Finite Group Databasebeta

← #25154169

Subgroups — #25154169

Order 11008.

842 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144461C211yesmaximal, characteristic, complemented
2#19144450C211yesmaximal, characteristic, complemented
3#19143128C211yesmaximal, characteristic
4#19143128C211yesmaximal, characteristic
5#19136143C211yesmaximal, characteristic, complemented
6#19135958C211yesmaximal, characteristic, complemented
7#19135958C211yesmaximal, characteristic, complemented
8#19135756C211yesmaximal, characteristic, complemented
9#19134868C211yesmaximal, characteristic, complemented
10#19134868C211yesmaximal, characteristic, complemented
11#19134830C211yesmaximal, characteristic, complemented
12#19134830C211yesmaximal, characteristic, complemented
13#19134657C211yesmaximal, characteristic, complemented
14#19134652C211yesmaximal, characteristic, complemented
15C43×#1793C211yesmaximal, characteristic
16C2×#13063332C2211yescharacteristic
17C2×#13063332C2211yescharacteristic
18#14583530C2211yescharacteristic
19#14583382C2211yescharacteristic
20#14583382C2211yescharacteristic
21#14583208—21no—
22#14583204—21no—
23#14583201—21no—
24#14583197—21no—
25#14583192—21no—
26#14583187—21no—
27#14583182—21no—
28#14583177—21no—
29C2×#13063260C2211yescharacteristic
30C2×#13063260C2211yescharacteristic
31#14582941C2211yescharacteristic
32#14582941C2211yescharacteristic
33C2×#13063258C2211yescharacteristic
34C2×#13063256C2211yescharacteristic
35#14582927C2211yescharacteristic
36#14582927C2211yescharacteristic
37#14582908—21no—
38#14582908—21no—
39#14582908C2211yescharacteristic
40#14582908C2211yescharacteristic
41#14582900—21no—
42#14582900—21no—
43#14582900C2211yescharacteristic
44#14582900C2211yescharacteristic
45#14582895—21no—
46#14582895—21no—
47#14582895C2211yescharacteristic
48#14582895C2211yescharacteristic
49C2×#13063249—21no—
50C2×#13063249—21no—
51C2×#13063249C2211yescharacteristic
52C2×#13063249C2211yescharacteristic
53#14582765C2211yescharacteristic
54#14582668C2211yescharacteristic
55#14582667C2211yescharacteristic
56#14582667C2211yescharacteristic
57#14582659C2211yescharacteristic
58#14582659C2211yescharacteristic
59#14582654C2211yescharacteristic
60C2×2−1+4×C43C2211yescharacteristic
61C2×#104×C43C2211yescharacteristic
62C2×#104×C43C2211yescharacteristic
63C43×#417C2211yescharacteristic
64C43×#417C2211yescharacteristic
65C2×#101×C43C2211yescharacteristic
66C43×#391C2211yescharacteristic
67#13063332—22no—
68#13063332—22no—
69#13063332—41no—
70#13063332—41no—
71C2×#10621877—21no—
72C2×#10621877—21no—
73C2×#10621877C2311yescharacteristic
74C2×#10621877C2311yescharacteristic
75C22×C43⋊C8—21no—
76C22×C43⋊C8—21no—
77C22×C43⋊C8C2311yescharacteristic
78C22×C43⋊C8C2311yescharacteristic
79#13063324—21no—
80#13063324—21no—
81#13063316—21no—
82#13063316—21no—
83#13063304—41no—
84Q8×Dic43—21no—
85Q8×Dic43—21no—
86D4×Dic43—21no—
87D4×Dic43—21no—
88D4×Dic43—41no—
89D4×Dic43—41no—
90#13063284—21no—
91#13063284—21no—
92#13063284C2311yescharacteristic
93#13063284C2311yescharacteristic
