Finite Group Databasebeta

← #25129576

Subgroups — #25129576

Order 11008.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583558C211yesmaximal, characteristic
2C2×#14583212C211yesmaximal, characteristic, complemented
3C22×#13063248C211yesmaximal, characteristic, complemented
4#19135836C212yesmaximal, complemented
5#19135798C212yesmaximal, complemented
6#19135021C211yesmaximal, characteristic, complemented
7#19134888C211yesmaximal, characteristic, complemented
8#19134848C212yesmaximal, complemented
9#19134533C212yesmaximal, complemented
10#19134529C211yesmaximal, characteristic, complemented
11C43×#2486C211yesmaximal, characteristic, complemented
12C22×#10621898C2211yescharacteristic
13C23×Dic86C2211yescharacteristic
14C23×C43⋊C8C2211yescharacteristic
15#14583558—24no—
16#14583558—24no—
17C2×#13063313C2211yescharacteristic
18C2×#13063311—21no—
19C2×#13063304C2211yescharacteristic
20C2×#13063303—21no—
21C2×#13063299C2211yescharacteristic
22C2×#13063297—21no—
23C2×C4×Dic86C2211yescharacteristic
24C2×(C22⋊D43)⋊C4—21no—
25#14583212—24no—
26C2×#13063258C2212yes—
27C2×#13063248—24no—
28C2×#13063248C2212yes—
29C2×#13063248C2212yes—
30C2×#13063248C2212yes—
31C2×#13063248C2212yes—
32C2×#13063245C2212yes—
33C2×#13063245C2212yes—
34C2×C22⋊(C43⋊C8)C2211yescharacteristic
35C2×C22⋊(C43⋊C8)C2212yes—
36#14582661C2212yes—
37#14582654C2212yes—
38C2×#123×C43—21no—
39C2×C4×D4×C43C2211yescharacteristic
40C22×C4⋊C4×C43C2211yescharacteristic
41C43×#398—21no—
42C43×#397—21no—
43C43×#389C2212yes—
44C43×#381—21no—
45C43×#380C2211yescharacteristic
46C43×#380C2212yes—
47C2×#10621898—22no—
48C2×#10621898—22no—
49C2×#10621898—24no—
50C2×#10621898—42no—
51C22×Dic86—24no—
52C22×Dic86—22no—
53C22×Dic86—21no—
54C22×Dic86—42no—
55C22×Dic86D414yes—
56C22×Dic86C2311yescharacteristic
57C22×Dic86C2311yescharacteristic
58C22×D172C2311yescharacteristic
59C22×C22⋊D43—21no—
60C22×C22⋊D43—21no—
61C23×Dic43—21no—
62C22×C4×D43—21no—
63C22×C43⋊C8—22no—
64C22×C43⋊C8—42no—
65C22×C43⋊C8C2312yes—
66C22×C43⋊C8C2312yes—
67#13063313—22no—
68#13063313—22no—
69#13063311—28no—
70#13063304—22no—
71#13063304—22no—
72#13063303—28no—
73#13063299—22no—
74#13063299D414yes—
75#13063297—28no—
76C4×Dic86—22no—
77C4×Dic86—22no—
78(C22⋊D43)⋊C4—28no—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×C22⋊Dic43—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×C4×Dic43—21no—
87#13063258—24no—
88#13063248—24no—
89#13063248—24no—
90#13063248—24no—
91#13063248—42no—
92#13063245—24no—
93#13063245—42no—
94C22⋊(C43⋊C8)—22no—
95C22⋊(C43⋊C8)—42no—
96C22×D4×C43D411yescharacteristic
97C23×C4×C43—21no—
98C23×C4×C43C2311yescharacteristic
99C23×C4×C43C2311yescharacteristic
100#123×C43—44no—
101C4×D4×C43—22no—
102C4×D4×C43—22no—
103C2×C4⋊C4×C43—22no—
104C2×C4⋊C4×C43—22no—
105C2×C4⋊C4×C43—21no—
106C2×C4⋊C4×C43—22no—
107C2×C4⋊C4×C43D411yescharacteristic
108C2×C4⋊C4×C43C2312yes—
109C2×C4⋊C4×C43C2312yes—
110C2×C22⋊C4×C43—21no—
111C2×C22⋊C4×C43—22no—
112C2×C22⋊C4×C43D411yescharacteristic
113C2×C22⋊C4×C43D411yescharacteristic
114C2×C42×C43C2311yescharacteristic
115#95×C43—22no—
116#95×C43—21no—
117#95×C43—22no—
118C24×C43C2×D411yescharacteristic
119C24×C43C2×D411yescharacteristic
120C23×D43—21no—
