Finite Group Databasebeta

← #25075168

Subgroups — #25075168

Order 11008.

1224 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145716C211yesmaximal, characteristic, complemented
2#19144595C211yesmaximal, characteristic, complemented
3#19144059C211yesmaximal, characteristic, complemented
4#19133830C214yesmaximal, complemented
5#19133684C214yesmaximal, complemented
6#19133267C211yesmaximal, characteristic, complemented
7#19133020C211yesmaximal, characteristic, complemented
8#19132986C211yesmaximal, characteristic, complemented
9C2×C43×#547C211yesmaximal, characteristic, complemented
10C2×#13063351—21no—
11#14583449C2211yescharacteristic
12#14583302C2212yescomplemented
13#14583247C2212yescomplemented
14#14583212—21no—
15#14583212—22no—
16#14583179C2212yescomplemented
17#14583176—22no—
18#14583114—22no—
19#14583104C2212yescomplemented
20#14583094C2212yescomplemented
21#14583094C2212yescomplemented
22#14583033—21no—
23C4×#10621882—22no—
24#14582952C2211yescharacteristic
25#14582952C2211yescharacteristic
26#14582923C2211yescharacteristic
27#14582898—21no—
28#14582645C2214yescomplemented
29D4⋊(C43⋊C8)C2214yescomplemented
30D4⋊(C43⋊C8)C2214yescomplemented
31C22×#41×C43—21no—
32C43×#547—22no—
33C43×#547—22no—
34C43×#547—22no—
35C43×#547C2214yes—
36C2×C4⋊Q8×C43C2211yescharacteristic
37C2×#123×C43—22no—
38C2×C22⋊Q8×C43—21no—
39C2×#121×C43—21no—
40C2×C4×D4×C43C2211yescharacteristic
41C2×C4×D4×C43C2211yescharacteristic
42C22×C4⋊C4×C43—21no—
43C2×#10621898—21no—
44#13063351—44no—
45C2×#10621883—22no—
46C2×#10621882—22no—
47C2×#10621882—22no—
48C2×#10621877—21no—
49#13063325C2311yescharacteristic
50#13063322C2311yescharacteristic
51#13063313—21no—
52#13063299—21no—
53#13063299—21no—
54C4×D172—21no—
55C4×D172—21no—
56#13063258—24no—
57#13063258—22no—
58C4⋊(C43⋊C8)—21no—
59C4⋊(C43⋊C8)—21no—
60C4⋊(C43⋊C8)C2312yes—
61C4⋊(C43⋊C8)C2312yes—
62#13063248—22no—
63#13063246—22no—
64#13063245—22no—
65#13063245—22no—
66#13063245—24no—
67C22⋊(C43⋊C8)—21no—
68C22⋊(C43⋊C8)—21no—
69C4×C43⋊C8—21no—
70C4×C43⋊C8C2312yes—
71C2×#41×C43—24no—
72C2×#41×C43—24no—
73C2×#41×C43—44no—
74C2×#41×C43—44no—
75C22×Q8×C43—21no—
76C22×D4×C43—22no—
77C22×D4×C43D411yescharacteristic
78C22×D4×C43D411yescharacteristic
79C23×C4×C43—22no—
80C23×C4×C43—21no—
81C23×C4×C43—21no—
82C4⋊Q8×C43—21no—
83C4⋊Q8×C43—21no—
84C4⋊Q8×C43D412yes—
85C4⋊Q8×C43C2312yes—
86#123×C43—28no—
87#123×C43—28no—
88C22⋊Q8×C43—24no—
89C22⋊Q8×C43—24no—
90#121×C43—24no—
91#121×C43—24no—
92C4×D4×C43—21no—
93C4×D4×C43—21no—
94C4×D4×C43—21no—
95C4×D4×C43—21no—
96C4×D4×C43D412yes—
97C4×D4×C43D412yes—
98C4×D4×C43C2312yes—
99C4×D4×C43C2312yes—
100C2×C4⋊C4×C43—22no—
101C2×C4⋊C4×C43—24no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—24no—
104C2×C4⋊C4×C43D411yescharacteristic
105C2×C4⋊C4×C43D411yescharacteristic
106C2×C4⋊C4×C43D412yes—
107C2×C22⋊C4×C43—22no—
108C2×C22⋊C4×C43—21no—
109C2×C22⋊C4×C43—21no—
110C2×C42×C43C2311yescharacteristic
