Finite Group Databasebeta

← #25066247

Subgroups — #25066247

Order 11008.

698 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19143702C211yesmaximal, characteristic, complemented
2#19143633C211yesmaximal, characteristic, complemented
3C4×#13063349C211yesmaximal, characteristic
4#19133476C211yesmaximal, characteristic, complemented
5#19133475C211yesmaximal, characteristic, complemented
6#19133456C211yesmaximal, characteristic, complemented
7#19133437C211yesmaximal, characteristic, complemented
8#19133436C211yesmaximal, characteristic, complemented
9#19133425C211yesmaximal, characteristic, complemented
10#19133424C211yesmaximal, characteristic, complemented
11#19133344C211yesmaximal, characteristic, complemented
12#19132974C211yesmaximal, characteristic, complemented
13D4⋊C8×D43C211yesmaximal, characteristic, complemented
14#19132881C211yesmaximal, characteristic, complemented
15C43×#1586C211yesmaximal, characteristic, complemented
16C2×#13063349C411yescharacteristic
17C4×#10621900C2211yescharacteristic
18C4×D4×D43C2211yescharacteristic
19C8×Dic86C2211yescharacteristic, complemented
20#14583023C2211yescharacteristic, complemented
21#14583011C411yescharacteristic
22#14583008C411yescharacteristic
23#14582998C411yescharacteristic
24#14582996C411yescharacteristic
25C4×#10621873C2211yescharacteristic
26C4×#10621883C2211yescharacteristic
27C4×#10621882C2211yescharacteristic
28#14582978C2211yescharacteristic
29#14582977C2211yescharacteristic
30#14582976C2211yescharacteristic
31#14582975—21no—
32#14582974—21no—
33D4×C43⋊C8C2211yescharacteristic, complemented
34C8×C22⋊D43—21no—
35(C22⋊D43)⋊C8—21no—
36#14582951C2211yescharacteristic
37#14582950C2211yescharacteristic
38#14582949—21no—
39#14582940C411yescharacteristic
40C2×Dic43⋊C8—21no—
41C4⋊C8×D43C2211yescharacteristic
42#14582902C411yescharacteristic
43#14582897C411yescharacteristic
44#14582862—21no—
45#14582861—21no—
46C4×#10621872C2211yescharacteristic
47C4×C8×D43C2211yescharacteristic
48#14582652C2211yescharacteristic, complemented
49#14582651C2211yescharacteristic, complemented
50#14582649C2211yescharacteristic, complemented
51#14582631C2211yescharacteristic, complemented
52#14582629C2211yescharacteristic, complemented
53#14582628C2211yescharacteristic
54D4⋊(C43⋊C8)C2211yescharacteristic
55D4⋊(C43⋊C8)C2211yescharacteristic, complemented
56#14582626C2211yescharacteristic
57#14582621C2211yescharacteristic, complemented
58#14582620C2211yescharacteristic, complemented
59#14582620C2211yescharacteristic, complemented
60C4×D8×C43C2211yescharacteristic
61C43×#436C2211yescharacteristic
62C8×D4×C43C2211yescharacteristic, complemented
63C43×#334C2211yescharacteristic, complemented
64C43×D4⋊C8C2211yescharacteristic
65C43×D4⋊C8C2211yescharacteristic, complemented
66C43×#322C2211yescharacteristic, complemented
67C2×#10621900C2×C411yescharacteristic
68C2×D4×D43C2×C411yescharacteristic
69C22×C4×D43—21no—
70#13063349—24no—
71C2×#10621873C2×C411yescharacteristic
72C2×#10621883C2×C411yescharacteristic
73C2×#10621882C2×C411yescharacteristic
74C2×#10621877—21no—
75C2×#10621872—21no—
76C2×#10621872—21no—
77C2×#10621872C2×C411yescharacteristic
