Finite Group Databasebeta

← #25132894

Subgroups — #25132894

Order 11008.

1092 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19148414C211yesmaximal, characteristic, complemented
2#19144392C211yesmaximal, characteristic, complemented
3#19139548C211yesmaximal, characteristic, complemented
4#19135938C212yesmaximal, complemented
5#19135749C212yesmaximal, complemented
6#19135584C211yesmaximal, characteristic, complemented
7#19135539C211yesmaximal, characteristic, complemented
8#19134973C212yesmaximal
9#19134819C212yesmaximal, complemented
10#19134615C211yesmaximal, characteristic, complemented
11C4×AGL(1,ℤ/8)×C43C211yesmaximal, characteristic
12Q8×Dic86—21no—
13#14583628—21no—
14#14583606—22no—
15#14583606—21no—
16D4×Dic86—22no—
17D4×Dic86—21no—
18#14583527—22no—
19#14583527C2212yes—
20#14583456—21no—
21#14583456C2211yescharacteristic
22C2×#13063325—21no—
23C2×#13063325C2211yescharacteristic
24C2×#13063304—22no—
25#14583382C2212yes—
26#14583199—22no—
27#14583189—21no—
28#14583179—21no—
29#14582939C2212yes—
30#14582923C2211yescharacteristic
31C2×#13063252C2212yes—
32#14582904C2212yes—
33#14582835—22no—
34#14582802C2212yes—
35#14582768C2212yes—
36C4×C4⋊Dic43C2211yescharacteristic
37#14582667C2212yes—
38#14582667C2212yes—
39#14582660C2212yes—
40#14582660C2212yes—
41#14582658C2212yes—
42C2×AGL(1,ℤ/8)×C43C2211yescharacteristic
43C4×#41×C43C2211yescharacteristic
44C2×C4×D4×C43C2211yescharacteristic
45C43×#442—21no—
46C43×#440—21no—
47C4×SD16×C43—21no—
48C4×D8×C43—21no—
49C43×#428C2211yescharacteristic
50C43×#418C2211yescharacteristic
51C43×#417C2211yescharacteristic
52C4×M16×C43C2211yescharacteristic
53C22×Dic86—22no—
54C22×Dic86D412yes—
55C2×#10621877C2312yes—
56#13063325—22no—
57#13063325—22no—
58#13063325—21no—
59#13063325—21no—
60#13063325—21no—
61#13063325—21no—
62#13063325—22no—
63#13063325D411yescharacteristic
64#13063325D411yescharacteristic
65#13063324—24no—
66#13063324—22no—
67#13063317—22no—
68#13063317D412yes—
69#13063316—24no—
70#13063316—22no—
71#13063304—22no—
72#13063304—22no—
73#13063304—21no—
74#13063304—21no—
75#13063304—48no—
76#13063302—24no—
77#13063302—24no—
78#13063302—24no—
79Q8×Dic43—22no—
80C4×Dic86—22no—
81C4×Dic86—22no—
82C4×Dic86—21no—
83C4×Dic86—21no—
84D4×Dic43—22no—
85D4×Dic43—24no—
86#13063284—22no—
87#13063284C2312yes—
88C2×C4⋊Dic43—22no—
89C2×C4⋊Dic43—21no—
90C2×C4⋊Dic43—22no—
91C2×C4⋊Dic43D412yes—
92C2×C4⋊Dic43C2312yes—
93C2×Dic43⋊C4—24no—
94C2×C22⋊Dic43—24no—
95C2×C4×Dic43C2312yes—
96#13063258—22no—
97#13063256—22no—
98C4⋊(C43⋊C8)—21no—
99C4⋊(C43⋊C8)—21no—
100#13063252—24no—
101#13063246—22no—
102#13063246—22no—
103C43⋊#95—21no—
104C2×#41×C43C2311yescharacteristic
105C22×D4×C43C2311yescharacteristic
106C23×C4×C43—21no—
107AGL(1,ℤ/8)×C43—22no—
108AGL(1,ℤ/8)×C43Q814yescomplemented
