Finite Group Databasebeta

← #25129314

Subgroups — #25129314

Order 11008.

1428 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063305C211yesmaximal, characteristic
2C2×#14583242C212yesmaximal, complemented
3#19136977C211yesmaximal, characteristic
4#19135800C212yesmaximal, complemented
5#19135794C212yesmaximal, complemented
6#19134906C212yesmaximal, complemented
7#19134788C211yesmaximal, characteristic, complemented
8#19134531C211yesmaximal, characteristic, complemented
9#19134529C211yesmaximal, characteristic, complemented
10#19134504C211yesmaximal, characteristic
11C43×#1852C211yesmaximal, characteristic, complemented
12C23×Dic86C2211yescharacteristic
13C23×C4×D43C2211yescharacteristic
14C2×#13063305—21no—
15C2×#13063305—21no—
16C2×#13063305—28no—
17C2×#13063305C2212yes—
18C2×#13063305C2212yes—
19C2×#13063283C2211yescharacteristic
20C2×#13063283C2211yescharacteristic
21C22×C4⋊Dic43—21no—
22C22×(C2×D43)⋊C4—21no—
23#14583242—28no—
24C2×#13063256C2212yescomplemented
25C2×#13063250C2212yescomplemented
26C2×#13063250C2212yescomplemented
27C2×#13063248C2212yescomplemented
28C2×#13063248C2212yescomplemented
29C2×C22⋊(C43⋊C8)C2211yescharacteristic
30C2×(C2×D43)⋊C8C2212yes—
31C2×(C2×D43)⋊C8C2212yes—
32#14582761—21no—
33#14582759C2211yescharacteristic
34#14582756C2211yescharacteristic
35#14582756C2211yescharacteristic
36#14582754—21no—
37C22⋊C4×Dic43—21no—
38#14582746—21no—
39#14582661C2212yescomplemented
40C22×C4⋊C4×C43C2211yescharacteristic
41C2×#107×C43C2212yescomplemented
42C2×C22⋊C8×C43C2211yescharacteristic
43C43×#380C2211yescharacteristic
44C43×#340C2212yescomplemented
45C24×D43D411yescharacteristic
46C22×Dic86—21no—
47C22×Dic86—24no—
48C22×Dic86—22no—
49C22×Dic86—22no—
50C22×Dic86—42no—
51C22×Dic86C2312yes—
52C23×Dic43—21no—
53C23×Dic43D411yescharacteristic
54C22×C4×D43—22no—
55C22×C4×D43—22no—
56C22×C4×D43—24no—
57C22×C4×D43—42no—
58C22×C4×D43D411yescharacteristic
59C22×C4×D43D411yescharacteristic
60C22×C4×D43C2311yescharacteristic
61C22×C4×D43C2311yescharacteristic
62C22×C43⋊C8C2312yes—
63#13063305—28no—
64#13063305—28no—
65#13063305—24no—
66#13063305—24no—
67#13063305—24no—
68#13063305D418yes—
69#13063283—24no—
70#13063283—42no—
71C2×C4⋊Dic43—28no—
72C2×C4⋊Dic43—22no—
73C2×C4⋊Dic43—22no—
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C22⋊Dic43—21no—
77C2×(C2×D43)⋊C4—28no—
78C2×(C2×D43)⋊C4—22no—
79C2×(C2×D43)⋊C4—22no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43—21no—
86#13063256—24no—
87#13063250—24no—
88#13063250—42no—
89#13063248—24no—
90#13063248—42no—
91C22⋊(C43⋊C8)—22no—
92(C2×D43)⋊C8—24no—
93(C2×D43)⋊C8—42no—
94C43⋊#95—21no—
95C43⋊#95—21no—
96C43⋊#95—21no—
97C23×C4×C43—21no—
98C23×C4×C43C2311yescharacteristic
99C22×C8×C43C2312yes—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—24no—
