Finite Group Databasebeta

← #25147039

Subgroups — #25147039

Order 11008.

716 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144995C211yesmaximal, characteristic
2C2×#14583206C211yesmaximal, characteristic, complemented
3#19143339C211yesmaximal, characteristic
4#19136043C211yesmaximal, characteristic, complemented
5#19136038C211yesmaximal, characteristic, complemented
6#19135891C211yesmaximal, characteristic, complemented
7#19135891C211yesmaximal, characteristic, complemented
8#19135587C211yesmaximal, characteristic, complemented
9#19135368C211yesmaximal, characteristic, complemented
10#19135112C211yesmaximal, characteristic, complemented
11#19135112C211yesmaximal, characteristic, complemented
12#19134657C211yesmaximal, characteristic, complemented
13#19134652C211yesmaximal, characteristic, complemented
14C4×#13063244C211yesmaximal, characteristic, complemented
15C43×#3037C211yesmaximal, characteristic
16C2×#13063332C2211yescharacteristic
17C2×#13063332C2211yescharacteristic
18C2×#13063317C2211yescharacteristic
19#14583267—21no—
20#14583263—21no—
21#14583253—21no—
22#14583253—21no—
23#14583206—21no—
24#14583206—21no—
25#14583206—21no—
26#14583206—21no—
27#14582977—21no—
28#14582971—21no—
29#14582971—21no—
30D4×C43⋊C8—21no—
31C2×#13063260C2211yescharacteristic
32C2×#13063260C2211yescharacteristic
33C2×#13063258C2211yescharacteristic
34C2×#13063256C2211yescharacteristic
35C2×C4⋊(C43⋊C8)C2211yescharacteristic
36#14582900C2211yescharacteristic
37#14582900C2211yescharacteristic
38#14582900C2211yescharacteristic
39#14582900C2211yescharacteristic
40C2×#13063249C2211yescharacteristic
41C2×#13063249C2211yescharacteristic
42C2×#13063244C2211yescharacteristic
43C2×#13063244C2211yescharacteristic
44#14582866C2211yescharacteristic
45#14582866C2211yescharacteristic
46C4×#10621877C2211yescharacteristic
47C4×#10621877C2211yescharacteristic
48#14582668C2211yescharacteristic
49#14582668C2211yescharacteristic
50#14582667C2211yescharacteristic
51#14582667C2211yescharacteristic
52#14582667C2211yescharacteristic
53#14582667C2211yescharacteristic
54#14582663C2211yescharacteristic
55#14582663C2211yescharacteristic
56C2×#137×C43C2211yescharacteristic
57C2×#137×C43C2211yescharacteristic
58C43×#532C2211yescharacteristic
59C2×C4×Q8×C43C2211yescharacteristic
60C43×#464—21no—
61C43×#464—21no—
62C43×#462—21no—
63C43×#460—21no—
64C43×#423C2211yescharacteristic
65C43×#417C2211yescharacteristic
66C43×#417C2211yescharacteristic
67#13063332—22no—
68#13063332—22no—
69#13063332—41no—
70#13063332—41no—
71C2×#10621877—21no—
72C2×#10621877—21no—
73C2×#10621877C2311yescharacteristic
74C2×#10621877C2311yescharacteristic
75C2×#10621877C2311yescharacteristic
76C2×#10621877C2311yescharacteristic
77C22×C43⋊C8—21no—
78C22×C43⋊C8—21no—
79C22×C43⋊C8C2311yescharacteristic
80C22×C43⋊C8C2311yescharacteristic
81#13063317—21no—
82#13063317—21no—
83#13063317—21no—
84#13063317—21no—
85C2×C4⋊Dic43C2311yescharacteristic
86C2×C4⋊Dic43C2311yescharacteristic
