Finite Group Databasebeta

← #25135321

Subgroups — #25135321

Order 11008.

1188 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583050C211yesmaximal, characteristic
2C2×#14582953C211yesmaximal, characteristic, complemented
3#19142935C211yesmaximal, characteristic, complemented
4C2×#14582802C211yesmaximal, characteristic
5#19135979C212yesmaximal, complemented
6#19135911C212yesmaximal, complemented
7#19135477C212yesmaximal, complemented
8#19135294C212yesmaximal, complemented
9#19134900C211yesmaximal, characteristic, complemented
10#19134782C211yesmaximal, characteristic, complemented
11C43×#1786C211yesmaximal, characteristic
12C22×#10621883C2212yes—
13C2×#13063304C2211yescharacteristic
14C22×C4⋊Dic43C2211yescharacteristic
15C22×C22⋊Dic43—21no—
16C22×C22⋊Dic43—21no—
17C22×C4×Dic43C2211yescharacteristic
18#14583050—24no—
19#14582953—24no—
20C2×#13063258C2212yescomplemented
21C2×Dic43⋊C8C2212yescomplemented
22C2×#13063250C2212yescomplemented
23C2×#13063246C2212yes—
24C2×C22⋊(C43⋊C8)C2211yescharacteristic
25C2×#13063235C2212yescomplemented
26#14582804C2211yescharacteristic, complemented
27#14582804C2211yescharacteristic, complemented
28#14582802—21no—
29#14582802—21no—
30#14582802C2212yes—
31#14582802C2212yes—
32#14582798C2211yescharacteristic, complemented
33#14582798C2211yescharacteristic, complemented
34#14582784—21no—
35#14582784—21no—
36#14582774—21no—
37#14582760—21no—
38#14582662C2212yescomplemented
39#14582661C2212yescomplemented
40C23×D4×C43C2211yescharacteristic
41C2×#102×C43C2212yes—
42C2×#102×C43C2212yes—
43C2×C22⋊C8×C43C2211yescharacteristic
44C43×#380C2211yescharacteristic
45C22×Dic86C2311yescharacteristic
46C23×Dic43—21no—
47C23×Dic43D411yescharacteristic
48C23×Dic43D411yescharacteristic
49C2×#10621883—24no—
50C2×#10621883—24no—
51C2×#10621883—42no—
52C22×C43⋊C8C2312yes—
53#13063304—22no—
54#13063304—22no—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43—21no—
57C2×C4⋊Dic43—21no—
58C2×C4⋊Dic43—42no—
59C2×C4⋊Dic43C2311yescharacteristic
60C2×C4⋊Dic43C2312yes—
61C2×Dic43⋊C4—21no—
62C2×C22⋊Dic43—21no—
63C2×C22⋊Dic43—22no—
64C2×C22⋊Dic43—22no—
65C2×C22⋊Dic43—22no—
66C2×C22⋊Dic43—22no—
67C2×C22⋊Dic43—44no—
68C2×C22⋊Dic43—44no—
69C2×C4×Dic43—22no—
70C2×C4×Dic43—22no—
71C2×C4×Dic43D411yescharacteristic
72C2×C4×Dic43D411yescharacteristic
73C2×C4×Dic43C2311yescharacteristic
74C2×C4×Dic43C2311yescharacteristic
75#13063258—42no—
76Dic43⋊C8—24no—
77#13063250—42no—
78#13063246—24no—
79C22⋊(C43⋊C8)D414yes—
80#13063235—24no—
81C43⋊#95—21no—
82C43⋊#95—21no—
83C43⋊#95—21no—
84C43⋊#95—21no—
85C43⋊#95—21no—
86C43⋊#95—21no—
87C25×C43—21no—
88C22×D4×C43—24no—
89C22×D4×C43—21no—
90C22×D4×C43—22no—
91C22×D4×C43—22no—
92C22×D4×C43—42no—
93C22×D4×C43C2312yes—
94C23×C4×C43C2311yescharacteristic
95C22×C8×C43C2312yes—
96C2×C4⋊C4×C43C2311yescharacteristic
97C2×C4⋊C4×C43C2311yescharacteristic
98C2×C22⋊C4×C43—21no—
99#102×C43—24no—
100#102×C43—42no—
101C22⋊C8×C43—22no—
102#95×C43—21no—
103C24×C43—24no—
104C24×C43—22no—
105C24×C43—42no—
106C24×C43—42no—
107C24×C43—42no—
108C24×C43—42no—
109C24×C43—42no—
110C24×C43—42no—