102C2×C4⋊C4×C43C2311yescharacteristic
103C2×C4⋊C4×C43C2311yescharacteristic
104C2×C4⋊C4×C43C2312yes—
105C2×C22⋊C4×C43—21no—
106#107×C43—24no—
107C22⋊C8×C43—22no—
108#95×C43—21no—
109C24×C43C2×D411yescharacteristic
110C23×D43—22no—
111C23×D43—22no—
112C23×D43—22no—
113C23×D43—22no—
114C23×D43—22no—
115C23×D43—41no—
116C23×D43—41no—
117C23×D43C2×D411yescharacteristic
118C23×D43C2×D411yescharacteristic
119C22×C4×C43—22no—
120C22×C4×C43—22no—
121C22×C4×C43—24no—
122C22×C4×C43—22no—
123C22×C4×C43—21no—
124C22×C4×C43—21no—
125C22×C4×C43—42no—
126C22×C4×C43—42no—
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43#4112yes—
129C22×C4×C43#4112yes—
130C22×C4×C43C2411yescharacteristic
131C22×Dic43—24no—
132C22×Dic43—22no—
133C22×Dic43—22no—
134C22×Dic43—22no—
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—41no—
138C22×Dic43—42no—
139C22×Dic43—42no—
140C22×Dic43C2×D411yescharacteristic
141C22×Dic43C2×D411yescharacteristic
142C2×C4×D43—21no—
143C2×C4×D43—21no—
144C2×C4×D43—22no—
145C2×C4×D43—22no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—21no—
149C2×C4×D43—21no—
150C2×C4×D43—22no—
151C2×C4×D43—22no—
152C2×C4×D43—22no—
153C2×C4×D43—41no—
154C2×C4×D43—41no—
155C2×C4×D43—48no—
156C2×C4×D43—48no—
157C2×C4×D43C2×D414yes—
158C2×Dic86—22no—
159C2×Dic86—22no—
160C2×Dic86—22no—
161C2×Dic86—22no—
162C2×Dic86—24no—
163C2×Dic86—22no—
164C2×Dic86—48no—
165C2×Dic86—48no—
166C2×Dic86—42no—
167C2×Dic86—42no—
168C2×Dic86C2×D414yes—
169C2×C8×C43—22no—
170C2×C8×C43—22no—
171C2×C8×C43—22no—
172C4⋊C4×C43—24no—
173C4⋊C4×C43—22no—
174C4⋊C4×C43—22no—
175C4⋊C4×C43—22no—
176C4⋊C4×C43C2×D414yes—
177C22⋊C4×C43—42no—
178C22⋊Dic43—42no—
179(C2×D43)⋊C4—28no—
180(C2×D43)⋊C4—28no—
181(C2×D43)⋊C4—42no—
182(C2×D43)⋊C4—42no—
183(C2×D43)⋊C4—42no—
184(C2×D43)⋊C4—42no—
185C4⋊Dic43—28no—
186C4⋊Dic43—28no—
187C4⋊Dic43—22no—
188C4⋊Dic43—22no—
189C4×Dic43—42no—
190C4×Dic43—42no—
191C2×C43⋊C8—22no—
192C2×C43⋊C8—22no—
193C2×C43⋊C8—22no—
194C23×C43—21no—
195C23×C43—21no—
196C23×C43—21no—
197C23×C43—21no—
198C23×C43#12312yes—
199C23×C43#13512yes—
200C23×C43#13712yes—
201C23×C43C22×D411yescharacteristic
202C22×D43—28no—
203C22×D43—21no—
204C22×D43—21no—
205C22×D43—21no—
206C22×D43—21no—
207C22×D43—21no—
208C22×D43—21no—
209C22×D43—21no—
210C22×D43—21no—
211C22×D43—21no—
212C22×D43—21no—
213C22×D43—21no—
214C22×D43—21no—
215C22×D43—44no—
216C22×D43—44no—
217C22×D43—44no—
218C22×D43—44no—
219C22×D43—44no—
220C22×D43—44no—
221C22×D43#12012yes—
222C22×D43#13712yes—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—24no—
226C2×C4×C43—24no—
227C2×C4×C43—24no—
228C2×C4×C43—48no—
229C2×C4×C43—42no—
230C2×C4×C43—42no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43#12312yes—
238C2×C4×C43C22×D412yes—
