Finite Group Databasebeta

← #25160064

Subgroups — #25160064

Order 11008.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144450C211yesmaximal, characteristic, complemented
2#19144422C211yesmaximal, characteristic, complemented
3#19143428C211yesmaximal, characteristic
4#19143190C211yesmaximal, characteristic, complemented
5#19136118C211yesmaximal, characteristic, complemented
6#19136107C211yesmaximal, characteristic, complemented
7#19135951C211yesmaximal, characteristic, complemented
8#19135946C211yesmaximal, characteristic, complemented
9#19135379C211yesmaximal, characteristic, complemented
10#19135193C211yesmaximal, characteristic, complemented
11#19134973C211yesmaximal, characteristic
12#19134870C211yesmaximal, characteristic, complemented
13#19134826C211yesmaximal, characteristic, complemented
14#19134807C211yesmaximal, characteristic, complemented
15C43×#1907C211yesmaximal, characteristic
16#14583530C2211yescharacteristic
17#14583382C2211yescharacteristic
18#14583382C2211yescharacteristic
19#14583374C2211yescharacteristic
20#14583204—21no—
21#14583201—21no—
22#14583201—21no—
23#14583197—21no—
24#14583192—21no—
25#14583187—21no—
26#14583177—21no—
27#14583177—21no—
28#14583011—21no—
29#14583000—21no—
30SD16×Dic43—21no—
31D8×Dic43—21no—
32C2×#13063260C2211yescharacteristic
33#14582941—21no—
34#14582941—21no—
35#14582941C2211yescharacteristic
36#14582941C2211yescharacteristic
37#14582941C2211yescharacteristic
38#14582941C2211yescharacteristic
39#14582939C2211yescharacteristic
40#14582937—21no—
41#14582937C2211yescharacteristic
42C2×#13063258C2211yescharacteristic
43C2×#13063257—21no—
44C2×#13063257C2211yescharacteristic
45#14582927C2211yescharacteristic
46#14582925C2211yescharacteristic
47C2×#13063252C2211yescharacteristic
48#14582904C2211yescharacteristic
49#14582895C2211yescharacteristic
50C2×#13063246C2211yescharacteristic
51#14582857C2211yescharacteristic
52#14582802C2211yescharacteristic
53#14582764C2211yescharacteristic
54#14582667C2211yescharacteristic
55#14582667C2211yescharacteristic
56#14582660C2211yescharacteristic
57#14582660C2211yescharacteristic
58#14582660C2211yescharacteristic
59#14582653C2211yescharacteristic
60C2×AGL(1,ℤ/8)×C43C2211yescharacteristic
61C2×#131×C43C2211yescharacteristic
62C43×#417C2211yescharacteristic
63C2×#102×C43C2211yescharacteristic
64C2×#100×C43C2211yescharacteristic
65C43×#343C2211yescharacteristic
66C43×#341C2211yescharacteristic
67C2×#10621877C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69#13063324—21no—
70#13063324—21no—
71#13063316—21no—
72#13063316—21no—
73#13063304—41no—
74Q8×Dic43—21no—
75Q8×Dic43—21no—
76D4×Dic43—21no—
77D4×Dic43—21no—
78D4×Dic43—21no—
79D4×Dic43—21no—
80D4×Dic43—21no—
81D4×Dic43—21no—
82D4×Dic43—41no—
83D4×Dic43—41no—
84#13063284—21no—
85#13063284—21no—
86#13063284C2311yescharacteristic
