Finite Group Databasebeta

← #25154176

Subgroups — #25154176

Order 11008.

850 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144457C211yesmaximal, characteristic, complemented
2#19144455C211yesmaximal, characteristic, complemented
3#19143128C211yesmaximal, characteristic
4#19136136C211yesmaximal, characteristic, complemented
5#19135962C211yesmaximal, characteristic, complemented
6#19135960C211yesmaximal, characteristic, complemented
7#19135754C211yesmaximal, characteristic, complemented
8#19134872C211yesmaximal, characteristic, complemented
9#19134870C211yesmaximal, characteristic, complemented
10#19134821C211yesmaximal, characteristic, complemented
11#19134819C211yesmaximal, characteristic, complemented
12#19134654C211yesmaximal, characteristic, complemented
13#19134652C211yesmaximal, characteristic, complemented
14#19134554C211yesmaximal, characteristic
15C43×#2898C211yesmaximal, characteristic
16C2×#13063332C2211yescharacteristic
17#14583527C2211yescharacteristic
18#14583382C2211yescharacteristic
19#14583206—21no—
20#14583204—21no—
21#14583201—21no—
22#14583199—21no—
23#14583189—21no—
24#14583187—21no—
25#14583184—21no—
26#14583182—21no—
27C2×#13063261C2211yescharacteristic
28C2×#13063260C2211yescharacteristic
29#14582942C2211yescharacteristic
30#14582939C2211yescharacteristic
31C2×#13063257C2211yescharacteristic
32C2×#13063256C2211yescharacteristic
33#14582926C2211yescharacteristic
34#14582925C2211yescharacteristic
35C2×#13063252C2211yescharacteristic
36C2×#13063252C2211yescharacteristic
37#14582908—21no—
38#14582908C2211yescharacteristic
39#14582900—21no—
40#14582900C2211yescharacteristic
41#14582900C2211yescharacteristic
42#14582900C2211yescharacteristic
43#14582895—21no—
44#14582895C2211yescharacteristic
45C2×#13063249—21no—
46C2×#13063249C2211yescharacteristic
47C2×#13063244C2211yescharacteristic
48#14582806C2211yescharacteristic
49#14582768C2211yescharacteristic
50#14582668C2211yescharacteristic
51#14582667C2211yescharacteristic
52#14582667C2211yescharacteristic
53#14582660C2211yescharacteristic
54#14582658C2211yescharacteristic
55#14582653C2211yescharacteristic
56C2×2−1+4×C43C2211yescharacteristic
57C2×#131×C43C2211yescharacteristic
58C43×#520C2211yescharacteristic
59C43×#419—21no—
60C43×#419C2211yescharacteristic
61C43×#417—21no—
62C43×#417C2211yescharacteristic
63C43×#416—21no—
64C43×#416C2211yescharacteristic
65C2×#103×C43—21no—
66C2×#103×C43C2211yescharacteristic
67#13063332—22no—
68#13063332—41no—
69C2×#10621877—21no—
70C2×#10621877C2311yescharacteristic
71C2×#10621877C2311yescharacteristic
72C2×#10621877C2311yescharacteristic
73C22×C43⋊C8—21no—
74C22×C43⋊C8C2311yescharacteristic
75#13063325—21no—
76#13063324—21no—
77#13063317—21no—
78#13063316—21no—
79#13063302—41no—
80Q8×Dic43—21no—
81D4×Dic43—21no—
82D4×Dic43—41no—
83#13063284—21no—
84#13063284C2311yescharacteristic
85C2×C4⋊Dic43—21no—
86C2×C4⋊Dic43C2311yescharacteristic
87C2×C4⋊Dic43C2311yescharacteristic
88C2×Dic43⋊C4—41no—