94C2×C4⋊Dic43—21no—
95C2×C4⋊Dic43—21no—
96C2×C4⋊Dic43C2311yescharacteristic
97C2×C4⋊Dic43C2311yescharacteristic
98#13063260—41no—
99#13063260—41no—
100#13063259—21no—
101#13063259—21no—
102#13063258—21no—
103#13063258—21no—
104#13063257—21no—
105#13063257—21no—
106#13063256—21no—
107#13063256—21no—
108Dic43⋊C8—21no—
109Dic43⋊C8—21no—
110Dic43⋊C8—21no—
111Dic43⋊C8—21no—
112#13063249—21no—
113#13063249—21no—
114#13063249—21no—
115#13063249—21no—
116#13063249—21no—
117#13063249—21no—
118#13063249—21no—
119#13063249—21no—
120#13063249—21no—
121#13063249—21no—
122#13063249D411yescharacteristic
123#13063249D411yescharacteristic
124#13063249D411yescharacteristic
125#13063249D411yescharacteristic
126#13063246—21no—
127#13063246—21no—
128#13063246D411yescharacteristic
129#13063246D411yescharacteristic
130#13063246D411yescharacteristic
131#13063246D411yescharacteristic
132C43⋊#95—41no—
1332−1+4×C43—22no—
1342−1+4×C43—22no—
1352−1+4×C43—41no—
1362−1+4×C43Q814yescomplemented
137C2×#41×C43—21no—
138C2×#41×C43—21no—
139C2×#41×C43—41no—
140C2×#41×C43C2311yescharacteristic
141C2×#41×C43C2311yescharacteristic
142C22×Q8×C43—21no—
143C22×Q8×C43—21no—
144C22×Q8×C43C2311yescharacteristic
145C2×M16×C43C2311yescharacteristic
146C2×M16×C43C2311yescharacteristic
147C2×C4⋊C4×C43C2311yescharacteristic
148C2×C42×C43C2311yescharacteristic
149#104×C43—22no—
150#104×C43—22no—
151#103×C43—21no—
152#103×C43—21no—
153#102×C43—21no—
154#102×C43—21no—
155#101×C43—41no—
156#95×C43—41no—
157#41×C43—21no—
158#41×C43—21no—
159#41×C43—21no—
160#41×C43—21no—
161#41×C43—24no—
162#41×C43—24no—
163#41×C43—44no—
164#41×C43—42no—
165#41×C43—42no—
166#41×C43—44no—
167#41×C43C2×Q812yes—
168#41×C43C2×Q812yes—
169#41×C43#4112yes—
170#41×C43#4112yes—
171C2×Q8×C43—21no—
172C2×Q8×C43—24no—
173C2×Q8×C43—24no—
174C2×Q8×C43—21no—
175C2×Q8×C43—21no—
176C2×Q8×C43—21no—
177C2×Q8×C43—21no—
178C2×Q8×C43—41no—
179C2×Q8×C43—42no—
180C2×Q8×C43—42no—
181C2×Q8×C43C2×D411yescharacteristic
182C2×Q8×C43C2×D411yescharacteristic
183C2×Q8×C43C2×Q812yes—
184C2×D4×C43—21no—
185C2×D4×C43—21no—
186C2×D4×C43—41no—
187C2×D4×C43C2×D411yescharacteristic
188C2×D4×C43C2×D411yescharacteristic
189C22×C4×C43—21no—
190C22×C4×C43—21no—
191C22×C4×C43—21no—
192C22×C4×C43—21no—
193C22×C4×C43—21no—
194C22×C4×C43—41no—
195C22×C4×C43—41no—
196C22×C4×C43C2411yescharacteristic
197C22×Dic43—21no—
198C22×Dic43—21no—
199C22×Dic43—41no—
200C22×Dic43—41no—
201C2×Dic86—41no—
202M16×C43—21no—
203M16×C43—21no—
204M16×C43—21no—
205M16×C43—21no—
206C2×C8×C43—21no—
207C2×C8×C43—21no—
208C4⋊C4×C43—21no—
209C4⋊C4×C43—21no—
210C42×C43—21no—
211C42×C43—21no—
212C22⋊Dic43—21no—
213C22⋊Dic43—21no—
214C22⋊Dic43—41no—
215C22⋊Dic43—41no—
216C4⋊Dic43—21no—
217C4⋊Dic43—21no—
218C4⋊Dic43—21no—
219C4⋊Dic43—21no—
220C4⋊Dic43—41no—
221C4⋊Dic43—41no—
222C4⋊Dic43—41no—
223C4⋊Dic43C2×D411yescharacteristic
224C4⋊Dic43C2×D411yescharacteristic
225C4⋊Dic43C2×D411yescharacteristic
226C4⋊Dic43C2×D411yescharacteristic
227Dic43⋊C4—41no—
228Dic43⋊C4—41no—
229C4×Dic43—21no—
230C4×Dic43—21no—
231C4×Dic43—41no—