109C2×SD16×C43—21no—
110C2×D8×C43—21no—
111C2×M16×C43C2311yescharacteristic
112C2×M16×C43C2311yescharacteristic
113C4×Q8×C43D411yescharacteristic
114C4×Q8×C43D411yescharacteristic
115C4×D4×C43—21no—
116C4×D4×C43—21no—
117C4×D4×C43—21no—
118C4×D4×C43—21no—
119C4×D4×C43—21no—
120C4×D4×C43—21no—
121C4×D4×C43D411yescharacteristic
122C4×D4×C43D411yescharacteristic
123#117×C43—21no—
124#117×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C22⋊C4×C43—21no—
127C2×C42×C43—21no—
128C2×C42×C43C2311yescharacteristic
129#107×C43—21no—
130#106×C43—21no—
131#103×C43—21no—
132#102×C43—21no—
133#102×C43—21no—
134#102×C43—21no—
135#97×C43—21no—
136C4×C8×C43—21no—
137C24×C43—21no—
138#41×C43—21no—
139#41×C43—21no—
140#41×C43C2×Q812yes—
141#41×C43#4112yes—
142C2×Q8×C43C2×D411yescharacteristic
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—21no—
148C2×D4×C43—21no—
149C2×D4×C43C2×D411yescharacteristic
150C2×D4×C43C2×Q812yescomplemented
151C22×C4×C43—21no—
152C22×C4×C43—21no—
153C22×C4×C43—21no—
154C22×C4×C43—21no—
155C22×C4×C43—44no—
156C22×C4×C43—41no—
157C22×C4×C43—41no—
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2411yescharacteristic
161C22×Dic43—24no—
162C22×Dic43—22no—
163C22×Dic43C2×D414yes—
164C2×Dic86—21no—
165C2×Dic86—22no—
166C2×Dic86—21no—
167C2×Dic86—24no—
168C2×Dic86—22no—
169C2×Dic86—44no—
170C2×Dic86—84no—
171SD16×C43—24no—
172D8×C43—24no—
173M16×C43—21no—
174M16×C43—21no—
175M16×C43—21no—
176M16×C43C2×Q812yescomplemented
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43C2×D411yescharacteristic
184C4⋊C4×C43C2×D411yescharacteristic
185C22⋊C4×C43—21no—
186C22⋊C4×C43—21no—
187C22⋊C4×C43—21no—
188C22⋊C4×C43—41no—
189C22⋊C4×C43—41no—
190C42×C43—21no—
191C42×C43—21no—
192C42×C43—21no—
193C42×C43C2×D411yescharacteristic
194C42×C43C2×D411yescharacteristic
195C42×C43C2×D411yescharacteristic
196C42×C43C2×D411yescharacteristic
197C22⋊Dic43—24no—
198C22⋊Dic43—22no—
199C22⋊Dic43—48no—
200C4⋊Dic43—24no—
201C4⋊Dic43—24no—
202C4⋊Dic43—24no—
203C4⋊Dic43—21no—
204C4⋊Dic43—22no—
205C4⋊Dic43—22no—
206C4⋊Dic43—21no—
207C4⋊Dic43—22no—
208C4⋊Dic43—22no—
209C4⋊Dic43C2×D412yes—
210C4⋊Dic43C2×D412yes—
211Dic43⋊C4—24no—
212Dic43⋊C4—24no—
213Dic43⋊C4—22no—
214Dic43⋊C4—22no—
215Dic43⋊C4—48no—
216C4×Dic43—24no—
217C4×Dic43—22no—
218C4×Dic43—22no—
219C4×Dic43—22no—
220#10621877—22no—
221#10621877—22no—
222C2×C43⋊C8—22no—
223C23×C43—21no—
224C23×C43—22no—
225C23×C43—42no—
226C23×C43—42no—
227C23×C43—41no—
228C23×C43—41no—
229C23×C43C22×D411yescharacteristic
230Q8×C43—21no—
231Q8×C43C22⋊Q812yescomplemented
232D4×C43—22no—
233D4×C43—22no—
234D4×C43—22no—
235D4×C43—21no—
236D4×C43—22no—
237D4×C43—41no—
238D4×C43—41no—
239D4×C43C22⋊Q812yescomplemented
240C2×C4×C43—21no—
241C2×C4×C43—22no—
242C2×C4×C43—22no—
