Finite Group Databasebeta

← #25160026

Subgroups — #25160026

Order 11008.

740 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
1#19144444C211yesmaximal, characteristic, complemented
2#19144427C211yesmaximal, characteristic, complemented
3#19143430C211yesmaximal, characteristic
4#19143195C211yesmaximal, characteristic, complemented
5#19136122C211yesmaximal, characteristic, complemented
6#19136100C211yesmaximal, characteristic, complemented
7#19135950C211yesmaximal, characteristic, complemented
8#19135939C211yesmaximal, characteristic, complemented
9#19135381C211yesmaximal, characteristic, complemented
10#19135195C211yesmaximal, characteristic, complemented
11#19135103C211yesmaximal, characteristic
12#19134969C211yesmaximal, characteristic
13#19134823C211yesmaximal, characteristic, complemented
14#19134806C211yesmaximal, characteristic, complemented
15C43×#1866C211yesmaximal, characteristic, complemented
16C2×#13063350C2211yescharacteristic
17#14583527C2211yescharacteristic
18#14583382C2211yescharacteristic
19#14583382C2211yescharacteristic
20#14583373C2211yescharacteristic
21#14583208—21no—
22#14583200—21no—
23#14583200—21no—
24#14583197—21no—
25#14583192—21no—
26#14583190—21no—
27#14583180—21no—
28#14583177—21no—
29#14583008—21no—
30#14583002—21no—
31#14582984—21no—
32#14582979—21no—
33#14582942C2211yescharacteristic
34#14582942C2211yescharacteristic
35#14582941—21no—
36#14582941—21no—
37#14582941C2211yescharacteristic
38#14582941C2211yescharacteristic
39#14582941C2211yescharacteristic
40#14582937—21no—
41#14582937C2211yescharacteristic
42C2×#13063259—21no—
43C2×#13063259C2211yescharacteristic
44C2×#13063259C2211yescharacteristic
45#14582925C2211yescharacteristic
46#14582905C2211yescharacteristic
47#14582900C2211yescharacteristic
48C2×#13063247C2211yescharacteristic
49C2×#13063247C2211yescharacteristic
50C2×#13063241C2211yescharacteristic
51#14582854C2211yescharacteristic
52#14582801C2211yescharacteristic
53#14582769C2211yescharacteristic
54#14582667C2211yescharacteristic
55#14582667C2211yescharacteristic
56#14582667C2211yescharacteristic
57#14582664C2211yescharacteristic
58#14582659C2211yescharacteristic
59#14582659C2211yescharacteristic
60C2×#131×C43C2211yescharacteristic
61C2×#108×C43C2211yescharacteristic
62C43×#424C2211yescharacteristic
63C2×#104×C43C2211yescharacteristic
64C43×#416C2211yescharacteristic
65C43×#339C2211yescharacteristic
66C43×#336C2211yescharacteristic
67C22×D172C2311yescharacteristic
68#13063350—22no—
69#13063350Q814yescomplemented
70C2×#10621884—21no—
71C2×#10621882—21no—
72C2×#10621877C2311yescharacteristic
73C2×#10621877C2311yescharacteristic
74#13063325—21no—
75#13063324—21no—
76#13063317—21no—
77#13063316—21no—
78#13063302—41no—
79Q8×Dic43—21no—
80Q8×Dic43—21no—
81C4×D172—21no—
82C4×D172—21no—
83D4×Dic43—21no—
84D4×Dic43—21no—
85D4×Dic43—41no—
86D4×Dic43—41no—
87#13063287—21no—
88#13063287—21no—
89#13063284—21no—