239C2×C4×C43C2×#4111yescharacteristic
240C2×C4×C43C2×#4111yescharacteristic
241C2×C4×C43H(5,𝔽2)12yes—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—21no—
245C2×Dic43—24no—
246C2×Dic43—21no—
247C2×Dic43—21no—
248C2×Dic43—44no—
249C2×Dic43—48no—
250C2×Dic43—42no—
251C2×Dic43—42no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43#12012yes—
259C4×D43—24no—
260C4×D43—24no—
261C4×D43—44no—
262C4×D43—44no—
263C4×D43—42no—
264C4×D43—42no—
265Dic86—24no—
266Dic86—24no—
267Dic86—48no—
268Dic86—42no—
269Dic86—42no—
270C8×C43—42no—
271C43⋊C8—42no—
272#24010—431nomaximal, Sylow
273C22×C43—22no—
274C22×C43—22no—
275C22×C43—22no—
276C22×C43—22no—
277C22×C43—22no—
278C22×C43—22no—
279C22×C43#48514yescomplemented
280C22×C43C2×#12011yescharacteristic
281C22×C43C2×#12311yescharacteristic
282C22×C43#53411yescharacteristic
283C22×C43C2×#13511yescharacteristic
284C22×C43C2×#13711yescharacteristic
285C22×C43C2×#13711yescharacteristic
286C22×C43#57511yescharacteristic
287C2×D43—22no—
288C2×D43—22no—
289C2×D43—22no—
290C2×D43—22no—
291C2×D43—22no—
292C2×D43—22no—
293C2×D43—416no—
294C2×D43—41no—
295C2×D43—41no—
296C2×D43—41no—
297C2×D43—41no—
298C2×D43—41no—
299C2×D43—41no—
300C2×D43—41no—
301C2×D43—41no—
302C4×C43—22no—
303C4×C43—44no—
304C4×C43—81no—
305C4×C43—81no—
306C4×C43#54612yes—
307C4×C43#54612yes—
308Dic43—22no—
309Dic43—44no—
310Dic43—41no—
311Dic43—81no—
312Dic43—81no—
313C22×C22⋊Q8—431no—
314C2×#485—432no—
315#2280—431no—
316#2036—432no—
317#2035—432no—
318#1873—432no—
319#1853—431no—
320#1852D4311yescharacteristic, complemented
321#1787—431no—
322#1787—431no—
323#1779—431no—
324C2×C43—24no—
325C2×C43#240511yescharacteristic
326C2×C43#300211yescharacteristic
327C2×C43#300411yescharacteristic
328C2×C43C2×#48511yescharacteristic
329C2×C43#309511yescharacteristic
330C2×C43#319011yescharacteristic
331C2×C43#319611yescharacteristic
332D43—42no—
333D43—42no—
334C23×Q8—431no—
335C24×C4—431no—
336C2×C22⋊Q8—432no—
337C2×C22⋊Q8—432no—
338C2×C22⋊Q8—868no—
339C2×C22⋊Q8—861no—
340C2×C22⋊Q8—861no—
341C2×#117—431no—
342C2×#117—431no—
343C22×C4⋊C4—861no—
344C22×C4⋊C4C2×D4311yescharacteristic
345C22×C22⋊C4—861no—
346#485—868no—
347C2×#107C2×D4312yescomplemented
348C2×#106—432no—
349C2×#103—432no—
350C2×#103—432no—
351C2×#103—432no—
352C2×#103—432no—
353C2×C22⋊C8—431no—
354C2×C22⋊C8—432no—
355C2×C22⋊C8—432no—
356C2×C22⋊C8C2×D4311yescharacteristic
357#381—861no—
358#380—431no—
359#380—431no—
360#380—431no—
361#380C2×D4311yescharacteristic
362#379—861no—
363C4×C22⋊C4—861no—
364C4×C22⋊C4—861no—
365#340—432no—
366#340C2×D4312yescomplemented
367C43#2401011yescharacteristic, complemented, Sylow
368C25—431no—
369C22×Q8—432no—
370C22×Q8—864no—
371C22×Q8—861no—
372C22×Q8—862no—
373C22×Q8—862no—
374C22×Q8—1722no—