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89#13063260—41no—
90#13063260—41no—
91#13063258—21no—
92#13063258—21no—
93#13063256—21no—
94#13063256—21no—
95C4⋊(C43⋊C8)D411yescharacteristic
96C4⋊(C43⋊C8)D411yescharacteristic
97C4⋊(C43⋊C8)D411yescharacteristic
98C4⋊(C43⋊C8)D411yescharacteristic
99#13063249—21no—
100#13063249—21no—
101#13063249—21no—
102#13063249—21no—
103#13063249—21no—
104#13063249—21no—
105#13063249—21no—
106#13063249—21no—
107#13063246—21no—
108#13063246—21no—
109#13063246—21no—
110#13063246—21no—
111#13063244—22no—
112#13063244—22no—
113C22⋊(C43⋊C8)—21no—
114C22⋊(C43⋊C8)—21no—
115C22⋊(C43⋊C8)—21no—
116C22⋊(C43⋊C8)—21no—
117#13063236—21no—
118#13063236—21no—
119C4×C43⋊C8—21no—
120C4×C43⋊C8—21no—
121C2×#41×C43C2311yescharacteristic
122C2×#41×C43C2311yescharacteristic
123C22×Q8×C43C2311yescharacteristic
124#137×C43—22no—
125#137×C43—22no—
126#137×C43—41no—
127#137×C43—41no—
128C2×Q16×C43—21no—
129C2×Q16×C43—21no—
130C2×SD16×C43—21no—
131C2×SD16×C43—21no—
132C2×M16×C43C2311yescharacteristic
133C2×M16×C43C2311yescharacteristic
134C4⋊Q8×C43D411yescharacteristic
135#127×C43D411yescharacteristic
136#124×C43D411yescharacteristic
137#124×C43D411yescharacteristic
138#121×C43—21no—
139#121×C43—21no—
140C4×Q8×C43—21no—
141C4×Q8×C43—21no—
142C4×Q8×C43—21no—
143C4×Q8×C43—21no—
144C4×D4×C43—21no—
145C4×D4×C43—21no—
146C2×C4⋊C4×C43—21no—
147C2×C4⋊C4×C43C2311yescharacteristic
148C2×C42×C43—21no—
149C2×C42×C43C2311yescharacteristic
150C4⋊C8×C43—21no—
151C4⋊C8×C43—21no—
152#103×C43—21no—
153#103×C43—21no—
154#102×C43—21no—
155#102×C43—21no—
156#41×C43—21no—
157#41×C43—21no—
158#41×C43—21no—
159#41×C43—21no—
160#41×C43—21no—
161#41×C43—21no—
162#41×C43—21no—
163#41×C43—21no—
164C2×Q8×C43—21no—
165C2×Q8×C43—21no—
166C2×Q8×C43—21no—
167C2×Q8×C43—21no—
168C2×Q8×C43—21no—
169C2×Q8×C43C2×D411yescharacteristic
170C2×Q8×C43C2×D411yescharacteristic
171C2×Q8×C43#4112yes—
172C2×D4×C43—21no—
173C2×D4×C43—21no—
174C2×D4×C43—21no—
175C2×D4×C43—21no—
176C2×D4×C43C2×D411yescharacteristic
177C2×D4×C43C2×D411yescharacteristic
178C22×C4×C43—21no—
179C22×C4×C43—21no—
180C22×C4×C43—21no—
181C22×C4×C43—21no—
182C22×C4×C43#4111yescharacteristic
183C22×C4×C43#4111yescharacteristic
184C22×C4×C43C2411yescharacteristic
185C22×Dic43—21no—
186C22×Dic43—21no—
187Q16×C43—42no—
188Q16×C43—42no—
189SD16×C43—42no—
190SD16×C43—42no—
191M16×C43—21no—
192M16×C43—21no—
193M16×C43—21no—
194M16×C43—21no—
195C2×C8×C43—21no—
196C2×C8×C43—21no—
197C4⋊C4×C43—21no—
198C4⋊C4×C43—21no—
199C4⋊C4×C43—21no—
200C4⋊C4×C43—21no—
201C4⋊C4×C43—41no—
202C4⋊C4×C43—41no—
203C22⋊C4×C43—21no—
204C22⋊C4×C43—21no—
205C22⋊C4×C43—21no—
206C22⋊C4×C43—21no—
207C42×C43—41no—
208C42×C43—41no—
209C42×C43C2×D411yescharacteristic
210C42×C43C2×D411yescharacteristic
211C42×C43C2×D411yescharacteristic
212C42×C43C2×D411yescharacteristic
213C4⋊Dic43—21no—
214C4⋊Dic43—21no—
215C4⋊Dic43—21no—
216C4⋊Dic43—21no—
217Dic43⋊C4—41no—
218Dic43⋊C4—41no—
219Dic43⋊C4—41no—
220Dic43⋊C4—41no—
221#10621877—21no—
222#10621877—21no—
223#10621877—21no—
