Finite Group Databasebeta

← #25131350

Subgroups — #25131350

Order 11008.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583091C211yesmaximal, characteristic
2C2×C4×#10621884C211yesmaximal, characteristic, complemented
3#19139513C211yesmaximal, characteristic, complemented
4#19135831C212yesmaximal, complemented
5#19135672C212yesmaximal, complemented
6#19135532C211yesmaximal, characteristic, complemented
7#19135402C211yesmaximal, characteristic, complemented
8#19134943C212yesmaximal, complemented
9#19134856C212yesmaximal, complemented
10#19134600C211yesmaximal, characteristic, complemented
11C2×C43×#559C211yesmaximal, characteristic, complemented
12C22×#10621884C2211yescharacteristic
13C22×#10621882C2212yes—
14C2×#13063301—21no—
15C2×C4×D172C2211yescharacteristic
16#14583091—24no—
17C4×#10621884—24no—
18C2×#13063256C2211yescharacteristic
19C2×C4⋊(C43⋊C8)C2211yescharacteristic
20C2×#13063249C2211yescharacteristic
21C2×#13063249C2212yes—
22C2×#13063246C2212yes—
23C2×#13063246C2212yes—
24C2×#13063246C2212yes—
25C2×#13063245C2211yescharacteristic
26C2×C4×C43⋊C8C2211yescharacteristic
27#14582821—22no—
28#14582801C2212yes—
29#14582797C2212yes—
30#14582769C2211yescharacteristic
31#14582756—21no—
32#14582661C2212yes—
33#14582654C2212yes—
34#14582654C2212yes—
35C43×#559—24no—
36C43×#559—24no—
37C2×#127×C43C2211yescharacteristic
38C2×#124×C43—21no—
39C2×#124×C43C2212yes—
40C2×#123×C43—22no—
41C2×C4×Q8×C43C2211yescharacteristic
42C2×C4×Q8×C43C2211yescharacteristic
43C2×C4×D4×C43—21no—
44C2×C4×D4×C43C2212yes—
45C22×D172C2311yescharacteristic
46C22×C4×D43—21no—
47C2×#10621884—22no—
48C2×#10621884—22no—
49C2×#10621884—22no—
50C2×#10621882—24no—
51C2×#10621882—24no—
52C2×#10621882—42no—
53C22×C43⋊C8C2311yescharacteristic
54C22×C43⋊C8C2312yes—
55C22×C43⋊C8C2311yescharacteristic
56#13063301—44no—
57C4×D172—22no—
58C4×D172—22no—
59C2×C4⋊Dic43—21no—
60C2×C4⋊Dic43—21no—
61C2×C4⋊Dic43C2311yescharacteristic
62C2×C4⋊Dic43C2312yes—
63C2×Dic43⋊C4—22no—
64C2×(C2×D43)⋊C4—22no—
65C2×(C2×D43)⋊C4—21no—
66C2×(C2×D43)⋊C4—22no—
67#13063256—22no—
68C4⋊(C43⋊C8)—22no—
69#13063249—22no—
70#13063249—42no—
71#13063246—24no—
72#13063246—24no—
73#13063246—42no—
74#13063245—22no—
75C4×C43⋊C8—22no—
76C43⋊#95—22no—
77C22×Q8×C43D411yescharacteristic
78C22×Q8×C43C2311yescharacteristic
79C22×D4×C43—21no—
80C22×D4×C43—21no—
81C22×D4×C43C2312yes—
82C23×C4×C43—22no—
83#127×C43—22no—
84#127×C43D414yes—
85#124×C43—28no—
86#124×C43—24no—
87#124×C43—24no—
88#123×C43—216no—
89C4×Q8×C43—22no—
90C4×Q8×C43—22no—
91C4×Q8×C43—22no—
92C4×Q8×C43D414yes—
93C4×D4×C43—28no—
94C4×D4×C43—24no—
95C4×D4×C43—24no—
96C2×C4⋊C4×C43—21no—
97C2×C4⋊C4×C43D411yescharacteristic
98C2×C4⋊C4×C43D412yes—
99C2×C4⋊C4×C43C2311yescharacteristic
100C2×C22⋊C4×C43—22no—
101C2×C22⋊C4×C43—22no—
102C2×C22⋊C4×C43—22no—
103C2×C42×C43—21no—
104C2×C42×C43C2312yes—
105C2×C42×C43C2311yescharacteristic
106C24×C43—22no—
107C23×D43—21no—
108C2×Q8×C43—22no—
109C2×Q8×C43—22no—
110C2×Q8×C43—22no—
111C2×Q8×C43—21no—
112C2×Q8×C43—21no—
113C2×Q8×C43—21no—
114C2×Q8×C43—21no—
115C2×Q8×C43—21no—
116C2×Q8×C43C2×D412yes—
117C2×D4×C43—24no—
