Finite Group Databasebeta

← #25131463

Subgroups — #25131463

Order 11008.

1074 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583618C211yesmaximal, characteristic
2C2×C4×#10621885C211yesmaximal, characteristic, complemented
3#19139957C211yesmaximal, characteristic, complemented
4#19135839C211yesmaximal, characteristic, complemented
5#19135834C211yesmaximal, characteristic
6#19135693C211yesmaximal, characteristic, complemented
7#19135687C211yesmaximal, characteristic
8#19135530C211yesmaximal, characteristic, complemented
9#19135407C211yesmaximal, characteristic
10#19134938C211yesmaximal, characteristic, complemented
11#19134931C211yesmaximal, characteristic
12#19134847C211yesmaximal, characteristic, complemented
13#19134840C211yesmaximal, characteristic
14#19134611C211yesmaximal, characteristic, complemented
15C2×C43×#464C211yesmaximal, characteristic
16C22×#10621885C2211yescharacteristic
17#14583618—24no—
18#14583618—24no—
19C2×#13063323C2211yescharacteristic
20C2×#13063318—21no—
21C2×#13063318—21no—
22C2×#13063312C2211yescharacteristic
23C2×#13063302—21no—
24C2×Q8×Dic43C2211yescharacteristic
25C2×C4×Dic86C2211yescharacteristic
26C2×C4×Dic86C2211yescharacteristic
27C2×D4×Dic43C2211yescharacteristic
28C2×#13063281—21no—
29C4×#10621885—24no—
30C2×#13063258C2211yescharacteristic
31C2×#13063250C2211yescharacteristic
32C2×#13063250C2211yescharacteristic
33C2×#13063249C2211yescharacteristic
34C2×#13063249C2211yescharacteristic
35C2×#13063249C2211yescharacteristic
36C2×#13063248C2211yescharacteristic
37C2×#13063246C2211yescharacteristic
38C2×#13063246C2211yescharacteristic
39C2×C4×C43⋊C8C2211yescharacteristic
40#14582838—21no—
41#14582819—21no—
42#14582806C2211yescharacteristic
43#14582806C2211yescharacteristic
44#14582795C2211yescharacteristic
45#14582795C2211yescharacteristic
46#14582783—21no—
47#14582783—21no—
48#14582765C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54#14582653C2211yescharacteristic
55C22×Q16×C43C2211yescharacteristic
56C22×SD16×C43C2211yescharacteristic
57C2×#124×C43C2211yescharacteristic
58C2×C4×Q8×C43C2211yescharacteristic
59C43×#464—24no—
60C2×C4⋊C8×C43C2211yescharacteristic
61C2×#103×C43C2211yescharacteristic
62C2×#102×C43C2211yescharacteristic
63C22×Dic86D411yescharacteristic
64C22×Dic86C2311yescharacteristic
65C23×Dic43—21no—
66C2×#10621885—22no—
67C2×#10621885—22no—
68C2×#10621885—22no—
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71#13063323—22no—
72#13063323—22no—
73#13063318—28no—
74#13063318—28no—
75#13063312—22no—
76#13063312—22no—
77#13063302—28no—
78Q8×Dic43—22no—
79Q8×Dic43—22no—
80C4×Dic86—22no—
81C4×Dic86—22no—
82C4×Dic86—22no—
83C4×Dic86D414yes—
84D4×Dic43—22no—
85D4×Dic43—22no—
86#13063281—28no—
87C2×C4⋊Dic43—21no—
88C2×C4⋊Dic43D411yescharacteristic
89C2×C4⋊Dic43C2311yescharacteristic
90C2×C4⋊Dic43C2311yescharacteristic
91C2×C4⋊Dic43C2311yescharacteristic
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C2×Dic43⋊C4—21no—
96C2×Dic43⋊C4—21no—
97C2×Dic43⋊C4D411yescharacteristic
98C2×Dic43⋊C4D411yescharacteristic
99C2×C22⋊Dic43—21no—
100C2×C22⋊Dic43—21no—
101C2×C22⋊Dic43—21no—
