Finite Group Databasebeta

← #25146153

Subgroups — #25146153

Order 11008.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583023C211yesmaximal, characteristic, complemented
2#19140084C211yesmaximal, characteristic
3#19139957C211yesmaximal, characteristic, complemented
4#19135981C211yesmaximal, characteristic
5#19135974C211yesmaximal, characteristic, complemented
6#19135915C211yesmaximal, characteristic, complemented
7#19135908C211yesmaximal, characteristic
8#19135457C211yesmaximal, characteristic, complemented
9#19135451C211yesmaximal, characteristic
10#19135407C211yesmaximal, characteristic
11#19135365C211yesmaximal, characteristic, complemented
12#19135275C211yesmaximal, characteristic
13#19135268C211yesmaximal, characteristic, complemented
14C4×#13063249C211yesmaximal, characteristic, complemented
15C2×C43×#464C211yesmaximal, characteristic
16C2×C4×Dic86C2211yescharacteristic
17#14583023—24no—
18C2×#13063259C2211yescharacteristic
19C2×#13063257C2211yescharacteristic
20C2×Dic43⋊C8C2211yescharacteristic
21C2×Dic43⋊C8C2211yescharacteristic
22C2×#13063249C2211yescharacteristic
23C2×#13063249C2211yescharacteristic
24C2×#13063249C2211yescharacteristic
25C2×#13063249C2211yescharacteristic
26C2×#13063246C2211yescharacteristic
27C2×#13063246C2211yescharacteristic
28C2×C4×C43⋊C8C2211yescharacteristic
29C2×C8×Dic43C2211yescharacteristic
30C2×C8×Dic43C2211yescharacteristic
31#14582838—21no—
32#14582824—21no—
33#14582819—21no—
34#14582811—21no—
35#14582806C2211yescharacteristic
36#14582806C2211yescharacteristic
37#14582806C2211yescharacteristic
38#14582806C2211yescharacteristic
39#14582802C2211yescharacteristic
40#14582802C2211yescharacteristic
41#14582795C2211yescharacteristic
42#14582795C2211yescharacteristic
43#14582793—21no—
44#14582793—21no—
45#14582783—21no—
46#14582783—21no—
47#14582765C2211yescharacteristic
48C4×C4⋊Dic43C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53C22×Q16×C43C2211yescharacteristic
54C22×SD16×C43C2211yescharacteristic
55C2×#124×C43C2211yescharacteristic
56C2×C4×Q8×C43C2211yescharacteristic
57C43×#464—24no—
58C2×C4⋊C8×C43C2211yescharacteristic
59C2×#103×C43C2211yescharacteristic
60C2×#102×C43C2211yescharacteristic
61C22×Dic86D411yescharacteristic
62C22×C43⋊C8C2311yescharacteristic
63C22×C43⋊C8C2311yescharacteristic
64C4×Dic86—22no—
65C4×Dic86—22no—
66C2×C4⋊Dic43D411yescharacteristic
67C2×C4⋊Dic43C2311yescharacteristic
68C2×C4⋊Dic43C2311yescharacteristic
69C2×C4⋊Dic43C2311yescharacteristic
70C2×C4⋊Dic43C2311yescharacteristic
71C2×Dic43⋊C4—21no—
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4D411yescharacteristic
74C2×Dic43⋊C4D411yescharacteristic
75C2×C22⋊Dic43—21no—
76C2×C22⋊Dic43—21no—
77C2×C22⋊Dic43—21no—
78C2×C22⋊Dic43—21no—
79C2×C4×Dic43—21no—
80C2×C4×Dic43C2311yescharacteristic
81C2×C4×Dic43C2311yescharacteristic
82#13063259—41no—
83#13063257—41no—
84Dic43⋊C8—22no—
85Dic43⋊C8—22no—
86#13063249—22no—
87#13063249—22no—
88#13063249—22no—
89#13063249—41no—
90#13063246—22no—
91#13063246—41no—
92C4×C43⋊C8—22no—
93C8×Dic43—22no—
94C8×Dic43—22no—
95C43⋊#95—21no—
96C43⋊#95—21no—
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—