375C23×C4—21no—
376C23×C4—431no—
377C23×C4—431no—
378C23×C4—431no—
379C23×C4—431no—
380C23×C4—431no—
381C23×C4—864no—
382C23×C4—861no—
383C23×C4—862no—
384C23×C4—862no—
385C23×C4—1722no—
386C23×C4C22×D4311yescharacteristic
387C22×C8—432no—
388C22×C8C22×D4312yes—
389C22⋊Q8—438no—
390C22⋊Q8—864no—
391C22⋊Q8—864no—
392C22⋊Q8—868no—
393C22⋊Q8—868no—
394C22⋊Q8—864no—
395#117—864no—
396#117—1722no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—431no—
400C2×C4⋊C4—431no—
401C2×C4⋊C4—862no—
402C2×C4⋊C4—862no—
403C2×C4⋊C4—868no—
404C2×C4⋊C4C22×D4311yescharacteristic
405C2×C4⋊C4C22×D4311yescharacteristic
406C2×C4⋊C4C22×D4312yes—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—862no—
409C2×C22⋊C4—862no—
410C2×C22⋊C4—868no—
411C2×C22⋊C4—861no—
412C2×C22⋊C4—861no—
413C2×C22⋊C4—861no—
414C2×C22⋊C4—861no—
415C2×C22⋊C4—861no—
416C2×C42—861no—
417C2×C42—861no—
418#107—24no—
419#106—864no—
420#103—864no—
421#103—864no—
422#103—1722no—
423#103—1722no—
424C22⋊C8—22no—
425C22⋊C8—864no—
426C22⋊C8—862no—
427C22⋊C8—1722no—
428#95—21no—
429#95—861no—
430#95—861no—
431#95—861no—
432C24—431no—
433C24—431no—
434C24—862no—
435C24—862no—
436C24—862no—
437C24—862no—
438C24—862no—
439C24—1721no—
440C24—1721no—
441C24D4×D4311yescharacteristic
442C2×Q8—434no—
443C2×Q8—864no—
444C2×Q8—862no—
445C2×Q8—862no—
446C2×Q8—862no—
447C2×Q8—862no—
448C2×Q8—862no—
449C2×Q8—1728no—
450C2×Q8—1728no—
451C2×Q8—1722no—
452C2×Q8—1722no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—431no—
462C22×C4—434no—
463C22×C4—431no—
464C22×C4—862no—
465C22×C4—862no—
466C22×C4—862no—
467C22×C4—861no—
468C22×C4—861no—
469C22×C4—862no—
470C22×C4—864no—
471C22×C4—862no—
472C22×C4—862no—
473C22×C4—862no—
474C22×C4—862no—
475C22×C4—862no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—862no—
479C22×C4—861no—
480C22×C4—861no—
481C22×C4—1721no—
482C22×C4—1721no—
483C22×C4—1728no—
484C22×C4—1728no—
485C22×C4—1721no—
486C22×C4—1722no—
487C22×C4—1722no—
488C22×C4#1062189812yes—
489C22×C4D4×D4311yescharacteristic
490C22×C4#1062190212yes—
491C22×C4C23×D4311yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—862no—
496C2×C8—862no—
497C2×C8—862no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—868no—
503C4⋊C4—868no—
504C4⋊C4—862no—
505C4⋊C4—862no—
506C4⋊C4D4×D4314yes—
507C22⋊C4—42no—
508C22⋊C4—868no—
509C22⋊C4—868no—
510C22⋊C4—1722no—
511C22⋊C4—1722no—
512C22⋊C4—1722no—
513C22⋊C4—1722no—
514C22⋊C4—1722no—
515C42—1722no—
516C42—1722no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—432no—
522C23—432no—
523C23—868no—
524C23—861no—
525C23—861no—
526C23—861no—
527C23—861no—
528C23—861no—
529C23—861no—
530C23—861no—
531C23—861no—
532C23—861no—
533C23—861no—
534C23—861no—
535C23—861no—
536C23—1724no—
537C23—1724no—
538C23—1724no—
539C23—1724no—