102C2×C4×Dic43—21no—
103C2×C4×Dic43—21no—
104C2×C4×Dic43C2311yescharacteristic
105C2×C4×Dic43C2311yescharacteristic
106#13063258—22no—
107#13063250—41no—
108#13063250—41no—
109#13063249—22no—
110#13063249—22no—
111#13063249—41no—
112#13063248—22no—
113#13063246—22no—
114#13063246—41no—
115C4×C43⋊C8—22no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95—21no—
119C43⋊#95—21no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C43⋊#95—21no—
124C22×Q8×C43C2311yescharacteristic
125C22×Q8×C43C2311yescharacteristic
126C22×D4×C43C2311yescharacteristic
127C2×Q16×C43—22no—
128C2×Q16×C43—22no—
129C2×Q16×C43—41no—
130C2×SD16×C43—22no—
131C2×SD16×C43—22no—
132C2×SD16×C43—41no—
133C22×C8×C43C2311yescharacteristic
134C22×C8×C43C2311yescharacteristic
135#124×C43—22no—
136#124×C43D414yes—
137C4×Q8×C43—22no—
138C4×Q8×C43—22no—
139C2×C4⋊C4×C43—21no—
140C2×C4⋊C4×C43C2311yescharacteristic
141C2×C22⋊C4×C43—21no—
142C2×C22⋊C4×C43—21no—
143C2×C42×C43—21no—
144C2×C42×C43C2311yescharacteristic
145C4⋊C8×C43—22no—
146#103×C43—22no—
147#102×C43—22no—
148C24×C43—21no—
149C2×Q8×C43—21no—
150C2×Q8×C43—21no—
151C2×Q8×C43—21no—
152C2×Q8×C43—21no—
153C2×Q8×C43—21no—
154C2×Q8×C43—21no—
155C2×Q8×C43—21no—
156C2×Q8×C43—21no—
157C2×Q8×C43—21no—
158C2×Q8×C43—21no—
159C2×Q8×C43C2×D412yes—
160C2×Q8×C43#4112yes—
161C2×D4×C43—21no—
162C2×D4×C43—21no—
163C2×D4×C43—21no—
164C2×D4×C43—21no—
165C2×D4×C43—21no—
166C2×D4×C43C2×D412yes—
167C22×C4×C43—21no—
168C22×C4×C43—21no—
169C22×C4×C43—21no—
170C22×C4×C43C2×D411yescharacteristic
171C22×C4×C43C2×D411yescharacteristic
172C22×C4×C43C2411yescharacteristic
173C22×Dic43—24no—
174C22×Dic43—21no—
175C22×Dic43—21no—
176C22×Dic43—21no—
177C22×Dic43—21no—
178C22×Dic43—42no—
179C22×Dic43—42no—
180C22×Dic43C2×D411yescharacteristic
181C22×Dic43C2×D411yescharacteristic
182C22×Dic43C2×D411yescharacteristic
183C22×Dic43C2×D411yescharacteristic
184C2×Dic86—21no—
185C2×Dic86—21no—
186C2×Dic86—21no—
187C2×Dic86—21no—
188C2×Dic86—22no—
189C2×Dic86—21no—
190C2×Dic86—22no—
191C2×Dic86—41no—
192C2×Dic86C2×D412yes—
193Q16×C43—44no—
194SD16×C43—44no—
195C2×C8×C43—21no—
196C2×C8×C43—21no—
197C2×C8×C43—21no—
198C2×C8×C43—21no—
199C2×C8×C43—21no—
200C2×C8×C43—21no—
201C4⋊C4×C43—21no—
202C4⋊C4×C43—21no—
203C4⋊C4×C43—42no—
204C22⋊C4×C43—24no—
205C22⋊C4×C43—24no—
206C42×C43—42no—
207C42×C43C2×D412yes—
208C42×C43C2×D412yes—
209C22⋊Dic43—24no—
210C22⋊Dic43—24no—
211C22⋊Dic43—24no—
212#10621885—28no—
213C4⋊Dic43—24no—
214C4⋊Dic43—22no—
215C4⋊Dic43—21no—
216C4⋊Dic43—21no—
217C4⋊Dic43—21no—
218C4⋊Dic43—21no—
219C4⋊Dic43—21no—
220C4⋊Dic43C2×D412yes—
221Dic43⋊C4—24no—
222Dic43⋊C4—24no—
223Dic43⋊C4—24no—
224Dic43⋊C4—22no—
225Dic43⋊C4—22no—
226Dic43⋊C4—42no—
227Dic43⋊C4—42no—
228C4×Dic43—24no—
229C4×Dic43—24no—
230C4×Dic43—21no—
231C4×Dic43—21no—
232C4×Dic43—21no—
233C4×Dic43—21no—
234C2×C43⋊C8—21no—
235C2×C43⋊C8—21no—
236C2×C43⋊C8—21no—
237C2×C43⋊C8—21no—
238C2×C43⋊C8—21no—
239C2×C43⋊C8#4112yes—
240C23×C43—22no—
241C23×C43—41no—
242C23×C43—41no—
243C23×C43—41no—
244C23×C43—41no—
245C23×C43—41no—
246C23×C43—41no—
247C23×C43C22×D411yescharacteristic
