Finite Group Databasebeta

← #25097487

Subgroups — #25097487

Order 11008.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147796C211yesmaximal, characteristic
2C2×#14582942C211yesmaximal, characteristic, complemented
3C2×#14582917C211yesmaximal, characteristic
4C2×#14582901C211yesmaximal, characteristic
5#19137218C211yesmaximal, characteristic, complemented
6#19134829C214yesmaximal, complemented
7#19134822C214yesmaximal, complemented
8#19134333C211yesmaximal, characteristic, complemented
9C43×#1733C211yesmaximal, characteristic, complemented
10C22×#10621898C2211yescharacteristic
11C22×#10621877C2211yescharacteristic
12#14583380—21no—
13#14583380—21no—
14#14583379C2212yes—
15#14583379C2212yes—
16#14583376—21no—
17#14583373—21no—
18#14583372—22no—
19C4×#10621898—22no—
20#14583367—22no—
21C2×#13063284C2211yescharacteristic
22#117×D43—22no—
23C22×C4⋊Dic43C2211yescharacteristic
24#14582942—22no—
25#14582942C2214yescomplemented
26C2×#13063258—21no—
27C2×#13063256—21no—
28#14582917C418yescomplemented
29C2×#13063253C2214yes—
30#14582901—22no—
31#14582901C2214yes—
32C2×#13063250—21no—
33C2×#13063247—21no—
34#14582798—22no—
35#14582774—22no—
36#14582769C2212yescomplemented
37#14582765C2212yescomplemented
38#14582667C2212yescomplemented
39#14582667C2212yescomplemented
40C22×M16×C43C2211yescharacteristic
41C2×#117×C43C2211yescharacteristic
42C2×#117×C43C2211yescharacteristic
43C43×#339C2214yescomplemented
44C2×#10621898—21no—
45C2×#10621898—21no—
46C2×#10621898—22no—
47C2×#10621898—22no—
48C2×#10621898C2×C414yes—
49C2×#10621898D412yes—
50C2×#10621898Q814yes—
51C2×#10621898C2312yes—
52C22×Dic86D411yescharacteristic
53C22×D172D411yescharacteristic
54C22×C22⋊D43—21no—
55C23×Dic43—21no—
56C22×C4×D43—21no—
57C2×#10621877—21no—
58C2×#10621877—22no—
59C2×#10621877C2×C414yescomplemented
60C2×#10621877C2312yes—
61C22×C43⋊C8—21no—
62C4×Dic86—22no—
63C4×Dic86—22no—
64#13063290—22no—
65#13063290—22no—
66C4×D172—22no—
67C4×D172—22no—
68#13063287—22no—
69#13063287—22no—
70C4×C22⋊D43—24no—
71C4×C22⋊D43—24no—
72(C22⋊D43)⋊C4—24no—
73(C22⋊D43)⋊C4—24no—
74#13063284—21no—
75#13063284—21no—
76#13063284—21no—
77#13063284—21no—
78#13063283—22no—
79#13063283—22no—
80#13063282—24no—
81#13063281—24no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43—21no—
84C2×C4⋊Dic43—21no—
85C2×C4⋊Dic43—21no—
86C2×C4⋊Dic43—21no—
87C2×C4⋊Dic43—24no—
88C2×C4⋊Dic43C2312yes—
89C2×C4⋊Dic43C2311yescharacteristic
90C2×C4⋊Dic43C2311yescharacteristic
91C4⋊C4×D43—22no—
92C4⋊C4×D43—22no—
93C2×C22⋊Dic43D412yes—
94C22⋊C4×D43—24no—
95C2×C4×Dic43D412yes—
96C42×D43—24no—
97#13063258—24no—
98#13063256—24no—
99#13063253C2×C4116yescomplemented
100#13063250—24no—
101#13063247—24no—
102C43⋊#95—24no—
103C23×C4×C43C2311yescharacteristic
104C2×M16×C43—21no—
105C2×M16×C43—22no—
106C2×M16×C43—22no—
107C2×M16×C43C2312yes—
