Finite Group Databasebeta

← #25986948

Subgroups — #25986948

Order 11008.

1660 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19149090C211yesmaximal, characteristic, complemented
2#19148396C211yesmaximal, characteristic
3#19147535C214yesmaximal, complemented
4#19147515C212yesmaximal, complemented
5#19147469C212yesmaximal, complemented
6#19147375C212yesmaximal, complemented
7#19147373C214yesmaximal, complemented
8#19147236C212yesmaximal, complemented
9#19145861C214yesmaximal, complemented
10#19145754C211yesmaximal, characteristic, complemented
11#19144843C214yesmaximal
12#19144695C211yesmaximal, characteristic
13#19144599C211yesmaximal, characteristic
14#19143692C211yesmaximal, characteristic, complemented
15C43×#3237C211yesmaximal, characteristic, complemented
16C43⋊#564—22no—
17#14583628—21no—
18#14583628C2212yescomplemented
19#14583627—21no—
20#14583627C2212yescomplemented
21#14583624C2212yescomplemented
22#14583624C2212yescomplemented
23#14583610C2212yes—
24#14583609—21no—
25#14583609—22no—
26#14583605C2214yes—
27Q8×D172—21no—
28#14583564C2212yes—
29#14583527C2212yes—
30#14583524—22no—
31#14583511—22no—
32#14583509—22no—
33#14583509C2214yes—
34#14583456C2211yescharacteristic
35#14583456C2211yescharacteristic
36C22⋊Q8×D43—22no—
37#14583373C2212yes—
38C2×C4×D172C2211yescharacteristic
39#14583306C2214yescomplemented
40#14583302C2211yescharacteristic, complemented
41#14583300C2214yescomplemented
42#14583295C2211yescharacteristic, complemented
43#14583266C2214yescomplemented
44#14583251C2211yescharacteristic, complemented
45#14583251C2211yescharacteristic, complemented
46#14583248C2214yescomplemented
47#14583240C2214yes—
48#14583235C2212yes—
49#14583227C2212yes—
50#14583208C2214yescomplemented
51#14583205C2214yescomplemented
52#14583203C2214yescomplemented
53#14583202C2214yes—
54#14583198C2214yes—
55#14583192C2214yescomplemented
56#14583186C2212yes—
57#14583183C2214yescomplemented
58#14583180C2212yes—
59#14583105C2212yes—
60#14583103C2214yes—
61#14583103C2214yes—
62#14583101C2212yes—
63#14583052C2212yes—
64#14583049C2212yes—
65#14583027C2214yescomplemented
66#14583026C2212yescomplemented
67#14583023C2212yescomplemented
68#14583002C2214yes—
69#14583001C2214yes—
70#14582998C2214yes—
71C4×#10621884C2212yes—
72C4×#10621884C2212yes—
73#14582986C2214yes—
74#14582984C2214yes—
75#14582957C2214yes—
76#14582952C2211yescharacteristic
77C43×#533C2211yescharacteristic
78C43×#533C2211yescharacteristic
79C43×#501C2212yescomplemented
80C43×#491C2212yescomplemented
81C43×#485C2214yes—
82C43×#477C2212yescomplemented
83C43×#475C2212yescomplemented
84C43×#432C2211yescharacteristic
85C2×Q8×D43—22no—
86C22×D172C2311yescharacteristic