121C2×D4×C43—22no—
122C2×D4×C43—22no—
123C2×D4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—22no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—22no—
129C22×C4×C43—22no—
130C22×C4×C43—22no—
131C22×C4×C43—22no—
132C22×C4×C43—42no—
133C22×C4×C43—42no—
134C22×C4×C43—42no—
135C22×C4×C43C2×D412yes—
136C22×C4×C43C2×D412yes—
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2×D411yescharacteristic
140C22×C4×C43C2×D411yescharacteristic
141C22×C4×C43C2411yescharacteristic
142C2×C22⋊D43—24no—
143C2×C22⋊D43—24no—
144C2×C22⋊D43—44no—
145C2×C22⋊D43—42no—
146C2×C22⋊D43—42no—
147C22×Dic43—24no—
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—44no—
152#10621898—48no—
153#10621898—44no—
154#10621898—44no—
155C2×D172—21no—
156C2×D172—22no—
157C2×D172—22no—
158C2×D172C2×D412yes—
159C2×C4×D43—24no—
160C2×C4×D43—44no—
161C2×Dic86—21no—
162C2×Dic86—22no—
163C2×Dic86—22no—
164C2×Dic86—22no—
165C2×Dic86—28no—
166C2×Dic86—24no—
167C2×Dic86—21no—
168C2×Dic86—44no—
169C2×Dic86—48no—
170C2×Dic86—44no—
171C2×Dic86C2×D412yes—
172C2×Dic86C2×D412yes—
173C2×Dic86#4114yes—
174C4⋊C4×C43—22no—
175C4⋊C4×C43—22no—
176C4⋊C4×C43—22no—
177C4⋊C4×C43—22no—
178C4⋊C4×C43—22no—
179C4⋊C4×C43—44no—
180C4⋊C4×C43—42no—
181C22⋊C4×C43—22no—
182C22⋊C4×C43—22no—
183C22⋊C4×C43—44no—
184C22⋊C4×C43—42no—
185C42×C43—21no—
186C42×C43—21no—
187C22⋊Dic43—24no—
188(C2×D43)⋊C4—24no—
189(C2×D43)⋊C4—24no—
190(C2×D43)⋊C4—24no—
191C4⋊Dic43—21no—
192C4⋊Dic43C2×D412yes—
193Dic43⋊C4—24no—
194Dic43⋊C4—24no—
195C4×Dic43—24no—
196C2×C43⋊C8—22no—
197C2×C43⋊C8—22no—
198C2×C43⋊C8—22no—
199C2×C43⋊C8—22no—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C2×C43⋊C8—44no—
203C23×C43—22no—
204C23×C43—22no—
205C23×C43—21no—
206C23×C43—22no—
207C23×C43—21no—
208C23×C43—21no—
209C23×C43—22no—
210C23×C43#12012yes—
211C23×C43#13512yes—
212C23×C43AGL(1,ℤ/8)12yes—
213C23×C43C22×D411yescharacteristic
214C22×D43—22no—
215C22×D43—42no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—42no—
219D4×C43—44no—
220C2×C4×C43—22no—
221C2×C4×C43—22no—
222C2×C4×C43—22no—
223C2×C4×C43—24no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—22no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—22no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43—42no—
239C2×C4×C43—42no—
240C2×C4×C43—84no—
241C2×C4×C43#12012yes—
242C2×C4×C43C22×D411yescharacteristic
243C2×C4×C43C22×D411yescharacteristic
244C2×C4×C43C2×#4112yes—
245C2×C4×C43H(5,𝔽2)12yes—
246C22⋊D43—48no—
247C22⋊D43—48no—
248C2×Dic43—22no—
249C2×Dic43—24no—
250C2×Dic43—22no—
251C2×Dic43—22no—
252C2×Dic43—48no—
253C2×Dic43—42no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256D172—24no—
257D172—41no—
258D172—41no—
259C4×D43—48no—
260Dic86—22no—
261Dic86—24no—
262Dic86—22no—
263Dic86—22no—
264Dic86—48no—
265Dic86—42no—
266Dic86#12314yescomplemented
267C43⋊C8—42no—
268C43⋊C8—42no—
269#24035—431nomaximal, Sylow
270C22×C43—22no—
271C22×C43—22no—
272C22×C43—24no—
273C22×C43—24no—
274C22×C43—41no—
275C22×C43—41no—
276C22×C43—41no—
277C22×C43—42no—
278C22×C43—42no—
279C22×C43#44914yes—
280C22×C43C2×#12011yescharacteristic
281C22×C43#54612yes—
282C22×C43C2×#13511yescharacteristic
283C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C43#57812yes—
