Finite Group Databasebeta

← #25130519

Subgroups — #25130519

Order 11008.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583619C211yesmaximal, characteristic, complemented
2C2×#14583179C211yesmaximal, characteristic, complemented
3C2×C4×#10621883C211yesmaximal, characteristic, complemented
4#19135798C212yesmaximal, complemented
5#19135598C212yesmaximal, complemented
6#19135555C211yesmaximal, characteristic, complemented
7#19135390C211yesmaximal, characteristic, complemented
8#19134918C212yesmaximal, complemented
9#19134792C212yesmaximal, complemented
10#19134597C211yesmaximal, characteristic, complemented
11C43×#2537C211yesmaximal, characteristic
12C22×#10621883C2211yescharacteristic
13#14583619—24no—
14#14583619—24no—
15C2×#13063325C2211yescharacteristic
16C2×#13063318—22no—
17C2×#13063312—21no—
18C2×#13063304C2212yes—
19C2×C4×Dic86C2211yescharacteristic
20C2×#13063290—21no—
21C2×#13063284C2212yes—
22#14583179—24no—
23C4×#10621883—24no—
24C2×#13063258C2212yes—
25C2×#13063256C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063248C2211yescharacteristic
28C2×#13063248C2212yes—
29C2×#13063246C2212yes—
30C2×#13063246C2211yescharacteristic
31C2×C22⋊(C43⋊C8)C2212yes—
32C2×C22⋊(C43⋊C8)C2212yes—
33C2×C4×C43⋊C8C2211yescharacteristic
34#14582661C2212yes—
35#14582661C2212yes—
36#14582661C2212yes—
37C2×#121×C43—21no—
38C2×#121×C43C2212yes—
39C2×C4×D4×C43C2211yescharacteristic
40C2×C4×D4×C43C2211yescharacteristic
41C43×#394—22no—
42C43×#384C2211yescharacteristic
43C43×#380—21no—
44C43×#380C2212yes—
45C22×Dic86—21no—
46C22×Dic86C2311yescharacteristic
47C2×#10621883—22no—
48C2×#10621883—22no—
49C2×#10621883—22no—
50C22×C43⋊C8C2311yescharacteristic
51C22×C43⋊C8C2312yes—
52C22×C43⋊C8C2311yescharacteristic
53#13063325—22no—
54#13063325D414yescomplemented
55#13063318—216no—
56#13063312—28no—
57#13063304—24no—
58#13063304—24no—
59C4×Dic86—22no—
60C4×Dic86D414yescomplemented
61#13063290—28no—
62#13063284—24no—
63#13063284—24no—
64C2×C4⋊Dic43—21no—
65C2×C4⋊Dic43C2311yescharacteristic
66C2×C4⋊Dic43C2312yes—
67C2×Dic43⋊C4—22no—
68C2×Dic43⋊C4—21no—
69C2×C22⋊Dic43—22no—
70C2×C22⋊Dic43—22no—
71C2×C4×Dic43—22no—
72C2×C4×Dic43—21no—
73#13063258—24no—
74#13063256—22no—
75C4⋊(C43⋊C8)—22no—
76#13063248—22no—
77#13063248—42no—
78#13063246—24no—
79#13063246—22no—
80C22⋊(C43⋊C8)—24no—
81C22⋊(C43⋊C8)—42no—
82C4×C43⋊C8—22no—
83C22×D4×C43—22no—
84C22×D4×C43D411yescharacteristic
85C22×D4×C43C2311yescharacteristic
86C23×C4×C43—21no—
87C23×C4×C43C2312yes—
88#121×C43—24no—
89#121×C43—42no—
90#121×C43—44no—
91C4×D4×C43—22no—
92C4×D4×C43—22no—
93C4×D4×C43—22no—
94C4×D4×C43—22no—
95C2×C4⋊C4×C43—21no—
96C2×C4⋊C4×C43—21no—
97C2×C4⋊C4×C43D411yescharacteristic
98C2×C4⋊C4×C43C2311yescharacteristic
99C2×C4⋊C4×C43C2312yes—
100C2×C22⋊C4×C43—22no—
101C2×C22⋊C4×C43—21no—
102C2×C22⋊C4×C43—22no—
103C2×C22⋊C4×C43D412yes—
104C2×C42×C43C2311yescharacteristic
105#95×C43—22no—
106C24×C43—21no—
107C24×C43C2×D412yes—
108C2×D4×C43—22no—
109C2×D4×C43—22no—
110C2×D4×C43—22no—
111C2×D4×C43—21no—