248Q8×C43—22no—
249Q8×C43—22no—
250Q8×C43—22no—
251Q8×C43—22no—
252Q8×C43—41no—
253Q8×C43—41no—
254Q8×C43—41no—
255Q8×C43—41no—
256D4×C43—22no—
257D4×C43—22no—
258D4×C43—41no—
259D4×C43—41no—
260C2×C4×C43—21no—
261C2×C4×C43—21no—
262C2×C4×C43—21no—
263C2×C4×C43—21no—
264C2×C4×C43—22no—
265C2×C4×C43—22no—
266C2×C4×C43—41no—
267C2×C4×C43—41no—
268C2×C4×C43—41no—
269C2×C4×C43—41no—
270C2×C4×C43—41no—
271C2×C4×C43—41no—
272C2×C4×C43—41no—
273C2×C4×C43#12012yes—
274C2×C4×C43#12012yes—
275C2×C4×C43C22×D411yescharacteristic
276C2×C4×C43C22×D411yescharacteristic
277C2×C4×C43C2×#4111yescharacteristic
278C2×C4×C43C2×#4111yescharacteristic
279C2×C4×C43H(5,𝔽2)11yescharacteristic
280C2×C4×C432−1+411yescharacteristic
281C2×Dic43—21no—
282C2×Dic43—21no—
283C2×Dic43—21no—
284C2×Dic43—22no—
285C2×Dic43—21no—
286C2×Dic43—22no—
287C2×Dic43—21no—
288C2×Dic43—21no—
289C2×Dic43—21no—
290C2×Dic43—21no—
291C2×Dic43—21no—
292C2×Dic43—21no—
293C2×Dic43—21no—
294C2×Dic43—48no—
295C2×Dic43—41no—
296C2×Dic43—41no—
297C2×Dic43—41no—
298C2×Dic43—41no—
299C2×Dic43—41no—
300C2×Dic43—41no—
301C2×Dic43—41no—
302C2×Dic43—41no—
303C2×Dic43—41no—
304C2×Dic43—41no—
305C2×Dic43#13512yes—
306Dic86—22no—
307Dic86—22no—
308Dic86—44no—
309Dic86—41no—
310Dic86—41no—
311C8×C43—41no—
312C8×C43—41no—
313C43⋊C8—22no—
314C43⋊C8—41no—
315#24195—431nomaximal, Sylow
316C22×C43—42no—
317C22×C43—42no—
318C22×C43—42no—
319C22×C43—81no—
320C22×C43—81no—
321C22×C43—81no—
322C22×C43—81no—
323C22×C43C2×#12011yescharacteristic
324C22×C43#54811yescharacteristic
325C22×C43#55011yescharacteristic
326C22×C43C2×#13511yescharacteristic
327C22×C43#57511yescharacteristic
328C22×C43#57811yescharacteristic
329C22×C43#57811yescharacteristic
330C4×C43—22no—
331C4×C43—22no—
332C4×C43—42no—
333C4×C43—42no—
334C4×C43—81no—
335C4×C43#52511yescharacteristic
336C4×C43#53711yescharacteristic
337C4×C43#54611yescharacteristic
338C4×C43#54711yescharacteristic
339Dic43—22no—
340Dic43—42no—
341Dic43—41no—
342Dic43—42no—
343Dic43—41no—
344Dic43—41no—
345Dic43—81no—
346Dic43—81no—
347C2×#556—431no—
348C2×#464D4311yescharacteristic
349C2×C4×Q16—431no—
350#2581—431no—
351#2043—431no—
352#2042—431no—
353#2016—431no—
354#2015—431no—
355#1984—431no—
356#1958—431no—
357#1878—431no—
358#1877—431no—
359#1863—431no—
360#1862—431no—
361#1805—431no—
362C2×C43—82no—
363C2×C43#240711yescharacteristic
364C2×C43#300211yescharacteristic
365C2×C43#301111yescharacteristic
366C2×C43#336411yescharacteristic
367C2×C43#337711yescharacteristic
368C2×C43#339311yescharacteristic
369C2×C43#340211yescharacteristic
370C22×Q16—431no—
371C22×Q16C2×D4311yescharacteristic
372C22×SD16C2×D4311yescharacteristic
373#556—864no—
374#556—864no—
375C2×C4⋊Q8—431no—
376C2×#126—861no—
377C2×#126—861no—
378C2×#124—431no—
379C2×#124C2×D4311yescharacteristic
380C2×C22⋊Q8—861no—
381C2×C4×Q8—431no—
382C2×C4×Q8—431no—
383C2×C4×Q8—431no—
384C2×C4×Q8C2×D4311yescharacteristic
385C2×C4×D4—431no—
386C2×#117—861no—
387#464—24no—
388C4×Q16—864no—
389C2×#107—431no—
390C2×C4⋊C8C2×D4311yescharacteristic
391C2×#103—431no—
392C2×#103—431no—
393C2×#103—431no—
394C2×#103—431no—