87C22×C4×D43—21no—
88C2×#10621873C2314yes—
89C2×#10621884C2312yes—
90C2×#10621884C2312yes—
91#13063325—22no—
92#13063325—22no—
93#13063325D411yescharacteristic
94#13063325D411yescharacteristic
95#13063325C2311yescharacteristic
96#13063325C2311yescharacteristic
97#13063324C2314yes—
98#13063323—22no—
99#13063323C2314yes—
100#13063318—24no—
101#13063318—24no—
102#13063317—21no—
103#13063317—22no—
104#13063317—22no—
105#13063317D412yes—
106#13063316—22no—
107#13063316—24no—
108#13063316—24no—
109#13063316C2314yes—
110#13063316C2314yes—
111#13063315—24no—
112#13063315—24no—
113#13063315—24no—
114#13063315—24no—
115#13063305—22no—
116#13063305—24no—
117#13063304—21no—
118#13063304D412yes—
119#13063304D412yes—
120#13063304C2312yes—
121#13063303—24no—
122#13063303—24no—
123#13063303—24no—
124#13063302—24no—
125#13063301—22no—
126#13063301—24no—
127Q8×Dic43—22no—
128Q8×Dic43C2314yes—
129C4×Dic86—22no—
130C4×Dic86—21no—
131C4×Dic86—22no—
132C4×Dic86C2312yes—
133C4×Dic86C2312yes—
134C4×Dic86C2312yes—
135#13063290—22no—
136#13063290C2314yes—
137C4×D172—21no—
138C4×D172—21no—
139C4×D172—22no—
140C4×D172—22no—
141C4×D172C2312yes—
142C4×D172C2312yes—
143#13063287C2314yes—
144#13063287C2314yes—
145#13063284—21no—
146#13063284C2312yes—
147C2×C4⋊Dic43C2311yescharacteristic
148C4⋊C4×D43—22no—
149C4⋊C4×D43—24no—
150C2×(C2×D43)⋊C4—21no—
151C22⋊C4×D43—24no—
152#13063259C2314yes—
153#13063258C2314yes—
154#13063256C2312yes—
155#13063256C2314yes—
156#13063256C2312yes—
157C4⋊(C43⋊C8)C2312yes—
158Dic43⋊C8C2314yes—
159Dic43⋊C8C2314yes—
160#13063250C2314yes—
161#13063249C2312yes—
162#13063249C2312yes—
163#13063248C2314yes—
164#13063248C2314yes—
165#13063247C2314yes—
166#13063245C2312yes—
167#13063245C2312yes—
168C22⋊(C43⋊C8)C2312yes—
169#13063235C2314yes—
170C4×C43⋊C8C2311yescharacteristic
171C4×C43⋊C8C2311yescharacteristic
172C8×Dic43C2314yes—
173C4⋊Q8×C43C2311yescharacteristic
174C4⋊Q8×C43C2311yescharacteristic
175C4⋊Q8×C43C2311yescharacteristic
176C4⋊Q8×C43C2311yescharacteristic
177#125×C43—21no—
178#125×C43C2312yes—
179C22⋊Q8×C43—21no—
180C22⋊Q8×C43—21no—
181C22⋊Q8×C43—21no—
182C22⋊Q8×C43C2312yes—
183C4×Q8×C43—21no—
184C4×Q8×C43—21no—
185C4×Q8×C43—21no—
186C4×Q8×C43C2312yes—
187#117×C43—21no—
188#117×C43C2312yes—
189C2×C42×C43C2311yescharacteristic
190#107×C43C2314yes—
191#106×C43C2314yes—
192C4⋊C8×C43C2312yes—
193#103×C43C2314yes—
194#103×C43C2314yes—
195C22⋊C8×C43C2312yes—
196#97×C43C2312yes—
197C23×D43—21no—
198C2×Q8×C43—21no—
199C2×Q8×C43—21no—
200C2×Q8×C43—21no—
201C2×Q8×C43C2412yes—
202C22×C4×C43C2×D411yescharacteristic
203C22×C4×C43C2×D411yescharacteristic
204C22×C4×C43C2411yescharacteristic
205C22×Dic43—21no—
206Q8×D43—84no—