232C4×Dic43—41no—
233#10621877—21no—
234#10621877—21no—
235#10621877—21no—
236#10621877—21no—
237#10621877—42no—
238#10621877—42no—
239C2×C43⋊C8—21no—
240C2×C43⋊C8—21no—
241C2×C43⋊C8—21no—
242C2×C43⋊C8—21no—
243C2×C43⋊C8—21no—
244C2×C43⋊C8—21no—
245C2×C43⋊C8—42no—
246C2×C43⋊C8—42no—
247C2×C43⋊C8C2×D411yescharacteristic
248C2×C43⋊C8C2×D411yescharacteristic
249C2×C43⋊C8C2×D411yescharacteristic
250C2×C43⋊C8C2×D411yescharacteristic
251C23×C43—21no—
252C23×C43—21no—
253C23×C43H(5,𝔽2)11yescharacteristic
254Q8×C43—21no—
255Q8×C43—21no—
256Q8×C43—22no—
257Q8×C43—22no—
258Q8×C43—44no—
259Q8×C43—41no—
260Q8×C43—41no—
261Q8×C43C22⋊Q812yes—
262Q8×C43C22⋊Q812yes—
263D4×C43—22no—
264D4×C43—22no—
265D4×C43—22no—
266D4×C43—22no—
267D4×C43—21no—
268D4×C43—21no—
269D4×C43—44no—
270D4×C43C22⋊Q812yes—
271D4×C43C22⋊Q812yes—
272C2×C4×C43—21no—
273C2×C4×C43—21no—
274C2×C4×C43—22no—
275C2×C4×C43—22no—
276C2×C4×C43—22no—
277C2×C4×C43—22no—
278C2×C4×C43—22no—
279C2×C4×C43—44no—
280C2×C4×C43—44no—
281C2×C4×C43—41no—
282C2×C4×C43—41no—
283C2×C4×C43—41no—
284C2×C4×C43—41no—
285C2×C4×C43—41no—
286C2×C4×C43—41no—
287C2×C4×C43—81no—
288C2×C4×C43C22×D411yescharacteristic
289C2×C4×C43C22×D411yescharacteristic
290C2×C4×C43C22×Q811yescharacteristic
291C2×C4×C43C2×#4111yescharacteristic
292C2×C4×C43C2×#4111yescharacteristic
293C2×C4×C43H(5,𝔽2)11yescharacteristic
294C2×Dic43—21no—
295C2×Dic43—21no—
296C2×Dic43—21no—
297C2×Dic43—21no—
298C2×Dic43—41no—
299C2×Dic43—41no—
300C2×Dic43—81no—
301C2×Dic43—81no—
302C2×Dic43—161no—
303C2×Dic43—161no—
304Dic86—161no—
305C8×C43—41no—
306C8×C43—41no—
307C43⋊C8—21no—
308C43⋊C8—21no—
309C43⋊C8—21no—
310C43⋊C8—21no—
311C43⋊C8—41no—
312C43⋊C8—41no—
313#25985—431nomaximal, Sylow
314C22×C43—21no—
315C22×C43—21no—
316C22×C43—22no—
317C22×C43—22no—
318C22×C43—41no—
319C22×C43—41no—
320C22×C43—41no—
321C22×C43—41no—
322C22×C43#54311yescharacteristic
323C22×C43D4211yescharacteristic
324C22×C43#55111yescharacteristic
325C4×C43—22no—
326C4×C43—22no—
327C4×C43—42no—
328C4×C43—81no—
329C4×C43—81no—
330C4×C43C2×C22⋊Q811yescharacteristic
331C4×C43C2×C22⋊Q811yescharacteristic
332C4×C43#54611yescharacteristic
333C4×C43#54611yescharacteristic
334Dic43—41no—
335Dic43—41no—
336Dic43—41no—
337Dic43—41no—
338Dic43—81no—
339Dic43—81no—
340#3084—431no—
341#3083—431no—
342#2898—431no—
343#2898—431no—
344#2097—431no—
345#2064—431no—
346#2064—431no—
347#2028—431no—
348#1866—431no—
349#1866—431no—
350#1860—431no—
351#1860—431no—
352#1816—431no—
353#1815—431no—
354#1793D4311yescharacteristic
355C2×C43—21no—
356C2×C43—21no—
357C2×C43—42no—
358C2×C43—42no—
359C2×C43#243711yescharacteristic
360C2×C43#328911yescharacteristic
361C2×C43#329011yescharacteristic
362C2×2−1+4C2×D4311yescharacteristic
363C2×#131—431no—
364C2×#131—431no—
365#544—431no—
366#520—431no—
367#520—431no—
368#479—861no—
369#479—861no—
370#478—861no—
371#478—861no—
372#477—861no—
373#476—861no—
374#475—861no—
375#474—861no—
376C2×#108—431no—
377C2×#108—431no—
378#428—431no—
379#428—431no—
380C2×#107—431no—
381C2×#106—431no—
382#424—431no—