108C22×C8×C43—21no—
109#117×C43—21no—
110#117×C43—21no—
111#117×C43—22no—
112#117×C43C2×C414yescomplemented
113#117×C43D412yes—
114#117×C43C2312yes—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43D411yescharacteristic
117C2×C4⋊C4×C43D411yescharacteristic
118C2×C22⋊C4×C43—21no—
119C2×C22⋊C4×C43—21no—
120C2×C42×C43—21no—
121C2×C42×C43C2312yes—
122C24×C43C2×D411yescharacteristic
123C23×D43—21no—
124C22×C4×C43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—22no—
127C22×C4×C43C22×C412yes—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×Q812yes—
134C22×C4×C43#4112yes—
135C22×C4×C43#4112yes—
136C22×C4×C43C2411yescharacteristic
137C2×C22⋊D43—24no—
138C2×C22⋊D43—22no—
139C2×C22⋊D43—24no—
140C2×C22⋊D43—22no—
141C22×Dic43—21no—
142C22×Dic43—22no—
143C22×Dic43—41no—
144C22×Dic43—41no—
145C22×Dic43—82no—
146C22×Dic43C2×D412yes—
147C22×Dic43C2×D412yes—
148#10621898—24no—
149#10621898—24no—
150#10621898—24no—
151#10621898—24no—
152#10621898—24no—
153#10621898—22no—
154#10621898—22no—
155#10621898C4⋊C418yescomplemented
156#10621898C22×C414yes—
157#10621898C2×Q814yes—
158C2×D172—21no—
159C2×D172—22no—
160C2×D172C4⋊C414yescomplemented
161C2×D172C2×D412yes—
162C2×C4×D43—24no—
163C2×C4×D43—22no—
164C2×C4×D43—24no—
165C2×C4×D43—22no—
166C2×C4×D43—24no—
167C2×C4×D43—24no—
168C2×Dic86—21no—
169C2×Dic86—22no—
170C2×Dic86C4⋊C414yescomplemented
171C2×Dic86C2×D412yes—
172M16×C43—22no—
173M16×C43—22no—
174M16×C43—42no—
175C2×C8×C43—22no—
176C2×C8×C43—42no—
177C4⋊C4×C43—24no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43C2×D412yes—
181C4⋊C4×C43C2×D412yes—
182C22⋊C4×C43—24no—
183C22⋊C4×C43—22no—
184C22⋊C4×C43—22no—
185C42×C43—22no—
186C42×C43—22no—
187C42×C43—22no—
188C42×C43C22×C414yes—
189C22⋊Dic43—22no—
190C22⋊Dic43—22no—
191(C2×D43)⋊C4—24no—
192(C2×D43)⋊C4—24no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—22no—
198C4⋊Dic43—22no—
199C4⋊Dic43—41no—
200C4⋊Dic43—41no—
201C4⋊Dic43—84no—
202Dic43⋊C4—24no—
203Dic43⋊C4—24no—
204C4×Dic43—24no—
205C4×Dic43—24no—
206C4×Dic43—22no—
207C4×Dic43—22no—
208#10621877—24no—
209#10621877—22no—
210#10621877C22×C414yes—
211C2×C43⋊C8—24no—
212C2×C43⋊C8—22no—
213C23×C43—41no—
214C23×C43—41no—
215C23×C43C4×D412yes—
216C23×C43#12011yescharacteristic
217C23×C43#12011yescharacteristic
218C23×C43C22⋊Q812yes—
219C23×C43C22×D411yescharacteristic
220C22×D43—22no—
221C22×D43—24no—
222C22×D43—81no—
223C22×D43—81no—
224C2×C4×C43—24no—
225C2×C4×C43—22no—
226C2×C4×C43—22no—
227C2×C4×C43—42no—
228C2×C4×C43—42no—
229C2×C4×C43—81no—
230C2×C4×C43—81no—
231C2×C4×C43C2×C4⋊C412yes—
232C2×C4×C43C2×C4⋊C412yes—
233C2×C4×C43C4×D412yes—
234C2×C4×C43C4×D412yes—
235C2×C4×C43C4×D412yes—
236C2×C4×C43#12011yescharacteristic
237C2×C4×C43#12011yescharacteristic
238C2×C4×C43C22⋊Q812yes—
239C2×C4×C43C22⋊Q812yes—
240C2×C4×C43C22⋊Q812yes—