111C24×C43—21no—
112C24×C43—21no—
113#41×C43—48no—
114#41×C43—416no—
115#41×C43—48no—
116C2×Q8×C43—22no—
117C2×Q8×C43—22no—
118C2×Q8×C43—42no—
119C2×Q8×C43—42no—
120C2×D4×C43—28no—
121C2×D4×C43—21no—
122C2×D4×C43—21no—
123C2×D4×C43—22no—
124C2×D4×C43—48no—
125C2×D4×C43—42no—
126C2×D4×C43D814yescomplemented
127C2×D4×C43C2×D412yes—
128C2×D4×C43C2×D412yes—
129C22×C4×C43—28no—
130C22×C4×C43—22no—
131C22×C4×C43—22no—
132C22×C4×C43—22no—
133C22×C4×C43—22no—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—22no—
137C22×C4×C43—48no—
138C22×C4×C43—44no—
139C22×C4×C43—42no—
140C22×C4×C43—42no—
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2×D411yescharacteristic
144C2×C22⋊D43—41no—
145#10621898—42no—
146#10621898—42no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×C4×D43—41no—
151C2×Dic86—21no—
152C4⋊C4×C43—28no—
153C4⋊C4×C43—21no—
154C4⋊C4×C43—22no—
155C4⋊C4×C43—21no—
156C4⋊C4×C43—22no—
157C4⋊C4×C43—22no—
158C4⋊C4×C43C2×D412yes—
159C4⋊C4×C43C2×D412yes—
160C4⋊C4×C43C2×D414yes—
161C22⋊C4×C43—28no—
162C22⋊C4×C43—22no—
163C22⋊C4×C43—22no—
164C22⋊C4×C43—22no—
165C22⋊C4×C43—22no—
166C42×C43C2×D411yescharacteristic
167C42×C43C2×D411yescharacteristic
168C42×C43C2×D411yescharacteristic
169C42×C43C2411yescharacteristic
170#10621883—44no—
171#10621882—44no—
172#10621882—44no—
173(C2×D43)⋊C4—41no—
174(C2×D43)⋊C4—41no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177#10621877—42no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8—22no—
180C2×C43⋊C8—22no—
181C23×C43—22no—
182C23×C43—24no—
183C23×C43—22no—
184C23×C43—42no—
185C23×C43—42no—
186C23×C43—42no—
187C23×C43—42no—
188C23×C43—42no—
189C23×C43#12011yescharacteristic
190C22×D43—41no—
191Q8×C43—42no—
192Q8×C43—42no—
193Q8×C43—44no—
194D4×C43—22no—
195D4×C43—22no—
196D4×C43—22no—
197D4×C43—24no—
198D4×C43—42no—
199D4×C43—416no—
200D4×C43C2×D814yes—
201C2×C4×C43—22no—
202C2×C4×C43—22no—
203C2×C4×C43—22no—
204C2×C4×C43—24no—
205C2×C4×C43—41no—
206C2×C4×C43—44no—
207C2×C4×C43—42no—
208C2×C4×C43—416no—
209C2×C4×C43—42no—
210C2×C4×C43—44no—
211C2×C4×C43—42no—
212C2×C4×C43—42no—
213C2×C4×C43—44no—
214C2×C4×C43—44no—
215C2×C4×C43#12011yescharacteristic
216C2×C4×C43#12011yescharacteristic
217C2×C4×C43#12011yescharacteristic
218C2×C4×C43C2×D812yes—
219C2×C4×C43AGL(1,ℤ/8)12yes—
220C2×C4×C43C22×D411yescharacteristic
221C2×C4×C43C22×D411yescharacteristic
222C2×C4×C43C22×D411yescharacteristic
223C22⋊D43—82no—
224C2×Dic43—41no—
225D172—42no—
226D172—41no—
227D172—41no—
228D172—42no—
229C4×D43—82no—
230Dic86—41no—
231Dic86—41no—
232C43⋊C8—42no—
233C43⋊C8—42no—
234C43⋊C8—42no—
235#18756—431nomaximal, Sylow
236C22×C43—21no—
237C22×C43—21no—
238C22×C43—24no—
239C22×C43—48no—
240C22×C43—42no—
241C22×C43—42no—
242C22×C43—42no—
243C22×C43—42no—
244C22×C43—42no—
245C22×C43—84no—
246C22×C43#44712yes—
247C22×C43C2×#12011yescharacteristic
248C2×D43—81no—
249C2×D43—81no—
250C2×D43—81no—
251C4×C43—21no—
252C4×C43—21no—