118C2×D4×C43—24no—
119C2×D4×C43—22no—
120C2×D4×C43—22no—
121C2×D4×C43—22no—
122C2×D4×C43—22no—
123C2×D4×C43—22no—
124C2×D4×C43—42no—
125C2×D4×C43—42no—
126C2×D4×C43—81no—
127C2×D4×C43—81no—
128C2×D4×C43C2×D414yes—
129C22×C4×C43—28no—
130C22×C4×C43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—82no—
133C22×C4×C43—82no—
134C22×C4×C43C2×D412yes—
135C22×C4×C43C2×D412yes—
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2411yescharacteristic
139C22×Dic43—21no—
140C22×Dic43—22no—
141C2×D172—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172—21no—
146C2×D172#4112yes—
147C2×C4×D43—42no—
148C2×C4×D43—42no—
149C2×C4×D43—42no—
150C4⋊C4×C43—24no—
151C4⋊C4×C43—22no—
152C4⋊C4×C43—24no—
153C4⋊C4×C43—21no—
154C4⋊C4×C43C2×D412yes—
155C22⋊C4×C43—28no—
156C22⋊C4×C43—28no—
157C22⋊C4×C43—28no—
158C42×C43—24no—
159C42×C43—22no—
160C42×C43—22no—
161C42×C43C2×D412yes—
162C42×C43C2×D412yes—
163#10621884—28no—
164#10621882—48no—
165(C2×D43)⋊C4—44no—
166(C2×D43)⋊C4—42no—
167(C2×D43)⋊C4—82no—
168C4⋊Dic43—21no—
169C4⋊Dic43—21no—
170C4⋊Dic43—22no—
171C4⋊Dic43—22no—
172C4⋊Dic43—42no—
173C4⋊Dic43—81no—
174Dic43⋊C4—44no—
175C2×C43⋊C8—21no—
176C2×C43⋊C8—21no—
177C2×C43⋊C8—21no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8—22no—
180C2×C43⋊C8—21no—
181C2×C43⋊C8—22no—
182C2×C43⋊C8—21no—
183C2×C43⋊C8#4112yes—
184C23×C43—24no—
185C23×C43—42no—
186C23×C43—42no—
187C23×C43—42no—
188C23×C43—42no—
189C23×C43—42no—
190C23×C43—42no—
191C23×C43C22×D411yescharacteristic
192C22×D43—22no—
193C22×D43—41no—
194C22×D43—41no—
195C22×D43—41no—
196C22×D43—41no—
197C22×D43—41no—
198C22×D43—41no—
199Q8×C43—22no—
200Q8×C43—22no—
201Q8×C43—41no—
202Q8×C43—41no—
203Q8×C43—44no—
204D4×C43—24no—
205D4×C43—24no—
206D4×C43—42no—
207D4×C43—42no—
208D4×C43—48no—
209D4×C43—84no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—22no—
213C2×C4×C43—22no—
214C2×C4×C43—22no—
215C2×C4×C43—22no—
216C2×C4×C43—22no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—22no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—88no—
225C2×C4×C43#12012yes—
226C2×C4×C43#12012yes—
227C2×C4×C43#13512yes—
228C2×C4×C43AGL(1,ℤ/8)12yes—
229C2×C4×C43C22×D411yescharacteristic
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C2×C4×C43C2×#4111yescharacteristic
233C2×C4×C43H(5,𝔽2)11yescharacteristic
234C2×C4×C432−1+411yescharacteristic
235C2×Dic43—41no—
236C2×Dic43—42no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—42no—
240C2×Dic43—42no—
241D172—22no—
242D172—22no—
243D172—41no—
244D172—41no—
245C4×D43—84no—
246C43⋊C8—22no—
247C43⋊C8—41no—
248C43⋊C8—42no—
249#24186—431nomaximal, Sylow
250C22×C43—44no—
251C22×C43—44no—
252C22×C43—44no—
253C22×C43—82no—
254C22×C43—82no—
255C22×C43—82no—
256C22×C43—82no—
257C22×C43C2×#12011yescharacteristic
258C22×C43#54811yescharacteristic
259C22×C43#55011yescharacteristic
260C22×C43C2×#13511yescharacteristic
261C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C43#57611yescharacteristic
263C22×C43#57711yescharacteristic
264C2×D43—42no—
265C2×D43—42no—
266C2×D43—42no—
267C2×D43—81no—
268C2×D43—81no—
269C2×D43—81no—
270C2×D43—81no—
271C4×C43—22no—
272C4×C43—42no—