87#13063284C2311yescharacteristic
88#13063284C2311yescharacteristic
89C2×C4⋊Dic43—21no—
90C2×C4⋊Dic43—21no—
91C2×C4⋊Dic43C2311yescharacteristic
92C2×C4⋊Dic43C2311yescharacteristic
93C2×C22⋊Dic43—41no—
94C2×C22⋊Dic43—41no—
95C2×C4×Dic43C2311yescharacteristic
96#13063260—22no—
97#13063259—21no—
98#13063259—21no—
99#13063259—21no—
100#13063259—21no—
101#13063259D411yescharacteristic
102#13063259D411yescharacteristic
103#13063258—21no—
104#13063258—21no—
105#13063258—21no—
106#13063257—21no—
107#13063257—21no—
108#13063257—21no—
109#13063257—21no—
110#13063257—21no—
111#13063257—21no—
112#13063257D411yescharacteristic
113#13063257D411yescharacteristic
114#13063256—21no—
115Dic43⋊C8—21no—
116Dic43⋊C8—21no—
117Dic43⋊C8—21no—
118Dic43⋊C8—21no—
119#13063252—22no—
120#13063249D411yescharacteristic
121#13063249D411yescharacteristic
122#13063246—21no—
123#13063246—21no—
124#13063246—21no—
125#13063246—21no—
126#13063246D411yescharacteristic
127#13063246D411yescharacteristic
128#13063235—21no—
129C8×Dic43—21no—
130C43⋊#95—41no—
131C2×#41×C43C2311yescharacteristic
132C22×D4×C43C2311yescharacteristic
133AGL(1,ℤ/8)×C43—22no—
134AGL(1,ℤ/8)×C43Q814yescomplemented
135C2×SD16×C43—21no—
136C2×D8×C43—21no—
137#131×C43—22no—
138#131×C43—22no—
139C2×M16×C43—21no—
140C2×M16×C43C2311yescharacteristic
141C2×M16×C43C2311yescharacteristic
142C2×M16×C43C2311yescharacteristic
143C22×C8×C43—21no—
144C22×C8×C43C2311yescharacteristic
145C2×C4⋊C4×C43C2311yescharacteristic
146#103×C43—21no—
147#102×C43—21no—
148#102×C43—21no—
149#102×C43—21no—
150#100×C43—22no—
151C24×C43—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×C43—21no—
159C2×D4×C43—21no—
160C2×D4×C43—21no—
161C2×D4×C43—21no—
162C2×D4×C43—21no—
163C2×D4×C43C2×D411yescharacteristic
164C2×D4×C43C2×Q812yes—
165C22×C4×C43—21no—
166C22×C4×C43—21no—
167C22×C4×C43C2411yescharacteristic
168C22×Dic43—21no—
169C22×Dic43—21no—
170C22×Dic43—21no—
171C22×Dic43—41no—
172C22×Dic43—41no—
173C22×Dic43—41no—
174C22×Dic43—41no—
175C2×Dic86—41no—
176SD16×C43—24no—
177D8×C43—24no—
178M16×C43—24no—
179M16×C43—21no—
180M16×C43—21no—
181M16×C43—21no—
182M16×C43—21no—
183M16×C43C2×Q812yes—
184M16×C43#4112yes—
185C2×C8×C43—24no—
186C2×C8×C43—21no—
187C2×C8×C43—21no—
188C2×C8×C43—21no—
189C2×C8×C43—21no—
190C2×C8×C43—21no—
191C2×C8×C43C2×D411yescharacteristic
192C2×C8×C43C2×D411yescharacteristic
193C4⋊C4×C43—21no—
194C4⋊C4×C43—21no—
195C22⋊Dic43—21no—
196C22⋊Dic43—21no—
197C22⋊Dic43—21no—
198C22⋊Dic43—41no—
199C22⋊Dic43—41no—
200C22⋊Dic43—41no—
201C22⋊Dic43—41no—
202C22⋊Dic43—82no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—41no—
208C4⋊Dic43—41no—
209C4⋊Dic43—41no—
210C4⋊Dic43C2×D411yescharacteristic
211C4⋊Dic43C2×D411yescharacteristic
212C4⋊Dic43C2×D411yescharacteristic
213C4⋊Dic43C2×D411yescharacteristic
214C4⋊Dic43C2×D411yescharacteristic