207C2×D172—21no—
208C2×D172—24no—
209C2×D172C2412yes—
210C2×C4×D43—22no—
211C2×C4×D43—22no—
212C2×C4×D43—24no—
213C2×C4×D43—24no—
214C2×C4×D43—82no—
215C2×Dic86—21no—
216C2×Dic86—24no—
217C2×Dic86C2×D412yes—
218C2×Dic86C2×D412yes—
219C2×Dic86C2412yes—
220C2×C8×C43C2414yes—
221C4⋊C4×C43—21no—
222C4⋊C4×C43—21no—
223C4⋊C4×C43—22no—
224C4⋊C4×C43—22no—
225C4⋊C4×C43—21no—
226C4⋊C4×C43—21no—
227C4⋊C4×C43—21no—
228C4⋊C4×C43—22no—
229C4⋊C4×C43—22no—
230C4⋊C4×C43C2412yes—
231C4⋊C4×C43C2412yes—
232C4⋊C4×C43C2412yes—
233C22⋊C4×C43—22no—
234C22⋊C4×C43—22no—
235C42×C43—22no—
236C42×C43—22no—
237C42×C43C2×D411yescharacteristic
238C42×C43C2×D411yescharacteristic
239C42×C43C2411yescharacteristic
240C42×C43C2411yescharacteristic
241C22⋊Dic43—22no—
242C22⋊Dic43C2×D414yes—
243#10621884—24no—
244#10621884—24no—
245(C2×D43)⋊C4—22no—
246(C2×D43)⋊C4—22no—
247(C2×D43)⋊C4—24no—
248(C2×D43)⋊C4—24no—
249C4⋊Dic43—21no—
250C4⋊Dic43—21no—
251C4⋊Dic43—21no—
252C4⋊Dic43—21no—
253C4⋊Dic43—24no—
254C4⋊Dic43—24no—
255C4⋊Dic43—24no—
256C4⋊Dic43C2×D412yes—
257C4⋊Dic43C2×D412yes—
258C4⋊Dic43C2×D412yes—
259C4⋊Dic43C2×D412yes—
260C4⋊Dic43C2412yes—
261C4⋊Dic43C2412yes—
262Dic43⋊C4—22no—
263Dic43⋊C4—22no—
264Dic43⋊C4—24no—
265Dic43⋊C4—24no—
266Dic43⋊C4—24no—
267Dic43⋊C4—24no—
268Dic43⋊C4—24no—
269Dic43⋊C4—24no—
270Dic43⋊C4C2×D414yes—
271Dic43⋊C4C2×D414yes—
272C4×Dic43—22no—
273C4×Dic43—24no—
274C4×Dic43—24no—
275C4×Dic43C2414yes—
276C2×C43⋊C8C2412yes—
277C2×C43⋊C8C2412yes—
278#10621873—28no—
279C23×C43C22×D411yescharacteristic
280C22×D43—22no—
281C22×D43—24no—
282C22×D43—81no—
283C22×D43—81no—
284Q8×C43—22no—
285Q8×C43—22no—
286Q8×C43—42no—
287Q8×C43—42no—
288Q8×C43—81no—
289C2×C4×C43—22no—
290C2×C4×C43—22no—
291C2×C4×C43—22no—
292C2×C4×C43—22no—
293C2×C4×C43—22no—
294C2×C4×C43—22no—
295C2×C4×C43—41no—
296C2×C4×C43C22×D411yescharacteristic
297C2×C4×C43C22×D411yescharacteristic
298C2×C4×C43C22×D411yescharacteristic
299C2×C4×C43C22×D411yescharacteristic
300C2×C4×C43C22×D411yescharacteristic
301C2×C4×C43C2511yescharacteristic
302C2×Dic43—22no—
303C2×Dic43—22no—
304C2×Dic43—22no—
305C2×Dic43—81no—
306C2×Dic43C22×D414yes—
307C2×Dic43C22×D414yes—
308D172—24no—
309D172—22no—
310D1722−1+414yes—
311C4×D43—44no—
312C4×D43—84no—
313C4×D43—84no—
314C4×D43—82no—
315Dic86—24no—
316Dic86—24no—
317Dic86—42no—
318Dic86—42no—
319Dic86—84no—
320C8×C43—24no—
321C43⋊C8—22no—
322C43⋊C8—22no—
323#61848—431nomaximal, Sylow
324C22×C43—21no—
325C22×C43—21no—
326C22×C43—21no—
327C22×C43C23×D411yescharacteristic
328C2×D43—24no—
329C2×D43—42no—
330C2×D43—42no—
331C2×D43—84no—
332C4×C43—21no—
333C4×C43—21no—
334C4×C43—21no—
335C4×C43—21no—
336C4×C43—42no—
337C4×C43—42no—
338C4×C43—42no—
