Finite Group Databasebeta

← #25129332

Subgroups — #25129332

Order 11008.

1340 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583477C211yesmaximal, characteristic
2C2×#14583217C211yesmaximal, characteristic, complemented
3C2×#14583229C211yesmaximal, characteristic, complemented
4#19135791C212yesmaximal, complemented
5#19135784C212yesmaximal, complemented
6#19134917C211yesmaximal, characteristic, complemented
7#19134907C211yesmaximal, characteristic, complemented
8#19134787C212yesmaximal, complemented
9#19134533C212yesmaximal, complemented
10#19134507C211yesmaximal, characteristic, complemented
11C43×#2311C211yesmaximal, characteristic, complemented
12C22×#10621898C2211yescharacteristic
13#14583477—24no—
14#14583477—24no—
15C2×#13063304C2211yescharacteristic
16C2×#13063304C2211yescharacteristic
17C2×#13063301—21no—
18C2×#13063299C2211yescharacteristic
19C2×#13063299C2211yescharacteristic
20C2×#13063297—21no—
21C2×#13063297—21no—
22C2×#13063296—21no—
23C22×C4⋊Dic43C2211yescharacteristic
24#14583229—24no—
25#14583217—24no—
26C2×#13063256C2212yes—
27C2×#13063256C2212yes—
28C2×#13063248C2212yes—
29C2×#13063248C2212yes—
30C2×#13063245C2212yes—
31C2×#13063245C2212yes—
32C2×C22⋊(C43⋊C8)C2211yescharacteristic
33C2×C22⋊(C43⋊C8)C2211yescharacteristic
34C2×C22⋊(C43⋊C8)C2212yes—
35C2×C22⋊(C43⋊C8)C2212yes—
36#14582661C2212yes—
37#14582661C2212yes—
38C22×#41×C43C2211yescharacteristic
39C2×#117×C43C2211yescharacteristic
40C22×C4⋊C4×C43C2211yescharacteristic
41C43×#391—21no—
42C43×#390—21no—
43C43×#386—21no—
44C43×#386—21no—
45C43×#380C2212yes—
46C43×#380C2212yes—
47C2×#10621898—22no—
48C2×#10621898—24no—
49C2×#10621898—22no—
50C2×#10621898—42no—
51C22×Dic86C2311yescharacteristic
52C22×D172C2311yescharacteristic
53C22×C22⋊D43—21no—
54C22×C22⋊D43—21no—
55C22×C22⋊D43—21no—
56C23×Dic43—21no—
57C22×C4×D43—21no—
58C22×C43⋊C8C2312yes—
59C22×C43⋊C8C2312yes—
60#13063304—22no—
61#13063304—22no—
62#13063304—22no—
63#13063304D414yes—
64#13063301—28no—
65#13063299—22no—
66#13063299—22no—
67#13063299—22no—
68#13063299D414yes—
69#13063297—28no—
70#13063297—28no—
71#13063296—28no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43—22no—
74C2×C4⋊Dic43—22no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—21no—
79C2×C22⋊Dic43—21no—
80C2×C22⋊Dic43—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83#13063256—24no—
84#13063256—24no—
85#13063248—24no—
86#13063248—42no—
87#13063245—24no—
88#13063245—42no—
89C22⋊(C43⋊C8)—22no—
90C22⋊(C43⋊C8)—22no—
91C22⋊(C43⋊C8)—24no—
92C22⋊(C43⋊C8)—42no—
93C2×#41×C43—22no—
94C2×#41×C43—22no—
95C2×#41×C43—24no—
96C2×#41×C43—42no—
97C22×Q8×C43D411yescharacteristic
98C22×D4×C43D411yescharacteristic
99C22×D4×C43D411yescharacteristic
100C22×D4×C43D411yescharacteristic
101C23×C4×C43—21no—
102C23×C4×C43C2311yescharacteristic
103C23×C4×C43C2311yescharacteristic
104C23×C4×C43C2311yescharacteristic
105#117×C43—42no—
106C2×C4⋊C4×C43—22no—
107C2×C4⋊C4×C43—22no—
108C2×C4⋊C4×C43—21no—
109C2×C4⋊C4×C43C2312yes—
110C2×C4⋊C4×C43C2312yes—
111C2×C22⋊C4×C43—21no—
112C2×C22⋊C4×C43—22no—
113C2×C22⋊C4×C43—22no—
114C2×C42×C43—21no—
115#95×C43—22no—
116#95×C43—22no—
117C24×C43C2×D411yescharacteristic
118C24×C43C2×D411yescharacteristic
