Finite Group Databasebeta

← #25078834

Subgroups — #25078834

Order 11008.

802 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145018C211yesmaximal, characteristic, complemented
2#19144597C211yesmaximal, characteristic, complemented
3#19144594C211yesmaximal, characteristic, complemented
4C8×D4×D43C211yesmaximal, characteristic, complemented
5#19133959C211yesmaximal, characteristic, complemented
6#19133958C211yesmaximal, characteristic, complemented
7#19133949C211yesmaximal, characteristic, complemented
8#19133948C211yesmaximal, characteristic, complemented
9#19133937C211yesmaximal, characteristic, complemented
10#19133936C211yesmaximal, characteristic, complemented
11#19133933C211yesmaximal, characteristic, complemented
12#19133340C211yesmaximal, characteristic, complemented
13#19133243C211yesmaximal, characteristic, complemented
14#19133241C211yesmaximal, characteristic, complemented
15C43×#1665C211yesmaximal, characteristic, complemented
16C2×#13063346—21no—
17C2×#13063346—21no—
18C22×C8×D43—21no—
19#14583450C2211yescharacteristic
20#14583447C2211yescharacteristic
21C4×D4×D43C2211yescharacteristic
22#14583280—21no—
23#14583280—21no—
24#14583278C2211yescharacteristic, complemented
25#14583275C2211yescharacteristic, complemented
26#14583272C2211yescharacteristic, complemented
27#14583214—21no—
28#14583211—21no—
29#14583180C2211yescharacteristic
30#14583179C2211yescharacteristic
31#14583176C2211yescharacteristic
32#14583095C2211yescharacteristic
33#14583094C2211yescharacteristic
34#14583091C2211yescharacteristic
35#14583034—21no—
36#14583031—21no—
37D8×Dic43—21no—
38#14582981—21no—
39#14582979—21no—
40#14582979—21no—
41C8×D172C2211yescharacteristic
42D4×C43⋊C8C2211yescharacteristic
43#14582964C2211yescharacteristic
44C8×C22⋊D43—21no—
45(C22⋊D43)⋊C8—21no—
46#14582951C2211yescharacteristic
47#14582951C2211yescharacteristic
48C4⋊C8×D43C2211yescharacteristic
49C22⋊C8×D43—21no—
50C4×C8×D43C2211yescharacteristic
51#14582651C2211yescharacteristic, complemented
52#14582651C2211yescharacteristic, complemented
53#14582650C2211yescharacteristic, complemented
54#14582646C2211yescharacteristic, complemented
55#14582645C2211yescharacteristic, complemented
56#14582644C2211yescharacteristic, complemented
57#14582643C2211yescharacteristic, complemented
58#14582642C2211yescharacteristic, complemented
59#14582642C2211yescharacteristic, complemented
60C43×#492C2211yescharacteristic, complemented
61C43×#474C2211yescharacteristic
62C43×#459C2211yescharacteristic
63C8×D4×C43C2211yescharacteristic
64C43×#334C2211yescharacteristic, complemented
65C43×#332C2211yescharacteristic, complemented
66C43×#331C2211yescharacteristic, complemented
67C2×#10621902—21no—
68C2×#10621900—21no—
69C2×D4×D43D411yescharacteristic
70C22×C4×D43—21no—
71#13063346—44no—
72#13063346—44no—
73C2×#10621885—21no—
74C2×#10621885—21no—
75C2×#10621873—21no—
76C2×#10621873—21no—
77C2×D344—21no—
78C2×#10621882—21no—
79C2×#10621882—21no—
80C2×#10621882—21no—
81C2×#10621882—21no—
82C22×C43⋊C8—21no—
83C2×C8×D43—21no—
84C2×C8×D43—21no—
85C2×C8×D43—21no—
86C2×C8×D43—44no—
87C2×C8×D43D411yescharacteristic
88C2×C8×D43D411yescharacteristic
89#13063325C2311yescharacteristic
90#13063322C2311yescharacteristic
91#13063314—21no—
92#13063312—21no—
93#13063300—21no—
94#13063300—21no—
95#13063296—21no—
96#13063296—21no—
97C4×D172C2311yescharacteristic
98D4×Dic43—21no—
99D4×Dic43—21no—
100D4×Dic43D411yescharacteristic
101#13063287—21no—
102#13063287—21no—
103#13063287D411yescharacteristic
104C4×C22⋊D43—21no—
105(C22⋊D43)⋊C4—21no—
106C4⋊C4×D43D411yescharacteristic
