Finite Group Databasebeta

← #25156748

Subgroups — #25156748

Order 11008.

868 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583102C211yesmaximal, characteristic
2#19138365C211yesmaximal, characteristic, complemented
3#19137500C211yesmaximal, characteristic
4#19136144C211yesmaximal, characteristic, complemented
5#19136094C211yesmaximal, characteristic, complemented
6#19136016C211yesmaximal, characteristic, complemented
7#19135839C211yesmaximal, characteristic, complemented
8#19135271C211yesmaximal, characteristic, complemented
9#19135152C211yesmaximal, characteristic, complemented
10#19134931C211yesmaximal, characteristic
11#19134929C211yesmaximal, characteristic
12#19134927C211yesmaximal, characteristic
13#19134849C211yesmaximal, characteristic, complemented
14#19134840C211yesmaximal, characteristic
15C2×C4×SD16×C43C211yesmaximal, characteristic
16C22×#10621883C2211yescharacteristic
17C2×#13063323C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×D4×Dic43C2211yescharacteristic
20#14583102—24no—
21C2×#13063258C2211yescharacteristic
22C2×Dic43⋊C8C2211yescharacteristic
23C2×#13063250C2211yescharacteristic
24C2×#13063250C2211yescharacteristic
25C2×#13063249C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063248C2211yescharacteristic
28C2×#13063246C2211yescharacteristic
29#14582837—21no—
30#14582830—21no—
31#14582806C2211yescharacteristic
32#14582805C2211yescharacteristic
33#14582804C2211yescharacteristic
34#14582797C2211yescharacteristic
35#14582795C2211yescharacteristic
36#14582795C2211yescharacteristic
37#14582783—21no—
38#14582783—21no—
39#14582772—21no—
40#14582771—21no—
41#14582764C2211yescharacteristic
42#14582757—21no—
43C4×Dic43⋊C4C2211yescharacteristic
44C2×C43⋊#95—21no—
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582653C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53C22×SD16×C43C2211yescharacteristic
54C2×C4×Q8×C43C2211yescharacteristic
55C2×C4×D4×C43C2211yescharacteristic
56C4×SD16×C43—24no—
57C2×#106×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C2×C4×C8×C43C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C23×Dic43—21no—
63C23×Dic43—21no—
64C2×#10621883—22no—
65C2×#10621883—22no—
66C2×#10621883—22no—
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69#13063323—22no—
70#13063323—41no—
71D4×Dic43—22no—
72D4×Dic43—22no—
73D4×Dic43—22no—
74D4×Dic43—41no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4D411yescharacteristic
84C2×Dic43⋊C4D411yescharacteristic
85C2×C22⋊Dic43—21no—
86C2×C22⋊Dic43—21no—
87C2×C4×Dic43C2311yescharacteristic
88C2×C4×Dic43C2311yescharacteristic
89#13063258—41no—
90Dic43⋊C8—22no—
91#13063250—22no—
92#13063250—22no—
93#13063249—22no—
94#13063249—22no—
95#13063248—41no—
96#13063246—22no—
97C43⋊#95—21no—
98C43⋊#95—21no—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95D411yescharacteristic
109C43⋊#95D411yescharacteristic
110C22×Q8×C43C2311yescharacteristic
111C22×D4×C43C2311yescharacteristic
112C23×C4×C43—21no—
113C2×SD16×C43—22no—
114C2×SD16×C43—22no—
115C2×SD16×C43D414yes—
116C22×C8×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C4×Q8×C43—22no—
119C4×Q8×C43—41no—
120C4×D4×C43—22no—
121C4×D4×C43—41no—
122C2×C4⋊C4×C43—21no—
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C22⋊C4×C43—21no—
126C2×C42×C43—21no—
127C2×C42×C43C2311yescharacteristic
128#106×C43—22no—
129#103×C43—22no—
130#102×C43—22no—
131C4×C8×C43—22no—
132C24×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43C2×D412yes—
139C2×D4×C43—21no—
140C2×D4×C43—21no—
141C2×D4×C43—21no—
142C2×D4×C43C2×D412yes—
143C2×D4×C43#4112yes—
144C2×D4×C43#4112yes—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43—42no—
149C22×C4×C43—42no—
150C22×C4×C43—81no—
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2411yescharacteristic
154C22×Dic43—24no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—42no—
160C22×Dic43—42no—
161C22×Dic43—81no—
162C22×Dic43—81no—
163C22×Dic43—81no—
164C22×Dic43C2×D411yescharacteristic
165C22×Dic43C2×D411yescharacteristic
166C22×Dic43C2×D411yescharacteristic
167C22×Dic43C2×D411yescharacteristic
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174SD16×C43—28no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43C2×D412yes—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—42no—
186C22⋊C4×C43—42no—
187C42×C43—21no—
