Finite Group Databasebeta

← #25129573

Subgroups — #25129573

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×#14583560C211yesmaximal, characteristic
2C2×#14583212C211yesmaximal, characteristic, complemented
3#19138897C211yesmaximal, characteristic, complemented
4#19135832C212yesmaximal, complemented
5#19135796C212yesmaximal, complemented
6#19135023C211yesmaximal, characteristic, complemented
7#19134887C211yesmaximal, characteristic, complemented
8#19134847C212yesmaximal, complemented
9#19134534C212yesmaximal, complemented
10#19134530C211yesmaximal, characteristic, complemented
11C22×#103×C43C211yesmaximal, characteristic, complemented
12C22×#10621902C2211yescharacteristic
13#14583560—24no—
14#14583560—24no—
15C2×#13063314—21no—
16C2×#13063314C2211yescharacteristic
17C2×#13063308—21no—
18C2×#13063301—21no—
19C2×#13063301C2211yescharacteristic
20C2×#13063299—21no—
21C2×#13063299C2211yescharacteristic
22C2×Q8×Dic43C2211yescharacteristic
23C2×C4×C22⋊D43—21no—
24C2×C4×C22⋊D43C2211yescharacteristic
25#14583212—24no—
26C2×#13063258C2212yes—
27C2×#13063249C2212yes—
28C2×#13063249C2212yes—
29C2×#13063247C2212yes—
30C2×#13063245C2212yes—
31C2×C22⋊(C43⋊C8)C2211yescharacteristic
32C2×(C2×D43)⋊C8C2212yes—
33#14582830—21no—
34#14582825—21no—
35#14582795C2212yes—
36#14582762—21no—
37#14582759C2211yescharacteristic
38#14582756C2212yes—
39#14582662C2212yes—
40#14582653C2212yes—
41C23×Q8×C43C2211yescharacteristic
42C23×C8×C43C2211yescharacteristic
43C22×C4⋊C4×C43C2211yescharacteristic
44C2×#103×C43—24no—
45C2×#103×C43C2212yes—
46C2×#103×C43C2212yes—
47C2×#10621902—22no—
48C2×#10621902—22no—
49C2×#10621902—24no—
50C2×#10621902—42no—
51C22×D172—21no—
52C22×D172C2311yescharacteristic
53C22×C22⋊D43—21no—
54C22×C22⋊D43D411yescharacteristic
55C22×C4×D43—21no—
56C22×C4×D43C2311yescharacteristic
57C22×C43⋊C8C2312yes—
58#13063314—28no—
59#13063314—22no—
60#13063314—22no—
61#13063308—44no—
62#13063301—28no—
63#13063301—22no—
64#13063301—22no—
65#13063299—28no—
66#13063299—22no—
67#13063299D414yes—
68Q8×Dic43—22no—
69Q8×Dic43—22no—
70C4×C22⋊D43—28no—
71C4×C22⋊D43—22no—
72C4×C22⋊D43—22no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43C2312yes—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4D411yescharacteristic
79C2×C22⋊Dic43—21no—
80C2×C22⋊Dic43D411yescharacteristic
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—22no—
86C2×(C2×D43)⋊C4D411yescharacteristic
87C2×C4×Dic43—21no—
88C2×C4×Dic43C2311yescharacteristic
89#13063258—24no—
90#13063249—24no—
91#13063249—42no—
92#13063247—42no—
93#13063245—24no—
94C22⋊(C43⋊C8)—22no—
95(C2×D43)⋊C8—42no—
96C43⋊#95—21no—
97C43⋊#95—22no—
98C43⋊#95—22no—
99C22×Q8×C43—22no—
100C22×Q8×C43—21no—
101C22×Q8×C43—24no—
102C22×Q8×C43—42no—
103C22×Q8×C43D414yes—
104C22×Q8×C43C2311yescharacteristic
105C22×Q8×C43C2311yescharacteristic
106C23×C4×C43—21no—
107C23×C4×C43—21no—
108C23×C4×C43C2311yescharacteristic
109C22×C8×C43—22no—
110C22×C8×C43—42no—
111C22×C8×C43C2312yes—
112C2×C4⋊C4×C43—22no—
113C2×C4⋊C4×C43—22no—
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43C2312yes—
116#103×C43—24no—
117#103×C43—24no—
118C24×C43C2×D411yescharacteristic
119C23×D43—21no—
120C23×D43C2×D411yescharacteristic
121C2×Q8×C43—24no—
122C2×Q8×C43—21no—