89C2×C4×Dic43C2311yescharacteristic
90#13063261—41no—
91#13063260—41no—
92#13063258—21no—
93#13063258—21no—
94#13063257—21no—
95#13063257—21no—
96#13063256—21no—
97#13063256—21no—
98#13063256—21no—
99#13063256—21no—
100C4⋊(C43⋊C8)—21no—
101C4⋊(C43⋊C8)—21no—
102Dic43⋊C8—21no—
103Dic43⋊C8—21no—
104#13063252—22no—
105#13063252—22no—
106#13063249—21no—
107#13063249—21no—
108#13063249—21no—
109#13063249—21no—
110#13063249—21no—
111#13063249—21no—
112#13063249—21no—
113#13063249D411yescharacteristic
114#13063249D411yescharacteristic
115#13063246—21no—
116#13063246—21no—
117#13063246—21no—
118#13063246D411yescharacteristic
119#13063246D411yescharacteristic
120#13063244—41no—
121C43⋊#95—41no—
1222−1+4×C43—22no—
1232−1+4×C43—22no—
1242−1+4×C43—41no—
1252−1+4×C43Q814yescomplemented
126C2×#41×C43—21no—
127C2×#41×C43—21no—
128C2×#41×C43—41no—
129C2×#41×C43C2311yescharacteristic
130C2×#41×C43C2311yescharacteristic
131C22×Q8×C43—21no—
132C22×Q8×C43—21no—
133C22×Q8×C43C2311yescharacteristic
134#131×C43—22no—
135#131×C43—41no—
136C2×M16×C43—21no—
137C2×M16×C43C2311yescharacteristic
138C22×C8×C43—21no—
139C22×C8×C43C2311yescharacteristic
140C4×Q8×C43—21no—
141C4×D4×C43—21no—
142C4×D4×C43—41no—
143#117×C43—21no—
144#117×C43C2311yescharacteristic
145C2×C4⋊C4×C43—21no—
146C2×C4⋊C4×C43C2311yescharacteristic
147#103×C43—21no—
148#103×C43—21no—
149#103×C43—21no—
150#103×C43—21no—
151#103×C43—21no—
152#103×C43D411yescharacteristic
153#103×C43D411yescharacteristic
154#102×C43—21no—
155#102×C43D411yescharacteristic
156#102×C43D411yescharacteristic
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—41no—
194C22×C4×C43—41no—
195C22×C4×C43—41no—
196C22×C4×C43C2411yescharacteristic
197C22×Dic43—21no—
198C22×Dic43—21no—
199C22×Dic43—21no—
200C22×Dic43—41no—
201C2×Dic86—41no—
202M16×C43—21no—
203M16×C43—21no—
204M16×C43—42no—
205C2×C8×C43—21no—
206C2×C8×C43—21no—
207C2×C8×C43—21no—
208C2×C8×C43—42no—
209C2×C8×C43C2×D411yescharacteristic
210C2×C8×C43C2×D411yescharacteristic
211C4⋊C4×C43—21no—
212C4⋊C4×C43—21no—
213C4⋊C4×C43—41no—
214C4⋊C4×C43C2×D411yescharacteristic
215C4⋊C4×C43C2×D411yescharacteristic
216C22⋊C4×C43—21no—
217C22⋊C4×C43—41no—
218C42×C43—21no—
219C42×C43—41no—
220C22⋊Dic43—21no—
221C22⋊Dic43—41no—
222C4⋊Dic43—21no—
223C4⋊Dic43—21no—
224C4⋊Dic43—21no—
225C4⋊Dic43—21no—
226C4⋊Dic43—41no—
227C4⋊Dic43—41no—
228C4⋊Dic43C2×D411yescharacteristic
229C4⋊Dic43C2×D411yescharacteristic
230Dic43⋊C4—41no—
231Dic43⋊C4—41no—
232Dic43⋊C4—82no—
233C4×Dic43—21no—
234C4×Dic43—21no—
235C4×Dic43—21no—
236C4×Dic43—41no—
237#10621877—21no—
238#10621877—21no—
239#10621877—21no—
240#10621877—21no—
241#10621877—21no—
242#10621877—21no—
243#10621877—42no—
244C2×C43⋊C8—21no—
245C2×C43⋊C8—21no—
246C2×C43⋊C8—21no—
247C2×C43⋊C8—21no—
248C2×C43⋊C8—21no—
249C2×C43⋊C8—42no—
250C2×C43⋊C8C2×D411yescharacteristic
251C2×C43⋊C8C2×D411yescharacteristic
252C23×C43—21no—
253C23×C43—21no—
254C23×C43H(5,𝔽2)11yescharacteristic
255Q8×C43—21no—
256Q8×C43—21no—