108C22×Q8×C43C2311yescharacteristic
109C22×Q8×C43C2311yescharacteristic
110C22×D4×C43C2311yescharacteristic
111C2×Q16×C43—22no—
112C2×Q16×C43—22no—
113C2×Q16×C43—41no—
114C2×SD16×C43—22no—
115C2×SD16×C43—22no—
116C2×SD16×C43—41no—
117C22×C8×C43C2311yescharacteristic
118C22×C8×C43C2311yescharacteristic
119#124×C43—22no—
120#124×C43—22no—
121C4×Q8×C43—22no—
122C4×Q8×C43—22no—
123C2×C4⋊C4×C43—21no—
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C22⋊C4×C43—21no—
126C2×C22⋊C4×C43—21no—
127C2×C42×C43—21no—
128C2×C42×C43C2311yescharacteristic
129C4⋊C8×C43D414yes—
130#103×C43—22no—
131#102×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×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43#4112yes—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—21no—
148C2×D4×C43—21no—
149C2×D4×C43—21no—
150C2×D4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43—21no—
153C22×C4×C43—21no—
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2411yescharacteristic
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43C2×D411yescharacteristic
162C22×Dic43C2×D411yescharacteristic
163C22×Dic43C2×D411yescharacteristic
164C22×Dic43C2×D411yescharacteristic
165C2×Dic86—22no—
166C2×Dic86—22no—
167C2×Dic86—22no—
168Q16×C43—44no—
169SD16×C43—44no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43C2×D412yes—
175C2×C8×C43C2×D412yes—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—42no—
179C22⋊C4×C43—42no—
180C22⋊C4×C43—42no—
181C42×C43—42no—
182C42×C43C2×D412yes—
183C42×C43#4112yes—
184C22⋊Dic43—42no—
185C22⋊Dic43—42no—
186C22⋊Dic43—81no—
187C22⋊Dic43—81no—
188C4⋊Dic43—22no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197Dic43⋊C4—22no—
198Dic43⋊C4—22no—
199Dic43⋊C4—42no—
200Dic43⋊C4—42no—
201C4×Dic43—21no—
202C4×Dic43—21no—
203C4×Dic43—21no—
204C4×Dic43—21no—
205C4×Dic43—81no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8#4112yes—
212C23×C43—41no—
213C23×C43—41no—
214C23×C43—41no—
215C23×C43—41no—
216C23×C43—41no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43C22×D411yescharacteristic
220Q8×C43—22no—
221Q8×C43—22no—
222Q8×C43—41no—
223Q8×C43—41no—
224Q8×C43—41no—
225Q8×C43—41no—
226Q8×C43—41no—
227Q8×C43—41no—
228D4×C43—41no—
229D4×C43—41no—
230D4×C43—41no—
231D4×C43—41no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—22no—
237C2×C4×C43—41no—
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#12112yes—
246C2×C4×C43#12112yes—
247C2×C4×C43C22×D411yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43H(5,𝔽2)11yescharacteristic
252C2×C4×C432−1+411yescharacteristic
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
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#13512yes—
276C2×Dic43#13512yes—
277Dic86—28no—
278C8×C43—22no—
279C8×C43—22no—
280C43⋊C8—22no—
281C43⋊C8—41no—
282#25297—431nomaximal, Sylow
283C22×C43—81no—
284C22×C43—81no—
285C22×C43—81no—
286C22×C43—81no—
287C22×C43—81no—
288C22×C43—81no—
289C22×C43—81no—
290C22×C43C2×#12111yescharacteristic
291C22×C43#54811yescharacteristic
292C22×C43#55011yescharacteristic
293C22×C43C2×#13511yescharacteristic
294C22×C43C2×#13511yescharacteristic
295C22×C43#57711yescharacteristic
296C22×C43#57811yescharacteristic
