Finite Group Databasebeta

← #25100803

Subgroups — #25100803

Order 11008.

898 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic43C211yesmaximal, characteristic
2#19137070C211yesmaximal, characteristic, complemented
3#19137057C211yesmaximal, characteristic
4#19135451C211yesmaximal, characteristic
5#19135323C211yesmaximal, characteristic, complemented
6#19135321C211yesmaximal, characteristic, complemented
7#19135281C211yesmaximal, characteristic
8#19135275C211yesmaximal, characteristic
9#19135274C211yesmaximal, characteristic, complemented
10#19135268C211yesmaximal, characteristic, complemented
11#106×Dic43C211yesmaximal, characteristic, complemented
12#103×Dic43C211yesmaximal, characteristic, complemented
13#102×Dic43C211yesmaximal, characteristic
14C4×C8×Dic43C211yesmaximal, characteristic, complemented
15C43×#1968C211yesmaximal, characteristic
16C2×Q8×Dic43C2211yescharacteristic
17C2×D4×Dic43C2211yescharacteristic
18C2×#13063281—21no—
19SD16×Dic43C418yescomplemented
20C2×#13063257C2211yescharacteristic
21C2×#13063257C2211yescharacteristic
22C2×#13063249C2211yescharacteristic
23C2×#13063249C2211yescharacteristic
24C2×#13063246C2211yescharacteristic
25C2×#13063246C2211yescharacteristic
26C2×C4×C43⋊C8C2211yescharacteristic
27C2×C8×Dic43C2211yescharacteristic
28C2×C8×Dic43C2211yescharacteristic
29C2×C8×Dic43C2211yescharacteristic
30C2×C8×Dic43C2211yescharacteristic
31#14582806C2211yescharacteristic
32#14582802C2211yescharacteristic
33#14582797C2211yescharacteristic
34#14582797C2211yescharacteristic
35#14582795C2211yescharacteristic
36#14582795C2211yescharacteristic
37#14582793—21no—
38#14582787—21no—
39#14582784—21no—
40#14582782—21no—
41C4⋊C4×Dic43C2211yescharacteristic
42C4⋊C4×Dic43C2211yescharacteristic
43#14582750—21no—
44C22⋊C4×Dic43—21no—
45#14582744—21no—
46C42×Dic43C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53C22×SD16×C43C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C2×C4×C8×C43C2211yescharacteristic
58C43×#391C2211yescharacteristic
59C43×#390C2211yescharacteristic
60C23×Dic43—21no—
61C22×C43⋊C8C2311yescharacteristic
62C22×C43⋊C8C2311yescharacteristic
63Q8×Dic43—22no—
64Q8×Dic43C2×C414yes—
65D4×Dic43—22no—
66D4×Dic43C2×C414yes—
67#13063281—44no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43C2311yescharacteristic
70C2×C4⋊Dic43C2311yescharacteristic
71C2×C4⋊Dic43C2311yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—21no—
76C2×Dic43⋊C4—21no—
77C2×C22⋊Dic43—21no—
78C2×C22⋊Dic43—21no—
79C2×C22⋊Dic43—21no—
80C2×C4×Dic43—21no—
81C2×C4×Dic43—21no—
82C2×C4×Dic43—21no—
83C2×C4×Dic43D411yescharacteristic
84C2×C4×Dic43D411yescharacteristic
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88C2×C4×Dic43C2311yescharacteristic
89#13063257—22no—
90#13063257C2×C414yes—
91#13063249—22no—
92#13063249C2×C414yes—
93#13063246—22no—
94#13063246C2×C414yes—
95C4×C43⋊C8—22no—
96C8×Dic43—22no—
97C8×Dic43—22no—
98C8×Dic43—22no—
99C8×Dic43C2×C414yes—
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×D4×C43C2311yescharacteristic
110C2×SD16×C43—22no—
111C2×SD16×C43—22no—
112C2×SD16×C43C2×C414yes—
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C22⋊C4×C43—21no—
118C2×C22⋊C4×C43—21no—
119C2×C42×C43C2311yescharacteristic
120#106×C43—22no—
121#103×C43—22no—
122#102×C43—22no—
123C4×C8×C43—22no—
124#95×C43—21no—
125#95×C43—21no—
126C24×C43—21no—
127C2×Q8×C43—21no—
128C2×Q8×C43—21no—
129C2×Q8×C43—21no—
130C2×Q8×C43—21no—
131C2×Q8×C43—21no—
132C2×Q8×C43C22×C412yes—
133C2×D4×C43—21no—
134C2×D4×C43—21no—
135C2×D4×C43—21no—
136C2×D4×C43—21no—
137C2×D4×C43—21no—
138C2×D4×C43C22×C412yes—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2411yescharacteristic
145C22×Dic43—24no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—42no—
151C22×Dic43—42no—
152C22×Dic43C2×D411yescharacteristic
153C22×Dic43C2×D411yescharacteristic
154C22×Dic43C2×D411yescharacteristic
155C22×Dic43C2×D411yescharacteristic
156SD16×C43—28no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43C22×C412yes—