107C22⋊C4×D43—21no—
108C42×D43C2311yescharacteristic
109#13063259—21no—
110#13063258—21no—
111#13063258—21no—
112C4⋊(C43⋊C8)C2311yescharacteristic
113C4⋊(C43⋊C8)C2311yescharacteristic
114C4⋊(C43⋊C8)C2311yescharacteristic
115Dic43⋊C8—21no—
116#13063247—21no—
117#13063247—21no—
118#13063246—21no—
119#13063246—21no—
120#13063245—21no—
121#13063245—21no—
122C22⋊(C43⋊C8)—21no—
123(C2×D43)⋊C8—21no—
124(C2×D43)⋊C8—21no—
125(C2×D43)⋊C8—21no—
126(C2×D43)⋊C8—21no—
127C4×C43⋊C8C2311yescharacteristic
128C8×Dic43—21no—
129C8×Dic43—21no—
130C8×Dic43D411yescharacteristic
131C8×Dic43D411yescharacteristic
132C2×SD16×C43—21no—
133C2×SD16×C43—21no—
134C2×D8×C43—21no—
135C22×C8×C43—21no—
136C4⋊Q8×C43C2311yescharacteristic
137#127×C43C2311yescharacteristic
138C4×D4×C43C2311yescharacteristic
139#107×C43—21no—
140C4⋊C8×C43C2311yescharacteristic
141C4⋊C8×C43C2311yescharacteristic
142C4⋊C8×C43C2311yescharacteristic
143#102×C43—21no—
144#102×C43—21no—
145C22⋊C8×C43—21no—
146C4×C8×C43C2311yescharacteristic
147C23×D43—21no—
148C2×Q8×C43—21no—
149C2×D4×C43—21no—
150C2×D4×C43—21no—
151C2×D4×C43—21no—
152C2×D4×C43C2×D411yescharacteristic
153C22×C4×C43—21no—
154C2×C22⋊D43—21no—
155C2×C22⋊D43—41no—
156C22×Dic43—21no—
157C22×Dic43—41no—
158#10621902—42no—
159#10621902—42no—
160#10621900—42no—
161#10621900—42no—
162D4×D43—22no—
163D4×D43—22no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172—21no—
167C2×D172—41no—
168C2×D172C2×D411yescharacteristic
169C2×C4×D43—21no—
170C2×C4×D43—21no—
171C2×C4×D43—21no—
172C2×C4×D43—42no—
173C2×C4×D43—42no—
174C2×C4×D43—41no—
175C2×C4×D43C2×D411yescharacteristic
176C2×C4×D43C2×D411yescharacteristic
177C2×C4×D43#4111yescharacteristic
178C2×Dic86—21no—
179SD16×C43—42no—
180SD16×C43—42no—
181D8×C43—42no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43—42no—
186C2×C8×C43C2×D411yescharacteristic
187C2×C8×C43C2×D411yescharacteristic
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43C2×D411yescharacteristic
191C22⋊C4×C43—21no—
192C42×C43C2411yescharacteristic
193C22⋊Dic43—21no—
194C22⋊Dic43—41no—
195C22⋊Dic43—41no—
196#10621885—42no—
197#10621885—42no—
198#10621882—42no—
199#10621882—42no—
200#10621882—42no—
201#10621882—42no—
202(C2×D43)⋊C4—21no—
203(C2×D43)⋊C4—21no—
204(C2×D43)⋊C4—41no—
205(C2×D43)⋊C4—41no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43C2×D411yescharacteristic
209Dic43⋊C4—21no—
210C4×Dic43C2×D411yescharacteristic
211C4×Dic43C2×D411yescharacteristic
212C4×Dic43#4111yescharacteristic
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—42no—
217C2×C43⋊C8C2×D411yescharacteristic
218C2×C43⋊C8C2×D411yescharacteristic
219D344—42no—
220#10621873—42no—
221#10621873—42no—
222C8×D43—22no—
223C8×D43—42no—
224C8×D43—42no—
225C8×D43—42no—
226C8×D43D814yescomplemented
227C23×C43—21no—
228C23×C43—41no—
229C22×D43—24no—
230C22×D43—21no—
231C22×D43—42no—
232C22×D43—42no—
233C22×D43—41no—
234C22×D43#12111yescharacteristic
235Q8×C43—41no—
236Q8×C43—41no—
237D4×C43—21no—
238D4×C43—41no—
239D4×C43—41no—
240D4×C43—41no—
241D4×C43—41no—
242D4×C43—81no—
243D4×C43#13512yescomplemented
244C2×C4×C43—21no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43C22×D411yescharacteristic
249C2×C4×C43C22×D411yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C22⋊D43—24no—
252C22⋊D43—82no—
253C2×Dic43—21no—
254C2×Dic43—42no—
255C2×Dic43—41no—
256C2×Dic43—82no—
257C2×Dic43#12111yescharacteristic
258C2×Dic43#12111yescharacteristic
259C2×Dic43#12111yescharacteristic
260D172—21no—
261D172—41no—
262D172—41no—
263D172—41no—
264D172—41no—
265D172—81no—
266D172—82no—
267D172#13512yescomplemented