297C4×C43—22no—
298C4×C43—22no—
299C4×C43—42no—
300C4×C43—81no—
301C4×C43—81no—
302C4×C43#52911yescharacteristic
303C4×C43#54211yescharacteristic
304C4×C43#54611yescharacteristic
305C4×C43#54811yescharacteristic
306Dic43—22no—
307Dic43—22no—
308Dic43—41no—
309Dic43—41no—
310Dic43—81no—
311Dic43—81no—
312Dic43—81no—
313Dic43—81no—
314C2×#464D4311yescharacteristic
315C2×C8×Q8—431no—
316#2592—431no—
317#2581—431no—
318#2067—431no—
319#2066—431no—
320#2056—431no—
321#2055—431no—
322#1969—431no—
323#1968—431no—
324#1958—431no—
325#1949—431no—
326#1934—431no—
327#1933—431no—
328C4×#103—431no—
329C2×C43—161no—
330C2×C43#242611yescharacteristic
331C2×C43#305011yescharacteristic
332C2×C43#306311yescharacteristic
333C2×C43#336411yescharacteristic
334C2×C43#336511yescharacteristic
335C2×C43#339311yescharacteristic
336C2×C43#339411yescharacteristic
337C22×Q16C2×D4311yescharacteristic
338C22×SD16C2×D4311yescharacteristic
339C2×#124C2×D4311yescharacteristic
340C2×C4×Q8—431no—
341C2×C4×Q8C2×D4311yescharacteristic
342#464—24no—
343C8×Q8—864no—
344C2×#107—431no—
345C2×#106—431no—
346C2×C4⋊C8—431no—
347C2×C4⋊C8—431no—
348C2×C4⋊C8C2×D4311yescharacteristic
349C2×#103—431no—
350C2×#103—431no—
351C2×#103—431no—
352C2×#103—431no—
353C2×#103C2×D4311yescharacteristic
354C2×#102—431no—
355C2×#102—431no—
356C2×#102C2×D4311yescharacteristic
357C2×C4×C8—431no—
358C2×C4×C8—431no—
359C2×C4×C8—431no—
360#400—861no—
361#397—861no—
362#395—861no—
363#393—861no—
364#391—431no—
365#391—431no—
366#391—431no—
367#391—431no—
368#390—431no—
369#390—431no—
370#389—431no—
371#389—431no—
372#388—861no—
373#388—861no—
374#387—861no—
375#387—861no—
376#382—431no—
377C4×C4⋊C4—431no—
378#340—431no—
379#340—431no—
380#340—431no—
381#340—431no—
382C43#2529711yescharacteristic, complemented, Sylow
383C22×Q8—431no—
384C22×Q8C22×D4311yescharacteristic
385C22×Q8C22×D4311yescharacteristic
386C22×D4C22×D4311yescharacteristic
387C2×Q16—22no—
388C2×Q16—22no—
389C2×Q16—41no—
390C2×SD16—22no—
391C2×SD16—22no—
392C2×SD16—41no—
393C22×C8—431no—
394C22×C8—431no—
395C22×C8C22×D4311yescharacteristic
396C22×C8C22×D4311yescharacteristic
397#124—22no—
398#124—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—862no—
402C4×Q8—862no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—431no—
405C2×C4⋊C4—431no—
406C2×C4⋊C4—431no—
407C2×C4⋊C4—431no—
408C2×C4⋊C4—431no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—861no—
412C2×C4⋊C4—861no—
413C2×C4⋊C4C22×D4311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—861no—
417C2×C22⋊C4—861no—
418C2×C22⋊C4—861no—
419C2×C22⋊C4—861no—
420C2×C42—21no—
421C2×C42—431no—
422C2×C42—431no—
423C2×C42—861no—
424C2×C42C22×D4311yescharacteristic
425#107—1721no—
426#106—1721no—
427C4⋊C8—862no—
428C4⋊C8—862no—
429C4⋊C8C22⋊D4314yes—
430#103—22no—
431#103—862no—
432#103—862no—
433#103—862no—
434#103—1721no—
435#102—22no—
436#102—862no—
437#102—1721no—
438C4×C8—862no—
439C4×C8—862no—
440C4×C8—862no—
441#95—861no—
442#95—861no—
443#95—861no—
444#95—861no—
445#95—861no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453#95—861no—
454C24—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—862no—