119C24×C43C2×D411yescharacteristic
120C23×D43—21no—
121#41×C43—48no—
122#41×C43—48no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43—41no—
126C2×D4×C43—22no—
127C2×D4×C43—22no—
128C2×D4×C43—22no—
129C2×D4×C43—22no—
130C2×D4×C43—22no—
131C2×D4×C43—22no—
132C2×D4×C43—41no—
133C2×D4×C43—41no—
134C2×D4×C43—41no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—22no—
138C22×C4×C43—22no—
139C22×C4×C43—22no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—22no—
143C22×C4×C43—22no—
144C22×C4×C43—22no—
145C22×C4×C43—22no—
146C22×C4×C43—42no—
147C22×C4×C43—82no—
148C22×C4×C43C2×D412yes—
149C22×C4×C43C2×D412yes—
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43#4114yes—
154C22×C4×C43C2411yescharacteristic
155C2×C22⋊D43—24no—
156C2×C22⋊D43—24no—
157C2×C22⋊D43—24no—
158C2×C22⋊D43—42no—
159C2×C22⋊D43—42no—
160C2×C22⋊D43—82no—
161C2×C22⋊D43—82no—
162C22×Dic43—24no—
163C22×Dic43—21no—
164C22×Dic43—21no—
165C22×Dic43—21no—
166C22×Dic43—82no—
167#10621898—48no—
168#10621898—48no—
169C2×D172—21no—
170C2×D172—22no—
171C2×D172—22no—
172C2×D172C2×D412yes—
173C2×C4×D43—24no—
174C2×C4×D43—42no—
175C2×C4×D43—42no—
176C2×Dic86—22no—
177C2×Dic86—22no—
178C2×Dic86—21no—
179C2×Dic86C2×D412yes—
180C4⋊C4×C43—22no—
181C4⋊C4×C43—22no—
182C4⋊C4×C43—22no—
183C4⋊C4×C43—22no—
184C4⋊C4×C43—42no—
185C22⋊C4×C43—42no—
186C22⋊C4×C43—82no—
187C22⋊C4×C43—82no—
188C42×C43—42no—
189C22⋊Dic43—24no—
190C22⋊Dic43—24no—
191(C2×D43)⋊C4—24no—
192(C2×D43)⋊C4—24no—
193C4⋊Dic43—24no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43C2×D412yes—
197C4⋊Dic43C2×D412yes—
198Dic43⋊C4—24no—
199Dic43⋊C4—24no—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C2×C43⋊C8—22no—
203C2×C43⋊C8—22no—
204C2×C43⋊C8—22no—
205C2×C43⋊C8—22no—
206C23×C43—21no—
207C23×C43—21no—
208C23×C43—21no—
209C23×C43—21no—
210C23×C43—22no—
211C23×C43—22no—
212C23×C43—22no—
213C23×C43—22no—
214C23×C43—21no—
215C23×C43—21no—
216C23×C43—21no—
217C23×C43—22no—
218C23×C43—22no—
219C23×C43#12012yes—
220C23×C43#12312yes—
221C23×C43C22×D411yescharacteristic
222C22×D43—22no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—42no—
226C22×D43—42no—
227Q8×C43—44no—
228D4×C43—44no—
229D4×C43—44no—
230D4×C43—44no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—24no—
236C2×C4×C43—24no—
237C2×C4×C43—24no—
238C2×C4×C43—24no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—42no—
243C2×C4×C43—42no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43—42no—
247C2×C4×C43—42no—
248C2×C4×C43—42no—
249C2×C4×C43—42no—
250C2×C4×C43—84no—
251C2×C4×C43#12012yes—
252C2×C4×C43#12312yes—
253C2×C4×C43C22×D411yescharacteristic
254C2×C4×C43C22×D411yescharacteristic
255C2×C4×C43C2×#4112yes—
256C2×C4×C43H(5,𝔽2)12yes—
257C22⋊D43—48no—
258C22⋊D43—84no—
259C22⋊D43—84no—
260C2×Dic43—22no—
261C2×Dic43—22no—
262C2×Dic43—22no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—42no—
266C2×Dic43—42no—
267C2×Dic43—84no—
268D172—24no—
269D172—42no—
270C4×D43—48no—
271Dic86—24no—
272Dic86—42no—
273C43⋊C8—42no—
274C43⋊C8—42no—
275#24012—431nomaximal, Sylow
276C22×C43—24no—
277C22×C43—22no—
278C22×C43—22no—
279C22×C43—22no—
280C22×C43—22no—
281C22×C43—42no—
282C22×C43—42no—
283C22×C43—42no—