215C4⋊Dic43C2×D411yescharacteristic
216Dic43⋊C4—41no—
217Dic43⋊C4—41no—
218C4×Dic43—21no—
219C4×Dic43—21no—
220C4×Dic43—21no—
221C4×Dic43—21no—
222C4×Dic43—21no—
223C4×Dic43—41no—
224C4×Dic43—41no—
225#10621877—21no—
226#10621877—21no—
227C2×C43⋊C8—21no—
228C2×C43⋊C8—21no—
229C2×C43⋊C8—21no—
230C2×C43⋊C8C2×D411yescharacteristic
231C2×C43⋊C8C2×D411yescharacteristic
232C23×C43—21no—
233C23×C43—22no—
234C23×C43—41no—
235C23×C43—41no—
236C23×C43—81no—
237C23×C43—81no—
238C23×C43H(5,𝔽2)11yescharacteristic
239Q8×C43—21no—
240Q8×C43C22⋊Q812yes—
241D4×C43—22no—
242D4×C43—22no—
243D4×C43—22no—
244D4×C43—21no—
245D4×C43—22no—
246D4×C43—41no—
247D4×C43—41no—
248D4×C43C22⋊Q812yes—
249C2×C4×C43—21no—
250C2×C4×C43—22no—
251C2×C4×C43—22no—
252C2×C4×C43—41no—
253C2×C4×C43—81no—
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—21no—
265C2×Dic43—21no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—81no—
272C2×Dic43—81no—
273C2×Dic43—161no—
274C2×Dic43—161no—
275C2×Dic43—161no—
276C2×Dic43—161no—
277Dic86—161no—
278C8×C43—22no—
279C8×C43—21no—
280C8×C43—22no—
281C8×C43—21no—
282C8×C43—41no—
283C43⋊C8—21no—
284C43⋊C8—21no—
285C43⋊C8—41no—
286#26402—431nomaximal, Sylow
287C22×C43—21no—
288C22×C43—21no—
289C22×C43—22no—
290C22×C43—41no—
291C22×C43—41no—
292C22×C43—42no—
293C22×C43—82no—
294C22×C43—81no—
295C22×C43—81no—
296C22×C43—161no—
297C22×C43#53911yescharacteristic
298C22×C43D4211yescharacteristic
299C22×C43#55411yescharacteristic
300C4×C43—22no—
301C4×C43—81no—
302C4×C43C2×C22⋊Q811yescharacteristic
303C4×C43#52511yescharacteristic
304C4×C43#54711yescharacteristic
305C4×C43D4×Q811yescharacteristic
306Dic43—41no—
307Dic43—41no—
308Dic43—41no—
309Dic43—41no—
310Dic43—41no—
311Dic43—41no—
312Dic43—81no—
313Dic43—81no—
314Dic43—81no—
315#3083—431no—
316#3081—431no—
317#2946—431no—
318#2912—431no—
319#2092—431no—
320#2090—431no—
321#2063—431no—
322#2061—431no—
323#1952—431no—
324#1922—431no—
325#1907D4311yescharacteristic
326#1883—431no—
327#1866—431no—
328#1860—431no—
329#1856—431no—
330C2×C43—21no—
331C2×C43—21no—
332C2×C43—42no—
333C2×C43—82no—
334C2×C43#244911yescharacteristic
335C2×C43#328811yescharacteristic
336C2×C43#329011yescharacteristic
337C2×AGL(1,ℤ/8)C2×D4311yescharacteristic
338C2×#131C2×D4311yescharacteristic
339#544—431no—
340#520—431no—
341#520—431no—
342#518—431no—
343#479—861no—
344#478—861no—
345#478—861no—
346#478—861no—
347#477—861no—
348#476—861no—
349#474—861no—
350#474—861no—
351#442—861no—
352#440—861no—
353C4×SD16—861no—
354C4×D8—861no—
355C2×#108—431no—
356#428—431no—
357#428—431no—
358#428—431no—
359#428—431no—
360#428—431no—
361#428—861no—
362#428—861no—
363#427—431no—
364#427—861no—
365C2×#107—431no—
366C2×#106—431no—
367C2×#106—861no—
368#424—431no—
369#424—431no—
370C2×#104—431no—
371#418—431no—
372#417C2×D4311yescharacteristic
373#416—431no—