253C4×C43—21no—
254C4×C43—42no—
255C4×C43—44no—
256C4×C43—42no—
257C4×C43#44712yes—
258C4×C43C22×D811yescharacteristic
259C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
260C4×C43#57512yes—
261Dic43—81no—
262C2×#547D4311yescharacteristic, complemented
263#3226—431no—
264#3100—431no—
265#3036—431no—
266#1647—434no—
267#1625—434no—
268#1546—431no—
269#1486—431no—
270#1478—431no—
271C2×C43—22no—
272C2×C43—44no—
273C2×C43—82no—
274C2×C43C2×#44711yescharacteristic
275C2×C43#300511yescharacteristic
276C2×C43#302011yescharacteristic
277D43—161no—
278C22×#41—21no—
279C2×AGL(1,ℤ/8)—861no—
280#547—22no—
281#547—22no—
282#547—22no—
283#547C2×D4314yes—
284#532—431no—
285C2×C4⋊Q8C2×D4311yescharacteristic
286C2×#123—22no—
287C2×C22⋊Q8—21no—
288C2×#121—21no—
289C2×C4×D4C2×D4311yescharacteristic
290C2×C4×D4C2×D4311yescharacteristic
291C22×C4⋊C4—21no—
292#500—432no—
293#486—432no—
294#480—861no—
295#480—862no—
296#474—432no—
297#474—862no—
298#463—862no—
299#461—432no—
300#459—432no—
301#459—432no—
302#447—861no—
303C4×D8—862no—
304#431—431no—
305#431—431no—
306#423—431no—
307#417—861no—
308#332—434no—
309D4⋊C8—434no—
310D4⋊C8—434no—
311C43#1875611yescharacteristic, complemented, Sylow
312C2×#41—24no—
313C2×#41—24no—
314C2×#41—44no—
315C2×#41—44no—
316C2×#41—861no—
317C22×Q8—21no—
318C22×D4—22no—
319C22×D4C22⋊D4311yescharacteristic
320C22×D4C22⋊D4311yescharacteristic
321C23×C4—22no—
322C23×C4—21no—
323C23×C4—21no—
324AGL(1,ℤ/8)—1724no—
325C2×SD16—862no—
326C2×D8—862no—
327C2×D8—862no—
328C2×M16—861no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4⋊Q8—431no—
332C4⋊Q8C22⋊D4312yes—
333C4⋊Q8C22×D4312yes—
334#127—431no—
335#124—861no—
336#123—28no—
337#123—28no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340#121—24no—
341#121—24no—
342#121—861no—
343#121—861no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—861no—
349C4×D4—861no—
350C4×D4C22⋊D4312yes—
351C4×D4C22⋊D4312yes—
352C4×D4C22×D4312yes—
353C4×D4C22×D4312yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4C22⋊D4311yescharacteristic
359C2×C4⋊C4C22⋊D4311yescharacteristic
360C2×C4⋊C4C22⋊D4312yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42C22×D4311yescharacteristic
365#107—862no—
366#107—864no—
367C4⋊C8—432no—
368C4⋊C8—432no—
369C4⋊C8—861no—
370C4⋊C8—861no—
371#103—862no—
372#102—862no—
373#102—862no—
374#102—862no—
375#102—864no—
376C22⋊C8—861no—
377C22⋊C8—861no—
378C4×C8—432no—
379C4×C8—861no—
380C24—21no—
381C24—21no—
382#41—48no—
383#41—416no—
384#41—48no—
385#41—1722no—
386#41—1722no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—861no—
392C2×D4—28no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—48no—
398C2×D4—861no—
399C2×D4—861no—
400C2×D4—861no—
401C2×D4—1721no—
402C2×D4#1062188214yescomplemented
403C2×D4C2×C22⋊D4312yes—
404C2×D4C2×C22⋊D4312yes—
405C22×C4—28no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—1721no—
418C22×C4C2×C22⋊D4311yescharacteristic