284C22×C43—41no—
285C22×C43—41no—
286C22×C43—42no—
287C22×C43—41no—
288C22×C43—42no—
289C22×C43—41no—
290C22×C43—41no—
291C22×C43C2×#12011yescharacteristic
292C22×C43C2×#12311yescharacteristic
293C22×C43#53411yescharacteristic
294C22×C43#57712yes—
295C22×C43#57712yes—
296C2×D43—42no—
297C2×D43—42no—
298C2×D43—42no—
299C2×D43—82no—
300C2×D43—82no—
301C4×C43—22no—
302C4×C43—22no—
303C4×C43—41no—
304C4×C43—82no—
305C4×C43—82no—
306C4×C43#54612yes—
307C4×C43#54612yes—
308Dic43—42no—
309Dic43—42no—
310Dic43—42no—
311C2×#537—431no—
312C2×#481—431no—
313C2×#483—431no—
314#2311D4311yescharacteristic, complemented
315#2035—432no—
316#2034—432no—
317#1875—431no—
318#1873—431no—
319#1853—432no—
320#1788—432no—
321#1780—431no—
322C2×C43—42no—
323C2×C43—42no—
324C2×C43—81no—
325C2×C43#240511yescharacteristic
326C2×C43#300911yescharacteristic
327C2×C43#301011yescharacteristic
328C2×C43#309912yes—
329C2×C43#320412yes—
330D43—82no—
331C22×#41—431no—
332C22×#41C2×D4311yescharacteristic
333#537—864no—
334#537—864no—
335C2×C22⋊Q8—431no—
336C2×C22⋊Q8—431no—
337C2×C22⋊Q8—861no—
338C2×#121—431no—
339C2×#121—431no—
340C2×#121—861no—
341C2×#121—861no—
342C2×#121—861no—
343C2×#117C2×D4311yescharacteristic
344C22×C4⋊C4—431no—
345C22×C4⋊C4C2×D4311yescharacteristic
346#483—864no—
347#481—864no—
348C2×#106—432no—
349C2×#106—432no—
350C2×#103—432no—
351C2×#103—432no—
352C2×#102—432no—
353C2×#102—432no—
354C2×C22⋊C8—431no—
355C2×C22⋊C8—431no—
356C2×C22⋊C8—432no—
357C2×C22⋊C8—432no—
358#391—21no—
359#390—21no—
360#386—21no—
361#386—21no—
362#380C2×D4312yes—
363#380C2×D4312yes—
364#340—432no—
365#340—432no—
366C43#2401211yescharacteristic, complemented, Sylow
367C2×#41—24no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—42no—
371C2×#41—862no—
372C2×#41—862no—
373C2×#41—864no—
374C2×#41—1722no—
375C22×Q8—431no—
376C22×Q8C22⋊D4311yescharacteristic
377C22×D4—431no—
378C22×D4—861no—
379C22×D4—861no—
380C22×D4—861no—
381C22×D4C22⋊D4311yescharacteristic
382C22×D4C22⋊D4311yescharacteristic
383C22×D4C22⋊D4311yescharacteristic
384C23×C4—21no—
385C23×C4—861no—
386C23×C4—861no—
387C23×C4C22×D4311yescharacteristic
388C23×C4C22×D4311yescharacteristic
389C23×C4C22×D4311yescharacteristic
390C22×C8—432no—
391C22×C8—432no—
392C22⋊Q8—434no—
393C22⋊Q8—862no—
394C22⋊Q8—862no—
395C22⋊Q8—868no—
396C22⋊Q8—862no—
397#121—434no—
398#121—862no—
399#121—862no—
400#121—868no—
401#121—868no—
402#121—868no—
403#121—862no—
404#117—42no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—431no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—862no—
411C2×C4⋊C4—862no—
412C2×C4⋊C4—861no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4C22×D4312yes—
416C2×C4⋊C4C22×D4312yes—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C42—21no—
425#106—864no—
426#106—864no—
427#103—864no—
428#103—1722no—
429#102—864no—
430#102—1722no—
431C22⋊C8—864no—
432C22⋊C8—862no—
433C22⋊C8—862no—
434C22⋊C8—1722no—
435#95—22no—
436#95—22no—
437C24—861no—
438C24C2×C22⋊D4311yescharacteristic
439C24C2×C22⋊D4311yescharacteristic
440C24C2×C22⋊D4311yescharacteristic
441#41—48no—
442#41—48no—
443#41—1728no—
444#41—1728no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C2×Q8—432no—