285C2×D43—42no—
286C2×D43—42no—
287C2×D43—42no—
288C2×D43—82no—
289C2×D43—82no—
290C4×C43—22no—
291C4×C43—41no—
292C4×C43—41no—
293C4×C43—41no—
294C4×C43—82no—
295C4×C43—82no—
296C4×C43C2×#12311yescharacteristic
297C4×C43#53411yescharacteristic
298C4×C43#54612yes—
299Dic43—24no—
300Dic43—42no—
301Dic43—42no—
302C2×#548—431no—
303C2×#480—431no—
304C22×#103—431no—
305#2486D4311yescharacteristic, complemented
306#2042—432no—
307#2036—432no—
308#1893—431no—
309#1869—431no—
310#1863—432no—
311#1788—432no—
312#1787—431no—
313C2×C43—24no—
314C2×C43—81no—
315C2×C43#240511yescharacteristic
316C2×C43C2×#44911yescharacteristic
317C2×C43#301111yescharacteristic
318C2×C43#330012yes—
319C2×C43#330812yes—
320D43—82no—
321C22×#41—431no—
322C23×Q8—431no—
323C23×C8—431no—
324#548—864no—
325#548—864no—
326C2×#124—431no—
327C2×#124—861no—
328C2×#123—21no—
329C2×C22⋊Q8—431no—
330C2×C22⋊Q8—861no—
331C2×#121—431no—
332C2×#121—861no—
333C2×C4×Q8—431no—
334C2×C4×D4—861no—
335C2×C4×D4C2×D4311yescharacteristic
336C22×C4⋊C4C2×D4311yescharacteristic
337#480—864no—
338C2×#107—432no—
339C2×#103—432no—
340C2×#103—432no—
341C2×#103—432no—
342C2×#103—432no—
343C2×#103—864no—
344C2×#102—432no—
345C2×#102—432no—
346C2×C22⋊C8—431no—
347C2×C22⋊C8—432no—
348#398—21no—
349#397—21no—
350#389C2×D4312yes—
351#381—21no—
352#380C2×D4311yescharacteristic
353#380C2×D4312yes—
354#340—432no—
355#336—432no—
356C43#2403511yescharacteristic, complemented, Sylow
357C2×#41—862no—
358C2×#41—862no—
359C2×#41—864no—
360C2×#41—1722no—
361C22×Q8—434no—
362C22×Q8—431no—
363C22×Q8—431no—
364C22×Q8—864no—
365C22×Q8—862no—
366C22×Q8—861no—
367C22×Q8—1722no—
368C22×D4—431no—
369C22×D4—861no—
370C22×D4—861no—
371C22×D4C22⋊D4311yescharacteristic
372C23×C4—21no—
373C23×C4—861no—
374C23×C4—861no—
375C23×C4C22×D4311yescharacteristic
376C23×C4C22×D4311yescharacteristic
377C22×C8—432no—
378C22×C8—432no—
379C22×C8—862no—
380C22×C8—1722no—
381#124—862no—
382#124—862no—
383#124—868no—
384#123—44no—
385C22⋊Q8—868no—
386C22⋊Q8—862no—
387C22⋊Q8—862no—
388#121—434no—
389#121—868no—
390#121—862no—
391C4×Q8—862no—
392C4×Q8—862no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—868no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—431no—
401C2×C4⋊C4—861no—
402C2×C4⋊C4—861no—
403C2×C4⋊C4C22⋊D4311yescharacteristic
404C2×C4⋊C4C22×D4312yes—
405C2×C4⋊C4C22×D4312yes—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—861no—
409C2×C22⋊C4—861no—
410C2×C22⋊C4—861no—
411C2×C22⋊C4—861no—
412C2×C22⋊C4C22⋊D4311yescharacteristic
413C2×C22⋊C4C22⋊D4311yescharacteristic
414C2×C42—861no—
415C2×C42C22×D4311yescharacteristic
416#107—864no—
417#103—864no—
418#103—864no—
419#103—864no—
420#103—1722no—
421#102—864no—
422#102—1722no—
423C22⋊C8—862no—
424C22⋊C8—1722no—
425#95—22no—
426#95—21no—
427#95—22no—
428C24—861no—
429C24C2×C22⋊D4311yescharacteristic
430C24C2×C22⋊D4311yescharacteristic
431#41—1728no—
432#41—1724no—
433#41—1724no—
434C2×Q8—432no—
435C2×Q8—432no—
436C2×Q8—434no—
437C2×Q8—868no—
438C2×Q8—864no—
439C2×Q8—861no—
440C2×Q8—861no—
441C2×Q8—862no—
442C2×Q8—862no—
443C2×Q8—862no—
444C2×Q8—1724no—
445C2×Q8—1724no—
446C2×Q8—1728no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—432no—
451C2×D4—864no—
