no—
391C23—5421no—
392C23—21681no—
393C23—21681no—
394C23—21681no—
395C23—21681no—
396C23D4×D27111yescharacteristic
397Q8—2712no—
398Q8—5422no—
399Q8—5421no—
400Q8—5421no—
401Q8—5421no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—21no—
407D4—81no—
408D4—2712no—
409D4—5422no—
410D4—5422no—
411D4—5424no—
412D4—5422no—
413D4—5424no—
414D4—5424no—
415D4—5424no—
416D4—5424no—
417D4—5421no—
418D4—10842no—
419D4—21681no—
420D4D4×D27112yes—
421D4D4×D27112yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—2711no—
426C2×C4—2712no—
427C2×C4—2712no—
428C2×C4—2711no—
429C2×C4—5424no—
430C2×C4—5424no—
431C2×C4—5424no—
432C2×C4—5421no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—10842no—
438C2×C4—21681no—
439C2×C4D4×D27111yescharacteristic
440C2×C4C23×D27111yescharacteristic
441C8—21no—
442C8—21no—
443C8—5421no—
444C8—5421no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—84no—
458C22—2712no—
459C22—5421no—
460C22—5421no—
461C22—5422no—
462C22—5421no—
463C22—5421no—
464C22—5421no—
465C22—10841no—
466C22—10841no—
467C22—21684no—
468C22—21684no—
469C22C2×D4×D27111yescharacteristic
470C4—21no—
471C4—41no—
472C4—2712no—
473C4—5421no—
474C4—5422no—
475C4—10841no—
476C4—10841no—
477C4C2×D4×D27111yescharacteristic
478C4C2×D4×D27111yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—5422no—
484C2—10841no—
485C2—10842no—
486C2#120×D27111yescharacteristic
487C2#3177469811yescharacteristic
488C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#447×C271C2
socleC22×C271C22×D4
hypercenter#447D271
hypocenterC271#3012
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×#22576179C2semidirect
4. #31774476C2semidirect
5. #31774061C2semidirect
6. #31774060C2semidirect
7. #31774049C2semidirect
8. #31774048C2semidirect
9. #31774040C2semidirect
10. #31774039C2semidirect
11. #31774029C2semidirect
12. #31773895C2semidirect
13. #102×D271C2semidirect
14. #31773862C2semidirect
15. #447×C271C2semidirect
23. C2×#16864653C22semidirect
31. C2×Dic1084C22semidirect
32. C2×#16864643C22semidirect
33. C2×#16864642C22semidirect
34. C2×#16864642C22semidirect
35. C2×#16864642C22semidirect
36. C2×#16864639C22semidirect
37. C2×#16864637C22semidirect
38. C2×C8×D271C22semidirect
39. #22576140C22semidirect
40. #22576139C22semidirect
45. #22576131C22semidirect
46. C4⋊C4×D271C22semidirect
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
57. C2×D8×C271C22semidirect
58. C2×D8×C271C22semidirect
59. #121×C271C22semidirect
60. #102×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
75. #16864653D4semidirect
96. D4×D271D4semidirect
244. C271#3012semidirect
257. #447D271semidirect
283. C2×D8C2×D271semidirect
284. C2×D8C2×D271semidirect
294. #121C2×D271semidirect
302. #102C2×D271semidirect
303. #102C2×D271semidirect
307. 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×#225761792
#317744762
#317740612
#317740602
#317740492
#317740482
#317740402
#317740392
#317740292
#317738952
#102×D2712
#317738622
#447×C2712
#3012271