449C2×Q8—861no—
450C2×Q8—862no—
451C2×Q8—862no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—432no—
462C2×D4—864no—
463C2×D4—864no—
464C2×D4—864no—
465C2×D4—861no—
466C2×D4—862no—
467C2×D4—862no—
468C2×D4—3442no—
469C2×D4—3442no—
470C2×D4—1722no—
471C2×D4—1722no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4—864no—
486C22×C4—864no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—3442no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4#1062189814yes—
494C22×C4D4×D4312yes—
495C22×C4D4×D4312yes—
496C22×C4C2×C22⋊D4311yescharacteristic
497C22×C4C2×C22⋊D4311yescharacteristic
498C22×C4C2×C22⋊D4311yescharacteristic
499C22×C4C23×D4311yescharacteristic
500C2×C8—862no—
501C2×C8—862no—
502C2×C8—862no—
503C2×C8—862no—
504C2×C8—862no—
505C2×C8—862no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—432no—
512C4⋊C4—432no—
513C4⋊C4—864no—
514C4⋊C4—864no—
515C4⋊C4—864no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C22⋊C4—42no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—864no—
522C22⋊C4—864no—
523C22⋊C4—864no—
524C22⋊C4—864no—
525C42—42no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—862no—
540C23—1721no—
541C23—1722no—
542C23—1722no—
543C23—1721no—
544C23#1306329312yes—
545C23#1306330812yes—
546C23C22×C22⋊D4311yescharacteristic
547Q8—44no—
548Q8—864no—
549Q8—1722no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—864no—
554D4—3444no—
555D4—3444no—
556D4—1728no—
557D4—1722no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—862no—
579C2×C4—862no—
580C2×C4—862no—
581C2×C4—3444no—
582C2×C4—1728no—
583C2×C4—1721no—
584C2×C4—1722no—
585C2×C4—1722no—
586C2×C4—1721no—
587C2×C4#1306329312yes—
588C2×C4#1306330812yes—
589C2×C4C2×D4×D4311yescharacteristic
590C2×C4C2×D4×D4311yescharacteristic
591C2×C4C2×#1062189812yes—
592C2×C4#1306337012yes—
593C8—1722no—
594C8—1722no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—3442no—
611C22—3442no—
612C22—1722no—
613C22—1722no—
614C22—1722no—
615C22C2×#1306329311yescharacteristic
616C22C2×#1306330811yescharacteristic
617C22#1458345811yescharacteristic
618C22#1458370712yes—
619C22#1458371012yes—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—82no—
624C4—82no—
625C4—1722no—
626C4—1722no—
627C4—1722no—
628C4#1458353912yes—
629C4#1458353912yes—
630C2—42no—
631C2—42no—
632C2—81no—
633C2—3442no—
634C2#1913792211yescharacteristic
635C2#1914387011yescharacteristic
636C2#1914387911yescharacteristic
637C2#1914458412yes—
638C2#1914550112yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2311C2
socleC23×C43C22×D4
prime radicalC2×#14583477C2
hypercenter#2311D43
hypocenterC43#24012
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#14583217C2semidirect
3. C2×#14583229C2semidirect
4. #19135791C2semidirect
5. #19135784C2semidirect
6. #19134917C2semidirect
7. #19134907C2semidirect
8. #19134787C2semidirect
9. #19134533C2semidirect
10. #19134507C2semidirect
11. C43×#2311C2semidirect
314. #2311D43semidirect
366. C43#24012semidirect

Maximal subgroups

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

subgroupindex
C2×#145834772
C2×#145832172
C2×#145832292
#191357912
#191357842
#191349172
#191349072
#191347872
#191345332
#191345072
C43×#23112
#2401243