419C22×C4C2×C22⋊D4311yescharacteristic
420C22×C4C2×C22⋊D4311yescharacteristic
421SD16—1724no—
422D8—1724no—
423D8—1724no—
424M16—1722no—
425C2×C8—862no—
426C2×C8—862no—
427C2×C8—862no—
428C4⋊C4—28no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—861no—
435C4⋊C4—861no—
436C4⋊C4C2×C22⋊D4312yes—
437C4⋊C4C2×C22⋊D4312yes—
438C4⋊C4C2×C22⋊D4314yes—
439C22⋊C4—28no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—1721no—
445C22⋊C4—1721no—
446C42C2×C22⋊D4311yescharacteristic
447C42C2×C22⋊D4311yescharacteristic
448C42C2×C22⋊D4311yescharacteristic
449C42C23×D4311yescharacteristic
450C23—22no—
451C23—24no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—1721no—
459C23#1306329411yescharacteristic
460Q8—42no—
461Q8—42no—
462Q8—44no—
463Q8—1721no—
464Q8—1721no—
465D4—22no—
466D4—22no—
467D4—22no—
468D4—24no—
469D4—42no—
470D4—416no—
471D4—3442no—
472D4—1722no—
473D4—1721no—
474D4—1721no—
475D4—1722no—
476D4C2×#1062188214yes—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—416no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—3442no—
492C2×C4—1721no—
493C2×C4#1306329411yescharacteristic
494C2×C4#1306329411yescharacteristic
495C2×C4#1306329411yescharacteristic
496C2×C4C2×#1062188212yes—
497C2×C4#1306335112yes—
498C2×C4C22×C22⋊D4311yescharacteristic
499C2×C4C22×C22⋊D4311yescharacteristic
500C2×C4C22×C22⋊D4311yescharacteristic
501C8—1722no—
502C8—1722no—
503C8—1722no—
504C22—21no—
505C22—21no—
506C22—24no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22—3441no—
515C22—3441no—
516C22—3441no—
517C22#1458303312yes—
518C22C2×#1306329411yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—44no—
524C4—42no—
525C4—3441no—
526C4#1458303312yes—
527C4C22×#1062188211yescharacteristic
528C4C2×#1306335111yescharacteristic
529C4#1458369712yes—
530C2—22no—
531C2—44no—
532C2—82no—
533C2—6881no—
534C2C2×#1458303311yescharacteristic
535C2#1914383711yescharacteristic
536C2#1914395911yescharacteristic
subgroupG/H
centerC22C2×#13063294
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×C43×#547C2
socleC22×C43C2×#120
hypercenterC2×#547D43
hypocenterC43#18756
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #19145716C2semidirect
2. #19144595C2semidirect
3. #19144059C2semidirect
4. #19133830C2semidirect
5. #19133684C2semidirect
6. #19133267C2semidirect
7. #19133020C2semidirect
8. #19132986C2semidirect
9. C2×C43×#547C2semidirect
12. #14583302C22semidirect
13. #14583247C22semidirect
16. #14583179C22semidirect
19. #14583104C22semidirect
20. #14583094C22semidirect
21. #14583094C22semidirect
28. #14582645C22semidirect
29. D4⋊(C43⋊C8)C22semidirect
30. D4⋊(C43⋊C8)C22semidirect
126. C2×D4×C43D8semidirect
262. C2×#547D43semidirect
311. C43#18756semidirect
402. C2×D4#10621882semidirect

Maximal subgroups

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

subgroupindex
#191457162
#191445952
#191440592
#191338302
#191336842
#191332672
#191330202
#191329862
C2×C43×#5472
#1875643