224#10621877—21no—
225#10621877—21no—
226#10621877—21no—
227#10621877—21no—
228#10621877—42no—
229#10621877—42no—
230#10621877#4112yes—
231C2×C43⋊C8—21no—
232C2×C43⋊C8—21no—
233C2×C43⋊C8—21no—
234C2×C43⋊C8—21no—
235C2×C43⋊C8—21no—
236C2×C43⋊C8—21no—
237C2×C43⋊C8—21no—
238C2×C43⋊C8—21no—
239C2×C43⋊C8—42no—
240C2×C43⋊C8—42no—
241C2×C43⋊C8C2×D411yescharacteristic
242C2×C43⋊C8C2×D411yescharacteristic
243C2×C43⋊C8C2×D411yescharacteristic
244C2×C43⋊C8C2×D411yescharacteristic
245C23×C43—21no—
246C23×C43—21no—
247C23×C43C2×#4111yescharacteristic
248Q8×C43—22no—
249Q8×C43—21no—
250Q8×C43—21no—
251Q8×C43—21no—
252Q8×C43—21no—
253Q8×C43—22no—
254Q8×C43—41no—
255Q8×C43—41no—
256D4×C43—21no—
257D4×C43—21no—
258D4×C43—21no—
259D4×C43—21no—
260D4×C43—42no—
261D4×C43—41no—
262D4×C43—41no—
263D4×C43—41no—
264D4×C43—41no—
265D4×C43—81no—
266C2×C4×C43—21no—
267C2×C4×C43—21no—
268C2×C4×C43—22no—
269C2×C4×C43—41no—
270C2×C4×C43—41no—
271C2×C4×C43—41no—
272C2×C4×C43—41no—
273C2×C4×C43—41no—
274C2×C4×C43—41no—
275C2×C4×C43—41no—
276C2×C4×C43—41no—
277C2×C4×C43—41no—
278C2×C4×C43—81no—
279C2×C4×C43#12111yescharacteristic
280C2×C4×C43#12111yescharacteristic
281C2×C4×C43#12111yescharacteristic
282C2×C4×C43#12111yescharacteristic
283C2×C4×C43C22×D411yescharacteristic
284C2×C4×C43C22×D411yescharacteristic
285C2×C4×C43C2×#4111yescharacteristic
286C2×C4×C43C2×#4111yescharacteristic
287C2×C4×C43H(5,𝔽2)11yescharacteristic
288C2×C4×C432−1+411yescharacteristic
289C2×Dic43—41no—
290C2×Dic43—41no—
291C2×Dic43—81no—
292C2×Dic43—81no—
293C8×C43—41no—
294C8×C43—41no—
295C43⋊C8—21no—
296C43⋊C8—21no—
297C43⋊C8—21no—
298C43⋊C8—21no—
299C43⋊C8—22no—
300C43⋊C8—41no—
301C43⋊C8—41no—
302C43⋊C8—41no—
303#25363—431nomaximal, Sylow
304C22×C43—21no—
305C22×C43—21no—
306C22×C43—41no—
307C22×C43—41no—
308C22×C43—41no—
309C22×C43—41no—
310C22×C43—41no—
311C22×C43—41no—
312C22×C43C2×#12111yescharacteristic
313C22×C43#52911yescharacteristic
314C22×C43#54211yescharacteristic
315C4×C43—21no—
316C4×C43—21no—
317C4×C43—21no—
318C4×C43—21no—
319C4×C43—41no—
320C4×C43—41no—
321C4×C43—42no—
322C4×C43—81no—
323C4×C43#54611yescharacteristic
324C4×C43#54611yescharacteristic
325C4×C43#54711yescharacteristic
326C4×C43#54811yescharacteristic
327Dic43—81no—
328Dic43—81no—
329#3142—431no—
330C2×#479—431no—
331#3037D4311yescharacteristic
332#2935—431no—
333#2079—431no—
334#2078—431no—
335#2052—431no—
336#2052—431no—
337#1998—431no—
338#1950—431no—
339#1908—431no—
340#1908—431no—
341#1816—431no—
342#1815—431no—
343C4×#101—431no—
344C2×C43—21no—
345C2×C43—21no—
346C2×C43—81no—
347C2×C43—81no—
348C2×C43#242711yescharacteristic
349C2×C43#306711yescharacteristic
350C2×C43#306911yescharacteristic
351C2×#137C2×D4311yescharacteristic
352C2×#137C2×D4311yescharacteristic
353C2×#131—431no—
354C2×#131—431no—
355#532C2×D4311yescharacteristic
356C2×#125—431no—
357C2×C4×Q8C2×D4311yescharacteristic
358#491—861no—
359#490—861no—
360#488—861no—
361#488—861no—
362#479—861no—
363#479—861no—
364#479—861no—
365#479—861no—
366#464—21no—