112C2×D4×C43—21no—
113C2×D4×C43—21no—
114C2×D4×C43—21no—
115C2×D4×C43—21no—
116C2×D4×C43—44no—
117C2×D4×C43—44no—
118C2×D4×C43—82no—
119C2×D4×C43#4112yes—
120C22×C4×C43—22no—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—22no—
126C22×C4×C43—21no—
127C22×C4×C43—22no—
128C22×C4×C43—42no—
129C22×C4×C43—42no—
130C22×C4×C43—42no—
131C22×C4×C43C2×D412yes—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2411yescharacteristic
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—22no—
138C2×Dic86—21no—
139C2×Dic86—21no—
140C2×Dic86—21no—
141C2×Dic86—21no—
142C2×Dic86—24no—
143C2×Dic86—21no—
144C2×Dic86—42no—
145C2×Dic86—42no—
146C2×Dic86C2×D412yes—
147C4⋊C4×C43—22no—
148C4⋊C4×C43—21no—
149C4⋊C4×C43—21no—
150C4⋊C4×C43—22no—
151C4⋊C4×C43—22no—
152C4⋊C4×C43—42no—
153C4⋊C4×C43—81no—
154C22⋊C4×C43—24no—
155C22⋊C4×C43—44no—
156C22⋊C4×C43—44no—
157C22⋊C4×C43—42no—
158C42×C43C2×D412yes—
159C42×C43C2×D412yes—
160C22⋊Dic43—28no—
161C22⋊Dic43—28no—
162#10621883—28no—
163C4⋊Dic43—24no—
164C4⋊Dic43—21no—
165C4⋊Dic43—22no—
166C4⋊Dic43C2×D412yes—
167C4⋊Dic43C2×D414yes—
168Dic43⋊C4—28no—
169Dic43⋊C4—24no—
170C4×Dic43—28no—
171C4×Dic43—24no—
172C2×C43⋊C8—21no—
173C2×C43⋊C8—21no—
174C2×C43⋊C8—21no—
175C2×C43⋊C8—22no—
176C2×C43⋊C8—22no—
177C2×C43⋊C8—21no—
178C2×C43⋊C8—22no—
179C2×C43⋊C8—21no—
180C2×C43⋊C8#4112yes—
181C23×C43—22no—
182C23×C43—22no—
183C23×C43—22no—
184C23×C43—22no—
185C23×C43—22no—
186C23×C43—22no—
187C23×C43—22no—
188C23×C43—22no—
189C23×C43—41no—
190C23×C43—41no—
191C23×C43—41no—
192C23×C43—41no—
193C23×C43—41no—
194C23×C43—41no—
195C23×C43C22×D411yescharacteristic
196D4×C43—22no—
197D4×C43—22no—
198D4×C43—41no—
199D4×C43—41no—
200D4×C43—44no—
201D4×C43—88no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—22no—
205C2×C4×C43—22no—
206C2×C4×C43—22no—
207C2×C4×C43—42no—
208C2×C4×C43—42no—
209C2×C4×C43—42no—
210C2×C4×C43—42no—
211C2×C4×C43—41no—
212C2×C4×C43—42no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—42no—
216C2×C4×C43—42no—
217C2×C4×C43—84no—
218C2×C4×C43#12012yes—
219C2×C4×C43#12012yes—
220C2×C4×C43#13512yes—
221C2×C4×C43AGL(1,ℤ/8)12yes—
222C2×C4×C43C22×D411yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43C2×#4111yescharacteristic
225C2×C4×C43C2×#4111yescharacteristic
226C2×C4×C43H(5,𝔽2)11yescharacteristic
227C2×C4×C432−1+411yescharacteristic
228C2×Dic43—22no—
229C2×Dic43—22no—
230C2×Dic43—24no—
231C2×Dic43—41no—
232C2×Dic43—41no—
233C2×Dic43—42no—
234C2×Dic43—42no—
235C2×Dic43—41no—
236C2×Dic43—41no—
237Dic86—22no—
238Dic86—22no—
239Dic86—41no—
240Dic86—41no—
241Dic86—48no—
242C43⋊C8—22no—
243C43⋊C8—41no—
244C43⋊C8—42no—
245#24119—431nomaximal, Sylow
246C22×C43—42no—
247C22×C43—42no—
248C22×C43—42no—
249C22×C43—42no—
250C22×C43—42no—
251C22×C43—42no—
252C22×C43—42no—
253C22×C43—42no—
254C22×C43—42no—
255C22×C43—42no—
256C22×C43—81no—
257C22×C43—81no—
258C22×C43—81no—
259C22×C43—81no—
260C22×C43C2×#12011yescharacteristic
261C22×C43#54611yescharacteristic
262C22×C43#54711yescharacteristic