339C4×C43—42no—
340C4×C43#57712yes—
341C4×C43C2×2−1+412yes—
342Dic43—24no—
343Dic43—24no—
344Dic43—42no—
345Dic43—42no—
346Dic43—42no—
347#3556—431no—
348#3501—431no—
349#3388—434no—
350#3386—432no—
351#3381—432no—
352#3372—434no—
353#3372—432no—
354#3359—432no—
355#3242—434no—
356#3237D4311yescharacteristic, complemented
357#3231—431no—
358#3123—434no—
359#3110—431no—
360#3101—431no—
361#2985—431no—
362C2×C43—41no—
363C2×C43#348611yescharacteristic
364C2×C43C2×#57711yescharacteristic
365C2×C43#358611yescharacteristic
366D43—44no—
367#564—862no—
368#557—432no—
369#557—432no—
370#557—432no—
371#557—432no—
372#557—861no—
373#557—861no—
374#554—432no—
375#554—434no—
376#554—862no—
377#554—861no—
378D4×Q8—432no—
379D4×Q8—861no—
380#544—432no—
381#543—862no—
382#541—434no—
383#541—862no—
384#541—862no—
385#533—431no—
386#533—431no—
387#533C2×D4311yescharacteristic
388#533C2×D4311yescharacteristic
389C2×C22⋊Q8—862no—
390#518—432no—
391C2×C4×D4—431no—
392#501—434no—
393#501C2×D4312yescomplemented
394#500—431no—
395#499—434no—
396#498—431no—
397#491—434no—
398#491C2×D4312yescomplemented
399#487—431no—
400#487—431no—
401#487—434no—
402#485—434no—
403#485C2×D4314yes—
404#484—432no—
405#483—432no—
406#479—434no—
407#479—434no—
408#479—434no—
409#478—434no—
410#478—434no—
411#477—434no—
412#477C2×D4312yescomplemented
413#476—432no—
414#475—434no—
415#475C2×D4312yescomplemented
416#474—432no—
417#461—432no—
418#461—432no—
419#461—434no—
420#461—434no—
421#450—432no—
422#450—432no—
423#446—432no—
424#446—434no—
425C8×Q8—432no—
426#440—434no—
427#440—434no—
428#440—434no—
429C4×SD16—432no—
430C4×SD16—432no—
431C4×SD16—434no—
432C4×SD16—434no—
433#433—434no—
434#432C2×D4311yescharacteristic
435#431—431no—
436C43#6184811yescharacteristic, complemented, Sylow
437C22×Q8—862no—
438C22×D4—431no—
439C23×C4—861no—
440C2×SD16—432no—
441C2×SD16—432no—
442C2×SD16—434no—
443C4⋊Q8—434no—
444C4⋊Q8—434no—
445C4⋊Q8—431no—
446C4⋊Q8—431no—
447C4⋊Q8—431no—
448C4⋊Q8—431no—
449C4⋊Q8—862no—
450C4⋊Q8—862no—
451C4⋊Q8—862no—
452C4⋊Q8C22×D4311yescharacteristic
453C4⋊Q8C22×D4311yescharacteristic
454C4⋊Q8C22×D4311yescharacteristic
455C4⋊Q8C22×D4311yescharacteristic
456#126—864no—
457#126—864no—
458#125—21no—
459#125—434no—
460#125—434no—
461#125—432no—
462#125—862no—
463#125—862no—
464#125—864no—
465#125—864no—
466#125—862no—
467#125—864no—
468#125—864no—
469#125—864no—
470#125—864no—
471#125—861no—
472#125C22×D4312yes—
473C22⋊Q8—21no—
474C22⋊Q8—21no—
475C22⋊Q8—21no—
476C22⋊Q8—432no—
477C22⋊Q8—432no—
478C22⋊Q8—432no—
479C22⋊Q8—864no—
480C22⋊Q8—864no—
481C22⋊Q8—864no—
482C22⋊Q8—864no—
483C22⋊Q8—862no—
484C22⋊Q8—864no—
485C22⋊Q8—862no—
486C22⋊Q8—864no—