188C42×C43—21no—
189C42×C43—42no—
190C22⋊Dic43—24no—
191C22⋊Dic43—42no—
192#10621883—44no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43#4112yes—
199Dic43⋊C4—22no—
200Dic43⋊C4—22no—
201Dic43⋊C4—42no—
202Dic43⋊C4—42no—
203Dic43⋊C4—42no—
204Dic43⋊C4—81no—
205Dic43⋊C4—81no—
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—21no—
209C4×Dic43—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C23×C43—22no—
217C23×C43—22no—
218C23×C43—22no—
219C23×C43—41no—
220C23×C43—41no—
221C23×C43—41no—
222C23×C43—41no—
223C23×C43C22×D411yescharacteristic
224Q8×C43—22no—
225Q8×C43—22no—
226Q8×C43—41no—
227Q8×C43—41no—
228D4×C43—22no—
229D4×C43—22no—
230D4×C43—22no—
231D4×C43#12314yes—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—22no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—84no—
247C2×C4×C43#13712yes—
248C2×C4×C43C22×D411yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43C2×#4111yescharacteristic
252C2×C4×C43H(5,𝔽2)11yescharacteristic
253C2×C4×C432−1+411yescharacteristic
254C2×Dic43—21no—
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—21no—
258C2×Dic43—22no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—84no—
278C2×Dic43—84no—
279C2×Dic43#13512yes—
280Dic86—41no—
281Dic86—41no—
282Dic86—41no—
283Dic86—41no—
284C8×C43—22no—
285C8×C43—41no—
286C43⋊C8—41no—
287C43⋊C8—41no—
288#26170—431nomaximal, Sylow
289C22×C43—24no—
290C22×C43—42no—
291C22×C43—42no—
292C22×C43—42no—
293C22×C43—42no—
294C22×C43—42no—
295C22×C43—42no—
296C22×C43#52511yescharacteristic
297C22×C43D4211yescharacteristic
298C22×C43#54811yescharacteristic
299C22×C43C2×#13511yescharacteristic
300C22×C43C2×#13711yescharacteristic
301C22×C43#57711yescharacteristic
302C22×C43#57711yescharacteristic
303C4×C43—22no—
304C4×C43—42no—
305C4×C43—41no—
306C4×C43—81no—
307C4×C43—81no—
308C4×C43C2×#12311yescharacteristic
309C4×C43#53911yescharacteristic
310C4×C43#54711yescharacteristic
311C4×C43#55311yescharacteristic
312Dic43—22no—
313Dic43—41no—
314Dic43—41no—
315Dic43—42no—
316Dic43—41no—
317Dic43—81no—
318Dic43—81no—
319Dic43—81no—
320C2×#461—431no—
321C2×C4×SD16D4311yescharacteristic
322#2446—431no—
323#2349—431no—
324#2097—431no—
325#2088—431no—
326#2074—431no—
327#2043—431no—
328#1933—431no—
329#1916—431no—
330#1877—431no—
331#1877—431no—
332#1877—431no—
333#1863—431no—
334#1862—431no—
335C2×C43—44no—
336C2×C43#244311yescharacteristic
337C2×C43#310711yescharacteristic
338C2×C43#312311yescharacteristic
339C2×C43#326611yescharacteristic
340C2×C43#328211yescharacteristic
341C2×C43#336011yescharacteristic
342C2×C43#337211yescharacteristic
343C22×SD16—431no—
344C22×SD16C2×D4311yescharacteristic
345C2×C4⋊Q8—431no—
346C2×C4×Q8C2×D4311yescharacteristic
347C2×C4×D4—431no—
348C2×C4×D4—431no—
349C2×C4×D4C2×D4311yescharacteristic
350#461—864no—
351C4×SD16—24no—
352C2×#107—431no—
353C2×#106C2×D4311yescharacteristic
354C2×C4⋊C8—431no—
355C2×#103—431no—
356C2×#103—431no—
357C2×#103—431no—
358C2×#103—431no—
359C2×#103—431no—
360C2×#103C2×D4311yescharacteristic
361C2×#102—431no—
362C2×#102C2×D4311yescharacteristic
363C2×C4×C8C2×D4311yescharacteristic
364#400—861no—
365#398—861no—
366#391—431no—
367#391—431no—
368#391—431no—
369#389—431no—
370#389—431no—
371#389—431no—
372#387—861no—
373#387—861no—
374#385—861no—
375#385—861no—
376#382—431no—
377#380—861no—
378C4×C4⋊C4—431no—
379C2×#95—861no—
380#340—431no—
381#340—431no—
382#340—431no—
383#340—431no—
384#336—431no—
385#336—431no—
386#336—431no—
387#336—431no—
388C43#2617011yescharacteristic, complemented, Sylow
389C22×Q8—431no—
390C22×Q8C22×D4311yescharacteristic
391C22×D4C22×D4311yescharacteristic
392C23×C4—21no—
393C23×C4—861no—
394C23×C4—861no—
395C2×SD16—22no—
396C2×SD16—22no—
397C2×SD16—862no—
398C2×SD16—862no—
399C2×SD16—862no—
400C2×SD16C22⋊D4314yes—
401C22×C8—431no—
402C22×C8—431no—
403C22×C8C22×D4311yescharacteristic
404C22×C8C22×D4311yescharacteristic
405C4⋊Q8—862no—
406C4⋊Q8—1721no—
407C4×Q8—22no—
408C4×Q8—41no—
409C4×D4—22no—
410C4×D4—41no—
411C4×D4—862no—
412C4×D4—862no—
413C4×D4—862no—
414C4×D4—1721no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—431no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—431no—
419C2×C4⋊C4—431no—
420C2×C4⋊C4—431no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4C22×D4311yescharacteristic