367#464—21no—
368#462—21no—
369#460—21no—
370#436—861no—
371#435—861no—
372#435—861no—
373C8×D4—861no—
374C2×#108—431no—
375C2×#108—431no—
376C2×#107—431no—
377C2×#106—431no—
378#423C2×D4311yescharacteristic
379C2×C4⋊C8—431no—
380#417—431no—
381#417—431no—
382#417—431no—
383#417—431no—
384#417C2×D4311yescharacteristic
385#417C2×D4311yescharacteristic
386C2×#103—431no—
387C2×#103—431no—
388C2×#101—431no—
389C2×#101—431no—
390#408—431no—
391#408—431no—
392C4×M16—431no—
393C4×M16—431no—
394#343—431no—
395#343—431no—
396#343—431no—
397#343—431no—
398#343—431no—
399#343—431no—
400#341—431no—
401#341—431no—
402C43#2536311yescharacteristic, complemented, Sylow
403C2×#41C22×D4311yescharacteristic
404C2×#41C22×D4311yescharacteristic
405C22×Q8C22×D4311yescharacteristic
406#137—22no—
407#137—22no—
408#137—41no—
409#137—41no—
410C2×Q16—21no—
411C2×Q16—21no—
412C2×SD16—21no—
413C2×SD16—21no—
414#131—862no—
415#131—862no—
416#131—1721no—
417#131—1721no—
418C2×M16—431no—
419C2×M16—431no—
420C2×M16—431no—
421C2×M16—431no—
422C2×M16—861no—
423C2×M16—861no—
424C2×M16C22×D4311yescharacteristic
425C2×M16C22×D4311yescharacteristic
426C22×C8—431no—
427C22×C8—431no—
428C22×C8—861no—
429C22×C8—861no—
430C4⋊Q8C22⋊D4311yescharacteristic
431#127C22⋊D4311yescharacteristic
432#125—861no—
433#125—861no—
434#125—861no—
435#125—861no—
436#124C22⋊D4311yescharacteristic
437#124C22⋊D4311yescharacteristic
438#121—21no—
439#121—21no—
440C4×Q8—21no—
441C4×Q8—21no—
442C4×Q8—21no—
443C4×Q8—21no—
444C4×D4—21no—
445C4×D4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—431no—
448C2×C4⋊C4—431no—
449C2×C4⋊C4—861no—
450C2×C4⋊C4—861no—
451C2×C4⋊C4C22×D4311yescharacteristic
452C2×C42—21no—
453C2×C42C22×D4311yescharacteristic
454#108—1721no—
455#108—1721no—
456#107—861no—
457#107—861no—
458#106—861no—
459#106—861no—
460C4⋊C8—21no—
461C4⋊C8—21no—
462C4⋊C8—431no—
463C4⋊C8—431no—
464C4⋊C8—431no—
465C4⋊C8—431no—
466#103—21no—
467#103—21no—
468#103—861no—
469#103—861no—
470#103—861no—
471#103—861no—
472#103—861no—
473#103—861no—
474#103—861no—
475#103—861no—
476#102—21no—
477#102—21no—
478#102—861no—
479#102—861no—
480#102—861no—
481#102—861no—
482#101—862no—
483#101—862no—
484C22⋊C8—861no—
485C22⋊C8—861no—
486C22⋊C8—861no—
487C22⋊C8—861no—
488#97—861no—
489#97—861no—
490C4×C8—861no—
491C4×C8—861no—
492#41—21no—
493#41—21no—
494#41—21no—
495#41—21no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8#1062190012yes—
506C2×Q8C2×C22⋊D4311yescharacteristic
507C2×Q8C2×C22⋊D4311yescharacteristic
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—21no—
512C2×D4C2×C22⋊D4311yescharacteristic
513C2×D4C2×C22⋊D4311yescharacteristic
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—861no—
519C22×C4—861no—
520C22×C4#1062190011yescharacteristic
521C22×C4#1062190011yescharacteristic
522C22×C4C23×D4311yescharacteristic
523Q16—42no—
524Q16—42no—
525SD16—42no—
526SD16—42no—
527M16—21no—
528M16—21no—
529M16—21no—
530M16—21no—
531M16—432no—
532M16—861no—
533M16—861no—
534M16—861no—
535M16—861no—