111C24×C43C2×D411yescharacteristic
112C2×D4×C43—24no—
113C2×D4×C43—22no—
114C2×D4×C43—22no—
115C2×D4×C43—22no—
116C2×D4×C43—22no—
117C2×D4×C43—22no—
118C2×D4×C43—48no—
119C2×D4×C43—48no—
120C2×D4×C43—42no—
121C2×D4×C43—42no—
122C2×D4×C43#4114yes—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43C2×D412yes—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43#4112yes—
130C22×C4×C43C2411yescharacteristic
131C22×Dic43—22no—
132C22×Dic43—22no—
133C22×Dic43—22no—
134C22×Dic43—22no—
135C22×Dic43—22no—
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—22no—
139C22×Dic43—41no—
140C22×Dic43—42no—
141C22×Dic43—42no—
142C22×Dic43—42no—
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C22×Dic43C2×D411yescharacteristic
146C22×Dic43C2×D411yescharacteristic
147C2×Dic86—21no—
148C2×Dic86—21no—
149C2×Dic86—22no—
150C2×Dic86—22no—
151C2×C8×C43—22no—
152C2×C8×C43—22no—
153C2×C8×C43—22no—
154C4⋊C4×C43—22no—
155C4⋊C4×C43—22no—
156C22⋊C4×C43—42no—
157C22⋊Dic43—44no—
158C22⋊Dic43—44no—
159C22⋊Dic43—44no—
160C22⋊Dic43—44no—
161C22⋊Dic43—42no—
162#10621883—48no—
163C4⋊Dic43—21no—
164C4⋊Dic43—21no—
165C4⋊Dic43—22no—
166C4⋊Dic43—22no—
167C4⋊Dic43—42no—
168C4⋊Dic43—42no—
169C4⋊Dic43—42no—
170Dic43⋊C4—42no—
171C4×Dic43—22no—
172C4×Dic43—22no—
173C4×Dic43—21no—
174C4×Dic43—21no—
175C4×Dic43—21no—
176C4×Dic43—21no—
177C2×C43⋊C8—22no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8C2×D414yes—
180C23×C43—21no—
181C23×C43—21no—
182C23×C43—21no—
183C23×C43—21no—
184C23×C43—24no—
185C23×C43—48no—
186C23×C43—48no—
187C23×C43—48no—
188C23×C43—42no—
189C23×C43—42no—
190C23×C43—42no—
191C23×C43—42no—
192C23×C43—42no—
193C23×C43—42no—
194C23×C43—84no—
195C23×C43—84no—
196C23×C43—84no—
197C23×C43—84no—
198C23×C43#12112yes—
199C23×C43AGL(1,ℤ/8)12yes—
200C23×C43#13712yes—
201C23×C43C22×D411yescharacteristic
202D4×C43—24no—
203D4×C43—24no—
204D4×C43—48no—
205D4×C43—42no—
206D4×C43—42no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—24no—
210C2×C4×C43—24no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43#12112yes—
218C2×C4×C43C22×D411yescharacteristic
219C2×C4×C43C2×#4112yes—
220C2×C4×C43C2×#4111yescharacteristic
221C2×C4×C43H(5,𝔽2)12yes—
222C2×Dic43—21no—
223C2×Dic43—21no—
224C2×Dic43—21no—
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—21no—
228C2×Dic43—21no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43—21no—
233C2×Dic43—44no—
234C2×Dic43—44no—
235C2×Dic43—41no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—42no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243C2×Dic43—41no—
244C2×Dic43—84no—
245C2×Dic43AGL(1,ℤ/8)12yes—
246Dic86—42no—
247Dic86—42no—
248C8×C43—42no—
249C43⋊C8—24no—
250#24507—431nomaximal, Sylow
251C22×C43—22no—
252C22×C43—22no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—22no—
256C22×C43—22no—
257C22×C43—44no—
258C22×C43—44no—
259C22×C43—44no—
260C22×C43—816no—
261C22×C43—82no—
262C22×C43—82no—
263C22×C43—82no—
264C22×C43—82no—
265C22×C43#46814yes—
266C22×C43C2×#12111yescharacteristic
267C22×C43#52511yescharacteristic
268C22×C43#53511yescharacteristic