263C22×C43C2×#13511yescharacteristic
264C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C43#57711yescharacteristic
266C22×C43#57811yescharacteristic
267C4×C43—22no—
268C4×C43—42no—
269C4×C43—81no—
270C4×C43—82no—
271C4×C43#44914yes—
272C4×C43#52511yescharacteristic
273C4×C43#53711yescharacteristic
274C4×C43#54811yescharacteristic
275C4×C43#55011yescharacteristic
276Dic43—42no—
277Dic43—44no—
278Dic43—42no—
279C2×#556—431no—
280C2×#474—431no—
281C2×C4×SD16—431no—
282#2537D4311yescharacteristic
283#2036—432no—
284#2000—432no—
285#1990—431no—
286#1954—431no—
287#1875—432no—
288#1854—432no—
289#1803—431no—
290C2×C43—82no—
291C2×C43—82no—
292C2×C43#240711yescharacteristic
293C2×C43C2×#44911yescharacteristic
294C2×C43#301311yescharacteristic
295C2×C43#330211yescharacteristic
296C2×C43#332411yescharacteristic
297C2×C43#334011yescharacteristic
298C2×C43#335311yescharacteristic
299C22×SD16—431no—
300#556—864no—
301#556—864no—
302C2×C4⋊Q8—431no—
303C2×#126—862no—
304C2×#124—861no—
305C2×C22⋊Q8—432no—
306C2×#121—21no—
307C2×#121C2×D4312yes—
308C2×C4×Q8—431no—
309C2×C4×Q8—861no—
310C2×C4×D4C2×D4311yescharacteristic
311C2×C4×D4C2×D4311yescharacteristic
312C2×#117—432no—
313#474—864no—
314C4×SD16—864no—
315C2×#107—432no—
316C2×#106—431no—
317C2×C4⋊C8—431no—
318C2×#103—431no—
319C2×#103—432no—
320C2×#102—431no—
321C2×#102—432no—
322C2×C22⋊C8—432no—
323C2×C22⋊C8—432no—
324C2×C4×C8—431no—
325#394—22no—
326#384C2×D4311yescharacteristic
327#380—21no—
328#380C2×D4312yes—
329#340—432no—
330#340—432no—
331#340—432no—
332C43#2411911yescharacteristic, complemented, Sylow
333C22×Q8—431no—
334C22×Q8—861no—
335C22×D4—22no—
336C22×D4C22⋊D4311yescharacteristic
337C22×D4C22×D4311yescharacteristic
338C23×C4—21no—
339C23×C4C22×D4312yes—
340C2×SD16—862no—
341C2×SD16—862no—
342C2×SD16—862no—
343C22×C8—431no—
344C22×C8—431no—
345C22×C8—432no—
346C4⋊Q8—434no—
347C4⋊Q8—862no—
348#126—8616no—
349#124—868no—
350C22⋊Q8—864no—
351C22⋊Q8—864no—
352#121—24no—
353#121—42no—
354#121—44no—
355C4×Q8—434no—
356C4×Q8—868no—
357C4×Q8—862no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362#117—864no—
363#117—864no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—431no—
367C2×C4⋊C4—432no—
368C2×C4⋊C4—861no—
369C2×C4⋊C4—862no—
370C2×C4⋊C4—861no—
371C2×C4⋊C4C22⋊D4311yescharacteristic
372C2×C4⋊C4C22×D4311yescharacteristic
373C2×C4⋊C4C22×D4312yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—862no—
378C2×C22⋊C4—862no—
379C2×C22⋊C4C22⋊D4312yes—
380C2×C42—862no—
381C2×C42—861no—
382C2×C42C22×D4311yescharacteristic
383#107—864no—
384#106—862no—
385C4⋊C8—862no—
386#103—862no—
387#103—1722no—
388#102—862no—
389#102—864no—
390C22⋊C8—864no—
391C22⋊C8—1722no—
392C4×C8—862no—
393#95—22no—
394C24—21no—
395C24C2×C22⋊D4312yes—
396C2×Q8—432no—
397C2×Q8—864no—
398C2×Q8—861no—
399C2×Q8—861no—
400C2×Q8—861no—
401C2×Q8—861no—
402C2×Q8—861no—
403C2×Q8—1722no—
404C2×Q8—1722no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4#1062190012yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—861no—