78C2×#10621872C2×C411yescharacteristic
79C22×C43⋊C8—21no—
80C2×C8×D43—21no—
81C2×C8×D43—21no—
82C4×Dic86C2311yescharacteristic
83#13063290C2×C411yescharacteristic
84C4×D172C2311yescharacteristic
85D4×Dic43C2×C411yescharacteristic
86D4×Dic43C2×C411yescharacteristic
87#13063287C2×C411yescharacteristic
88C4×C22⋊D43—21no—
89C4×C22⋊D43—21no—
90(C22⋊D43)⋊C4—21no—
91(C22⋊D43)⋊C4—21no—
92#13063283C2×C411yescharacteristic
93#13063281—21no—
94C4⋊C4×D43C2×C411yescharacteristic
95C22⋊C4×D43—21no—
96C2×C4×Dic43—21no—
97C42×D43C2311yescharacteristic
98#13063258C2×C411yescharacteristic
99#13063257C2×C411yescharacteristic
100#13063256C2×C411yescharacteristic
101C4⋊(C43⋊C8)C2311yescharacteristic
102C4⋊(C43⋊C8)C2311yescharacteristic
103Dic43⋊C8—21no—
104Dic43⋊C8—21no—
105Dic43⋊C8—21no—
106Dic43⋊C8D411yescharacteristic
107Dic43⋊C8D411yescharacteristic
108#13063250C2×C411yescharacteristic
109#13063248C2×C411yescharacteristic
110#13063247C2×C411yescharacteristic
111#13063246C2×C411yescharacteristic
112#13063246C2×C411yescharacteristic
113#13063245C2×C411yescharacteristic
114C22⋊(C43⋊C8)—21no—
115C22⋊(C43⋊C8)—21no—
116(C2×D43)⋊C8—21no—
117(C2×D43)⋊C8—21no—
118(C2×D43)⋊C8—21no—
119(C2×D43)⋊C8D411yescharacteristic
120(C2×D43)⋊C8D411yescharacteristic
121#13063235C2×C411yescharacteristic
122#13063235C2×C411yescharacteristic
123C4×C43⋊C8C2311yescharacteristic
124C4×C43⋊C8C2311yescharacteristic
125C4×C43⋊C8C2311yescharacteristic
126C8×Dic43—21no—
127C8×Dic43—21no—
128C8×Dic43—21no—
129C8×Dic43—21no—
130C2×D8×C43C2×C411yescharacteristic
131C2×M16×C43—21no—
132C22×C8×C43—21no—
133C4×D4×C43C2311yescharacteristic
134C4×D4×C43C2311yescharacteristic
135#107×C43C2×C411yescharacteristic
136C4⋊C8×C43C2311yescharacteristic
137C4⋊C8×C43C2311yescharacteristic
138#102×C43C2×C411yescharacteristic
139#102×C43C2×C411yescharacteristic
140C22⋊C8×C43—21no—
141C22⋊C8×C43—21no—
142C4×C8×C43C2311yescharacteristic
143C4×C8×C43C2311yescharacteristic
144C4×C8×C43C2311yescharacteristic
145C23×D43—21no—
146C2×D4×C43C22×C411yescharacteristic
147C2×D4×C43C22×C411yescharacteristic
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C2×C22⋊D43—21no—
151C2×C22⋊D43—21no—
152C22×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155#10621900—22no—
156#10621900M1614yescomplemented
157D4×D43—22no—
158D4×D43—22no—
159C2×D172C22×C411yescharacteristic
160C2×C4×D43—21no—
161C2×C4×D43—21no—
162C2×C4×D43—21no—
163C2×C4×D43—81no—
164C2×C4×D43—81no—
165C2×C4×D43C22×C411yescharacteristic
166C2×C4×D43C2×D411yescharacteristic
167C2×C4×D43#4111yescharacteristic
168C2×Dic86C22×C411yescharacteristic
169D8×C43—22no—
170M16×C43—81no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—81no—
175C2×C8×C43C22×C411yescharacteristic
176C2×C8×C43C22×C411yescharacteristic
177C2×C8×C43C2×D411yescharacteristic
178C2×C8×C43C2×D411yescharacteristic
179C4⋊C4×C43C22×C411yescharacteristic
180C4⋊C4×C43C22×C411yescharacteristic
181C22⋊C4×C43—21no—
182C22⋊C4×C43—21no—
183C42×C43C2411yescharacteristic
184C22⋊Dic43—21no—
185C22⋊Dic43—21no—
186#10621883—22no—
187#10621882—22no—
188(C2×D43)⋊C4—21no—
189(C2×D43)⋊C4—21no—
190(C2×D43)⋊C4—21no—
191C4⋊Dic43C22×C411yescharacteristic