268C4×D43—21no—
269C4×D43—21no—
270C4×D43—41no—
271C4×D43—42no—
272C4×D43—41no—
273C4×D43—41no—
274C4×D43—42no—
275C4×D43—82no—
276C4×D43C2×D812yes—
277C4×D43AGL(1,ℤ/8)12yes—
278Dic86—41no—
279Dic86—41no—
280C8×C43—21no—
281C8×C43—41no—
282C8×C43—41no—
283C8×C43—41no—
284C8×C43C2×D812yes—
285C43⋊C8—21no—
286C43⋊C8—41no—
287C43⋊C8—41no—
288C43⋊C8—41no—
289C43⋊C8C2×D812yes—
290#19101—431nomaximal, Sylow
291C22×C43—22no—
292C22×C43—41no—
293C22×C43—41no—
294C22×C43—81no—
295C22×C43—81no—
296C22×C43—81no—
297C22×C43C2×#12111yescharacteristic
298C2×D43—21no—
299C2×D43—21no—
300C2×D43—22no—
301C2×D43—48no—
302C2×D43—41no—
303C2×D43—41no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43#45912yes—
308C4×C43—21no—
309C4×C43—41no—
310C4×C43—81no—
311C4×C43C22×D811yescharacteristic
312C4×C43C2×#13511yescharacteristic
313C4×C43C2×#13511yescharacteristic
314C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
315Dic43—21no—
316Dic43—21no—
317Dic43—41no—
318Dic43—81no—
319Dic43#45912yes—
320#3147—431no—
321#3100—431no—
322#3100—431no—
323C2×C8×D4—431no—
324#1669—431no—
325#1669—431no—
326#1667—431no—
327#1667—431no—
328#1665—431no—
329#1665—431no—
330#1665—431no—
331#1665D4311yescharacteristic, complemented
332#1564—431no—
333#1540—431no—
334#1540—431no—
335C2×C43—42no—
336C2×C43—161no—
337C2×C43C2×#45911yescharacteristic
338C2×C43#305011yescharacteristic
339C2×C43#306711yescharacteristic
340D43—22no—
341D43—42no—
342D43—161no—
343C2×#135—861no—
344C2×#135—861no—
345C23×C8—861no—
346#532—431no—
347#532—431no—
348C2×C4×D4—431no—
349#495—861no—
350#495—861no—
351#494—431no—
352#493—431no—
353#492—431no—
354#492C2×D4311yescharacteristic, complemented
355#480—861no—
356#480—861no—
357#474—431no—
358#474—431no—
359#474—431no—
360#474C2×D4311yescharacteristic
361#459—431no—
362#459—431no—
363#459—431no—
364#459C2×D4311yescharacteristic
365#447—861no—
366#447—861no—
367C4×D8—861no—
368C4×D8—861no—
369C4×D8—861no—
370C4×D8—861no—
371C8×D4—431no—
372C8×D4—431no—
373C8×D4—431no—
374C8×D4—861no—
375C8×D4—861no—
376C8×D4C2×D4311yescharacteristic
377#431—431no—
378#431—431no—
379C2×C4⋊C8—431no—
380C2×C22⋊C8—861no—
381C2×C4×C8—431no—
382#335—431no—
383#335—431no—
384#334—431no—
385#334C2×D4311yescharacteristic, complemented
386#332—431no—
387#332—431no—
388#332—431no—
389#332C2×D4311yescharacteristic, complemented
390#331—431no—
391#331—431no—
392#331—431no—
393#331C2×D4311yescharacteristic, complemented
394C43#1910111yescharacteristic, complemented, Sylow
395C2×#41—861no—
396C2×#41—861no—
397C22×D4—431no—
398C23×C4—861no—
399#135—1724no—
400#135—1724no—
401C2×Q16—861no—
402C2×Q16—861no—
403C2×SD16—21no—
404C2×SD16—21no—
405C2×SD16—861no—
406C2×SD16—861no—
407C2×D8—21no—
408C2×D8—861no—
409C2×D8—861no—
410C2×D8—861no—
411C2×D8—861no—
412C2×D8—861no—
413C22×C8—21no—
414C22×C8—431no—
415C22×C8—431no—
416C22×C8—861no—
417C22×C8—861no—
418C22×C8—861no—
419C22×C8—861no—
420C22×C8—1724no—
421C4⋊Q8—431no—
422C4⋊Q8C22×D4311yescharacteristic
423#127—431no—
424#127C22×D4311yescharacteristic
425#124—861no—
426#124—861no—
427#121—861no—
428#121—861no—
429#121—861no—
430#121—861no—
431C4×D4—431no—
432C4×D4—431no—
433C4×D4—431no—
434C4×D4—861no—
435C4×D4—861no—
436C4×D4—861no—
437C4×D4—861no—
438C4×D4—861no—
439C4×D4—861no—
440C4×D4C22×D4311yescharacteristic
441C2×C4⋊C4—431no—
442C2×C22⋊C4—861no—
443C2×C42—431no—
444#107—21no—
445#107—861no—
446#107—861no—
447#107—861no—
448C4⋊C8—431no—
449C4⋊C8—431no—
450C4⋊C8—431no—
451C4⋊C8—861no—
452C4⋊C8C22×D4311yescharacteristic
453C4⋊C8C22×D4311yescharacteristic