395C2×#103—431no—
396C2×#103—431no—
397C2×#103C2×D4311yescharacteristic
398C2×#102—431no—
399C2×#102—431no—
400C2×#102C2×D4311yescharacteristic
401C2×C4×C8—431no—
402#400—861no—
403#395—861no—
404#391—431no—
405#391—431no—
406#389—431no—
407#389—431no—
408#387—861no—
409#387—861no—
410#382—431no—
411#340—431no—
412#340—431no—
413#336—431no—
414#336—431no—
415#336—431no—
416#336—431no—
417C43#2419511yescharacteristic, complemented, Sylow
418C22×Q8—431no—
419C22×Q8—431no—
420C22×Q8C22×D4311yescharacteristic
421C22×Q8C22×D4311yescharacteristic
422C22×D4C22×D4311yescharacteristic
423C23×C4—861no—
424C2×Q16—22no—
425C2×Q16—22no—
426C2×Q16—41no—
427C2×Q16—862no—
428C2×Q16—862no—
429C2×Q16—862no—
430C2×SD16—22no—
431C2×SD16—22no—
432C2×SD16—41no—
433C22×C8—431no—
434C22×C8—431no—
435C22×C8C22×D4311yescharacteristic
436C22×C8C22×D4311yescharacteristic
437C4⋊Q8—862no—
438C4⋊Q8—862no—
439#126—868no—
440#126—868no—
441#124—22no—
442#124—862no—
443#124—862no—
444#124C22⋊D4314yes—
445C22⋊Q8—868no—
446C4×Q8—22no—
447C4×Q8—22no—
448C4×Q8—434no—
449C4×Q8—862no—
450C4×Q8—862no—
451C4×Q8—862no—
452C4×Q8—862no—
453C4×Q8—862no—
454C4×D4—862no—
455C4×D4—862no—
456#117—868no—
457C2×C4⋊C4—21no—
458C2×C4⋊C4—431no—
459C2×C4⋊C4—431no—
460C2×C4⋊C4—431no—
461C2×C4⋊C4—431no—
462C2×C4⋊C4—431no—
463C2×C4⋊C4—431no—
464C2×C4⋊C4—861no—
465C2×C4⋊C4—861no—
466C2×C4⋊C4—861no—
467C2×C4⋊C4—861no—
468C2×C4⋊C4—861no—
469C2×C4⋊C4—861no—
470C2×C4⋊C4C22×D4311yescharacteristic
471C2×C22⋊C4—21no—
472C2×C22⋊C4—21no—
473C2×C22⋊C4—861no—
474C2×C22⋊C4—861no—
475C2×C22⋊C4—861no—
476C2×C42—21no—
477C2×C42—431no—
478C2×C42—431no—
479C2×C42—861no—
480C2×C42—861no—
481C2×C42C22×D4311yescharacteristic
482#107—862no—
483C4⋊C8—22no—
484#103—22no—
485#103—862no—
486#103—862no—
487#103—862no—
488#103—1721no—
489#103—1721no—
490#103—1721no—
491#102—22no—
492#102—862no—
493#102—1721no—
494C4×C8—862no—
495#95—861no—
496#95—861no—
497#95—861no—
498#95—861no—
499#95—861no—
500#95—861no—
501#95—861no—
502#95—861no—
503C24—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—21no—
508C2×Q8—21no—
509C2×Q8—21no—
510C2×Q8—21no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—21no—
514C2×Q8—432no—
515C2×Q8—862no—
516C2×Q8—862no—
517C2×Q8—861no—
518C2×Q8—861no—
519C2×Q8—861no—
520C2×Q8—861no—
521C2×Q8—861no—
522C2×Q8—1721no—
523C2×Q8#1062190012yes—
524C2×Q8C2×C22⋊D4312yes—
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—21no—
528C2×D4—21no—
529C2×D4—21no—
530C2×D4C2×C22⋊D4312yes—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—431no—
535C22×C4—431no—
536C22×C4—431no—
537C22×C4—431no—
538C22×C4—864no—
539C22×C4—861no—
540C22×C4—861no—
541C22×C4—861no—
542C22×C4—861no—
543C22×C4—1722no—
544C22×C4—1722no—
545C22×C4D4×D4311yescharacteristic
546C22×C4D4×D4311yescharacteristic
547C22×C4C23×D4311yescharacteristic
548Q16—44no—
549Q16—868no—
550SD16—44no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8—21no—
555C2×C8—21no—
556C2×C8—21no—
557C2×C8—432no—
558C2×C8—861no—
559C2×C8—861no—
560C2×C8—861no—
561C2×C8—861no—
562C2×C8—861no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—42no—