192C4⋊Dic43C22×C411yescharacteristic
193Dic43⋊C4—21no—
194Dic43⋊C4—21no—
195Dic43⋊C4—21no—
196C4×Dic43—21no—
197C4×Dic43—21no—
198C4×Dic43—21no—
199C4×Dic43C22×C411yescharacteristic
200C4×Dic43C2×D411yescharacteristic
201C4×Dic43#4111yescharacteristic
202#10621877—81no—
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—81no—
207C2×C43⋊C8C22×C411yescharacteristic
208C2×C43⋊C8C22×C411yescharacteristic
209C2×C43⋊C8C2×D411yescharacteristic
210C2×C43⋊C8C2×D411yescharacteristic
211#10621873—22no—
212#10621872—22no—
213#10621872—22no—
214#10621872—81no—
215#10621872—81no—
216C8×D43—42no—
217C8×D43—81no—
218C23×C43—21no—
219C23×C43—21no—
220C22×D43—21no—
221C22×D43—42no—
222C22×D43—81no—
223C22×D43—81no—
224C22×D43C4×D411yescharacteristic
225D4×C43—21no—
226D4×C43—21no—
227D4×C43—21no—
228D4×C43C2×M1612yes—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43C23×C411yescharacteristic
236C2×C4×C43C22×D411yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C22⋊D43—24no—
239C22⋊D43—42no—
240C2×Dic43—21no—
241C2×Dic43—24no—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—81no—
245C2×Dic43—81no—
246C2×Dic43C4×D411yescharacteristic
247C2×Dic43C4×D411yescharacteristic
248C2×Dic43C4×D411yescharacteristic
249D172—21no—
250D172—21no—
251C4×D43—21no—
252C4×D43—21no—
253C4×D43—41no—
254C4×D43—41no—
255C4×D43—41no—
256C4×D43—81no—
257C4×D43—81no—
258C4×D43C2×M1612yes—
259C4×D43#13112yes—
260Dic86—21no—
261Dic86C2×M1612yes—
262C8×C43—22no—
263C8×C43—21no—
264C8×C43—21no—
265C8×C43—21no—
266C8×C43—21no—
267C8×C43—41no—
268C8×C43—41no—
269C43⋊C8—21no—
270C43⋊C8—21no—
271C43⋊C8—21no—
272C43⋊C8—21no—
273C43⋊C8—41no—
274C43⋊C8—41no—
275C43⋊C8—41no—
276#17875—431nomaximal, Sylow
277C22×C43—22no—
278C22×C43—41no—
279C22×C43—41no—
280C22×C43—41no—
281C22×C43—41no—
282C22×C43—41no—
283C22×C43C2×C4×D411yescharacteristic
284C2×D43—21no—
285C2×D43—21no—
286C2×D43—41no—
287C2×D43—41no—
288C2×D43—41no—
289C2×D43—84no—
290C2×D43#43512yes—
291C4×C43—21no—
292C4×C43—41no—
293C4×C43—41no—
294C4×C43C22×M1611yescharacteristic
295C4×C43C2×#13111yescharacteristic
296C4×C43#57611yescharacteristic
297C4×C43#57711yescharacteristic
298Dic43—22no—
299Dic43—21no—
300Dic43—21no—
301Dic43—41no—
302Dic43—41no—
303Dic43#43512yes—
304#2988—431no—
305#2975—431no—
306C4×AGL(1,ℤ/8)—431no—
307#1588—431no—
308#1588—431no—
309#1586—431no—
310#1586D4311yescharacteristic, complemented
311#1583—431no—
312#1583—431no—
313#1581—431no—
314#1581—431no—
315#1565—431no—
316#1476—431no—
317C2×D4⋊C8—431no—
318#1451—431no—
319C2×C43—42no—
320C2×C43—81no—
321C2×C43C2×#43511yescharacteristic
322C2×C43#294611yescharacteristic
323C2×C43#295611yescharacteristic
324D43—22no—
325D43—81no—
326C2×AGL(1,ℤ/8)—431no—
327C4×#41—431no—
328C2×C4×D4—431no—
329C8×Q8—431no—
330C8×Q8—431no—
331#442—431no—
332#442—431no—
333#440—431no—
334#440—431no—
335C4×SD16—431no—
336C4×SD16—431no—
337C4×D8—431no—
338C4×D8C2×D4311yescharacteristic
339#436—431no—
340#436—431no—
341#436—431no—
342#436—861no—
343#436—861no—
344#436C2×D4311yescharacteristic
345C8×D4—431no—
346C8×D4—861no—
347C8×D4—861no—
348C8×D4C2×D4311yescharacteristic, complemented
349#431—431no—
350#431—431no—