374C2×#102—431no—
375C2×#102C2×D4311yescharacteristic
376C2×#100C2×D4311yescharacteristic
377#405—431no—
378#390—431no—
379#382—431no—
380#343—431no—
381#343—431no—
382#343C2×D4311yescharacteristic
383#341C2×D4311yescharacteristic
384#339—431no—
385#339—431no—
386#339—431no—
387#336—431no—
388C43#2640211yescharacteristic, complemented, Sylow
389C2×#41C22×D4311yescharacteristic
390C22×D4C22×D4311yescharacteristic
391AGL(1,ℤ/8)—22no—
392AGL(1,ℤ/8)Dic8614yescomplemented
393C2×SD16—21no—
394C2×D8—21no—
395#131—22no—
396#131—22no—
397C2×M16—21no—
398C2×M16—431no—
399C2×M16C22×D4311yescharacteristic
400C2×M16C22×D4311yescharacteristic
401C2×M16C22×D4311yescharacteristic
402C22×C8—21no—
403C22×C8—431no—
404C22×C8C22×D4311yescharacteristic
405C4⋊Q8—861no—
406C4⋊Q8—861no—
407#125—861no—
408#125—861no—
409C22⋊Q8—1721no—
410C4×Q8—861no—
411C4×Q8—861no—
412C4×D4—861no—
413C4×D4—861no—
414C4×D4—861no—
415C4×D4—861no—
416C4×D4—861no—
417C4×D4—861no—
418C4×D4—1721no—
419C4×D4—1721no—
420#117—431no—
421#117—431no—
422#117—431no—
423#117—861no—
424#117—861no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—431no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4C22×D4311yescharacteristic
430C2×C22⋊C4—1721no—
431C2×C22⋊C4—1721no—
432C2×C42—431no—
433#108—862no—
434#107—431no—
435#107—431no—
436#107—861no—
437#107—861no—
438#107—861no—
439#107—861no—
440#107—861no—
441#107—861no—
442#107—861no—
443#106—431no—
444#106—431no—
445#106—861no—
446#106—861no—
447#106—861no—
448#106—861no—
449#106—861no—
450#106—861no—
451#106—861no—
452C4⋊C8—861no—
453C4⋊C8—861no—
454C4⋊C8—861no—
455C4⋊C8—861no—
456#104—862no—
457#103—21no—
458#103—431no—
459#103—431no—
460#102—21no—
461#102—21no—
462#102—21no—
463#102—431no—
464#102—431no—
465#102—861no—
466#102—861no—
467#102—861no—
468#102—861no—
469#100—22no—
470#97—861no—
471C4×C8—861no—
472#95—1721no—
473C24—21no—
474#41—21no—
475#41—21no—
476#41C2×Dic8612yes—
477#41#1062190012yes—
478C2×Q8—1721no—
479C2×Q8D4×D4311yescharacteristic
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×Dic8612yes—
487C2×D4D4×D4311yescharacteristic
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—861no—
491C22×C4—861no—
492C22×C4—861no—
493C22×C4—1721no—
494C22×C4—1721no—
495C22×C4—1721no—
496C22×C4—1721no—
497C22×C4C23×D4311yescharacteristic
498SD16—24no—
499D8—24no—
500M16—24no—
501M16—21no—
502M16—21no—
503M16—21no—
504M16—21no—
505M16—861no—
506M16—861no—
507M16C2×Dic8612yes—
508M16#1062190012yes—
509C2×C8—24no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—431no—
516C2×C8—431no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C2×C8D4×D4311yescharacteristic
521C2×C8D4×D4311yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—431no—
525C4⋊C4—431no—
526C4⋊C4—431no—
527C4⋊C4—431no—
528C4⋊C4—431no—