540C23—1724no—
541C23—1724no—
542C23#1306331012yes—
543C23#1306334412yes—
544C23#1306335712yes—
545C23C2×D4×D4311yescharacteristic
546Q8—864no—
547Q8—864no—
548Q8—1728no—
549Q8—1722no—
550Q8—1722no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—432no—
566C2×C4—861no—
567C2×C4—861no—
568C2×C4—864no—
569C2×C4—864no—
570C2×C4—864no—
571C2×C4—861no—
572C2×C4—861no—
573C2×C4—861no—
574C2×C4—1724no—
575C2×C4—1724no—
576C2×C4—1724no—
577C2×C4—1721no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1728no—
584C2×C4—1722no—
585C2×C4—1722no—
586C2×C4—1722no—
587C2×C4—1722no—
588C2×C4#1306331012yes—
589C2×C4C2×D4×D4312yes—
590C2×C4C2×#1062189811yescharacteristic
591C2×C4C2×#1062190211yescharacteristic
592C2×C4#1306337112yes—
593C8—42no—
594C8—1722no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—862no—
602C22—862no—
603C22—862no—
604C22—862no—
605C22—862no—
606C22—862no—
607C22—17216no—
608C22—1721no—
609C22—1721no—
610C22—1721no—
611C22—1721no—
612C22—1721no—
613C22—1721no—
614C22—1721no—
615C22—1721no—
616C22#1458324214yescomplemented
617C22#120×D4311yescharacteristic
618C22C2×#1306331011yescharacteristic
619C22#1458346011yescharacteristic
620C22C2×#1306334411yescharacteristic
621C22#137×D4311yescharacteristic
622C22C2×#1306335711yescharacteristic
623C22#1458369511yescharacteristic
624C4—22no—
625C4—44no—
626C4—81no—
627C4—81no—
628C4—862no—
629C4—3441no—
630C4—3441no—
631C4—1721no—
632C4—1724no—
633C4#1458354012yes—
634C4#1458354512yes—
635C2—24no—
636C2—1722no—
637C2—1722no—
638C2#1913792411yescharacteristic
639C2#1914380811yescharacteristic
640C2#1914383011yescharacteristic
641C2C2×#1458324211yescharacteristic
642C2#1914454511yescharacteristic
643C2#1914537511yescharacteristic
644C2#1914543011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1852C2
socleC23×C43C22×D4
prime radicalC24×D43D4
hypercenter#1852D43
hypocenterC43#24010
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#14583242C2semidirect
4. #19135800C2semidirect
5. #19135794C2semidirect
6. #19134906C2semidirect
7. #19134788C2semidirect
8. #19134531C2semidirect
9. #19134529C2semidirect
11. C43×#1852C2semidirect
24. C2×#13063256C22semidirect
25. C2×#13063250C22semidirect
26. C2×#13063250C22semidirect
27. C2×#13063248C22semidirect
28. C2×#13063248C22semidirect
39. #14582661C22semidirect
41. C2×#107×C43C22semidirect
44. C43×#340C22semidirect
279. C22×C43#485semidirect
320. #1852D43semidirect
347. C2×#107C2×D43semidirect
366. #340C2×D43semidirect
367. C43#24010semidirect
616. C22#14583242semidirect

Maximal subgroups

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

subgroupindex
C22×#130633052
C2×#145832422
#191369772
#191358002
#191357942
#191349062
#191347882
#191345312
#191345292
#191345042
C43×#18522
#2401043