90#13063284—21no—
91#13063284C2311yescharacteristic
92#13063284C2311yescharacteristic
93C2×C4⋊Dic43—21no—
94C2×C4⋊Dic43—21no—
95C2×C4⋊Dic43—41no—
96C2×C4⋊Dic43C2311yescharacteristic
97C2×C4⋊Dic43C2311yescharacteristic
98C2×C4⋊Dic43C2311yescharacteristic
99C2×(C2×D43)⋊C4—41no—
100C22⋊C4×D43—41no—
101#13063259—21no—
102#13063259—21no—
103#13063259—21no—
104#13063259—21no—
105#13063259—21no—
106#13063259—21no—
107#13063259—21no—
108#13063259D411yescharacteristic
109#13063259D411yescharacteristic
110#13063258—21no—
111#13063258—21no—
112#13063257—21no—
113#13063257—21no—
114#13063257—21no—
115#13063257D411yescharacteristic
116#13063257D411yescharacteristic
117#13063256—21no—
118#13063256—21no—
119Dic43⋊C8—21no—
120Dic43⋊C8—21no—
121#13063249—21no—
122#13063247—21no—
123#13063247—21no—
124#13063247—21no—
125#13063247—21no—
126#13063247—21no—
127#13063247—21no—
128#13063246—21no—
129#13063241—22no—
130#13063235—21no—
131C8×Dic43—21no—
132C2×#41×C43C2311yescharacteristic
133#131×C43—22no—
134#131×C43—22no—
135C2×M16×C43—21no—
136C2×M16×C43C2311yescharacteristic
137C2×M16×C43C2311yescharacteristic
138C22×C8×C43—21no—
139C22×C8×C43C2311yescharacteristic
140C22×C8×C43C2311yescharacteristic
141#117×C43C2311yescharacteristic
142C2×C42×C43C2311yescharacteristic
143#108×C43—22no—
144C4⋊C8×C43—21no—
145C4⋊C8×C43—21no—
146#104×C43—22no—
147#103×C43D411yescharacteristic
148#103×C43D411yescharacteristic
149#102×C43D411yescharacteristic
150#102×C43D411yescharacteristic
151C23×D43—21no—
152#41×C43—21no—
153#41×C43—21no—
154#41×C43C2×Q812yes—
155#41×C43#4112yes—
156C2×Q8×C43C2×D411yescharacteristic
157C2×D4×C43—21no—
158C2×D4×C43C2×D411yescharacteristic
159C22×C4×C43—21no—
160C22×C4×C43—21no—
161C22×C4×C43C2411yescharacteristic
162C22×Dic43—21no—
163C22×Dic43—21no—
164C22×Dic43—21no—
165C22×Dic43—41no—
166C22×Dic43—41no—
167C2×D172—21no—
168C2×D172—21no—
169C2×D172—21no—
170C2×D172—21no—
171C2×D172—21no—
172C2×D172C2×Q812yes—
173C2×C4×D43—41no—
174C2×C4×D43—41no—
175C2×Dic86—41no—
176M16×C43—24no—
177M16×C43—21no—
178M16×C43—21no—
179M16×C43—21no—
180M16×C43#4112yes—
181C2×C8×C43—24no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43—21no—
186C2×C8×C43—21no—
187C2×C8×C43—21no—
188C2×C8×C43C2×D411yescharacteristic
189C2×C8×C43C2×D411yescharacteristic
190C2×C8×C43C2×D411yescharacteristic
191C2×C8×C43C2×D411yescharacteristic
192C4⋊C4×C43C2×D411yescharacteristic
193C4⋊C4×C43C2×D411yescharacteristic
194C22⋊C4×C43—21no—
195C42×C43—21no—
196C42×C43—21no—
197C42×C43—21no—
198C22⋊Dic43—21no—
199C22⋊Dic43—21no—
200C22⋊Dic43—41no—
201C22⋊Dic43—41no—
202#10621884—24no—
203#10621882—24no—
204(C2×D43)⋊C4—41no—
205(C2×D43)⋊C4—41no—
206(C2×D43)⋊C4—82no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213C4⋊Dic43—41no—
214C4⋊Dic43—41no—
215C4⋊Dic43—41no—
216C4⋊Dic43—82no—
217C4⋊Dic43C2×D411yescharacteristic
218C4⋊Dic43C2×D411yescharacteristic
219C4⋊Dic43C2×D411yescharacteristic
