Finite Group Databasebeta

← #25159691

Subgroups — #25159691

Order 11008.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583196C211yesmaximal, characteristic
2C2×#14582986C211yesmaximal, characteristic
3#19138452C211yesmaximal, characteristic, complemented
4#19137504C211yesmaximal, characteristic
5#19136140C211yesmaximal, characteristic, complemented
6#19136080C211yesmaximal, characteristic, complemented
7#19135973C211yesmaximal, characteristic, complemented
8#19135322C211yesmaximal, characteristic, complemented
9#19135155C211yesmaximal, characteristic, complemented
10#19134930C211yesmaximal, characteristic
11#19134928C211yesmaximal, characteristic
12#19134857C211yesmaximal, characteristic
13#19134850C211yesmaximal, characteristic, complemented
14#19134839C211yesmaximal, characteristic
15C43×#2059C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C2×#13063290C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19C2×#13063287C2211yescharacteristic
20C2×#13063287C2211yescharacteristic
21#14583196—24no—
22#14582986—24no—
23C2×#13063258C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063250C2211yescharacteristic
28C2×#13063248C2211yescharacteristic
29C2×#13063248C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31C2×#13063245C2211yescharacteristic
32C2×#13063245C2211yescharacteristic
33C2×C8×Dic43C2211yescharacteristic
34#14582837—21no—
35#14582832—21no—
36#14582805C2211yescharacteristic
37#14582804C2211yescharacteristic
38#14582798C2211yescharacteristic
39#14582798C2211yescharacteristic
40#14582795—21no—
41#14582795C2211yescharacteristic
42#14582780—21no—
43#14582773—21no—
44#14582770—21no—
45#14582764C2211yescharacteristic
46#14582756—21no—
47C4×Dic43⋊C4C2211yescharacteristic
48#95×D43—21no—
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#125×C43C2211yescharacteristic
55C2×#107×C43C2211yescharacteristic
56C2×#106×C43C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×D172C2311yescharacteristic
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C2×#10621873—22no—
67C2×#10621873—22no—
68C2×#10621873Q814yes—
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71#13063290—22no—
72#13063290—41no—
73C4×D172—22no—
74C4×D172—41no—
75#13063287—22no—
76#13063287—22no—
77#13063287—22no—
78#13063287—41no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4D411yescharacteristic
86C2×Dic43⋊C4D411yescharacteristic
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×C4×Dic43—21no—
91C2×C4×Dic43C2311yescharacteristic
92C2×C4×Dic43C2311yescharacteristic
93#13063258—22no—
94#13063256—22no—
95#13063256—22no—
96C4⋊(C43⋊C8)—22no—
97#13063250—41no—
98#13063248—22no—
99#13063248—22no—
100#13063247—22no—
101#13063245—22no—
102#13063245D414yes—
103C8×Dic43—22no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95D411yescharacteristic
114C43⋊#95D411yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C22×C8×C43C2311yescharacteristic
117#125×C43—22no—
118#125×C43—41no—
119C2×C4⋊C4×C43—21no—
120C2×C4⋊C4×C43—21no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C42×C43C2311yescharacteristic
127#107×C43—41no—