241C2×C4×C43#12312yes—
242C2×C4×C43#12312yes—
243C2×C4×C43C23×C411yescharacteristic
244C2×C4×C43C22×D411yescharacteristic
245C2×C4×C43C22×D411yescharacteristic
246C2×C4×C43C22×Q811yescharacteristic
247C2×C4×C43C2×#4111yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C22⋊D43—24no—
250C22⋊D43—24no—
251C22⋊D43—28no—
252C2×Dic43—22no—
253C2×Dic43—24no—
254C2×Dic43—22no—
255C2×Dic43—22no—
256C2×Dic43—42no—
257C2×Dic43—42no—
258C2×Dic43—44no—
259C2×Dic43—42no—
260C2×Dic43—82no—
261C2×Dic43—82no—
262D172—22no—
263D172—24no—
264D172C2×C4⋊C414yes—
265C4×D43—24no—
266C4×D43—24no—
267C4×D43—28no—
268C4×D43—84no—
269C4×D43—84no—
270Dic86—22no—
271Dic86—24no—
272Dic86C2×C4⋊C414yes—
273C8×C43—42no—
274C43⋊C8—24no—
275#20865—431nomaximal, Sylow
276C22×C43—21no—
277C22×C43—21no—
278C22×C43—21no—
279C22×C43—21no—
280C22×C43—21no—
281C22×C43—21no—
282C22×C43—42no—
283C22×C43—42no—
284C22×C43#38512yes—
285C22×C43#38512yes—
286C22×C43C2×C4×D411yescharacteristic
287C22×C43C2×#12011yescharacteristic
288C22×C43C2×C22⋊Q811yescharacteristic
289C2×D43—24no—
290C2×D43—42no—
291C2×D43—42no—
292C2×D43—84no—
293C4×C43—44no—
294C4×C43—42no—
295C4×C43—42no—
296C4×C43#38512yes—
297C4×C43#38512yes—
298C4×C43C22×C4⋊C411yescharacteristic
299C4×C43C2×C4×D411yescharacteristic
300C4×C43C2×C22⋊Q811yescharacteristic
301C4×C43C2×#12311yescharacteristic
302Dic43—24no—
303Dic43—42no—
304Dic43—42no—
305Dic43—82no—
306#3428—431no—
307C2×#428—431no—
308C2×#421—431no—
309C2×#417—431no—
310#2317—431no—
311#1860—434no—
312#1859—434no—
313#1742—431no—
314#1733D4311yescharacteristic, complemented
315C2×C43—21no—
316C2×C43—22no—
317C2×C43—21no—
318C2×C43—22no—
319C2×C43C2×#38511yescharacteristic
320C2×C43#302111yescharacteristic
321C2×C43#302211yescharacteristic
322D43—44no—
323C22×#41—431no—
324C22×M16—431no—
325C22×M16C2×D4311yescharacteristic
326#520—432no—
327#520—432no—
328#520—861no—
329#520—861no—
330#519—861no—
331#518—862no—
332#518—861no—
333C4×#41—862no—
334C4×#41—862no—
335C2×#117—431no—
336C2×#117—862no—
337C2×#117C2×D4311yescharacteristic
338C2×#117C2×D4311yescharacteristic
339C22×C4⋊C4—431no—
340#428—434no—
341#428—862no—
342C2×#107—861no—
343C2×#106—861no—
344#421—438no—
345C2×#104—434no—
346#417—434no—
347#417—862no—
348C2×#103—861no—
349C2×#102—861no—
350#389—862no—
351#385—862no—
352#384—432no—
353#382—432no—
354#343—432no—
355#343—432no—
356#339C2×D4314yescomplemented
357C43#2086511yescharacteristic, complemented, Sylow
358C2×#41—434no—
359C2×#41—432no—
360C2×#41—434no—
361C2×#41—432no—
362C2×#41—861no—
363C2×#41—862no—
364C2×#41—861no—
365C2×#41—862no—
366C22×Q8—431no—
367C22×D4—431no—
368C22×D4—861no—
369C23×C4—861no—
370C23×C4—861no—
371C23×C4C22×D4311yescharacteristic
372C2×M16—21no—
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—434no—
376C2×M16—432no—
377C2×M16—861no—