243C2×C4×C43—22no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—41no—
247C2×C4×C43—42no—
248C2×C4×C43—42no—
249C2×C4×C43—42no—
250C2×C4×C43—41no—
251C2×C4×C43—82no—
252C2×C4×C43—82no—
253C2×C4×C43#12011yescharacteristic
254C2×C4×C43#12011yescharacteristic
255C2×C4×C43#12011yescharacteristic
256C2×C4×C43#12011yescharacteristic
257C2×C4×C43C22×D411yescharacteristic
258C2×C4×C43C22×D411yescharacteristic
259C2×C4×C43C22×Q811yescharacteristic
260C2×C4×C43C2×#4111yescharacteristic
261C2×C4×C432−1+411yescharacteristic
262C2×C4×C432−1+411yescharacteristic
263C2×Dic43—24no—
264C2×Dic43—22no—
265C2×Dic43—24no—
266C2×Dic43—22no—
267C2×Dic43—42no—
268C2×Dic43—42no—
269C2×Dic43—44no—
270C2×Dic43—84no—
271Dic86—42no—
272Dic86—44no—
273Dic86—42no—
274Dic86—82no—
275Dic86—84no—
276C8×C43—22no—
277C8×C43—41no—
278C43⋊C8—42no—
279#24320—431nomaximal, Sylow
280C22×C43—21no—
281C22×C43—21no—
282C22×C43—22no—
283C22×C43—44no—
284C22×C43—41no—
285C22×C43—41no—
286C22×C43—42no—
287C22×C43—82no—
288C22×C43—81no—
289C22×C43—81no—
290C22×C43C2×#12011yescharacteristic
291C22×C43#54811yescharacteristic
292C22×C43D4×Q811yescharacteristic
293C4×C43—22no—
294C4×C43—21no—
295C4×C43—21no—
296C4×C43—21no—
297C4×C43—41no—
298C4×C43—41no—
299C4×C43—81no—
300C4×C43#45312yes—
301C4×C43C2×C22⋊Q811yescharacteristic
302C4×C43#53611yescharacteristic
303C4×C43#54811yescharacteristic
304C4×C43D4×Q811yescharacteristic
305Dic43—44no—
306Dic43—44no—
307Dic43—42no—
308Dic43—42no—
309#3503—431no—
310#3077—431no—
311C4×AGL(1,ℤ/8)D4311yescharacteristic
312#2542—431no—
313#2060—432no—
314#2027—432no—
315#1997—431no—
316#1986—431no—
317#1883—432no—
318#1859—432no—
319#1806—431no—
320C2×C43—21no—
321C2×C43—21no—
322C2×C43—42no—
323C2×C43—82no—
324C2×C43#240911yescharacteristic
325C2×C43C2×#45311yescharacteristic
326C2×C43#302011yescharacteristic
327C2×AGL(1,ℤ/8)C2×D4311yescharacteristic
328Q82—861no—
329#557—861no—
330#554—862no—
331#554—861no—
332D4×Q8—862no—
333D4×Q8—861no—
334#544—432no—
335#544—862no—
336#533—431no—
337#533—861no—
338C2×C4⋊Q8—431no—
339C2×C4⋊Q8—861no—
340C2×C22⋊Q8—862no—
341#520—432no—
342C4×#41C2×D4311yescharacteristic
343C2×C4×D4C2×D4311yescharacteristic
344#478—862no—
345#476—861no—
346#474—861no—
347#442—21no—
348#440—21no—
349C4×SD16—21no—
350C4×D8—21no—
351#428—432no—
352#428C2×D4311yescharacteristic
353#423—431no—
354C2×#104—432no—
355#418—432no—
356#418C2×D4311yescharacteristic
357#417C2×D4311yescharacteristic
358C4×M16C2×D4311yescharacteristic
359#399—862no—
360#390—432no—
361#384—432no—
362C4×C4⋊C4—431no—
363#343—432no—
364#343—432no—
365#339—432no—
366#339—432no—
367#339—432no—
368C43#2432011yescharacteristic, complemented, Sylow
369C2×#41C22×D4311yescharacteristic
370C22×Q8—432no—
371C22×Q8—862no—
372C22×D4C22×D4311yescharacteristic