487C22⋊Q8—861no—
488C22⋊Q8C22×D4312yes—
489C4×Q8—21no—
490C4×Q8—21no—
491C4×Q8—21no—
492C4×Q8—432no—
493C4×Q8—432no—
494C4×Q8—434no—
495C4×Q8—434no—
496C4×Q8—432no—
497C4×Q8—862no—
498C4×Q8—862no—
499C4×Q8—862no—
500C4×Q8—862no—
501C4×Q8—861no—
502C4×Q8C22×D4312yes—
503C4×D4—434no—
504C4×D4—434no—
505C4×D4—432no—
506C4×D4—432no—
507C4×D4—862no—
508C4×D4—862no—
509C4×D4—861no—
510C4×D4—861no—
511#117—21no—
512#117—432no—
513#117—861no—
514#117C22×D4312yes—
515C2×C4⋊C4—431no—
516C2×C4⋊C4—864no—
517C2×C4⋊C4—862no—
518C2×C22⋊C4—864no—
519C2×C22⋊C4—861no—
520C2×C42C22×D4311yescharacteristic
521#107—434no—
522#107—434no—
523#107C22×D4314yes—
524#106—432no—
525#106—432no—
526#106—434no—
527#106C22×D4314yes—
528C4⋊C8—434no—
529C4⋊C8—434no—
530C4⋊C8—432no—
531C4⋊C8C22×D4312yes—
532#103—432no—
533#103—432no—
534#103—434no—
535#103—434no—
536#103—434no—
537#103C22×D4314yes—
538#103C22×D4314yes—
539#102—434no—
540#102—432no—
541#102—432no—
542C22⋊C8—432no—
543C22⋊C8C22×D4312yes—
544#97—434no—
545#97C22×D4312yes—
546C4×C8—434no—
547C4×C8—431no—
548C4×C8—431no—
549C24—861no—
550C2×Q8—21no—
551C2×Q8—21no—
552C2×Q8—21no—
553C2×Q8—432no—
554C2×Q8—432no—
555C2×Q8—432no—
556C2×Q8—864no—
557C2×Q8—861no—
558C2×Q8—3444no—
559C2×Q8C23×D4312yes—
560C2×D4—432no—
561C2×D4—864no—
562C2×D4—861no—
563C22×C4—864no—
564C22×C4—864no—
565C22×C4—861no—
566C22×C4—862no—
567C22×C4—862no—
568C22×C4—3442no—
569C22×C4D4×D4311yescharacteristic
570C22×C4D4×D4311yescharacteristic
571C22×C4C23×D4311yescharacteristic
572SD16—868no—
573SD16—864no—
574SD16—864no—
575C2×C8—432no—
576C2×C8—432no—
577C2×C8C23×D4314yes—
578C4⋊C4—22no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—22no—
584C4⋊C4—21no—
585C4⋊C4—22no—
586C4⋊C4—22no—
587C4⋊C4—432no—
588C4⋊C4—434no—
589C4⋊C4—432no—
590C4⋊C4—432no—
591C4⋊C4—432no—
592C4⋊C4—434no—
593C4⋊C4—432no—
594C4⋊C4—432no—
595C4⋊C4—864no—
596C4⋊C4—864no—
597C4⋊C4—864no—
598C4⋊C4—864no—
599C4⋊C4—864no—
600C4⋊C4—864no—
601C4⋊C4—864no—
602C4⋊C4—864no—
603C4⋊C4—864no—
604C4⋊C4—861no—
605C4⋊C4—861no—
606C4⋊C4—861no—
607C4⋊C4—862no—
608C4⋊C4—861no—
609C4⋊C4—862no—
610C4⋊C4C23×D4312yes—
611C4⋊C4C23×D4312yes—
612C4⋊C4C23×D4312yes—
613C22⋊C4—22no—
614C22⋊C4—22no—
615C22⋊C4—434no—
616C22⋊C4—864no—
617C22⋊C4—864no—
618C22⋊C4—862no—
619C22⋊C4—862no—
620C22⋊C4—862no—
621C42—22no—
622C42—22no—
623C42—434no—
624C42—864no—
625C42—864no—
626C42—862no—
627C42D4×D4311yescharacteristic
628C42D4×D4311yescharacteristic
629C42C23×D4311yescharacteristic
630C42C23×D4311yescharacteristic
631C23—864no—
632C23—862no—
633C23—3441no—
634C23—3441no—
635C23C2×D4×D4311yescharacteristic
636Q8—22no—
637Q8—22no—
638Q8—42no—
639Q8—42no—
640Q8—81no—
641Q8—864no—
642Q8—864no—