383#424—431no—
384C2×#104C2×D4311yescharacteristic
385C2×#104C2×D4311yescharacteristic
386#419—431no—
387#419—431no—
388#419—861no—
389#419—861no—
390#417—431no—
391#417—431no—
392#417—861no—
393#417—861no—
394#417C2×D4311yescharacteristic
395#417C2×D4311yescharacteristic
396#416—431no—
397#416—431no—
398#416—861no—
399#416—861no—
400C2×#103—431no—
401C2×#103—431no—
402C2×#103—861no—
403C2×#103—861no—
404C2×#101C2×D4311yescharacteristic
405#391C2×D4311yescharacteristic
406#382—431no—
407#343—431no—
408#343—431no—
409#343—431no—
410#339—431no—
411#339—431no—
412#336—431no—
413C43#2598511yescharacteristic, complemented, Sylow
4142−1+4—22no—
4152−1+4—22no—
4162−1+4—41no—
4172−1+4Dic8614yescomplemented
418C2×#41—21no—
419C2×#41—21no—
420C2×#41—41no—
421C2×#41C22×D4311yescharacteristic
422C2×#41C22×D4311yescharacteristic
423C22×Q8—21no—
424C22×Q8—21no—
425C22×Q8C22×D4311yescharacteristic
426#131—862no—
427#131—862no—
428#131—1721no—
429#131—1721no—
430C2×M16—431no—
431C2×M16—431no—
432C2×M16—861no—
433C2×M16—861no—
434C2×M16C22×D4311yescharacteristic
435C2×M16C22×D4311yescharacteristic
436C22×C8—431no—
437C22×C8—431no—
438C22×C8—861no—
439C22×C8—861no—
440C4⋊Q8—861no—
441C4⋊Q8—861no—
442#125—861no—
443#125—861no—
444C22⋊Q8—1721no—
445C4×Q8—861no—
446C4×Q8—861no—
447C4×D4—861no—
448C4×D4—861no—
449C4×D4—1721no—
450C4×D4—1721no—
451#117—431no—
452#117—431no—
453#117—861no—
454#117—861no—
455C2×C4⋊C4—431no—
456C2×C4⋊C4—431no—
457C2×C4⋊C4—861no—
458C2×C4⋊C4—861no—
459C2×C4⋊C4C22×D4311yescharacteristic
460C2×C42C22×D4311yescharacteristic
461#108—1721no—
462#108—1721no—
463#107—861no—
464#107—861no—
465#107—861no—
466#107—861no—
467#106—861no—
468#106—861no—
469#106—861no—
470#106—861no—
471C4⋊C8—861no—
472C4⋊C8—861no—
473C4⋊C8—861no—
474C4⋊C8—861no—
475#104—22no—
476#104—22no—
477#103—21no—
478#103—21no—
479#103—431no—
480#103—431no—
481#103—431no—
482#103—431no—
483#103—861no—
484#103—861no—
485#103—861no—
486#103—861no—
487#103—861no—
488#103—861no—
489#103—861no—
490#103—861no—
491#103—861no—
492#103—861no—
493#102—21no—
494#102—21no—
495#102—431no—
496#102—431no—
497#102—431no—
498#102—431no—
499#102—861no—
500#102—861no—
501#101—41no—
502#95—41no—
503#95—1721no—
504#41—21no—
505#41—21no—
506#41—21no—
507#41—21no—
508#41—24no—
509#41—24no—
510#41—44no—
511#41—42no—
512#41—42no—
513#41—44no—
514#41C2×Dic8612yes—
515#41C2×Dic8612yes—
516#41#1062190012yes—
517#41#1062190012yes—
518C2×Q8—21no—
519C2×Q8—24no—
520C2×Q8—24no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—21no—
524C2×Q8—21no—
525C2×Q8—41no—
526C2×Q8—42no—
527C2×Q8—42no—
528C2×Q8—1721no—
529C2×Q8C2×Dic8612yes—
530C2×Q8D4×D4311yescharacteristic
531C2×Q8D4×D4311yescharacteristic
532C2×D4—21no—
533C2×D4—21no—
534C2×D4—41no—
535C2×D4D4×D4311yescharacteristic
536C2×D4D4×D4311yescharacteristic
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—41no—
543C22×C4—41no—
544C22×C4—861no—
545C22×C4—861no—
546C22×C4—1721no—
547C22×C4—1721no—
548C22×C4C23×D4311yescharacteristic
549M16—21no—
550M16—21no—
551M16—21no—
552M16—21no—