273C4×C43—42no—
274C4×C43—42no—
275C4×C43—42no—
276C4×C43#44914yes—
277C4×C43#52511yescharacteristic
278C4×C43#53711yescharacteristic
279C4×C43#54611yescharacteristic
280C4×C43#54711yescharacteristic
281Dic43—82no—
282Dic43—81no—
283C2×#559D4311yescharacteristic, complemented
284C2×#459—431no—
285C2×C4×SD16—431no—
286#2538—431no—
287#2042—432no—
288#2012—432no—
289#1984—431no—
290#1957—431no—
291#1879—432no—
292#1864—432no—
293#1803—431no—
294C2×C43—84no—
295C2×C43#240711yescharacteristic
296C2×C43C2×#44911yescharacteristic
297C2×C43#301211yescharacteristic
298C2×C43#336111yescharacteristic
299C2×C43#337311yescharacteristic
300C2×C43#339411yescharacteristic
301C2×C43#339911yescharacteristic
302D43—82no—
303C22×SD16—431no—
304C22×D8—432no—
305#559—24no—
306#559—24no—
307C2×#127C2×D4311yescharacteristic
308C2×#124—21no—
309C2×#124C2×D4312yes—
310C2×#123—22no—
311C2×C22⋊Q8—861no—
312C2×C4×Q8C2×D4311yescharacteristic
313C2×C4×Q8C2×D4311yescharacteristic
314C2×C4×D4—21no—
315C2×C4×D4—431no—
316C2×C4×D4C2×D4312yes—
317#459—864no—
318C4×SD16—864no—
319C2×#106—431no—
320C2×C4⋊C8—431no—
321C2×#103—431no—
322C2×#103—432no—
323C2×#102—432no—
324C2×#102—432no—
325C2×#102—431no—
326C2×#102—432no—
327C2×C4×C8—431no—
328#396—862no—
329#390—432no—
330#389—432no—
331#384—431no—
332#380—861no—
333#340—432no—
334#336—432no—
335#336—432no—
336C43#2418611yescharacteristic, complemented, Sylow
337C22×Q8C22⋊D4311yescharacteristic
338C22×Q8C22×D4311yescharacteristic
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—431no—
342C22×D4C22×D4312yes—
343C23×C4—22no—
344C23×C4—861no—
345C2×SD16—862no—
346C2×SD16—862no—
347C2×SD16—862no—
348C2×D8—864no—
349C2×D8—864no—
350C2×D8—1722no—
351C22×C8—431no—
352C22×C8—432no—
353C22×C8—431no—
354#127—22no—
355#127C22⋊D4314yes—
356#124—28no—
357#124—24no—
358#124—24no—
359#123—216no—
360C22⋊Q8—1724no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8C22⋊D4314yes—
365C4×D4—28no—
366C4×D4—24no—
367C4×D4—24no—
368C4×D4—862no—
369C4×D4—862no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—431no—
372C2×C4⋊C4—432no—
373C2×C4⋊C4—861no—
374C2×C4⋊C4—862no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4C22⋊D4311yescharacteristic
377C2×C4⋊C4C22⋊D4312yes—
378C2×C4⋊C4C22×D4311yescharacteristic
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—862no—
383C2×C22⋊C4—861no—
384C2×C22⋊C4—862no—
385C2×C42—21no—
386C2×C42C22×D4312yes—
387C2×C42C22×D4311yescharacteristic
388#106—862no—
389C4⋊C8—862no—
390#103—862no—
391#103—1722no—
392#102—864no—
393#102—864no—
394#102—862no—
395#102—1722no—
396C4×C8—862no—
397#95—862no—
398C24—22no—
399C24—861no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×C22⋊D4312yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—432no—
421C2×D4—861no—
422C2×D4—861no—
423C2×D4—861no—
424C2×D4—861no—
425C2×D4—861no—
426C2×D4C2×C22⋊D4314yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—28no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—861no—
433C22×C4—862no—
434C22×C4—1722no—
435C22×C4—1722no—
436C22×C4—1722no—
437C22×C4C2×C22⋊D4312yes—
438C22×C4C2×C22⋊D4312yes—
439C22×C4C2×C22⋊D4311yescharacteristic