378C2×M16—862no—
379C2×M16C22×D4312yes—
380C22×C8—21no—
381C22×C8—861no—
382C4×Q8—862no—
383C4×Q8—862no—
384C4×Q8—862no—
385C4×Q8—862no—
386C4×D4—862no—
387C4×D4—862no—
388C4×D4—864no—
389C4×D4—864no—
390C4×D4—862no—
391C4×D4—862no—
392C4×D4—864no—
393C4×D4—864no—
394#117—21no—
395#117—21no—
396#117—22no—
397#117—864no—
398#117—864no—
399#117—861no—
400#117—861no—
401#117—861no—
402#117—861no—
403#117—862no—
404#117—862no—
405#117C4×D4314yescomplemented
406#117D17212yes—
407#117C22×D4312yes—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—432no—
412C2×C4⋊C4—862no—
413C2×C4⋊C4—862no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—864no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4D17211yescharacteristic
421C2×C4⋊C4D17211yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—432no—
425C2×C22⋊C4—864no—
426C2×C42—21no—
427C2×C42—432no—
428C2×C42—864no—
429C2×C42C22×D4312yes—
430#107—864no—
431#106—864no—
432#104—4316no—
433#103—864no—
434#102—864no—
435#95—864no—
436C24—861no—
437C24D4×D4311yescharacteristic
438#41—438no—
439#41—434no—
440#41—434no—
441#41—864no—
442#41—864no—
443#41—864no—
444#41—864no—
445#41—864no—
446#41—862no—
447#41—862no—
448C2×Q8—434no—
449C2×Q8—432no—
450C2×Q8—861no—
451C2×Q8—862no—
452C2×D4—434no—
453C2×D4—432no—
454C2×D4—864no—
455C2×D4—862no—
456C2×D4—864no—
457C2×D4—862no—
458C2×D4—861no—
459C2×D4—862no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—432no—
464C22×C4—432no—
465C22×C4—864no—
466C22×C4—864no—
467C22×C4—861no—
468C22×C4—864no—
469C22×C4—862no—
470C22×C4—864no—
471C22×C4—862no—
472C22×C4—862no—
473C22×C4—3442no—
474C22×C4—1721no—
475C22×C4—1721no—
476C22×C4C2×C4×D4312yes—
477C22×C4C2×D17211yescharacteristic
478C22×C4C2×D17211yescharacteristic
479C22×C4#1062189812yes—
480C22×C4D4×D4311yescharacteristic
481C22×C4D4×D4311yescharacteristic
482C22×C4D4×D4311yescharacteristic
483C22×C4#1062190012yes—
484C22×C4Q8×D4312yes—
485C22×C4C23×D4311yescharacteristic
486M16—22no—
487M16—22no—
488M16—42no—
489M16—434no—
490M16—864no—
491M16—862no—
492C2×C8—22no—
493C2×C8—42no—
494C2×C8—864no—
495C2×C8—862no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—864no—
500C4⋊C4—864no—
501C4⋊C4—861no—
502C4⋊C4—861no—
503C4⋊C4—861no—
504C4⋊C4—861no—
505C4⋊C4—862no—
506C4⋊C4—862no—
507C4⋊C4—3444no—
508C4⋊C4—1721no—
509C4⋊C4—1721no—
510C4⋊C4C2×D17212yes—
511C4⋊C4C2×D17212yes—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—864no—
516C22⋊C4—864no—
517C22⋊C4—862no—
518C22⋊C4—862no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—864no—
523C42—864no—
524C42—862no—
525C42—862no—
526C42C2×C4×D4314yes—
527C23—41no—
528C23—41no—
529C23—864no—
530C23—862no—
531C23—3441no—
532C23—3441no—
533C23(C22⋊D43)⋊C412yes—
534C23#1306329511yescharacteristic
535C23#1306329511yescharacteristic
536C23#1306330312yes—