220C4⋊Dic43C2×D411yescharacteristic
221Dic43⋊C4—41no—
222Dic43⋊C4—41no—
223C4×Dic43—21no—
224C4×Dic43—21no—
225C4×Dic43—41no—
226C4×Dic43—41no—
227#10621877—21no—
228#10621877—21no—
229#10621877—21no—
230#10621877C2×Q812yes—
231C2×C43⋊C8—21no—
232C2×C43⋊C8—21no—
233C23×C43—21no—
234C23×C43H(5,𝔽2)11yescharacteristic
235C22×D43—22no—
236C22×D43—41no—
237C22×D43—41no—
238C22×D43—81no—
239C22×D43—81no—
240Q8×C43—21no—
241Q8×C43C22⋊Q812yes—
242D4×C43—22no—
243D4×C43—22no—
244D4×C43—21no—
245D4×C43C22⋊Q812yes—
246C2×C4×C43—21no—
247C2×C4×C43—22no—
248C2×C4×C43—22no—
249C2×C4×C43—22no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43C22×D411yescharacteristic
255C2×C4×C43C22×D411yescharacteristic
256C2×C4×C43C22×Q811yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×C4×C432−1+411yescharacteristic
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—81no—
268C2×Dic43—81no—
269C2×Dic43—81no—
270C2×Dic43—161no—
271C2×Dic43—161no—
272D172—22no—
273D172—22no—
274D172—41no—
275D172—41no—
276C4×D43—161no—
277C4×D43—161no—
278Dic86—161no—
279C8×C43—21no—
280C8×C43—22no—
281C8×C43—21no—
282C8×C43—21no—
283C8×C43—21no—
284C8×C43—41no—
285C43⋊C8—22no—
286C43⋊C8—41no—
287#26400—431nomaximal, Sylow
288C22×C43—21no—
289C22×C43—21no—
290C22×C43—22no—
291C22×C43—41no—
292C22×C43—41no—
293C22×C43#53911yescharacteristic
294C22×C43D4211yescharacteristic
295C22×C43#55411yescharacteristic
296C2×D43—42no—
297C2×D43—81no—
298C2×D43—81no—
299C2×D43—82no—
300C2×D43—161no—
301C4×C43—22no—
302C4×C43—42no—
303C4×C43—41no—
304C4×C43—41no—
305C4×C43C2×C22⋊Q811yescharacteristic
306C4×C43#52511yescharacteristic
307C4×C43#54711yescharacteristic
308C4×C43D4×Q811yescharacteristic
309Dic43—41no—
310Dic43—41no—
311Dic43—41no—
312Dic43—41no—
313Dic43—81no—
314Dic43—81no—
315Dic43—81no—
316#3083—431no—
317#3081—431no—
318#2946—431no—
319#2913—431no—
320#2093—431no—
321#2089—431no—
322#2063—431no—
323#2060—431no—
324#1952—431no—
325#1922—431no—
326#1907—431no—
327#1883—431no—
328#1866D4311yescharacteristic, complemented
329#1859—431no—
330#1856—431no—
331C2×C43—21no—
332C2×C43—21no—
333C2×C43—42no—
334C2×C43#244911yescharacteristic
335C2×C43#328911yescharacteristic
336C2×C43#329211yescharacteristic
337D43—82no—
338C2×AGL(1,ℤ/8)—431no—
339C2×#131C2×D4311yescharacteristic
340#544—431no—
341#520—431no—
342#520—431no—
343#518—431no—
344#479—861no—
345#478—861no—
346#478—861no—
347#478—861no—
348#477—861no—
349#476—861no—
350#474—861no—
351#474—861no—
352#442—861no—
353#440—861no—
354C4×SD16—861no—
355C4×D8—861no—
356C2×#108C2×D4311yescharacteristic
357#428—431no—
358#428—431no—
359#428—431no—
360#428—431no—
361#428—431no—
362#428—861no—
363#428—861no—
364#427—431no—
365#427—861no—
366C2×#107—431no—
367C2×#107—431no—
368C2×#107—861no—
369#424—431no—
370#424C2×D4311yescharacteristic
371C2×#104C2×D4311yescharacteristic
372#418—431no—
373#417—431no—
374#416C2×D4311yescharacteristic