643Q8—3444no—
644Q8—1722no—
645Q8—1722no—
646D4—434no—
647D4—864no—
648D4—862no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—41no—
656C2×C4—434no—
657C2×C4—434no—
658C2×C4—862no—
659C2×C4—862no—
660C2×C4—862no—
661C2×C4—3442no—
662C2×C4—3444no—
663C2×C4—3444no—
664C2×C4—3441no—
665C2×C4—1724no—
666C2×C4C2×D4×D4311yescharacteristic
667C2×C4C2×D4×D4311yescharacteristic
668C2×C4C2×D4×D4311yescharacteristic
669C2×C4C2×D4×D4311yescharacteristic
670C2×C4C2×D4×D4311yescharacteristic
671C2×C4C24×D4311yescharacteristic
672C8—24no—
673C8—862no—
674C8—862no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—864no—
679C22—3444no—
680C22—1722no—
681C22—1722no—
682C22C22×D4×D4311yescharacteristic
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—864no—
692C4—864no—
693C4—1722no—
694C4—1722no—
695C4—1722no—
696C4#1458370812yes—
697C42−1+4×D4312yes—
698C2—41no—
699C2—1724no—
700C2#1914824411yescharacteristic
701C2C2×#1458370811yescharacteristic
702C2#1914937911yescharacteristic
subgroupG/H
centerC22C22×D4×D43
commutator, cosocleC2×C4×C43C25
FrattiniC2×C4C24×D43
FittingC43×#3237C2
socleC22×C43C23×D4
prime radicalC22×D172C23
hypercenter#3237D43
hypocenterC43#61848
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3237

Complements

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

NH ≅ G/Ntype
1. #19149090C2semidirect
3. #19147535C2semidirect
4. #19147515C2semidirect
5. #19147469C2semidirect
6. #19147375C2semidirect
7. #19147373C2semidirect
8. #19147236C2semidirect
9. #19145861C2semidirect
10. #19145754C2semidirect
14. #19143692C2semidirect
15. C43×#3237C2semidirect
18. #14583628C22semidirect
20. #14583627C22semidirect
21. #14583624C22semidirect
22. #14583624C22semidirect
39. #14583306C22semidirect
40. #14583302C22semidirect
41. #14583300C22semidirect
42. #14583295C22semidirect
43. #14583266C22semidirect
44. #14583251C22semidirect
45. #14583251C22semidirect
46. #14583248C22semidirect
50. #14583208C22semidirect
51. #14583205C22semidirect
52. #14583203C22semidirect
55. #14583192C22semidirect
57. #14583183C22semidirect
65. #14583027C22semidirect
66. #14583026C22semidirect
67. #14583023C22semidirect
79. C43×#501C22semidirect
80. C43×#491C22semidirect
82. C43×#477C22semidirect
83. C43×#475C22semidirect
356. #3237D43semidirect
393. #501C2×D43semidirect
398. #491C2×D43semidirect
412. #477C2×D43semidirect
415. #475C2×D43semidirect
436. C43#61848semidirect

Maximal subgroups

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

subgroupindex
#191490902
#191483962
#191475352
#191475152
#191474692
#191473752
#191473732
#191472362
#191458612
#191457542
#191448432
#191446952
#191445992
#191436922
C43×#32372
#6184843