454C4⋊C8C22×D4311yescharacteristic
455#102—21no—
456#102—21no—
457#102—861no—
458#102—861no—
459#102—861no—
460#102—861no—
461#102—861no—
462#102—861no—
463C22⋊C8—21no—
464C22⋊C8—861no—
465C22⋊C8—861no—
466C22⋊C8—861no—
467C22⋊C8—861no—
468C22⋊C8—861no—
469C4×C8—431no—
470C4×C8—431no—
471C4×C8—431no—
472C4×C8—861no—
473C4×C8—861no—
474C4×C8C22×D4311yescharacteristic
475C24—861no—
476#41—1722no—
477#41—1722no—
478#41—1722no—
479#41—1722no—
480C2×Q8—21no—
481C2×Q8—861no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—431no—
486C2×D4—861no—
487C2×D4—862no—
488C2×D4—862no—
489C2×D4—861no—
490C2×D4—861no—
491C2×D4—861no—
492C2×D4—1721no—
493C2×D4—1721no—
494C2×D4D4×D4311yescharacteristic
495C22×C4—21no—
496C22×C4—431no—
497C22×C4—431no—
498C22×C4—431no—
499C22×C4—861no—
500C22×C4—861no—
501C22×C4—861no—
502C22×C4—861no—
503C22×C4—1721no—
504C22×C4—1722no—
505C22×C4—1722no—
506C22×C4—1721no—
507Q16—1722no—
508Q16—1722no—
509SD16—42no—
510SD16—42no—
511SD16—1722no—
512SD16—1722no—
513D8—42no—
514D8—1722no—
515D8—1722no—
516D8—1722no—
517D8—1722no—
518D8—1722no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—42no—
523C2×C8—431no—
524C2×C8—434no—
525C2×C8—431no—
526C2×C8—862no—
527C2×C8—861no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—1722no—
531C2×C8—1722no—
532C2×C8—1722no—
533C2×C8—1722no—
534C2×C8D4×D4311yescharacteristic
535C2×C8D4×D4311yescharacteristic
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—431no—
539C4⋊C4—861no—
540C4⋊C4—861no—
541C4⋊C4—861no—
542C4⋊C4D4×D4311yescharacteristic
543C22⋊C4—21no—
544C22⋊C4—861no—
545C22⋊C4—861no—
546C22⋊C4—861no—
547C22⋊C4—1721no—
548C22⋊C4—1721no—
549C22⋊C4—1721no—
550C22⋊C4—1721no—
551C42—431no—
552C42—431no—
553C42—431no—
554C42C23×D4311yescharacteristic
555C23—21no—
556C23—41no—
557C23—431no—
558C23—864no—
559C23—861no—
560C23—1722no—
561C23—1722no—
562C23—1721no—
563Q8—41no—
564Q8—41no—
565Q8—1721no—
566Q8—1721no—
567D4—21no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—81no—
573D4—432no—
574D4—864no—
575D4—861no—
576D4—3441no—
577D4—3442no—
578D4—3442no—
579D4—1721no—
580D4—1721no—
581D4—1721no—
582D4—1721no—
583D4#1306334612yescomplemented
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—431no—
589C2×C4—432no—
590C2×C4—432no—
591C2×C4—431no—
592C2×C4—431no—
593C2×C4—861no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—3442no—
597C2×C4—3442no—
598C2×C4—1721no—
599C2×C4—1722no—
600C2×C4—1722no—
601C2×C4—1722no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4C2×D4×D4311yescharacteristic
606C2×C4C2×D4×D4311yescharacteristic
607C2×C4#41×D4311yescharacteristic
608C8—21no—
609C8—41no—
610C8—41no—
611C8—41no—
612C8—432no—
613C8—861no—
614C8—1721no—
615C8—1721no—
616C8—1721no—
617C8D8×D4312yes—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—432no—
625C22—861no—
626C22—861no—
627C22—862no—
628C22—3441no—
629C22—3441no—
630C22—3441no—
631C22—1728no—
632C22—1721no—
633C22—1721no—
634C22#121×D4311yescharacteristic
635C4—21no—
636C4—41no—
637C4—81no—
638C4—432no—
639C4—861no—
640C4—861no—
641C4—3441no—
642C4—1721no—
643C4C2×D8×D4311yescharacteristic
644C4#135×D4311yescharacteristic
645C4C2×#1306334611yescharacteristic
646C4AGL(1,ℤ/8)×D4311yescharacteristic
647C2—42no—
648C2—161no—
649C2—862no—
650C2—1722no—
651C2—6881no—
652C2#459×D4311yescharacteristic
653C2#1914417111yescharacteristic
654C2#1914431011yescharacteristic
subgroupG/H
centerC22#121×D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#1665C2
socleC22×C43C2×#121
hypercenter#1665D43
hypocenterC43#19101
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #1665