529C4⋊C4—431no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—1721no—
535C4⋊C4—1721no—
536C4⋊C4—1721no—
537C4⋊C4—1721no—
538C4⋊C4—1721no—
539C22⋊C4—861no—
540C22⋊C4—861no—
541C22⋊C4—861no—
542C22⋊C4—3442no—
543C22⋊C4—1721no—
544C22⋊C4—1721no—
545C22⋊C4—1721no—
546C22⋊C4—1721no—
547C42—861no—
548C42—861no—
549C42—861no—
550C42—861no—
551C42—861no—
552C42—1721no—
553C42—1721no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—81no—
559C23—81no—
560C23#1306337111yescharacteristic
561Q8—21no—
562Q8—6881no—
563Q8#1306330212yes—
564D4—22no—
565D4—22no—
566D4—22no—
567D4—22no—
568D4—21no—
569D4—41no—
570D4—41no—
571D4#1306330212yes—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—81no—
577C2×C4—861no—
578C2×C4—861no—
579C2×C4—861no—
580C2×C4—861no—
581C2×C4—861no—
582C2×C4—861no—
583C2×C4—3441no—
584C2×C4—3441no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—6881no—
591C2×C4—6881no—
592C2×C4—6881no—
593C2×C4—6881no—
594C2×C4C2×D4×D4311yescharacteristic
595C2×C4C2×D4×D4311yescharacteristic
596C2×C4C22×Dic8611yescharacteristic
597C2×C4C2×#1062190011yescharacteristic
598C2×C4C2×#1062190011yescharacteristic
599C2×C4#1306337211yescharacteristic
600C8—22no—
601C8—21no—
602C8—22no—
603C8—21no—
604C8—41no—
605C8—861no—
606C8—861no—
607C8—1721no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—161no—
618C22#1458349311yescharacteristic
619C22#1458353311yescharacteristic
620C22#1458360611yescharacteristic
621C4—22no—
622C4—81no—
623C4—3441no—
624C4—3441no—
625C4—3441no—
626C4—1721no—
627C4—1721no—
628C4—1721no—
629C4—1721no—
630C4—1721no—
631C4—1721no—
632C4C2×#1306330211yescharacteristic
633C4#1458340511yescharacteristic
634C4#1458354911yescharacteristic
635C4D4×Dic8611yescharacteristic
636C2—21no—
637C2—21no—
638C2—42no—
639C2—82no—
640C2#1913841011yescharacteristic
641C2#1914635811yescharacteristic
642C2#1914637811yescharacteristic
subgroupG/H
centerC22#14583533
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1907C2
socleC22×C43D42
prime radicalC2×AGL(1,ℤ/8)×C43C22
hypercenter#1907D43
hypocenterC43#26402
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1907

Complements

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

NH ≅ G/Ntype
1. #19144450C2semidirect
2. #19144422C2semidirect
4. #19143190C2semidirect
5. #19136118C2semidirect
6. #19136107C2semidirect
7. #19135951C2semidirect
8. #19135946C2semidirect
9. #19135379C2semidirect
10. #19135193C2semidirect
12. #19134870C2semidirect
13. #19134826C2semidirect
14. #19134807C2semidirect
134. AGL(1,ℤ/8)×C43Q8semidirect
388. C43#26402semidirect
392. AGL(1,ℤ/8)Dic86semidirect

Maximal subgroups

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

subgroupindex
#191444502
#191444222
#191434282
#191431902
#191361182
#191361072
#191359512
#191359462
#191353792
#191351932
#191349732
#191348702
#191348262
#191348072
C43×#19072
#2640243