467C2×Q8—862no—
468C2×Q8—862no—
469C2×Q8#1062190012yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—431no—
480C22×C4—431no—
481C22×C4—431no—
482C22×C4—431no—
483C22×C4—861no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4D4×D4311yescharacteristic
488C22×C4D4×D4311yescharacteristic
489C22×C4C23×D4311yescharacteristic
490Q16—44no—
491SD16—44no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—432no—
497C2×C8—861no—
498C2×C8—861no—
499C2×C8—861no—
500C2×C8—861no—
501C2×C8—861no—
502C2×C8C2×C22⋊D4312yes—
503C2×C8C2×C22⋊D4312yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—862no—
508C4⋊C4—862no—
509C4⋊C4—862no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—1722no—
519C4⋊C4—1722no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—3441no—
523C22⋊C4—3441no—
524C22⋊C4—1722no—
525C22⋊C4—1722no—
526C42—42no—
527C42—861no—
528C42—861no—
529C42—861no—
530C42—861no—
531C42—3441no—
532C42#1062190012yes—
533C42C2×C22⋊D4312yes—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D4311yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—868no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—432no—
569C2×C4—432no—
570C2×C4—861no—
571C2×C4—861no—
572C2×C4—861no—
573C2×C4—861no—
574C2×C4—861no—
575C2×C4—861no—
576C2×C4—861no—
577C2×C4—861no—
578C2×C4—861no—
579C2×C4—861no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4#1306329612yes—
593C2×C4#1306329612yes—
594C2×C4C22×C22⋊D4311yescharacteristic
595C2×C4#41×D4311yescharacteristic
596C2×C4C2×#1062190011yescharacteristic
597C2×C4C2×#1062190011yescharacteristic
598C2×C4#1306337011yescharacteristic
599C2×C4#1306337211yescharacteristic
600C8—22no—
601C8—22no—
602C8—862no—
603C8—1721no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1306329611yescharacteristic
612C22#1458355511yescharacteristic
613C22#1458356711yescharacteristic
614C22#135×D4311yescharacteristic
615C22#135×D4311yescharacteristic
616C22#1458370811yescharacteristic
617C22#1458371511yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—862no—
624C4—862no—
625C4—3441no—
626C4—3441no—
627C4—3441no—
628C4—3441no—
629C4—1721no—
630C4—1721no—
631C4#1458343011yescharacteristic
632C4#1458351611yescharacteristic
633C4#1458354311yescharacteristic
634C4#1458355611yescharacteristic
635C2—161no—
636C2#1913815711yescharacteristic
637C2#1914417211yescharacteristic
638C2#1914428011yescharacteristic
639C2#1914727511yescharacteristic
640C2#1914728611yescharacteristic
641C2#1914757811yescharacteristic
642C2#1914758611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#464C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenterC2×#464D43
hypocenterC43#25297
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×#14583023C2semidirect
3. #19139957C2semidirect
5. #19135974C2semidirect
6. #19135915C2semidirect
8. #19135457C2semidirect
11. #19135365C2semidirect
13. #19135268C2semidirect
14. C4×#13063249C2semidirect
382. C43#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#145830232
#191400842
#191399572
#191359812
#191359742
#191359152
#191359082
#191354572
#191354512
#191354072
#191353652
#191352752
#191352682
C4×#130632492
C2×C43×#4642
#2529743