566C4⋊C4—432no—
567C4⋊C4—864no—
568C4⋊C4—864no—
569C4⋊C4—864no—
570C4⋊C4—862no—
571C4⋊C4—862no—
572C4⋊C4—862no—
573C4⋊C4—864no—
574C4⋊C4—861no—
575C4⋊C4—861no—
576C4⋊C4—861no—
577C4⋊C4—861no—
578C4⋊C4—861no—
579C4⋊C4—1722no—
580C4⋊C4—1722no—
581C22⋊C4—24no—
582C22⋊C4—24no—
583C22⋊C4—864no—
584C22⋊C4—864no—
585C22⋊C4—864no—
586C42—42no—
587C42—864no—
588C42—864no—
589C42—861no—
590C42—861no—
591C42—861no—
592C42—861no—
593C42D4×D4312yes—
594C42C2×C22⋊D4312yes—
595C23—22no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23C2×D4×D4311yescharacteristic
603Q8—22no—
604Q8—22no—
605Q8—22no—
606Q8—22no—
607Q8—41no—
608Q8—41no—
609Q8—41no—
610Q8—41no—
611Q8—862no—
612Q8—862no—
613Q8—1724no—
614Q8—1721no—
615Q8—1721no—
616D4—22no—
617D4—22no—
618D4—41no—
619D4—41no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—432no—
634C2×C4—861no—
635C2×C4—861no—
636C2×C4—861no—
637C2×C4—862no—
638C2×C4—861no—
639C2×C4—861no—
640C2×C4—862no—
641C2×C4—861no—
642C2×C4—861no—
643C2×C4—861no—
644C2×C4—861no—
645C2×C4—861no—
646C2×C4—861no—
647C2×C4—1728no—
648C2×C4—1721no—
649C2×C4—1721no—
650C2×C4—1721no—
651C2×C4—1721no—
652C2×C4—1721no—
653C2×C4—1721no—
654C2×C4—1721no—
655C2×C4—1721no—
656C2×C4—1721no—
657C2×C4—1721no—
658C2×C4#1306329312yes—
659C2×C4#1306329312yes—
660C2×C4C2×D4×D4311yescharacteristic
661C2×C4C22×C22⋊D4311yescharacteristic
662C2×C4#41×D4311yescharacteristic
663C2×C4C2×#1062190011yescharacteristic
664C2×C4#1306337011yescharacteristic
665C2×C4#1306337211yescharacteristic
666C8—41no—
667C8—41no—
668C8—862no—
669C8—1721no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22C2×#1306329311yescharacteristic
678C22#1458355511yescharacteristic
679C22#1458356711yescharacteristic
680C22#135×D4311yescharacteristic
681C22#1458369611yescharacteristic
682C22#1458371511yescharacteristic
683C22#1458371511yescharacteristic
684C4—22no—
685C4—22no—
686C4—42no—
687C4—42no—
688C4—81no—
689C4—862no—
690C4—3441no—
691C4—3441no—
692C4—1721no—
693C4—1722no—
694C4—1722no—
695C4—1721no—
696C4—1721no—
697C4#1458340911yescharacteristic
698C4#1458347811yescharacteristic
699C4#1458354311yescharacteristic
700C4#1458354811yescharacteristic
701C2—82no—
702C2#1913793611yescharacteristic
703C2#1914380911yescharacteristic
704C2#1914388911yescharacteristic
705C2#1914727511yescharacteristic
706C2#1914744111yescharacteristic
707C2#1914757811yescharacteristic
708C2#1914767411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#464C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenterC2×#464D43
hypocenterC43#24195
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10621885C2semidirect
3. #19139957C2semidirect
4. #19135839C2semidirect
6. #19135693C2semidirect
8. #19135530C2semidirect
10. #19134938C2semidirect
12. #19134847C2semidirect
14. #19134611C2semidirect
417. C43#24195semidirect

Maximal subgroups

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

subgroupindex
C2×#145836182
C2×C4×#106218852
#191399572
#191358392
#191358342
#191356932
#191356872
#191355302
#191354072
#191349382
#191349312
#191348472
#191348402
#191346112
C2×C43×#4642
#2419543