128#106×C43—22no—
129#95×C43—21no—
130C23×D43—21no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2411yescharacteristic
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43C2×D411yescharacteristic
142C22×Dic43C2×D411yescharacteristic
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172C2×D412yes—
149C2×D172C2×Q812yes—
150C2×D172#4112yes—
151C2×C4×D43—24no—
152C2×C4×D43—42no—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×C4×D43—42no—
156C2×C4×D43—81no—
157C2×C4×D43—81no—
158C2×C4×D43—81no—
159C2×C4×D43—81no—
160C2×Dic86—21no—
161C2×Dic86—21no—
162C2×Dic86—21no—
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86C2×Q812yes—
166C2×C8×C43—21no—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43C2×Q812yes—
172C4⋊C4×C43—21no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—42no—
179C4⋊C4×C43—42no—
180C4⋊C4×C43—81no—
181C4⋊C4×C43C2×D412yes—
182C4⋊C4×C43#4112yes—
183C42×C43—21no—
184C42×C43—21no—
185(C2×D43)⋊C4—24no—
186(C2×D43)⋊C4—42no—
187(C2×D43)⋊C4—42no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—42no—
193C4⋊Dic43—81no—
194Dic43⋊C4—22no—
195Dic43⋊C4—41no—
196Dic43⋊C4—42no—
197Dic43⋊C4—42no—
198C4×Dic43—21no—
199C4×Dic43—21no—
200C4×Dic43—21no—
201C4×Dic43—21no—
202C4×Dic43—42no—
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8C2×D412yes—
209#10621873—28no—
210C23×C43C22×D411yescharacteristic
211C22×D43—22no—
212C22×D43—22no—
213C22×D43—22no—
214C22×D43—41no—
215C22×D43—41no—
216C22×D43—41no—
217C22×D43—41no—
218C2×C4×C43—22no—
219C2×C4×C43—22no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43C22×D411yescharacteristic
237C2×C4×C43C22×Q811yescharacteristic
238C2×C4×C43C2×#4111yescharacteristic
239C2×C4×C43C2×#4111yescharacteristic
240C2×C4×C43H(5,𝔽2)11yescharacteristic
241C2×C4×C432−1+411yescharacteristic
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—21no—
247C2×Dic43—22no—
248C2×Dic43—21no—
249C2×Dic43—21no—
250C2×Dic43—21no—
251C2×Dic43—21no—
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43#13512yes—
263D172—22no—
264D172—22no—
265D172—22no—
266D172C22⋊Q814yes—
267C4×D43—84no—
268C4×D43—84no—
269C4×D43—84no—
270Dic86—22no—
271Dic86—22no—
272Dic86—41no—
273Dic86—41no—
274C8×C43—22no—
275C8×C43—41no—
276C43⋊C8—22no—
277C43⋊C8—41no—
278#26378—431nomaximal, Sylow
279C22×C43#52511yescharacteristic
280C22×C43D4211yescharacteristic
281C22×C43D4×Q811yescharacteristic
282C22×C43C2×#13511yescharacteristic
283C22×C43#57511yescharacteristic
284C22×C43#57711yescharacteristic
285C22×C43#57711yescharacteristic
286C2×D43—24no—
287C2×D43—42no—
288C2×D43—42no—
289C2×D43—42no—
290C2×D43—42no—
291C2×D43—42no—
292C2×D43—42no—
293C4×C43—42no—
294C4×C43—41no—
295C4×C43—42no—
296C4×C43—41no—
297C4×C43—81no—
298C4×C43—81no—
299C4×C43C2×C22⋊Q811yescharacteristic
300C4×C43#53911yescharacteristic
301C4×C43#54711yescharacteristic
302C4×C43#55411yescharacteristic
303Dic43—22no—
304Dic43—41no—
305Dic43—41no—
306Dic43—42no—
307Dic43—41no—
308Dic43—81no—
309Dic43—81no—
310C2×#478—431no—
311C2×C4×SD16—431no—
312#2453—431no—
313#2349—431no—
314#2097—431no—
315#2085—431no—
316#2066—431no—