452C2×D4—864no—
453C2×D4—861no—
454C2×D4—862no—
455C2×D4—862no—
456C2×D4—1724no—
457C2×D4—1722no—
458C2×D4—1722no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—864no—
471C22×C4—861no—
472C22×C4—864no—
473C22×C4—861no—
474C22×C4—861no—
475C22×C4—1724no—
476C22×C4—1724no—
477C22×C4D4×D4312yes—
478C22×C4D4×D4312yes—
479C22×C4C2×C22⋊D4311yescharacteristic
480C22×C4C2×C22⋊D4311yescharacteristic
481C22×C4C2×C22⋊D4311yescharacteristic
482C22×C4C2×C22⋊D4311yescharacteristic
483C22×C4C23×D4311yescharacteristic
484C2×C8—862no—
485C2×C8—862no—
486C2×C8—862no—
487C2×C8—862no—
488C2×C8—862no—
489C2×C8—862no—
490C2×C8—1724no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—432no—
499C4⋊C4—864no—
500C4⋊C4—864no—
501C4⋊C4—861no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—864no—
507C22⋊C4—864no—
508C22⋊C4—864no—
509C22⋊C4—864no—
510C42—21no—
511C42—21no—
512C42—864no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—862no—
521C23—1721no—
522C23—1721no—
523C23—1722no—
524C23—1722no—
525C23#1306329312yes—
526C23#1306334512yes—
527C23#1306335112yes—
528C23C22×C22⋊D4311yescharacteristic
529Q8—434no—
530Q8—862no—
531Q8—864no—
532Q8—862no—
533Q8—862no—
534Q8—1728no—
535Q8—1722no—
536D4—44no—
537D4—864no—
538D4—1728no—
539D4—1728no—
540D4—1721no—
541D4—1721no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—84no—
563C2×C4—862no—
564C2×C4—864no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—1728no—
568C2×C4—1728no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4—1722no—
572C2×C4#1306329312yes—
573C2×C4C2×D4×D4311yescharacteristic
574C2×C4C2×D4×D4311yescharacteristic
575C2×C4#41×D4312yes—
576C2×C4#1306337112yes—
577C8—1722no—
578C8—1722no—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—24no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—42no—
588C22—3442no—
589C22—3442no—
590C22—1722no—
591C22—1722no—
592C22—1722no—
593C22#1458304214yes—
594C22C2×#1306329311yescharacteristic
595C22#1458353712yes—
596C22C2×#1306334511yescharacteristic
597C22C2×#1306335111yescharacteristic
598C22#1458371412yes—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4—82no—
605C4—864no—
606C4—1722no—
607C4—1722no—
608C4#123×D4311yescharacteristic
609C4#1458346011yescharacteristic
610C4#1458354112yes—
611C2—24no—
612C2—81no—
613C2—3442no—
614C2#1913791911yescharacteristic
615C2C2×#1458304211yescharacteristic
616C2#1914388811yescharacteristic
617C2#1914647612yes—
618C2#1914658012yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2486C2
socleC23×C43C22×D4
prime radicalC2×#14583558C2
hypercenter#2486D43
hypocenterC43#24035
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C2×#14583212C2semidirect
3. C22×#13063248C2semidirect
4. #19135836C2semidirect
5. #19135798C2semidirect
6. #19135021C2semidirect
7. #19134888C2semidirect
8. #19134848C2semidirect
9. #19134533C2semidirect
10. #19134529C2semidirect
11. C43×#2486C2semidirect
266. Dic86#123semidirect
305. #2486D43semidirect
356. C43#24035semidirect

Maximal subgroups

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

subgroupindex
C2×#145835582
C2×#145832122
C22×#130632482
#191358362
#191357982
#191350212
#191348882
#191348482
#191345332
#191345292
C43×#24862
#2403543