Finite Group Databasebeta

← C2×#19133607 #25068837

Subgroups — C2×#19133607 #25068837

Order 11008.

1156 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×#14583487C211yesmaximal, characteristic
2C2×#14583229C211yesmaximal, characteristic, complemented
3C2×#14583212C211yesmaximal, characteristic, complemented
4#19133607C218yesmaximal, direct product factor
5C2×#14582635C211yesmaximal, characteristic, complemented
6C2×#14582632C211yesmaximal, characteristic, complemented
7C2×#14582623C211yesmaximal, characteristic, complemented
8C2×C43×#397C211yesmaximal, characteristic, complemented
9#14583487—22no—
10#14583487—22no—
11#14583487—22no—
12#14583487C2214yes—
13C2×#13063318—21no—
14C2×#13063311—21no—
15C2×#13063307—21no—
16C2×#13063304C2211yescharacteristic
17C2×#13063299C2211yescharacteristic
18C2×#13063298—21no—
19C2×#13063294—21no—
20C2×C4×C22⋊D43—21no—
21#14583229—22no—
22#14583229C2214yescomplemented
23#14583212—22no—
24#14583212C2214yescomplemented
25C2×#13063258—21no—
26C2×#13063256—21no—
27C2×#13063248—21no—
28C2×#13063245—21no—
29C2×C22⋊(C43⋊C8)C2211yescharacteristic
30C2×C22⋊(C43⋊C8)C2211yescharacteristic
31#14582635C2214yescomplemented
32#14582632C2214yescomplemented
33#14582623C2214yescomplemented
34C22×C4⋊C4×C43C2211yescharacteristic
35C22×C22⋊C4×C43—21no—
36C22×C22⋊C4×C43C2211yescharacteristic
37C43×#397—22no—
38C43×#397C2214yescomplemented
39C2×#95×C43—21no—
40C2×#95×C43C2211yescharacteristic
41C22×Dic86D411yescharacteristic
42C22×D172D411yescharacteristic
43C22×C22⋊D43—21no—
44C22×C22⋊D43—21no—
45C22×C22⋊D43—21no—
46C22×C4×D43—21no—
47C22×C43⋊C8—21no—
48C22×C43⋊C8—21no—
49#13063318—24no—
50#13063318—24no—
51#13063311—24no—
52#13063311—24no—
53#13063307—24no—
54#13063307—24no—
55#13063304—21no—
56#13063304—21no—
57#13063304D412yes—
58#13063304C2312yes—
59#13063299—21no—
60#13063299—21no—
61#13063299D412yes—
62#13063299C2312yes—
63#13063298—24no—
64#13063298—24no—
65#13063294—24no—
66#13063294—24no—
67C4×C22⋊D43—24no—
68C4×C22⋊D43—24no—
69C2×C4⋊Dic43D411yescharacteristic
70C2×C4⋊Dic43D411yescharacteristic
71C2×Dic43⋊C4—21no—
72C2×Dic43⋊C4—21no—
73C2×C22⋊Dic43—21no—
74C2×C22⋊Dic43—21no—
75C2×C22⋊Dic43—21no—
76C2×(C2×D43)⋊C4—21no—
77C2×(C2×D43)⋊C4—21no—
78C2×C4×Dic43—21no—
79#13063258—24no—
80#13063256—24no—
81#13063248—24no—
82#13063245—24no—
83C22⋊(C43⋊C8)—21no—
84C22⋊(C43⋊C8)—21no—
85C22⋊(C43⋊C8)C2312yes—
86C22⋊(C43⋊C8)C2312yes—
87C25×C43D411yescharacteristic
88C23×C4×C43—21no—
89C23×C4×C43—21no—
90C23×C4×C43D411yescharacteristic
91C23×C4×C43C2311yescharacteristic
92C2×C4⋊C4×C43—21no—
93C2×C4⋊C4×C43—21no—
94C2×C4⋊C4×C43—21no—
95C2×C4⋊C4×C43—21no—
96C2×C4⋊C4×C43—21no—
97C2×C4⋊C4×C43C2312yes—
98C2×C22⋊C4×C43—21no—
99C2×C22⋊C4×C43—21no—
100C2×C22⋊C4×C43—21no—
101C2×C22⋊C4×C43—21no—
102C2×C22⋊C4×C43—24no—
103C2×C22⋊C4×C43—44no—
104C2×C22⋊C4×C43D412yes—
105C2×C22⋊C4×C43C2312yes—
106#95×C43—24no—
107#95×C43—21no—
108#95×C43C2312yes—
109C24×C43—21no—
110C24×C43—22no—
111C24×C43—21no—
112C24×C43—22no—
113C24×C43—42no—
114C24×C43—42no—