317#2059D4311yescharacteristic, complemented
318#1941—431no—
319#1916—431no—
320#1877—431no—
321#1877—431no—
322#1864—431no—
323#1863—431no—
324#1862—431no—
325C2×C43#244911yescharacteristic
326C2×C43#310211yescharacteristic
327C2×C43#312311yescharacteristic
328C2×C43#326711yescharacteristic
329C2×C43#328211yescharacteristic
330C2×C43#338111yescharacteristic
331C2×C43#338811yescharacteristic
332D43—44no—
333C22×SD16—431no—
334C2×#125C2×D4311yescharacteristic
335C2×C4×Q8—431no—
336C2×C4×D4—431no—
337C2×C4×D4—431no—
338C2×C4×D4—431no—
339#478—864no—
340C4×SD16—864no—
341C2×#107—431no—
342C2×#107C2×D4311yescharacteristic
343C2×#106—431no—
344C2×#106—431no—
345C2×#106C2×D4311yescharacteristic
346C2×C4⋊C8—431no—
347C2×#103—431no—
348C2×#103—431no—
349C2×#103—431no—
350C2×#102—431no—
351C2×#102—431no—
352C2×#102—431no—
353C2×C4×C8—431no—
354#400—861no—
355#398—861no—
356#391—431no—
357#391—431no—
358#389—431no—
359#389—431no—
360#389—431no—
361#389—861no—
362#389C2×D4311yescharacteristic
363#387—861no—
364#385—861no—
365#385—861no—
366#382—431no—
367#380—861no—
368C4×C4⋊C4—431no—
369C2×#95—861no—
370#340—431no—
371#340—431no—
372#340C2×D4311yescharacteristic
373#340C2×D4311yescharacteristic
374#336—431no—
375#336—431no—
376#336—431no—
377#336C2×D4311yescharacteristic
378C43#2637811yescharacteristic, complemented, Sylow
379C22×Q8—431no—
380C22×D4—431no—
381C23×C4—861no—
382C23×C4—861no—
383C23×C4—861no—
384C2×SD16—434no—
385C2×SD16—862no—
386C2×SD16—862no—
387C22×C8—431no—
388C22×C8—431no—
389C22×C8C22×D4311yescharacteristic
390C22×C8C22×D4311yescharacteristic
391#125—22no—
392#125—41no—
393C4×Q8—862no—
394C4×Q8—1721no—
395C4×D4—862no—
396C4×D4—862no—
397C4×D4—862no—
398C4×D4—862no—
399C4×D4—1721no—
400C4×D4—1721no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—431no—
405C2×C4⋊C4—431no—
406C2×C4⋊C4—431no—
407C2×C4⋊C4—431no—
408C2×C4⋊C4—861no—
409C2×C4⋊C4—861no—
410C2×C4⋊C4—861no—
411C2×C4⋊C4—861no—
412C2×C4⋊C4C22×D4311yescharacteristic
413C2×C4⋊C4C22×D4311yescharacteristic
414C2×C4⋊C4C22×D4311yescharacteristic
415C2×C4⋊C4C22×D4311yescharacteristic
416C2×C22⋊C4—861no—
417C2×C22⋊C4—861no—
418C2×C22⋊C4—861no—
419C2×C42—431no—
420C2×C42—431no—
421C2×C42—861no—
422C2×C42C22×D4311yescharacteristic
423#107—41no—
424#107—862no—
425#106—22no—
426#106—862no—
427#106—862no—
428C4⋊C8—862no—
429#103—862no—
430#103—862no—
431#103—1721no—
432#102—434no—
433#102—862no—
434#102—862no—
435C4×C8—862no—
436#95—21no—
437#95—431no—
438#95—431no—
439#95—861no—
440#95—861no—
441#95—861no—
442#95—861no—
443#95—861no—
444#95—861no—
445#95—861no—
446#95—861no—
447#95—861no—
448C24—861no—
449C2×Q8—432no—
450C2×Q8—861no—
451C2×Q8—861no—
452C2×Q8—861no—
453C2×Q8—861no—
454C2×Q8—861no—
455C2×D4—432no—
456C2×D4—432no—
457C2×D4—432no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—431no—
466C22×C4—431no—
467C22×C4—431no—
468C22×C4—431no—
469C22×C4—864no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—861no—
473C22×C4—3441no—
474C22×C4—3441no—
475C22×C4—3441no—
476C22×C4—3441no—
477C22×C4—1722no—
478C22×C4—1722no—
479C22×C4—1722no—
480C22×C4—1722no—