375C2×#102—431no—
376C2×#102—431no—
377C2×#100—431no—
378#405—431no—
379#390—431no—
380#384—431no—
381#343—431no—
382#343—431no—
383#343—431no—
384#341—431no—
385#339—431no—
386#339—431no—
387#339C2×D4311yescharacteristic
388#336C2×D4311yescharacteristic
389C43#2640011yescharacteristic, complemented, Sylow
390C2×#41C22×D4311yescharacteristic
391C22×D4—431no—
392AGL(1,ℤ/8)—434no—
393AGL(1,ℤ/8)—862no—
394C2×SD16—861no—
395C2×D8—861no—
396#131—22no—
397#131—22no—
398C2×M16—21no—
399C2×M16—431no—
400C2×M16—431no—
401C2×M16C22×D4311yescharacteristic
402C2×M16C22×D4311yescharacteristic
403C22×C8—21no—
404C22×C8C22×D4311yescharacteristic
405C22×C8C22×D4311yescharacteristic
406C4⋊Q8—861no—
407C4⋊Q8—861no—
408#125—861no—
409#125—861no—
410C22⋊Q8—1721no—
411C4×Q8—861no—
412C4×Q8—861no—
413C4×D4—861no—
414C4×D4—861no—
415C4×D4—861no—
416C4×D4—861no—
417C4×D4—861no—
418C4×D4—861no—
419C4×D4—1721no—
420C4×D4—1721no—
421#117—431no—
422#117—431no—
423#117—861no—
424#117—861no—
425#117C22×D4311yescharacteristic
426C2×C4⋊C4—431no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—431no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4—861no—
431C2×C4⋊C4—1721no—
432C2×C22⋊C4—1721no—
433C2×C22⋊C4—1721no—
434C2×C42C22×D4311yescharacteristic
435#108—22no—
436#107—431no—
437#107—431no—
438#107—861no—
439#107—861no—
440#107—861no—
441#107—861no—
442#107—861no—
443#107—861no—
444#107—861no—
445#107—861no—
446#107—861no—
447#106—431no—
448#106—431no—
449#106—861no—
450#106—861no—
451#106—861no—
452#106—861no—
453#106—861no—
454C4⋊C8—21no—
455C4⋊C8—21no—
456C4⋊C8—861no—
457C4⋊C8—861no—
458#104—22no—
459#103—861no—
460#103D17211yescharacteristic
461#103D17211yescharacteristic
462#102—861no—
463#102—861no—
464#102—861no—
465#102—861no—
466#102—861no—
467#102—861no—
468#102—861no—
469#102D17211yescharacteristic
470#102D17211yescharacteristic
471#100—862no—
472#97—861no—
473C4×C8—861no—
474C24—861no—
475#41—21no—
476#41—21no—
477#41#1062190012yes—
478#41Q8×D4312yes—
479C2×Q8—1721no—
480C2×Q8C2×D17211yescharacteristic
481C2×D4—21no—
482C2×D4—432no—
483C2×D4—861no—
484C2×D4—861no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—861no—
488C2×D4C2×D17211yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—861no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—1721no—
495C22×C4—1721no—
496C22×C4—1721no—
497C22×C4—1721no—
498C22×C4C23×D4311yescharacteristic
499SD16—864no—
500D8—864no—
501M16—24no—
502M16—21no—
503M16—21no—
504M16—21no—
505M16—432no—
506M16—861no—
507M16—861no—
508M16—861no—
509M16#1062190012yes—
510C2×C8—24no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8C2×D17211yescharacteristic
520C2×C8C2×D17211yescharacteristic
521C2×C8D4×D4311yescharacteristic
522C2×C8D4×D4311yescharacteristic
523C4⋊C4—431no—
524C4⋊C4—431no—
525C4⋊C4—431no—
526C4⋊C4—431no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—861no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—3442no—