115C24×C43C2×D411yescharacteristic
116C24×C43C2×D412yes—
117C23×D43—21no—
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43—22no—
121C22×C4×C43—22no—
122C22×C4×C43—42no—
123C22×C4×C43—42no—
124C22×C4×C43—42no—
125C22×C4×C43—42no—
126C22×C4×C43—41no—
127C22×C4×C43—41no—
128C22×C4×C43—42no—
129C22×C4×C43—42no—
130C22×C4×C43—42no—
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2×D412yes—
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2411yescharacteristic
138C2×C22⋊D43—24no—
139C2×C22⋊D43—24no—
140C2×C22⋊D43—22no—
141C2×C22⋊D43—22no—
142C2×C22⋊D43—42no—
143C2×C22⋊D43—42no—
144C2×C22⋊D43—82no—
145C2×C22⋊D43—82no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43—21no—
149C2×D172—21no—
150C2×D172—42no—
151C2×D172C2×D412yes—
152C2×C4×D43—24no—
153C2×C4×D43—82no—
154C2×Dic86—21no—
155C2×Dic86—42no—
156C2×Dic86C2×D412yes—
157C4⋊C4×C43—22no—
158C4⋊C4×C43—22no—
159C4⋊C4×C43—41no—
160C4⋊C4×C43—41no—
161C22⋊C4×C43—22no—
162C22⋊C4×C43—22no—
163C22⋊C4×C43—22no—
164C22⋊C4×C43—22no—
165C22⋊C4×C43—48no—
166C22⋊Dic43—24no—
167C22⋊Dic43—24no—
168C22⋊Dic43—22no—
169C22⋊Dic43—22no—
170(C2×D43)⋊C4—24no—
171(C2×D43)⋊C4—22no—
172(C2×D43)⋊C4—22no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43C2×D412yes—
176C4⋊Dic43C2×D412yes—
177Dic43⋊C4—24no—
178Dic43⋊C4—22no—
179Dic43⋊C4—22no—
180C4×Dic43—24no—
181C2×C43⋊C8—22no—
182C2×C43⋊C8—22no—
183C2×C43⋊C8—42no—
184C2×C43⋊C8—42no—
185C23×C43—22no—
186C23×C43—22no—
187C23×C43—22no—
188C23×C43—22no—
189C23×C43—22no—
190C23×C43—42no—
191C23×C43—42no—
192C23×C43—48no—
193C23×C43—42no—
194C23×C43—42no—
195C23×C43—42no—
196C23×C43—42no—
197C23×C43—81no—
198C23×C43—81no—
199C23×C43—84no—
200C23×C43#12011yescharacteristic
201C23×C43#12011yescharacteristic
202C23×C43#13512yes—
203C23×C43AGL(1,ℤ/8)12yes—
204C23×C43C22×D411yescharacteristic
205C22×D43—22no—
206C22×D43—42no—
207C22×D43—42no—
208C22×D43—42no—
209C2×C4×C43—22no—
210C2×C4×C43—22no—
211C2×C4×C43—22no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—42no—
217C2×C4×C43—42no—
218C2×C4×C43—42no—
219C2×C4×C43—42no—
220C2×C4×C43—42no—
221C2×C4×C43—42no—
222C2×C4×C43—42no—
223C2×C4×C43—42no—
224C2×C4×C43—81no—
225C2×C4×C43—84no—
226C2×C4×C43—82no—
227C2×C4×C43—82no—
228C2×C4×C43#12011yescharacteristic
229C2×C4×C43#12011yescharacteristic
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C22×D411yescharacteristic
232C22⋊D43—42no—
233C22⋊D43—42no—
234C22⋊D43—44no—
235C22⋊D43—84no—
236C22⋊D43—84no—
237C2×Dic43—22no—
238C2×Dic43—22no—
239C2×Dic43—22no—
240C2×Dic43—42no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243D172—42no—
244D172—81no—
245C4×D43—84no—
246Dic86—42no—
247Dic86—81no—
248C43⋊C8—42no—
249C43⋊C8—42no—
250C2×#1612—431nomaximal, Sylow
251C22×C43—24no—
252C22×C43—24no—
253C22×C43—21no—
254C22×C43—21no—
255C22×C43—21no—
256C22×C43—21no—
257C22×C43—44no—
258C22×C43—42no—
259C22×C43—42no—
260C22×C43—42no—
261C22×C43—82no—
262C22×C43—84no—
263C22×C43—84no—
264C22×C43#44912yes—
265C22×C43#44912yes—
266C22×C43#45512yes—
267C22×C43#45812yes—
268C22×C43C2×#12011yescharacteristic