481C22×C4D4×D4311yescharacteristic
482C22×C4D4×D4311yescharacteristic
483C22×C4C23×D4311yescharacteristic
484SD16—868no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—432no—
491C2×C8—861no—
492C2×C8—861no—
493C2×C8—861no—
494C2×C8—861no—
495C2×C8—861no—
496C2×C8Q8×D4312yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—862no—
507C4⋊C4—861no—
508C4⋊C4—861no—
509C4⋊C4—861no—
510C4⋊C4—861no—
511C4⋊C4—3441no—
512C4⋊C4—1721no—
513C4⋊C4—1722no—
514C4⋊C4—1722no—
515C4⋊C4—1722no—
516C4⋊C4D4×D4312yes—
517C4⋊C4#1062190212yes—
518C22⋊C4—864no—
519C22⋊C4—1722no—
520C22⋊C4—1722no—
521C42—21no—
522C42—21no—
523C42—861no—
524C42—861no—
525C42—861no—
526C42—861no—
527C42—1722no—
528C23—862no—
529C23—862no—
530C23—862no—
531C23—1721no—
532C23—1721no—
533C23—1721no—
534C23—1721no—
535C23C2×D4×D4311yescharacteristic
536Q8—862no—
537Q8—862no—
538Q8—1721no—
539Q8—1721no—
540D4—434no—
541D4—862no—
542D4—862no—
543D4—862no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—432no—
563C2×C4—861no—
564C2×C4—861no—
565C2×C4—862no—
566C2×C4—861no—
567C2×C4—861no—
568C2×C4—861no—
569C2×C4—861no—
570C2×C4—861no—
571C2×C4—861no—
572C2×C4—861no—
573C2×C4—861no—
574C2×C4—861no—
575C2×C4—3444no—
576C2×C4—3444no—
577C2×C4—3444no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4C2×D4×D4311yescharacteristic
587C2×C4C2×Q8×D4311yescharacteristic
588C2×C4#41×D4311yescharacteristic
589C2×C4C2×#1062190211yescharacteristic
590C2×C4#1306337011yescharacteristic
591C2×C4#1306337311yescharacteristic
592C8—22no—
593C8—41no—
594C8—862no—
595C8—1721no—
596C22—864no—
597C22—1722no—
598C22—1722no—
599C22—1722no—
600C22—1722no—
601C22—1722no—
602C22—1722no—
603C22#1458341211yescharacteristic
604C22#1458353111yescharacteristic
605C22#1458356411yescharacteristic
606C22#135×D4311yescharacteristic
607C22#1458369811yescharacteristic
608C22#1458370611yescharacteristic
609C22#1458370911yescharacteristic
610C4—41no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—862no—
617C4—3441no—
618C4—3441no—
619C4—1722no—
620C4—1721no—
621C4—1721no—
622C4—1721no—
623C4C22⋊Q8×D4311yescharacteristic
624C4#1458349511yescharacteristic
625C4#1458355211yescharacteristic
626C4#1458361011yescharacteristic
627C2—1724no—
628C2#1913839711yescharacteristic
629C2#1914462011yescharacteristic
630C2#1914484911yescharacteristic
631C2#1914612311yescharacteristic
632C2#1914630511yescharacteristic
633C2#1914746711yescharacteristic
634C2#1914753911yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2059C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2059D43
hypocenterC43#26378
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
3. #19138452C2semidirect
5. #19136140C2semidirect
6. #19136080C2semidirect
7. #19135973C2semidirect
8. #19135322C2semidirect
9. #19135155C2semidirect
13. #19134850C2semidirect
15. C43×#2059C2semidirect
317. #2059D43semidirect
378. C43#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#145831962
C2×#145829862
#191384522
#191375042
#191361402
#191360802
#191359732
#191353222
#191351552
#191349302
#191349282
#191348572
#191348502
#191348392
C43×#20592
#2637843