536M16—861no—
537M16—861no—
538M16—861no—
539M16—1722no—
540M16—1722no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—431no—
544C2×C8—431no—
545C2×C8—431no—
546C2×C8—431no—
547C2×C8—861no—
548C2×C8—861no—
549C2×C8—861no—
550C2×C8—861no—
551C2×C8—861no—
552C2×C8—861no—
553C2×C8—861no—
554C2×C8—861no—
555C2×C8—1722no—
556C2×C8—1722no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—41no—
562C4⋊C4—41no—
563C4⋊C4—861no—
564C4⋊C4—861no—
565C4⋊C4—861no—
566C4⋊C4—861no—
567C4⋊C4—1721no—
568C4⋊C4—1721no—
569C4⋊C4—1721no—
570C4⋊C4—1721no—
571C22⋊C4—21no—
572C22⋊C4—21no—
573C22⋊C4—21no—
574C22⋊C4—21no—
575C42—41no—
576C42—41no—
577C42D4×D4311yescharacteristic
578C42D4×D4311yescharacteristic
579C42C2×C22⋊D4311yescharacteristic
580C42C2×C22⋊D4311yescharacteristic
581C23—21no—
582C23—21no—
583C23C2×#1062190011yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—21no—
587Q8—21no—
588Q8—21no—
589Q8—21no—
590Q8—41no—
591Q8—41no—
592D4—21no—
593D4—21no—
594D4—21no—
595D4—21no—
596D4—42no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—81no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—81no—
615C2×C4—3441no—
616C2×C4—3441no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4#1306329711yescharacteristic
620C2×C4#1306329711yescharacteristic
621C2×C4#1306329711yescharacteristic
622C2×C4#1306329711yescharacteristic
623C2×C4C2×D4×D4311yescharacteristic
624C2×C4C22×C22⋊D4311yescharacteristic
625C2×C4#41×D4311yescharacteristic
626C2×C4C2×#1062190011yescharacteristic
627C2×C4#1306337011yescharacteristic
628C2×C4#1306337211yescharacteristic
629C8—41no—
630C8—41no—
631C8—861no—
632C8—861no—
633C8—861no—
634C8—861no—
635C8—862no—
636C8—1721no—
637C8—1721no—
638C8—1721no—
639C22—21no—
640C22—21no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22C2×#1306329711yescharacteristic
648C22#1458343011yescharacteristic
649C22#1458351611yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—81no—
658C4—3441no—
659C4—3441no—
660C4#1458354311yescharacteristic
661C4#1458354611yescharacteristic
662C4#1458354811yescharacteristic
663C4#1458355511yescharacteristic
664C2—21no—
665C2—21no—
666C2—81no—
667C2—81no—
668C2#1913817411yescharacteristic
669C2#1914431211yescharacteristic
670C2#1914432911yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#3037C2
socleC22×C43C2×#121
prime radicalC43×#532C22
hypercenter#3037D43
hypocenterC43#25363
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. C2×#14583206C2semidirect
4. #19136043C2semidirect
5. #19136038C2semidirect
6. #19135891C2semidirect
7. #19135891C2semidirect
8. #19135587C2semidirect
9. #19135368C2semidirect
10. #19135112C2semidirect
11. #19135112C2semidirect
12. #19134657C2semidirect
13. #19134652C2semidirect
14. C4×#13063244C2semidirect
402. C43#25363semidirect

Maximal subgroups

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

subgroupindex
#191449952
C2×#145832062
#191433392
#191360432
#191360382
#191358912
#191355872
#191353682
#191351122
#191346572
#191346522
C4×#130632442
C43×#30372
#2536343