553M16—861no—
554M16—861no—
555M16—861no—
556M16—861no—
557M16—1722no—
558M16—1722no—
559C2×C8—21no—
560C2×C8—21no—
561C2×C8—431no—
562C2×C8—431no—
563C2×C8—431no—
564C2×C8—431no—
565C2×C8—861no—
566C2×C8—861no—
567C2×C8—861no—
568C2×C8—861no—
569C2×C8—861no—
570C2×C8—861no—
571C2×C8—1722no—
572C2×C8—1722no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—431no—
576C4⋊C4—431no—
577C4⋊C4—431no—
578C4⋊C4—431no—
579C4⋊C4—861no—
580C4⋊C4—861no—
581C4⋊C4—861no—
582C4⋊C4—861no—
583C4⋊C4—1721no—
584C4⋊C4—1721no—
585C4⋊C4—1721no—
586C4⋊C4—1721no—
587C4⋊C4—1721no—
588C22⋊C4—861no—
589C22⋊C4—861no—
590C22⋊C4—1721no—
591C22⋊C4—1721no—
592C42—21no—
593C42—21no—
594C42—861no—
595C42—861no—
596C42—1721no—
597C42—1721no—
598C23—21no—
599C23—21no—
600C23#1306337011yescharacteristic
601Q8—21no—
602Q8—21no—
603Q8—22no—
604Q8—22no—
605Q8—44no—
606Q8—41no—
607Q8—41no—
608Q8—6881no—
609Q8#1306330212yes—
610Q8#1306330212yes—
611D4—22no—
612D4—22no—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—21no—
617D4—44no—
618D4#1306330212yes—
619D4#1306330212yes—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—81no—
636C2×C4—861no—
637C2×C4—861no—
638C2×C4—861no—
639C2×C4—861no—
640C2×C4—3441no—
641C2×C4—3441no—
642C2×C4—1721no—
643C2×C4—1721no—
644C2×C4—6881no—
645C2×C4—6881no—
646C2×C4C2×D4×D4311yescharacteristic
647C2×C4C2×D4×D4311yescharacteristic
648C2×C4C22×Dic8611yescharacteristic
649C2×C4C2×#1062190011yescharacteristic
650C2×C4C2×#1062190011yescharacteristic
651C2×C4#1306337011yescharacteristic
652C8—41no—
653C8—41no—
654C8—861no—
655C8—861no—
656C8—861no—
657C8—861no—
658C8—1721no—
659C8—1721no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#1458352511yescharacteristic
669C22#1458353111yescharacteristic
670C22#1458358011yescharacteristic
671C4—22no—
672C4—22no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—3441no—
677C4—3441no—
678C4—1721no—
679C4—1721no—
680C4—1721no—
681C4—1721no—
682C4C2×#1306330211yescharacteristic
683C4C2×#1306330211yescharacteristic
684C4#1458353611yescharacteristic
685C4#1458353611yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—42no—
690C2#1913826111yescharacteristic
691C2#1914636511yescharacteristic
692C2#1914637411yescharacteristic
subgroupG/H
centerC22#14583531
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1793C2
socleC22×C43D42
prime radicalC2×2−1+4×C43C22
hypercenter#1793D43
hypocenterC43#25985
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. #19144461C2semidirect
2. #19144450C2semidirect
5. #19136143C2semidirect
6. #19135958C2semidirect
7. #19135958C2semidirect
8. #19135756C2semidirect
9. #19134868C2semidirect
10. #19134868C2semidirect
11. #19134830C2semidirect
12. #19134830C2semidirect
13. #19134657C2semidirect
14. #19134652C2semidirect
136. 2−1+4×C43Q8semidirect
413. C43#25985semidirect
417. 2−1+4Dic86semidirect

Maximal subgroups

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

subgroupindex
#191444612
#191444502
#191431282
#191361432
#191359582
#191357562
#191348682
#191348302
#191346572
#191346522
C43×#17932
#2598543