537C23C2×D4×D4311yescharacteristic
538Q8—434no—
539Q8—862no—
540Q8—864no—
541D4—434no—
542D4—864no—
543D4—864no—
544D4—868no—
545D4—862no—
546D4—864no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—862no—
555C2×C4—864no—
556C2×C4—864no—
557C2×C4—864no—
558C2×C4—868no—
559C2×C4—862no—
560C2×C4—862no—
561C2×C4—3444no—
562C2×C4—3444no—
563C2×C4—3442no—
564C2×C4—3442no—
565C2×C4—1724no—
566C2×C4—1722no—
567C2×C4—1722no—
568C2×C4—1722no—
569C2×C4C4⋊C4×D4312yes—
570C2×C4C4⋊C4×D4312yes—
571C2×C4(C22⋊D43)⋊C412yes—
572C2×C4C4×D17212yes—
573C2×C4C4×D17212yes—
574C2×C4#1306329511yescharacteristic
575C2×C4#1306329511yescharacteristic
576C2×C4#1306330312yes—
577C2×C4#1306330512yes—
578C2×C4#1306330512yes—
579C2×C4#1306330712yes—
580C2×C4#1306330712yes—
581C2×C4C22×C4×D4311yescharacteristic
582C2×C4C2×D4×D4311yescharacteristic
583C2×C4C22×D17211yescharacteristic
584C2×C4C2×Q8×D4311yescharacteristic
585C2×C4C2×#1062190011yescharacteristic
586C2×C4C2×#1062189811yescharacteristic
587C8—42no—
588C8—864no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—42no—
596C22—42no—
597C22—864no—
598C22—3444no—
599C22—1722no—
600C22—1722no—
601C22#1458277312yes—
602C22#1458277312yes—
603C22C2×(C22⋊D43)⋊C411yescharacteristic
604C22C2×#1306329511yescharacteristic
605C22C2×#1306330311yescharacteristic
606C4—44no—
607C4—42no—
608C4—42no—
609C4—864no—
610C4—3442no—
611C4—1722no—
612C4—1722no—
613C4#1458277312yes—
614C4#1458277312yes—
615C4C2×C4⋊C4×D4311yescharacteristic
616C4C2×C4×D17211yescharacteristic
617C4C2×#1306330511yescharacteristic
618C4C2×#1306330711yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—1724no—
624C2C2×#1458277311yescharacteristic
625C2#1914396811yescharacteristic
626C2#1914397511yescharacteristic
subgroupG/H
centerC22C2×#13063295
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1733C2
socleC22×C43C2×#120
prime radicalC22×#10621898C22
cosocleC22×C4×C43C24
hypercenter#1733D43
hypocenterC43#20865
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. C2×#14582942C2semidirect
5. #19137218C2semidirect
6. #19134829C2semidirect
7. #19134822C2semidirect
8. #19134333C2semidirect
9. C43×#1733C2semidirect
25. #14582942C22semidirect
28. #14582917C4semidirect
36. #14582769C22semidirect
37. #14582765C22semidirect
38. #14582667C22semidirect
39. #14582667C22semidirect
43. C43×#339C22semidirect
59. C2×#10621877C2×C4semidirect
99. #13063253C2×C4semidirect
112. #117×C43C2×C4semidirect
155. #10621898C4⋊C4semidirect
160. C2×D172C4⋊C4semidirect
170. C2×Dic86C4⋊C4semidirect
314. #1733D43semidirect
356. #339C2×D43semidirect
357. C43#20865semidirect
405. #117C4×D43semidirect

Maximal subgroups

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

subgroupindex
#191477962
C2×#145829422
C2×#145829172
C2×#145829012
#191372182
#191348292
#191348222
#191343332
C43×#17332
#2086543