269C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C43C2×#13711yescharacteristic
272C22×C43#57511yescharacteristic
273C4×C43—22no—
274C4×C43—81no—
275C4×C43—81no—
276C4×C43#54612yes—
277C4×C43#55112yes—
278Dic43—22no—
279Dic43—41no—
280Dic43—41no—
281Dic43—41no—
282Dic43—81no—
283Dic43—82no—
284Dic43—81no—
285C2×#450—431no—
286C2×#432—431no—
287#2855—431no—
288C2×#390—431no—
289#2067—432no—
290#2055—432no—
291#1973—432no—
292#1936—432no—
293#1872—431no—
294#1852—431no—
295#1786D4311yescharacteristic
296C2×C43—24no—
297C2×C43—84no—
298C2×C43#241111yescharacteristic
299C2×C43C2×#46811yescharacteristic
300C2×C43#305511yescharacteristic
301C2×C43#311011yescharacteristic
302C2×C43#311411yescharacteristic
303C2×C43#321511yescharacteristic
304C2×C43#321711yescharacteristic
305C23×D4C2×D4311yescharacteristic
306C22×SD16—432no—
307C2×C22⋊Q8—431no—
308C22×C4⋊C4—431no—
309C22×C22⋊C4—861no—
310C22×C22⋊C4—861no—
311C22×C42—431no—
312#450—864no—
313#432—864no—
314C2×#107—432no—
315C2×C4⋊C8—432no—
316C2×#103—432no—
317C2×#102—432no—
318C2×#102C2×D4312yes—
319C2×#102C2×D4312yes—
320C2×C22⋊C8—431no—
321C2×C22⋊C8C2×D4311yescharacteristic
322C2×#97—432no—
323#391—431no—
324#391—431no—
325#390—432no—
326#390—432no—
327#390—861no—
328#390—861no—
329#389—431no—
330#389—431no—
331#387—861no—
332#387—861no—
333#385—861no—
334#381—861no—
335#380C2×D4311yescharacteristic
336#340—432no—
337#340—432no—
338C43#2450711yescharacteristic, complemented, Sylow
339C25—21no—
340C22×Q8—431no—
341C22×D4—24no—
342C22×D4—22no—
343C22×D4—21no—
344C22×D4—22no—
345C22×D4—42no—
346C22×D4C22×D4312yes—
347C23×C4—431no—
348C23×C4—431no—
349C23×C4—861no—
350C23×C4C22×D4311yescharacteristic
351C2×SD16—864no—
352C2×SD16—864no—
353C2×SD16—1722no—
354C22×C8—432no—
355C22×C8C22×D4312yes—
356C22⋊Q8—862no—
357C22⋊Q8—862no—
358C2×C4⋊C4—432no—
359C2×C4⋊C4—431no—
360C2×C4⋊C4—861no—
361C2×C4⋊C4—861no—
362C2×C4⋊C4—861no—
363C2×C4⋊C4—861no—
364C2×C4⋊C4—1722no—
365C2×C4⋊C4C22×D4311yescharacteristic
366C2×C4⋊C4C22×D4311yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—862no—
369C2×C22⋊C4—862no—
370C2×C22⋊C4—862no—
371C2×C22⋊C4—862no—
372C2×C22⋊C4—861no—
373C2×C22⋊C4—1724no—
374C2×C22⋊C4—1724no—
375C2×C42—431no—
376C2×C42—431no—
377C2×C42—431no—
378C2×C42—431no—
379C2×C42—862no—
380C2×C42—862no—
381#107—1722no—
382C4⋊C8—864no—
383#103—1722no—
384#102—24no—
385#102—42no—
386#102—864no—
387C22⋊C8—22no—
388C22⋊C8—434no—
389#97—864no—
390#95—21no—
391#95—861no—
392#95—861no—
393#95—861no—
394#95—861no—
395#95—861no—
396#95—861no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24D4×D4311yescharacteristic
406C2×Q8—861no—
407C2×Q8—862no—
408C2×Q8—861no—
409C2×Q8—862no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4#1062190014yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—431no—
426C22×C4—431no—
427C22×C4—431no—
428C22×C4—431no—
429C22×C4—862no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—862no—
433C22×C4—862no—
434C22×C4—861no—
435C22×C4—861no—
436C22×C4—862no—
437C22×C4—1721no—
438C22×C4—1722no—
439C22×C4—1722no—
440C22×C4—1722no—
441C22×C4#1062189812yes—
442C22×C4D4×D4311yescharacteristic