257Q8×C43—22no—
258Q8×C43—22no—
259Q8×C43—44no—
260Q8×C43—41no—
261Q8×C43—41no—
262Q8×C43C22⋊Q812yes—
263Q8×C43C22⋊Q812yes—
264D4×C43—22no—
265D4×C43—22no—
266D4×C43—22no—
267D4×C43—22no—
268D4×C43—21no—
269D4×C43—21no—
270D4×C43—44no—
271D4×C43C22⋊Q812yes—
272D4×C43C22⋊Q812yes—
273C2×C4×C43—21no—
274C2×C4×C43—21no—
275C2×C4×C43—22no—
276C2×C4×C43—22no—
277C2×C4×C43—22no—
278C2×C4×C43—22no—
279C2×C4×C43—22no—
280C2×C4×C43—21no—
281C2×C4×C43—21no—
282C2×C4×C43—44no—
283C2×C4×C43—44no—
284C2×C4×C43—41no—
285C2×C4×C43—41no—
286C2×C4×C43—41no—
287C2×C4×C43—81no—
288C2×C4×C43—161no—
289C2×C4×C43C22×D411yescharacteristic
290C2×C4×C43C22×D411yescharacteristic
291C2×C4×C43C22×Q811yescharacteristic
292C2×C4×C43C2×#4111yescharacteristic
293C2×C4×C43C2×#4111yescharacteristic
294C2×C4×C43H(5,𝔽2)11yescharacteristic
295C2×Dic43—22no—
296C2×Dic43—21no—
297C2×Dic43—21no—
298C2×Dic43—41no—
299C2×Dic43—41no—
300C2×Dic43—41no—
301C2×Dic43—41no—
302C2×Dic43—81no—
303C2×Dic43—81no—
304C2×Dic43—161no—
305Dic86—161no—
306C8×C43—21no—
307C8×C43—21no—
308C8×C43—41no—
309C43⋊C8—21no—
310C43⋊C8—21no—
311C43⋊C8—41no—
312C43⋊C8—41no—
313C43⋊C8—41no—
314#25986—431nomaximal, Sylow
315C22×C43—21no—
316C22×C43—21no—
317C22×C43—22no—
318C22×C43—22no—
319C22×C43—41no—
320C22×C43—41no—
321C22×C43—41no—
322C22×C43—41no—
323C22×C43#54311yescharacteristic
324C22×C43D4211yescharacteristic
325C22×C43#55111yescharacteristic
326C4×C43—22no—
327C4×C43—22no—
328C4×C43—42no—
329C4×C43—41no—
330C4×C43—41no—
331C4×C43—81no—
332C4×C43C2×C22⋊Q811yescharacteristic
333C4×C43C2×C22⋊Q811yescharacteristic
334C4×C43#54611yescharacteristic
335C4×C43#54611yescharacteristic
336Dic43—42no—
337Dic43—41no—
338Dic43—41no—
339Dic43—81no—
340Dic43—81no—
341#3084—431no—
342#3084—431no—
343#2898—431no—
344#2898D4311yescharacteristic
345#2096—431no—
346#2064—431no—
347#2064—431no—
348#2028—431no—
349#1866—431no—
350#1866—431no—
351#1859—431no—
352#1859—431no—
353#1815—431no—
354#1815—431no—
355#1793—431no—
356C2×C43—21no—
357C2×C43—21no—
358C2×C43—42no—
359C2×C43—42no—
360C2×C43#243711yescharacteristic
361C2×C43#328911yescharacteristic
362C2×C43#329111yescharacteristic
363C2×2−1+4C2×D4311yescharacteristic
364C2×#131—431no—
365C2×#131C2×D4311yescharacteristic
366#544—431no—
367#520—431no—
368#520C2×D4311yescharacteristic
369#479—861no—
370#479—861no—
371#478—861no—
372#478—861no—
373#476—861no—
374#476—861no—
375#475—861no—
376#475—861no—
377C2×#108—431no—
378C2×#108—431no—
379#428—431no—
380#428—431no—
381C2×#106—431no—
382C2×#106—431no—
383#424—431no—
384#424—431no—
385C2×#104—431no—
386C2×#104—431no—
387#419—21no—
388#419—431no—
389#419—861no—
390#419C2×D4311yescharacteristic
391#417—21no—
392#417—431no—
393#417—431no—
394#417—431no—
395#417—861no—
396#417C2×D4311yescharacteristic
397#416—21no—
398#416—431no—
399#416—861no—
400#416C2×D4311yescharacteristic
401C2×#103—21no—
402C2×#103—431no—
403C2×#103—861no—
404C2×#103C2×D4311yescharacteristic
405C2×#101—431no—
406#391—431no—