429C22×C4—862no—
430C22×C4—861no—
431C22×C4C2×C22⋊D4312yes—
432C22×C4C2×C22⋊D4311yescharacteristic
433C22×C4C2×C22⋊D4311yescharacteristic
434C22×C4C23×D4311yescharacteristic
435SD16—868no—
436C2×C8—432no—
437C2×C8—861no—
438C2×C8—861no—
439C2×C8—861no—
440C2×C8—862no—
441C2×C8—862no—
442C2×C8—861no—
443C2×C8—862no—
444C2×C8—861no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—432no—
453C4⋊C4—434no—
454C4⋊C4—868no—
455C4⋊C4—864no—
456C4⋊C4—864no—
457C4⋊C4—861no—
458C4⋊C4—862no—
459C22⋊C4—24no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—868no—
464C22⋊C4—868no—
465C42—868no—
466C42—864no—
467C42D4×D4312yes—
468C42D4×D4312yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23C22×C22⋊D4311yescharacteristic
484Q8—862no—
485Q8—862no—
486Q8—1728no—
487Q8—1721no—
488Q8—1721no—
489D4—22no—
490D4—22no—
491D4—44no—
492D4—41no—
493D4—41no—
494D4—88no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—862no—
512C2×C4—864no—
513C2×C4—862no—
514C2×C4—1722no—
515C2×C4—1722no—
516C2×C4—1721no—
517C2×C4—1721no—
518C2×C4—1721no—
519C2×C4—1721no—
520C2×C4#1306329312yes—
521C2×C4#1306329312yes—
522C2×C4#1306334512yes—
523C2×C4#1306335112yes—
524C2×C4C2×D4×D4311yescharacteristic
525C2×C4C2×D4×D4311yescharacteristic
526C2×C4#41×D4311yescharacteristic
527C2×C4C2×#1062190011yescharacteristic
528C2×C4#1306337011yescharacteristic
529C2×C4#1306337211yescharacteristic
530C8—862no—
531C8—1722no—
532C8—1721no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#1306329311yescharacteristic
548C22#1458354311yescharacteristic
549C22#1458354811yescharacteristic
550C22C2×#1306334511yescharacteristic
551C22C2×#1306335111yescharacteristic
552C22#1458371011yescharacteristic
553C22#1458371411yescharacteristic
554C4—22no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—1724no—
559C4—1722no—
560C4—1722no—
561C4#1458304214yes—
562C4#1458340911yescharacteristic
563C4#1458347811yescharacteristic
564C4#1458355511yescharacteristic
565C4#1458356711yescharacteristic
566C2—82no—
567C2—82no—
568C2#1913793611yescharacteristic
569C2C2×#1458304211yescharacteristic
570C2#1914390711yescharacteristic
571C2#1914651211yescharacteristic
572C2#1914682611yescharacteristic
573C2#1914700811yescharacteristic
574C2#1914717911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
Fitting, prime radicalC43×#2537C2
socleC23×C43C22×D4
hypercenter#2537D43
hypocenterC43#24119
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#14583619C2semidirect
2. C2×#14583179C2semidirect
3. C2×C4×#10621883C2semidirect
4. #19135798C2semidirect
5. #19135598C2semidirect
6. #19135555C2semidirect
7. #19135390C2semidirect
8. #19134918C2semidirect
9. #19134792C2semidirect
10. #19134597C2semidirect
54. #13063325D4semidirect
60. C4×Dic86D4semidirect
332. C43#24119semidirect

Maximal subgroups

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

subgroupindex
C2×#145836192
C2×#145831792
C2×C4×#106218832
#191357982
#191355982
#191355552
#191353902
#191349182
#191347922
#191345972
C43×#25372
#2411943