534C4⋊C4—1721no—
535C4⋊C4—1721no—
536C4⋊C4—1721no—
537C4⋊C4—1721no—
538C4⋊C4—1721no—
539C4⋊C4C2×D17211yescharacteristic
540C4⋊C4C2×D17211yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—861no—
543C22⋊C4—861no—
544C22⋊C4—3442no—
545C22⋊C4—1721no—
546C22⋊C4—1721no—
547C22⋊C4—1721no—
548C22⋊C4—1721no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—861no—
553C42—861no—
554C42—1721no—
555C42—1721no—
556C23—21no—
557C23—862no—
558C23—3441no—
559C23—3441no—
560C23—1721no—
561C23—1721no—
562C23#1306337111yescharacteristic
563Q8—21no—
564Q8—6881no—
565Q8#1306330512yes—
566D4—22no—
567D4—22no—
568D4—21no—
569D4—862no—
570D4—862no—
571D4—1721no—
572D4—1721no—
573D4#1306330512yes—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—861no—
585C2×C4—861no—
586C2×C4—3441no—
587C2×C4—3441no—
588C2×C4—3441no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—6881no—
593C2×C4—6881no—
594C2×C4—6881no—
595C2×C4—6881no—
596C2×C4C2×D4×D4311yescharacteristic
597C2×C4C22×D17211yescharacteristic
598C2×C4C2×Q8×D4311yescharacteristic
599C2×C4C2×#1062190011yescharacteristic
600C2×C4C2×#1062190011yescharacteristic
601C2×C4#1306337211yescharacteristic
602C8—21no—
603C8—21no—
604C8—21no—
605C8—22no—
606C8—21no—
607C8—41no—
608C8—862no—
609C8—1721no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—3442no—
616C22—3441no—
617C22—3441no—
618C22—1722no—
619C22—6881no—
620C22#1458349311yescharacteristic
621C22D4×D17211yescharacteristic
622C22#1458360711yescharacteristic
623C4—22no—
624C4—42no—
625C4—41no—
626C4—41no—
627C4—3441no—
628C4—3441no—
629C4—3441no—
630C4—1721no—
631C4—1721no—
632C4—1721no—
633C4—1721no—
634C4C2×#1306330511yescharacteristic
635C4#1458340511yescharacteristic
636C4#1458355311yescharacteristic
637C4#1458356211yescharacteristic
638C2—21no—
639C2—21no—
640C2—42no—
641C2—3442no—
642C2#1913840211yescharacteristic
643C2#1914637111yescharacteristic
644C2#1914639511yescharacteristic
subgroupG/H
centerC22D4×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1866C2
socleC22×C43D42
prime radicalC2×#13063350C22
hypercenter#1866D43
hypocenterC43#26400
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1866

Complements

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

NH ≅ G/Ntype
1. #19144444C2semidirect
2. #19144427C2semidirect
4. #19143195C2semidirect
5. #19136122C2semidirect
6. #19136100C2semidirect
7. #19135950C2semidirect
8. #19135939C2semidirect
9. #19135381C2semidirect
10. #19135195C2semidirect
13. #19134823C2semidirect
14. #19134806C2semidirect
15. C43×#1866C2semidirect
69. #13063350Q8semidirect
328. #1866D43semidirect
389. C43#26400semidirect

Maximal subgroups

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

subgroupindex
#191444442
#191444272
#191434302
#191431952
#191361222
#191361002
#191359502
#191359392
#191353812
#191351952
#191351032
#191349692
#191348232
#191348062
C43×#18662
#2640043