269C22×C43C2×#13511yescharacteristic
270C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
271C2×D43—42no—
272C2×D43—42no—
273C2×D43—42no—
274C2×D43—84no—
275C4×C43—21no—
276C4×C43—21no—
277C4×C43—21no—
278C4×C43—21no—
279C4×C43—42no—
280C4×C43—82no—
281C4×C43—82no—
282Dic43—42no—
283Dic43—42no—
284Dic43—42no—
285C2×#538—431no—
286C2×#483—431no—
287C2×#480—431no—
288C2×#397D4311yescharacteristic, complemented
289#1612—438no—
290C2×#328—431no—
291C2×#327—431no—
292C2×#323—431no—
293C2×C43—21no—
294C2×C43—21no—
295C2×C43—22no—
296C2×C43—82no—
297C2×C43#161214yescomplemented
298C2×C43C2×#44911yescharacteristic
299C2×C43C2×#45511yescharacteristic
300C2×C43C2×#45811yescharacteristic
301D43—82no—
302#538—434no—
303#538—862no—
304#538—862no—
305#538—862no—
306C2×#126—861no—
307C2×#124—861no—
308C2×#123—861no—
309C2×C22⋊Q8—431no—
310C2×#121—431no—
311C2×#121—861no—
312C2×#120—861no—
313C2×C4×D4—861no—
314C22×C4⋊C4C2×D4311yescharacteristic
315C22×C22⋊C4—21no—
316C22×C22⋊C4C2×D4311yescharacteristic
317#483—434no—
318#483—862no—
319#480—434no—
320#480—862no—
321C2×#107—861no—
322C2×#106—861no—
323C2×#103—861no—
324C2×#102—861no—
325C2×C22⋊C8—431no—
326C2×C22⋊C8—431no—
327#397—22no—
328#397C2×D4314yescomplemented
329C2×#95—21no—
330C2×#95C2×D4311yescharacteristic
331#328—434no—
332#327—434no—
333#323—434no—
334C43C2×#161211yescharacteristic, complemented, Sylow
335C25C22⋊D4311yescharacteristic
336C22×Q8—431no—
337C22×D4—431no—
338C22×D4—861no—
339C22×D4—861no—
340C22×D4—861no—
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—861no—
344C23×C4C22⋊D4311yescharacteristic
345C23×C4C22×D4311yescharacteristic
346C22×C8—861no—
347C22×C8—861no—
348#126—864no—
349#126—864no—
350#124—864no—
351#124—864no—
352#123—864no—
353#123—864no—
354C22⋊Q8—432no—
355C22⋊Q8—432no—
356C22⋊Q8—861no—
357C22⋊Q8—861no—
358#121—432no—
359#121—432no—
360#121—864no—
361#121—864no—
362#121—861no—
363#121—861no—
364#120—864no—
365#120—864no—
366C4×D4—864no—
367C4×D4—864no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—431no—
374C2×C4⋊C4—431no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—861no—
377C2×C4⋊C4C22×D4312yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—44no—
384C2×C22⋊C4—861no—
385C2×C22⋊C4—861no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C22⋊C4—861no—
389C2×C22⋊C4C22⋊D4312yes—
390C2×C22⋊C4C22×D4312yes—
391C2×C42—861no—
392#107—864no—
393#106—864no—
394#103—864no—
395#102—864no—
396C22⋊C8—432no—
397C22⋊C8—432no—
398C22⋊C8—861no—
399C22⋊C8—861no—
400#95—24no—
401#95—21no—
402#95C22×D4312yes—
403C24—21no—
404C24—22no—
405C24—21no—
406C24—22no—
407C24—42no—
408C24—42no—
409C24—861no—
410C24C2×C22⋊D4311yescharacteristic
411C24C2×C22⋊D4312yes—
412C2×Q8—432no—
413C2×Q8—861no—
414C2×Q8—1722no—
415C2×D4—432no—
416C2×D4—864no—
417C2×D4—864no—
418C2×D4—862no—
419C2×D4—862no—
420C2×D4—861no—
421C2×D4—3442no—
422C2×D4—3442no—
423C2×D4—1722no—
424C2×D4—1722no—
425C2×D4—1722no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—864no—
440C22×C4—861no—
441C22×C4—861no—