123C2×Q8×C43—21no—
124C2×Q8×C43—22no—
125C2×Q8×C43—28no—
126C2×Q8×C43—22no—
127C2×Q8×C43—22no—
128C2×Q8×C43—44no—
129C2×Q8×C43—44no—
130C2×Q8×C43—48no—
131C2×Q8×C43C2×D412yes—
132C2×Q8×C43C2×D412yes—
133C2×Q8×C43#4114yes—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—24no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—22no—
140C22×C4×C43—44no—
141C22×C4×C43—42no—
142C22×C4×C43—42no—
143C22×C4×C43—42no—
144C22×C4×C43C2×D412yes—
145C22×C4×C43C2×D412yes—
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C2×C22⋊D43—22no—
149C2×C22⋊D43—24no—
150C2×C22⋊D43—22no—
151C2×C22⋊D43—22no—
152C2×C22⋊D43—42no—
153C2×C22⋊D43—42no—
154C22×Dic43—21no—
155C22×Dic43—22no—
156C22×Dic43C2×D411yescharacteristic
157C22×Dic43C2×D411yescharacteristic
158C22×Dic43C2×D411yescharacteristic
159#10621902—48no—
160#10621902—44no—
161#10621902—44no—
162C2×D172—24no—
163C2×D172—21no—
164C2×D172—22no—
165C2×D172—22no—
166C2×D172—44no—
167C2×D172C2×D412yes—
168C2×C4×D43—24no—
169C2×C4×D43—21no—
170C2×C4×D43—21no—
171C2×C4×D43—22no—
172C2×C4×D43—22no—
173C2×C4×D43—44no—
174C2×C8×C43—22no—
175C2×C8×C43—22no—
176C2×C8×C43—22no—
177C2×C8×C43—44no—
178C4⋊C4×C43—22no—
179C4⋊C4×C43—22no—
180C4⋊C4×C43—42no—
181C22⋊Dic43—22no—
182C22⋊Dic43—42no—
183(C2×D43)⋊C4—24no—
184(C2×D43)⋊C4—24no—
185(C2×D43)⋊C4—24no—
186(C2×D43)⋊C4—24no—
187(C2×D43)⋊C4—22no—
188(C2×D43)⋊C4—44no—
189C4⋊Dic43—24no—
190C4⋊Dic43—21no—
191C4⋊Dic43—22no—
192C4⋊Dic43—22no—
193C4⋊Dic43C2×D412yes—
194Dic43⋊C4—24no—
195Dic43⋊C4—22no—
196Dic43⋊C4—44no—
197C4×Dic43—24no—
198C4×Dic43—21no—
199C4×Dic43—21no—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C2×C43⋊C8—22no—
203C23×C43—22no—
204C23×C43—22no—
205C23×C43#12012yes—
206C23×C43#13512yes—
207C23×C43AGL(1,ℤ/8)12yes—
208C23×C43C22×D411yescharacteristic
209C22×D43—21no—
210C22×D43—22no—
211C22×D43—22no—
212C22×D43—21no—
213C22×D43—21no—
214C22×D43—22no—
215C22×D43—42no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—42no—
219Q8×C43—22no—
220Q8×C43—24no—
221Q8×C43—22no—
222Q8×C43—22no—
223Q8×C43—48no—
224Q8×C43—42no—
225Q8×C43#12314yescomplemented
226C2×C4×C43—22no—
227C2×C4×C43—24no—
228C2×C4×C43—22no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—24no—
232C2×C4×C43—22no—
233C2×C4×C43—48no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43—84no—
239C2×C4×C43#12012yes—
240C2×C4×C43C22×D411yescharacteristic
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43C2×#4112yes—
243C2×C4×C43H(5,𝔽2)12yes—
244C22⋊D43—44no—
245C22⋊D43—48no—
246C2×Dic43—21no—
247C2×Dic43—21no—
248C2×Dic43—22no—
249C2×Dic43—21no—
250C2×Dic43—22no—
251C2×Dic43—21no—
252C2×Dic43—21no—
253C2×Dic43—22no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—42no—
257C2×Dic43—42no—
258C2×Dic43—42no—
259D172—24no—
260D172—48no—
261D172—41no—
262D172—41no—
263C4×D43—48no—
264C4×D43—41no—
265C4×D43—41no—
266C4×D43—42no—
267C8×C43—42no—
268C43⋊C8—42no—
269#24035—431nomaximal, Sylow
270C22×C43—22no—
271C22×C43—22no—
272C22×C43—24no—
273C22×C43—24no—
274C22×C43#44914yes—
275C22×C43C2×#12011yescharacteristic
276C22×C43#54612yes—
277C22×C43C2×#13511yescharacteristic
278C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C43#57812yes—
280C2×D43—41no—
281C2×D43—42no—
282C2×D43—41no—