407#384—431no—
408#343—431no—
409#343—431no—
410#343—431no—
411#339—431no—
412#339—431no—
413#336—431no—
414C43#2598611yescharacteristic, complemented, Sylow
4152−1+4—22no—
4162−1+4—22no—
4172−1+4—41no—
4182−1+4Dic8614yescomplemented
419C2×#41—21no—
420C2×#41—21no—
421C2×#41—41no—
422C2×#41C22×D4311yescharacteristic
423C2×#41C22×D4311yescharacteristic
424C22×Q8—21no—
425C22×Q8—21no—
426C22×Q8C22×D4311yescharacteristic
427#131—22no—
428#131—41no—
429#131—862no—
430#131—1721no—
431C2×M16—21no—
432C2×M16—431no—
433C2×M16—431no—
434C2×M16—431no—
435C2×M16—861no—
436C2×M16C22×D4311yescharacteristic
437C22×C8—21no—
438C22×C8—431no—
439C22×C8—861no—
440C22×C8C22×D4311yescharacteristic
441C4⋊Q8—861no—
442C4⋊Q8—861no—
443#125—861no—
444#125—861no—
445C22⋊Q8—1721no—
446C4×Q8—21no—
447C4×Q8—861no—
448C4×D4—21no—
449C4×D4—41no—
450C4×D4—861no—
451C4×D4—1721no—
452#117—21no—
453#117—431no—
454#117—861no—
455#117C22×D4311yescharacteristic
456C2×C4⋊C4—21no—
457C2×C4⋊C4—431no—
458C2×C4⋊C4—431no—
459C2×C4⋊C4—861no—
460C2×C4⋊C4—1721no—
461C2×C4⋊C4C22×D4311yescharacteristic
462C2×C42—431no—
463#108—1721no—
464#108—1721no—
465#107—861no—
466#107—861no—
467#106—861no—
468#106—861no—
469#106—861no—
470#106—861no—
471#106—861no—
472#106—861no—
473C4⋊C8—861no—
474C4⋊C8—861no—
475C4⋊C8—861no—
476C4⋊C8—861no—
477#104—862no—
478#104—862no—
479#103—21no—
480#103—21no—
481#103—21no—
482#103—21no—
483#103—21no—
484#103—431no—
485#103—431no—
486#103—861no—
487#103—861no—
488#103—861no—
489#103—861no—
490#103—861no—
491#103—861no—
492#103—861no—
493#103C22⋊D4311yescharacteristic
494#103C22⋊D4311yescharacteristic
495#102—21no—
496#102—431no—
497#102—431no—
498#102—861no—
499#102—861no—
500#102—861no—
501#102C22⋊D4311yescharacteristic
502#102C22⋊D4311yescharacteristic
503#101—1721no—
504#95—1721no—
505#41—21no—
506#41—21no—
507#41—21no—
508#41—21no—
509#41—24no—
510#41—24no—
511#41—44no—
512#41—42no—
513#41—42no—
514#41—44no—
515#41C2×Dic8612yes—
516#41C2×Dic8612yes—
517#41#1062189812yes—
518#41#1062190012yes—
519C2×Q8—21no—
520C2×Q8—24no—
521C2×Q8—24no—
522C2×Q8—21no—
523C2×Q8—21no—
524C2×Q8—21no—
525C2×Q8—21no—
526C2×Q8—41no—
527C2×Q8—42no—
528C2×Q8—42no—
529C2×Q8—1721no—
530C2×Q8C2×Dic8612yes—
531C2×Q8D4×D4311yescharacteristic
532C2×Q8C2×C22⋊D4311yescharacteristic
533C2×D4—21no—
534C2×D4—21no—
535C2×D4—41no—
536C2×D4D4×D4311yescharacteristic
537C2×D4C2×C22⋊D4311yescharacteristic
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—41no—
543C22×C4—41no—
544C22×C4—41no—
545C22×C4—861no—
546C22×C4—861no—
547C22×C4—861no—
548C22×C4—1721no—
549C22×C4C23×D4311yescharacteristic
550M16—21no—
551M16—21no—
552M16—42no—
553M16—861no—
554M16—861no—
555M16—861no—
556M16—861no—
557M16—861no—
558M16—861no—
559M16—1722no—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—42no—
564C2×C8—431no—
565C2×C8—431no—
566C2×C8—861no—
567C2×C8—861no—
568C2×C8—861no—
569C2×C8—861no—
570C2×C8—861no—
571C2×C8—1722no—
572C2×C8C2×C22⋊D4311yescharacteristic