351#431—861no—
352#428—431no—
353C2×C4⋊C8—431no—
354C2×C4⋊C8—861no—
355#418—431no—
356#417—431no—
357#407—861no—
358#407—861no—
359C4×M16—431no—
360C2×C4×C8—431no—
361#335—431no—
362#335—431no—
363#334—431no—
364#334C2×D4311yescharacteristic, complemented
365Q8⋊C8—431no—
366Q8⋊C8—431no—
367D4⋊C8—431no—
368D4⋊C8—431no—
369D4⋊C8—431no—
370D4⋊C8—431no—
371D4⋊C8C2×D4311yescharacteristic
372D4⋊C8C2×D4311yescharacteristic, complemented
373#322—431no—
374#322—431no—
375#322—431no—
376#322C2×D4311yescharacteristic, complemented
377C43#1787511yescharacteristic, complemented, Sylow
378C2×#41—431no—
379C22×D4—431no—
380C23×C4—861no—
381AGL(1,ℤ/8)—864no—
382C2×SD16—431no—
383C2×SD16—431no—
384C2×D8—431no—
385C2×D8C4×D4311yescharacteristic
386C2×M16—21no—
387C2×M16—431no—
388C2×M16—431no—
389C2×M16—861no—
390C2×M16—861no—
391C2×M16—861no—
392C22×C8—21no—
393C22×C8—861no—
394C22×C8—861no—
395C22×C8—861no—
396C4×Q8—431no—
397C4×Q8—431no—
398C4×D4—431no—
399C4×D4—431no—
400C4×D4—431no—
401C4×D4—431no—
402C4×D4—861no—
403C4×D4—861no—
404C4×D4—861no—
405C4×D4—861no—
406C4×D4C22×D4311yescharacteristic
407C4×D4C22×D4311yescharacteristic
408#117—431no—
409#117—861no—
410C2×C4⋊C4—431no—
411C2×C22⋊C4—861no—
412C2×C42—431no—
413C2×C42—861no—
414#107—431no—
415#107C4×D4311yescharacteristic
416#106—431no—
417#106—431no—
418C4⋊C8—431no—
419C4⋊C8—431no—
420C4⋊C8—431no—
421C4⋊C8—431no—
422C4⋊C8—861no—
423C4⋊C8—861no—
424C4⋊C8—861no—
425C4⋊C8C22×D4311yescharacteristic
426C4⋊C8C22×D4311yescharacteristic
427#103—431no—
428#103—431no—
429#102—431no—
430#102—431no—
431#102—431no—
432#102—431no—
433#102C4×D4311yescharacteristic
434#102C4×D4311yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8—431no—
438C22⋊C8—431no—
439C22⋊C8—861no—
440C22⋊C8—861no—
441C22⋊C8—861no—
442C22⋊C8—861no—
443C22⋊C8—861no—
444#97—431no—
445#97—431no—
446C4×C8—431no—
447C4×C8—431no—
448C4×C8—431no—
449C4×C8—861no—
450C4×C8—861no—
451C4×C8—861no—
452C4×C8—861no—
453C4×C8C22×D4311yescharacteristic
454C4×C8C22×D4311yescharacteristic
455C4×C8C22×D4311yescharacteristic
456C24—861no—
457#41—434no—
458#41—862no—
459C2×Q8—431no—
460C2×D4—431no—
461C2×D4—862no—
462C2×D4—862no—
463C2×D4—861no—
464C2×D4—861no—
465C2×D4C2×C4×D4311yescharacteristic
466C2×D4C2×C4×D4311yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—431no—
470C22×C4—431no—
471C22×C4—431no—
472C22×C4—861no—
473C22×C4—861no—
474C22×C4—861no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—3441no—
479C22×C4—3441no—
480SD16—862no—
481SD16—862no—
482D8—22no—
483D8—862no—
484M16—81no—
485M16—862no—
486M16—862no—
487M16—3441no—
488M16—3441no—
489M16—3441no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—81no—
494C2×C8—431no—
495C2×C8—431no—
496C2×C8—431no—
497C2×C8—431no—
498C2×C8—861no—
499C2×C8—861no—
500C2×C8—861no—
501C2×C8—3441no—
502C2×C8—3441no—
503C2×C8—1722no—
504C2×C8C2×C4×D4311yescharacteristic
505C2×C8C2×C4×D4311yescharacteristic
506C2×C8D4×D4311yescharacteristic
507C2×C8D4×D4311yescharacteristic
508C4⋊C4—431no—
509C4⋊C4—431no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4C2×C4×D4311yescharacteristic
514C4⋊C4C2×C4×D4311yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—861no—