283C2×D43—42no—
284C2×D43—41no—
285C2×D43—42no—
286C2×D43—42no—
287C2×D43—42no—
288C2×D43—82no—
289C2×D43—82no—
290C4×C43—24no—
291C4×C43—22no—
292C4×C43—42no—
293C4×C43—82no—
294C4×C43C2×#12311yescharacteristic
295C4×C43#53411yescharacteristic
296C4×C43#54612yes—
297Dic43—41no—
298Dic43—41no—
299Dic43—42no—
300Dic43—41no—
301Dic43—82no—
302C2×#548—431no—
303C2×#480—431no—
304C22×#103D4311yescharacteristic, complemented
305#2486—431no—
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#240511yescharacteristic
315C2×C43C2×#44911yescharacteristic
316C2×C43#301111yescharacteristic
317C2×C43#330012yes—
318C2×C43#330812yes—
319D43—81no—
320D43—82no—
321C22×#41—431no—
322C23×Q8C2×D4311yescharacteristic
323C23×C8C2×D4311yescharacteristic
324#548—864no—
325#548—864no—
326C2×#124—431no—
327C2×#124—861no—
328C2×#123—861no—
329C2×C22⋊Q8—431no—
330C2×C22⋊Q8—861no—
331C2×#121—431no—
332C2×#121—861no—
333C2×C4×Q8—431no—
334C2×C4×D4—431no—
335C2×C4×D4—861no—
336C22×C4⋊C4C2×D4311yescharacteristic
337#480—864no—
338C2×#107—432no—
339C2×#103—24no—
340C2×#103—432no—
341C2×#103—432no—
342C2×#103C2×D4312yes—
343C2×#103C2×D4312yes—
344C2×#102—432no—
345C2×#102—432no—
346C2×C22⋊C8—431no—
347C2×C22⋊C8—432no—
348#398—861no—
349#397—861no—
350#389—432no—
351#381—861no—
352#380—431no—
353#380—432no—
354#340—432no—
355#336—432no—
356C43#2403511yescharacteristic, complemented, Sylow
357C2×#41—862no—
358C2×#41—862no—
359C2×#41—864no—
360C2×#41—1722no—
361C22×Q8—21no—
362C22×Q8—22no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8C22⋊D4314yes—
366C22×Q8C22×D4311yescharacteristic
367C22×Q8C22×D4311yescharacteristic
368C22×D4—431no—
369C22×D4—431no—
370C22×D4—861no—
371C22×D4—861no—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—431no—
375C23×C4—861no—
376C23×C4C22×D4311yescharacteristic
377C22×C8—22no—
378C22×C8—42no—
379C22×C8—432no—
380C22×C8C22×D4312yes—
381#124—862no—
382#124—868no—
383#124—862no—
384#123—1724no—
385C22⋊Q8—862no—
386C22⋊Q8—868no—
387C22⋊Q8—862no—
388#121—434no—
389#121—868no—
390#121—862no—
391C4×Q8—862no—
392C4×Q8—862no—
393C4×D4—868no—
394C4×D4—862no—
395C4×D4—862no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—431no—
400C2×C4⋊C4—431no—
401C2×C4⋊C4—432no—
402C2×C4⋊C4—861no—
403C2×C4⋊C4—861no—
404C2×C4⋊C4—862no—
405C2×C4⋊C4C22×D4312yes—
406C2×C22⋊C4—431no—
407C2×C22⋊C4—431no—
408C2×C22⋊C4—861no—
409C2×C22⋊C4—861no—
410C2×C22⋊C4—861no—
411C2×C22⋊C4—861no—
412C2×C22⋊C4—862no—
413C2×C22⋊C4—861no—
414C2×C42—431no—
415C2×C42—861no—
416#107—864no—
417#103—24no—
418#103—24no—
419#103—864no—
420#103—1722no—
421#102—864no—
422#102—1722no—
423C22⋊C8—862no—
424C22⋊C8—1722no—
425#95—862no—
426#95—862no—
427#95—861no—
428C24—431no—
429C24—861no—
430C24D4×D4311yescharacteristic
431#41—1728no—
432#41—1724no—
433#41—1724no—
434C2×Q8—24no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8#1062189814yes—
445C2×Q8C2×C22⋊D4312yes—
446C2×Q8C2×C22⋊D4312yes—
447C2×D4—432no—
448C2×D4—864no—
449C2×D4—864no—
450C2×D4—862no—
451C2×D4—862no—
452C2×D4—862no—
453C2×D4—861no—
454C2×D4—862no—
455C2×D4—862no—
456C2×D4—1724no—
457C2×D4—1722no—
458C2×D4—1722no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—431no—
470C22×C4—431no—
471C22×C4—431no—
472C22×C4—864no—
473C22×C4—861no—