573C2×C8C2×C22⋊D4311yescharacteristic
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—41no—
577C4⋊C4—431no—
578C4⋊C4—431no—
579C4⋊C4—861no—
580C4⋊C4—861no—
581C4⋊C4—861no—
582C4⋊C4—861no—
583C4⋊C4—3442no—
584C4⋊C4—1721no—
585C4⋊C4—1721no—
586C4⋊C4—1721no—
587C4⋊C4—1721no—
588C4⋊C4C2×C22⋊D4311yescharacteristic
589C4⋊C4C2×C22⋊D4311yescharacteristic
590C22⋊C4—21no—
591C22⋊C4—41no—
592C22⋊C4—861no—
593C22⋊C4—1721no—
594C42—21no—
595C42—41no—
596C42—861no—
597C42—861no—
598C42—861no—
599C42—1721no—
600C23—21no—
601C23—21no—
602C23#1306337011yescharacteristic
603Q8—21no—
604Q8—21no—
605Q8—22no—
606Q8—22no—
607Q8—44no—
608Q8—41no—
609Q8—41no—
610Q8—6881no—
611Q8#1306330212yes—
612Q8#1306330412yes—
613D4—22no—
614D4—22no—
615D4—22no—
616D4—22no—
617D4—21no—
618D4—21no—
619D4—44no—
620D4#1306330212yes—
621D4#1306330412yes—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—81no—
637C2×C4—161no—
638C2×C4—862no—
639C2×C4—861no—
640C2×C4—861no—
641C2×C4—3441no—
642C2×C4—3441no—
643C2×C4—1721no—
644C2×C4—1721no—
645C2×C4—1721no—
646C2×C4—1721no—
647C2×C4—6881no—
648C2×C4C2×D4×D4311yescharacteristic
649C2×C4C22×C22⋊D4311yescharacteristic
650C2×C4C22×Dic8611yescharacteristic
651C2×C4C2×#1062190011yescharacteristic
652C2×C4C2×#1062189811yescharacteristic
653C2×C4#1306337011yescharacteristic
654C8—21no—
655C8—21no—
656C8—41no—
657C8—861no—
658C8—861no—
659C8—1721no—
660C8—1721no—
661C8—1721no—
662C22—21no—
663C22—21no—
664C22—22no—
665C22—22no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22#1458352511yescharacteristic
671C22D4×C22⋊D4311yescharacteristic
672C22#1458357511yescharacteristic
673C4—22no—
674C4—22no—
675C4—41no—
676C4—41no—
677C4—42no—
678C4—81no—
679C4—3441no—
680C4—3441no—
681C4—1722no—
682C4—1721no—
683C4—1721no—
684C4C2×#1306330211yescharacteristic
685C4C2×#1306330411yescharacteristic
686C4#1458353911yescharacteristic
687C4#1458354311yescharacteristic
688C2—21no—
689C2—21no—
690C2—42no—
691C2—42no—
692C2#1913825611yescharacteristic
693C2#1914636611yescharacteristic
694C2#1914638611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2898C2
socleC22×C43D42
prime radicalC2×2−1+4×C43C22
hypercenter#2898D43
hypocenterC43#25986
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2898

Complements

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

NH ≅ G/Ntype
1. #19144457C2semidirect
2. #19144455C2semidirect
4. #19136136C2semidirect
5. #19135962C2semidirect
6. #19135960C2semidirect
7. #19135754C2semidirect
8. #19134872C2semidirect
9. #19134870C2semidirect
10. #19134821C2semidirect
11. #19134819C2semidirect
12. #19134654C2semidirect
13. #19134652C2semidirect
125. 2−1+4×C43Q8semidirect
414. C43#25986semidirect
418. 2−1+4Dic86semidirect

Maximal subgroups

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

subgroupindex
#191444572
#191444552
#191431282
#191361362
#191359622
#191359602
#191357542
#191348722
#191348702
#191348212
#191348192
#191346542
#191346522
#191345542
C43×#28982
#2598643