427C2×C4⋊C4C22×D4311yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—861no—
430C2×C22⋊C4—861no—
431C2×C42—21no—
432C2×C42—431no—
433C2×C42—431no—
434C2×C42C22×D4311yescharacteristic
435#107—1721no—
436#106—22no—
437C4⋊C8—862no—
438#103—22no—
439#103—862no—
440#103—862no—
441#103—862no—
442#103—862no—
443#103—1721no—
444#102—22no—
445#102—862no—
446C4×C8—22no—
447#95—431no—
448#95—431no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453#95—861no—
454#95—861no—
455#95—861no—
456#95—861no—
457#95—861no—
458#95—861no—
459#95—861no—
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—861no—
467C2×Q8—861no—
468C2×Q8—861no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×Q8—861no—
472C2×Q8C2×C22⋊D4312yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4#1062189812yes—
477C2×D4#1062189812yes—
478C2×D4C2×C22⋊D4312yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—431no—
486C22×C4—431no—
487C22×C4—431no—
488C22×C4—431no—
489C22×C4—864no—
490C22×C4—861no—
491C22×C4—861no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—3441no—
495C22×C4—3441no—
496C22×C4—3441no—
497C22×C4—1722no—
498C22×C4—1722no—
499C22×C4D4×D4311yescharacteristic
500C22×C4D4×D4311yescharacteristic
501C22×C4C23×D4311yescharacteristic
502SD16—28no—
503SD16—1724no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—861no—
510C2×C8—861no—
511C2×C8—861no—
512C2×C8—861no—
513C2×C8—861no—
514C2×C8—861no—
515C2×C8C2×C22⋊D4312yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—432no—
522C4⋊C4—862no—
523C4⋊C4—862no—
524C4⋊C4—861no—
525C4⋊C4—861no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—3441no—
530C4⋊C4—3441no—
531C4⋊C4—1722no—
532C4⋊C4—1722no—
533C4⋊C4—1722no—
534C22⋊C4—42no—
535C22⋊C4—864no—
536C22⋊C4—1722no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—861no—
541C42—861no—
542C42—861no—
543C42—861no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×D4×D4311yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556Q8—1721no—
557Q8—1721no—
558Q8—1721no—
559Q8—1721no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1306330814yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—432no—
580C2×C4—861no—
581C2×C4—861no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—862no—
585C2×C4—861no—
586C2×C4—861no—
587C2×C4—861no—
588C2×C4—861no—
589C2×C4—861no—
590C2×C4—861no—
591C2×C4—861no—
592C2×C4—3444no—
593C2×C4—3444no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4—1721no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4#1306335412yes—
606C2×C4C22×C22⋊D4311yescharacteristic
607C2×C4#41×D4311yescharacteristic
608C2×C4C2×#1062189811yescharacteristic
609C2×C4C2×#1062189811yescharacteristic
610C2×C4#1306337011yescharacteristic
611C2×C4#1306337211yescharacteristic
612C8—22no—
613C8—41no—
614C8—1721no—
615C8—1721no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22#1458340811yescharacteristic
624C22D4×C22⋊D4311yescharacteristic
625C22#1458355811yescharacteristic
626C22#135×D4311yescharacteristic
627C22C2×#1306335411yescharacteristic
628C22#1458370511yescharacteristic
629C22#1458370611yescharacteristic
630C4—22no—
631C4—42no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—862no—
636C4—3441no—
637C4—3441no—
638C4—3441no—
639C4—1721no—
640C4—1721no—
641C4—1722no—
642C4—1721no—
643C4C2×#1306330811yescharacteristic
644C4#1458349111yescharacteristic
645C4#1458354811yescharacteristic
646C4#1458359811yescharacteristic
647C2—44no—
648C2#1913832411yescharacteristic
649C2#1914466211yescharacteristic
650C2#1914484411yescharacteristic
651C2#1914610511yescharacteristic
652C2#1914629611yescharacteristic
653C2#1914724511yescharacteristic
654C2#1914737411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C4×SD16×C43C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenterC2×C4×SD16D43
hypocenterC43#26170
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. #19138365C2semidirect
4. #19136144C2semidirect
5. #19136094C2semidirect
6. #19136016C2semidirect
7. #19135839C2semidirect
8. #19135271C2semidirect
9. #19135152C2semidirect
13. #19134849C2semidirect
388. C43#26170semidirect

Maximal subgroups

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

subgroupindex
C2×#145831022
#191383652
#191375002
#191361442
#191360942
#191360162
#191358392
#191352712
#191351522
#191349312
#191349292
#191349272
#191348492
#191348402
C2×C4×SD16×C432
#2617043