474C22×C4—861no—
475C22×C4—861no—
476C22×C4—862no—
477C22×C4—862no—
478C22×C4—862no—
479C22×C4—1724no—
480C22×C4D4×D4311yescharacteristic
481C22×C4D4×D4312yes—
482C22×C4C2×C22⋊D4312yes—
483C22×C4C23×D4311yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C2×C8—862no—
489C2×C8—862no—
490C2×C8—862no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—432no—
495C4⋊C4—864no—
496C4⋊C4—864no—
497C4⋊C4—862no—
498C4⋊C4—861no—
499C4⋊C4—862no—
500C4⋊C4—862no—
501C4⋊C4—1724no—
502C22⋊C4—864no—
503C22⋊C4—864no—
504C22⋊C4—864no—
505C22⋊C4—862no—
506C22⋊C4—862no—
507C22⋊C4—864no—
508C22⋊C4—1724no—
509C22⋊C4—1722no—
510C42—864no—
511C42—861no—
512C42—861no—
513C23—22no—
514C23—22no—
515C23—862no—
516C23—861no—
517C23—861no—
518C23—861no—
519C23—862no—
520C23—862no—
521C23—1721no—
522C23—1721no—
523C23—1722no—
524C23—1722no—
525C23#1306329312yes—
526C23#1306334612yes—
527C23#1306335212yes—
528C23C2×D4×D4311yescharacteristic
529Q8—22no—
530Q8—24no—
531Q8—22no—
532Q8—22no—
533Q8—48no—
534Q8—42no—
535Q8#1306330814yescomplemented
536D4—864no—
537D4—1728no—
538D4—1724no—
539D4—1728no—
540D4—1721no—
541D4—1721no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4—861no—
556C2×C4—862no—
557C2×C4—861no—
558C2×C4—861no—
559C2×C4—861no—
560C2×C4—861no—
561C2×C4—862no—
562C2×C4—862no—
563C2×C4—1728no—
564C2×C4—1722no—
565C2×C4—1722no—
566C2×C4—1722no—
567C2×C4—1721no—
568C2×C4—1721no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4—1722no—
572C2×C4#1306329312yes—
573C2×C4C2×D4×D4311yescharacteristic
574C2×C4C22×C22⋊D4311yescharacteristic
575C2×C4C2×#1062189812yes—
576C2×C4#1306337012yes—
577C8—42no—
578C8—1722no—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—24no—
583C22—3442no—
584C22—3442no—
585C22—1722no—
586C22—1721no—
587C22—1722no—
588C22—1722no—
589C22—1721no—
590C22—1721no—
591C22—1722no—
592C22—1722no—
593C22#1458304314yes—
594C22C2×#1306329311yescharacteristic
595C22#1458353912yes—
596C22C2×#1306334611yescharacteristic
597C22C2×#1306335211yescharacteristic
598C22#1458371212yes—
599C4—22no—
600C4—24no—
601C4—42no—
602C4—82no—
603C4—3442no—
604C4—1721no—
605C4—1722no—
606C4—1721no—
607C4—1721no—
608C4C2×#1306330811yescharacteristic
609C4#1458345811yescharacteristic
610C4#1458353912yes—
611C2—24no—
612C2—3442no—
613C2—3441no—
614C2#1913792211yescharacteristic
615C2C2×#1458304311yescharacteristic
616C2#1914388611yescharacteristic
617C2#1914647812yes—
618C2#1914658212yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC22×#103×C43C2
socleC23×C43C22×D4
prime radicalC2×#14583560C2
hypercenterC22×#103D43
hypocenterC43#24035
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C2×#14583212C2semidirect
3. #19138897C2semidirect
4. #19135832C2semidirect
5. #19135796C2semidirect
6. #19135023C2semidirect
7. #19134887C2semidirect
8. #19134847C2semidirect
9. #19134534C2semidirect
10. #19134530C2semidirect
11. C22×#103×C43C2semidirect
225. Q8×C43#123semidirect
304. C22×#103D43semidirect
356. C43#24035semidirect
535. Q8#13063308semidirect

Maximal subgroups

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

subgroupindex
C2×#145835602
C2×#145832122
#191388972
#191358322
#191357962
#191350232
#191348872
#191348472
#191345342
#191345302
C22×#103×C432
#2403543