440C22×C4C2×C22⋊D4311yescharacteristic
441C22×C4C23×D4311yescharacteristic
442SD16—868no—
443D8—1728no—
444C2×C8—432no—
445C2×C8—861no—
446C2×C8—861no—
447C2×C8—861no—
448C2×C8—862no—
449C2×C8—862no—
450C2×C8—861no—
451C2×C8—862no—
452C2×C8—861no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—861no—
458C4⋊C4—861no—
459C4⋊C4—862no—
460C4⋊C4—862no—
461C4⋊C4—3441no—
462C4⋊C4—1724no—
463C4⋊C4—1722no—
464C4⋊C4C2×C22⋊D4312yes—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—3442no—
469C22⋊C4—1724no—
470C22⋊C4—1722no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42C2×C22⋊D4312yes—
475C42C2×C22⋊D4312yes—
476C23—24no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—862no—
484C23—1721no—
485C23—1721no—
486C23—1721no—
487C23—1721no—
488C23—1721no—
489C23—1721no—
490C23C22×C22⋊D4311yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—44no—
494Q8—41no—
495Q8—41no—
496D4—24no—
497D4—24no—
498D4—48no—
499D4—42no—
500D4—42no—
501D4—84no—
502D4—862no—
503D4—862no—
504D4—1721no—
505D4—1721no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—88no—
521C2×C4—3444no—
522C2×C4—1722no—
523C2×C4—1721no—
524C2×C4—1721no—
525C2×C4—1722no—
526C2×C4—1721no—
527C2×C4—1722no—
528C2×C4#1306329412yes—
529C2×C4#1306329412yes—
530C2×C4#1306334512yes—
531C2×C4#1306335012yes—
532C2×C4C22×C22⋊D4311yescharacteristic
533C2×C4C22×C22⋊D4311yescharacteristic
534C2×C4#41×D4311yescharacteristic
535C2×C4#41×D4311yescharacteristic
536C2×C4#1306337111yescharacteristic
537C2×C4#1306337211yescharacteristic
538C8—862no—
539C8—1721no—
540C8—1722no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—3441no—
549C22—3441no—
550C22—3441no—
551C22—3441no—
552C22—1722no—
553C22—1722no—
554C22—1722no—
555C22C2×#1306329411yescharacteristic
556C22#1458355611yescharacteristic
557C22#1458356811yescharacteristic
558C22C2×#1306334511yescharacteristic
559C22C2×#1306335011yescharacteristic
560C22#1458370211yescharacteristic
561C22#1458371011yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—3442no—
568C4—3441no—
569C4#1458304414yes—
570C4#1458340611yescharacteristic
571C4#1458347711yescharacteristic
572C4#1458353711yescharacteristic
573C4#1458354811yescharacteristic
574C2—84no—
575C2—3442no—
576C2#1913793411yescharacteristic
577C2C2×#1458304411yescharacteristic
578C2#1914389911yescharacteristic
579C2#1914725111yescharacteristic
580C2#1914738311yescharacteristic
581C2#1914758711yescharacteristic
582C2#1914763311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#559C2
socleC23×C43C22×D4
prime radicalC2×#14583091C2
hypercenterC2×#559D43
hypocenterC43#24186
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10621884C2semidirect
3. #19139513C2semidirect
4. #19135831C2semidirect
5. #19135672C2semidirect
6. #19135532C2semidirect
7. #19135402C2semidirect
8. #19134943C2semidirect
9. #19134856C2semidirect
10. #19134600C2semidirect
11. C2×C43×#559C2semidirect
283. C2×#559D43semidirect
336. C43#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#145830912
C2×C4×#106218842
#191395132
#191358312
#191356722
#191355322
#191354022
#191349432
#191348562
#191346002
C2×C43×#5592
#2418643