Complements

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

NH ≅ G/Ntype
1. #19145018C2semidirect
2. #19144597C2semidirect
3. #19144594C2semidirect
4. C8×D4×D43C2semidirect
5. #19133959C2semidirect
6. #19133958C2semidirect
7. #19133949C2semidirect
8. #19133948C2semidirect
9. #19133937C2semidirect
10. #19133936C2semidirect
11. #19133933C2semidirect
12. #19133340C2semidirect
13. #19133243C2semidirect
14. #19133241C2semidirect
15. C43×#1665C2semidirect
24. #14583278C22semidirect
25. #14583275C22semidirect
26. #14583272C22semidirect
51. #14582651C22semidirect
52. #14582651C22semidirect
53. #14582650C22semidirect
54. #14582646C22semidirect
55. #14582645C22semidirect
56. #14582644C22semidirect
57. #14582643C22semidirect
58. #14582642C22semidirect
59. #14582642C22semidirect
60. C43×#492C22semidirect
64. C43×#334C22semidirect
65. C43×#332C22semidirect
66. C43×#331C22semidirect
226. C8×D43D8semidirect
243. D4×C43#135semidirect
267. D172#135semidirect
331. #1665D43semidirect
354. #492C2×D43semidirect
385. #334C2×D43semidirect
389. #332C2×D43semidirect
393. #331C2×D43semidirect
394. C43#19101semidirect
583. D4#13063346semidirect

Maximal subgroups

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

subgroupindex
#191450182
#191445972
#191445942
C8×D4×D432
#191339592
#191339582
#191339492
#191339482
#191339372
#191339362
#191339332
#191333402
#191332432
#191332412
C43×#16652
#1910143