518C22⋊C4—861no—
519C22⋊C4—861no—
520C22⋊C4—861no—
521C22⋊C4—861no—
522C42—431no—
523C42—431no—
524C42—431no—
525C42—861no—
526C42—861no—
527C42—861no—
528C42C23×D4311yescharacteristic
529C23—21no—
530C23—21no—
531C23—431no—
532C23—861no—
533C23—3441no—
534C23—3441no—
535C23—1722no—
536Q8—432no—
537Q8—861no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—864no—
542D4—861no—
543D4—861no—
544D4—1722no—
545D4M16×D4312yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—431no—
553C2×C4—432no—
554C2×C4—432no—
555C2×C4—431no—
556C2×C4—431no—
557C2×C4—864no—
558C2×C4—861no—
559C2×C4—861no—
560C2×C4—861no—
561C2×C4—861no—
562C2×C4—861no—
563C2×C4—3441no—
564C2×C4—3441no—
565C2×C4—3441no—
566C2×C4—3441no—
567C2×C4—1721no—
568C2×C4—1721no—
569C2×C4—1721no—
570C2×C4C22×C4×D4311yescharacteristic
571C2×C4C2×D4×D4311yescharacteristic
572C2×C4#41×D4311yescharacteristic
573C8—22no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—21no—
578C8—41no—
579C8—41no—
580C8—861no—
581C8—861no—
582C8—861no—
583C8—861no—
584C8—1721no—
585C8—1721no—
586C8—1721no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—432no—
594C22—861no—
595C22—861no—
596C22—3444no—
597C22—1721no—
598C22—1721no—
599C22—1721no—
600C22C4×D4×D4311yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—432no—
605C4—861no—
606C4—861no—
607C4—862no—
608C4—1721no—
609C4—1721no—
610C4C2×M16×D4311yescharacteristic
611C4#131×D4311yescharacteristic
612C4#1458370311yescharacteristic
613C4#1458370911yescharacteristic
614C2—42no—
615C2—81no—
616C2—862no—
617C2—3441no—
618C2#435×D4311yescharacteristic
619C2#1914342411yescharacteristic
620C2#1914350011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC42C23×D43
FittingC43×#1586C2
socleC22×C43C2×C4×D4
prime radicalC2×#13063349C4
cosocleC42×C43C24
hypercenter#1586D43
hypocenterC43#17875
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
1. #19143702C2semidirect
2. #19143633C2semidirect
4. #19133476C2semidirect
5. #19133475C2semidirect
6. #19133456C2semidirect
7. #19133437C2semidirect
8. #19133436C2semidirect
9. #19133425C2semidirect
10. #19133424C2semidirect
11. #19133344C2semidirect
12. #19132974C2semidirect
13. D4⋊C8×D43C2semidirect
14. #19132881C2semidirect
15. C43×#1586C2semidirect
19. C8×Dic86C22semidirect
20. #14583023C22semidirect
33. D4×C43⋊C8C22semidirect
48. #14582652C22semidirect
49. #14582651C22semidirect
50. #14582649C22semidirect
51. #14582631C22semidirect
52. #14582629C22semidirect
55. D4⋊(C43⋊C8)C22semidirect
57. #14582621C22semidirect
58. #14582620C22semidirect
59. #14582620C22semidirect
62. C8×D4×C43C22semidirect
63. C43×#334C22semidirect
65. C43×D4⋊C8C22semidirect
66. C43×#322C22semidirect
156. #10621900M16semidirect
310. #1586D43semidirect
348. C8×D4C2×D43semidirect
364. #334C2×D43semidirect
372. D4⋊C8C2×D43semidirect
376. #322C2×D43semidirect
377. C43#17875semidirect

Maximal subgroups

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

subgroupindex
#191437022
#191436332
C4×#130633492
#191334762
#191334752
#191334562
#191334372
#191334362
#191334252
#191334242
#191333442
#191329742
D4⋊C8×D432
#191328812
C43×#15862
#1787543