373C23×C4—21no—
374AGL(1,ℤ/8)—22no—
375AGL(1,ℤ/8)Dic8614yescomplemented
376C2×SD16—21no—
377C2×D8—21no—
378C2×M16—432no—
379C2×M16C22×D4311yescharacteristic
380C2×M16C22×D4311yescharacteristic
381C4⋊Q8—431no—
382C4⋊Q8—431no—
383C4⋊Q8—862no—
384C4⋊Q8—862no—
385C4⋊Q8—862no—
386C4⋊Q8—864no—
387C4⋊Q8—862no—
388C4⋊Q8—861no—
389C4⋊Q8—861no—
390C4⋊Q8—861no—
391C4⋊Q8—861no—
392#125—432no—
393#125—862no—
394#125—864no—
395#125—862no—
396C22⋊Q8—864no—
397C22⋊Q8—864no—
398C22⋊Q8—864no—
399C22⋊Q8—862no—
400C22⋊Q8—862no—
401C22⋊Q8—861no—
402C22⋊Q8—861no—
403C22⋊Q8—1728no—
404C4×Q8—862no—
405C4×Q8—862no—
406C4×Q8—862no—
407C4×Q8—861no—
408C4×Q8—861no—
409C4×Q8C22⋊D4311yescharacteristic
410C4×Q8C22⋊D4311yescharacteristic
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—864no—
418C4×D4—862no—
419C4×D4C22⋊D4311yescharacteristic
420C4×D4C22⋊D4311yescharacteristic
421#117—21no—
422#117—432no—
423#117—862no—
424#117C22×D4311yescharacteristic
425C2×C4⋊C4—432no—
426C2×C4⋊C4—432no—
427C2×C4⋊C4—862no—
428C2×C4⋊C4—862no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4—864no—
431C2×C4⋊C4C22×D4311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—864no—
434C2×C42—21no—
435C2×C42—432no—
436C2×C42C22×D4311yescharacteristic
437#107—21no—
438#107—862no—
439#106—21no—
440#106—862no—
441C4⋊C8—861no—
442C4⋊C8—861no—
443#104—864no—
444#103—21no—
445#102—21no—
446#102—21no—
447#102—21no—
448#102—862no—
449#102—862no—
450#97—21no—
451C4×C8—21no—
452#95—861no—
453C24—21no—
454#41—21no—
455#41—21no—
456#41C2×Dic8612yes—
457#41#1062189812yes—
458C2×Q8—864no—
459C2×Q8—862no—
460C2×Q8—861no—
461C2×Q8—861no—
462C2×Q8—862no—
463C2×Q8—3444no—
464C2×Q8—1724no—
465C2×Q8C2×C22⋊D4311yescharacteristic
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×Dic8612yescomplemented
473C2×D4C2×C22⋊D4311yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—44no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—434no—
482C22×C4—864no—
483C22×C4—862no—
484C22×C4D4×D4311yescharacteristic
485C22×C4D4×D4311yescharacteristic
486C22×C4C23×D4311yescharacteristic
487SD16—24no—
488D8—24no—
489M16—21no—
490M16—21no—
491M16—21no—
492M16—862no—
493M16—862no—
494M16C2×Dic8612yescomplemented
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—862no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—432no—
503C4⋊C4—432no—
504C4⋊C4—864no—
505C4⋊C4—864no—
506C4⋊C4—864no—
507C4⋊C4—864no—
508C4⋊C4—864no—
509C4⋊C4—862no—
510C4⋊C4—862no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—862no—
514C4⋊C4—862no—
515C4⋊C4—862no—
516C4⋊C4—862no—
517C4⋊C4—1728no—
518C4⋊C4C2×C22⋊D4311yescharacteristic
519C4⋊C4C2×C22⋊D4311yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—864no—
526C22⋊C4—862no—
527C22⋊C4—1728no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—864no—