443C22×C4D4×D4312yes—
444C22×C4C23×D4311yescharacteristic
445SD16—1728no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—434no—
450C2×C8—862no—
451C2×C8—862no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—861no—
455C4⋊C4—861no—
456C4⋊C4—862no—
457C4⋊C4—862no—
458C4⋊C4—1722no—
459C4⋊C4—1722no—
460C4⋊C4—1722no—
461C4⋊C4—1722no—
462C22⋊C4—42no—
463C22⋊C4—1724no—
464C22⋊C4—1724no—
465C22⋊C4—1724no—
466C22⋊C4—1724no—
467C22⋊C4—1722no—
468C42—862no—
469C42—862no—
470C42—861no—
471C42—861no—
472C42—861no—
473C42—861no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23#1306329812yes—
493C23#1306334912yes—
494C23#1306335412yes—
495C23C2×D4×D4311yescharacteristic
496Q8—1722no—
497Q8—1722no—
498D4—24no—
499D4—24no—
500D4—48no—
501D4—42no—
502D4—42no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—432no—
514C2×C4—861no—
515C2×C4—861no—
516C2×C4—861no—
517C2×C4—861no—
518C2×C4—861no—
519C2×C4—861no—
520C2×C4—861no—
521C2×C4—861no—
522C2×C4—861no—
523C2×C4—861no—
524C2×C4—861no—
525C2×C4—3444no—
526C2×C4—1724no—
527C2×C4—1721no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1722no—
531C2×C4—1721no—
532C2×C4—1722no—
533C2×C4—1721no—
534C2×C4—1722no—
535C2×C4—1721no—
536C2×C4—1724no—
537C2×C4#1306329812yes—
538C2×C4C2×D4×D4311yescharacteristic
539C2×C4C2×#1062190012yes—
540C2×C4C2×#1062189811yescharacteristic
541C2×C4#1306337012yes—
542C8—42no—
543C8—864no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22#1458314414yes—
559C22C2×#1306329811yescharacteristic
560C22#1458340511yescharacteristic
561C22#1458346411yescharacteristic
562C22AGL(1,ℤ/8)×D4311yescharacteristic
563C22C2×#1306334911yescharacteristic
564C22C2×#1306335411yescharacteristic
565C22#1458369511yescharacteristic
566C4—22no—
567C4—81no—
568C4—81no—
569C4—862no—
570C4—3441no—
571C4—3442no—
572C4—3441no—
573C4—1721no—
574C4—1721no—
575C4—1721no—
576C4#1458353612yes—
577C4#1458357512yes—
578C2—24no—
579C2—84no—
580C2#1913798211yescharacteristic
581C2C2×#1458314411yescharacteristic
582C2#1914421211yescharacteristic
583C2#1914468911yescharacteristic
584C2#1914472811yescharacteristic
585C2#1914559511yescharacteristic
586C2#1914561111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1786C2
socleC23×C43C22×D4
prime radicalC23×D4×C43C22
hypercenter#1786D43
hypocenterC43#24507
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#14582953C2semidirect
3. #19142935C2semidirect
5. #19135979C2semidirect
6. #19135911C2semidirect
7. #19135477C2semidirect
8. #19135294C2semidirect
9. #19134900C2semidirect
10. #19134782C2semidirect
20. C2×#13063258C22semidirect
21. C2×Dic43⋊C8C22semidirect
22. C2×#13063250C22semidirect
25. C2×#13063235C22semidirect
26. #14582804C22semidirect
27. #14582804C22semidirect
32. #14582798C22semidirect
33. #14582798C22semidirect
38. #14582662C22semidirect
39. #14582661C22semidirect
338. C43#24507semidirect

Maximal subgroups

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

subgroupindex
C2×#145830502
C2×#145829532
#191429352
C2×#145828022
#191359792
#191359112
#191354772
#191352942
#191349002
#191347822
C43×#17862
#2450743