442C22×C4—861no—
443C22×C4—3442no—
444C22×C4D4×D4311yescharacteristic
445C22×C4D4×D4311yescharacteristic
446C22×C4D4×D4311yescharacteristic
447C22×C4D4×D4311yescharacteristic
448C22×C4C2×C22⋊D4312yes—
449C22×C4C2×C22⋊D4311yescharacteristic
450C22×C4C23×D4311yescharacteristic
451C2×C8—862no—
452C2×C8—862no—
453C2×C8—1722no—
454C2×C8—1722no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—432no—
460C4⋊C4—432no—
461C4⋊C4—864no—
462C4⋊C4—861no—
463C4⋊C4—861no—
464C4⋊C4—862no—
465C4⋊C4—862no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—48no—
471C22⋊C4—864no—
472C22⋊C4—864no—
473C22⋊C4—864no—
474C22⋊C4—862no—
475C22⋊C4—862no—
476C22⋊C4—862no—
477C22⋊C4—862no—
478C42—864no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—81no—
492C23—81no—
493C23—84no—
494C23—862no—
495C23—1722no—
496C23—1722no—
497C23—1722no—
498C23#1306329311yescharacteristic
499C23#1306329311yescharacteristic
500C23#1306334512yes—
501C23#1306335112yes—
502C23C22×C22⋊D4311yescharacteristic
503Q8—3441no—
504Q8—1722no—
505D4—3441no—
506D4—3444no—
507D4—3444no—
508D4—1722no—
509D4—1724no—
510D4—1722no—
511D4—1722no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—862no—
532C2×C4—862no—
533C2×C4—862no—
534C2×C4—3444no—
535C2×C4—1722no—
536C2×C4—1722no—
537C2×C4—1722no—
538C2×C4#1306329311yescharacteristic
539C2×C4#1306329311yescharacteristic
540C2×C4C2×D4×D4311yescharacteristic
541C2×C4C2×D4×D4311yescharacteristic
542C8—1722no—
543C8—1722no—
544C22—21no—
545C22—21no—
546C22—24no—
547C22—24no—
548C22—21no—
549C22—21no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22—84no—
556C22—84no—
557C22—3444no—
558C22—1722no—
559C22—1722no—
560C22—1722no—
561C22#1458304212yes—
562C22#1458304212yes—
563C22#1458307612yes—
564C22#1458308812yes—
565C22C2×#1306329311yescharacteristic
566C22C2×#1306334511yescharacteristic
567C22C2×#1306335111yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—1722no—
576C4—1722no—
577C4—1722no—
578C2—21no—
579C2—22no—
580C2—21no—
581C2—82no—
582C2—3442no—
583C2#1913360714yesdirect product factor
584C2C2×#1458304211yescharacteristic
585C2C2×#1458307611yescharacteristic
586C2C2×#1458308811yescharacteristic
subgroupG/H
centerC23#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#397C2
socleC23×C43#120
prime radicalC2×#14583487C2
hypercenterC2×#397D43
hypocenterC43C2×#1612
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C2×#14583229C2semidirect
3. C2×#14583212C2semidirect
4. #19133607C2direct (N × H)
5. C2×#14582635C2semidirect
6. C2×#14582632C2semidirect
7. C2×#14582623C2semidirect
8. C2×C43×#397C2semidirect
22. #14583229C22semidirect
24. #14583212C22semidirect
31. #14582635C22semidirect
32. #14582632C22semidirect
33. #14582623C22semidirect
38. C43×#397C22semidirect
288. C2×#397D43semidirect
297. C2×C43#1612semidirect
328. #397C2×D43semidirect
334. C43C2×#1612semidirect
583. C2#19133607direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145834872
C2×#145832292
C2×#145832122
#191336072
C2×#145826352
C2×#145826322
C2×#145826232
C2×C43×#3972
C2×#161243