532C42—862no—
533C42—862no—
534C42—862no—
535C42D4×D4311yescharacteristic
536C42D4×D4311yescharacteristic
537C42C2×C22⋊D4311yescharacteristic
538C42C2×C22⋊D4311yescharacteristic
539C23—21no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23C2×D4×D4311yescharacteristic
546Q8—21no—
547Q8—3442no—
548Q8—3444no—
549Q8—1722no—
550Q8—1722no—
551Q8—1724no—
552Q8#1306330412yescomplemented
553D4—22no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—21no—
558D4—41no—
559D4—41no—
560D4#1306330412yescomplemented
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—864no—
575C2×C4—862no—
576C2×C4—862no—
577C2×C4—864no—
578C2×C4—3444no—
579C2×C4—1722no—
580C2×C4—1722no—
581C2×C4—1724no—
582C2×C4#1306329311yescharacteristic
583C2×C4#1306329311yescharacteristic
584C2×C4#1306329311yescharacteristic
585C2×C4#1306329311yescharacteristic
586C2×C4C2×D4×D4311yescharacteristic
587C2×C4C22×C22⋊D4311yescharacteristic
588C2×C4C22×Dic8611yescharacteristic
589C2×C4C2×#1062189811yescharacteristic
590C2×C4#1306337211yescharacteristic
591C2×C4#1306337211yescharacteristic
592C8—22no—
593C8—41no—
594C8—1722no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—44no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×#1306329311yescharacteristic
606C22#1458355811yescharacteristic
607C22D4×Dic8611yescharacteristic
608C4—21no—
609C4—21no—
610C4—22no—
611C4—21no—
612C4—41no—
613C4—41no—
614C4—81no—
615C4—1724no—
616C4—1722no—
617C4—1722no—
618C4—1724no—
619C4#1458306512yes—
620C4C2×#1306330411yescharacteristic
621C4#1458347011yescharacteristic
622C4#1458355811yescharacteristic
623C4D4×Dic8611yescharacteristic
624C2—21no—
625C2—21no—
626C2—42no—
627C2—82no—
628C2#1913796011yescharacteristic
629C2C2×#1458306511yescharacteristic
630C2#1914395711yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC4×AGL(1,ℤ/8)×C43C2
socleC22×C43C2×#120
prime radicalC2×AGL(1,ℤ/8)×C43C22
hypercenterC4×AGL(1,ℤ/8)D43
hypocenterC43#24320
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #19148414C2semidirect
2. #19144392C2semidirect
3. #19139548C2semidirect
4. #19135938C2semidirect
5. #19135749C2semidirect
6. #19135584C2semidirect
7. #19135539C2semidirect
9. #19134819C2semidirect
10. #19134615C2semidirect
108. AGL(1,ℤ/8)×C43Q8semidirect
150. C2×D4×C43C2×Q8semidirect
176. M16×C43C2×Q8semidirect
231. Q8×C43C22⋊Q8semidirect
239. D4×C43C22⋊Q8semidirect
368. C43#24320semidirect
375. AGL(1,ℤ/8)Dic86semidirect
472. C2×D4C2×Dic86semidirect
494. M16C2×Dic86semidirect
552. Q8#13063304semidirect
560. D4#13063304semidirect

Maximal subgroups

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

subgroupindex
#191484142
#191443922
#191395482
#191359382
#191357492
#191355842
#191355392
#191349732
#191348192
#191346152
C4×AGL(1,ℤ/8)×C432
#2432043