160C2×C8×C43#4112yes—
161C2×C8×C43#4112yes—
162C2×C8×C43#4112yes—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C22⋊C4×C43—42no—
168C22⋊C4×C43—42no—
169C42×C43—21no—
170C42×C43—21no—
171C22⋊Dic43—24no—
172C22⋊Dic43—42no—
173C22⋊Dic43—42no—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43—24no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43C22×C412yes—
182C4⋊Dic43C22×C412yes—
183Dic43⋊C4—42no—
184Dic43⋊C4—42no—
185Dic43⋊C4—42no—
186Dic43⋊C4—42no—
187C4×Dic43—22no—
188C4×Dic43—21no—
189C4×Dic43—21no—
190C4×Dic43—24no—
191C4×Dic43—22no—
192C4×Dic43—22no—
193C4×Dic43—42no—
194C4×Dic43—42no—
195C4×Dic43SD1614yescomplemented
196C4×Dic43C22×C412yes—
197C4×Dic43#4112yes—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8C22×C412yes—
203C2×C43⋊C8#4112yes—
204C23×C43—22no—
205C23×C43—41no—
206C23×C43—41no—
207C23×C43—41no—
208C23×C43—41no—
209C23×C43—41no—
210C23×C43—41no—
211C23×C43C22×D411yescharacteristic
212Q8×C43—22no—
213Q8×C43—22no—
214Q8×C43—41no—
215Q8×C43—41no—
216D4×C43—22no—
217D4×C43—22no—
218D4×C43—41no—
219D4×C43—41no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—22no—
225C2×C4×C43—41no—
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×C43C2×SD1612yes—
234C2×C4×C43#13512yes—
235C2×C4×C43C23×C411yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C2×C4×C43C2×#4111yescharacteristic
239C2×C4×C43C2×#4111yescharacteristic
240C2×C4×C43C2×#4111yescharacteristic
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—21no—
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—48no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43C4×D412yes—
261C2×Dic43C4×D412yes—
262C2×Dic43C2×SD1612yes—
263C2×Dic43C2×SD1612yes—
264C2×Dic43#13512yes—
265C2×Dic43#13512yes—
266C8×C43—22no—
267C8×C43#11714yescomplemented
268C43⋊C8—22no—
269C43⋊C8—22no—
270#21274—431nomaximal, Sylow
271C22×C43—42no—
272C22×C43—42no—
273C22×C43—42no—
274C22×C43—81no—
275C22×C43—81no—
276C22×C43—81no—
277C22×C43—81no—
278C22×C43C2×C4×D411yescharacteristic
279C22×C43#53211yescharacteristic
280C22×C43#53211yescharacteristic
281C22×C43C22×SD1611yescharacteristic
282C22×C43C2×#13511yescharacteristic
283C22×C43C2×#13511yescharacteristic
284C22×C43C2×#13511yescharacteristic
285C4×C43—22no—
286C4×C43—22no—
287C4×C43—42no—
288C4×C43—81no—
289C4×C43—81no—
290C4×C43C2×#11711yescharacteristic
291C4×C43C4×#4111yescharacteristic
292C4×C43#52911yescharacteristic
293C4×C43#52911yescharacteristic
294Dic43—22no—
295Dic43—22no—
296Dic43—22no—
297Dic43—42no—
298Dic43—42no—
299Dic43—81no—
300Dic43—81no—
301Dic43C4×SD1614yes—
302C2×C4×SD16—431no—
303#2295—431no—
304#2293—431no—
305#1968—431no—
306#1968D4311yescharacteristic
307#1941—431no—
308#1941—431no—
309#1934—431no—
310#1934—431no—
311#1933—431no—
312#1933—431no—
313C4×#106—431no—
314C4×#103—431no—
315C4×#102—431no—
316C42×C8—431no—
317C2×C43—82no—
318C2×C43#229311yescharacteristic
319C2×C43C2×C4×SD1611yescharacteristic
320C2×C43C4×#13511yescharacteristic
321C2×C43#316711yescharacteristic
322C2×C43#316711yescharacteristic
323C2×C43#316811yescharacteristic
324C2×C43#316811yescharacteristic
325C22×SD16C2×D4311yescharacteristic
326C2×C4×Q8—431no—
327C2×C4×D4—431no—
328C2×#117—861no—
329C4×SD16—438no—
330C2×#106—431no—
331C2×#106—431no—
332C2×#106C2×D4311yescharacteristic
333C2×#103—431no—
334C2×#103—431no—
335C2×#103C2×D4311yescharacteristic
336C2×#102—431no—
337C2×#102—431no—
338C2×#102C2×D4311yescharacteristic
339C2×C4×C8—431no—
340C2×C4×C8—431no—
341C2×C4×C8—431no—
342C2×C4×C8—431no—
343C2×C4×C8—431no—
344C2×C4×C8C2×D4311yescharacteristic
345#391—431no—
346#391C2×D4311yescharacteristic
347#390—431no—
348#390C2×D4311yescharacteristic
349#389—431no—
350#389—431no—
351#389—431no—
352#389—431no—
353#388—861no—
354#388—861no—
355#387—861no—
356#387—861no—
357C4×C4⋊C4—431no—
358C4×C4⋊C4—431no—
359C4×C4⋊C4—861no—
360C4×C22⋊C4—861no—
361#376—861no—
362C43—431no—
363#340—431no—
364#340—431no—
365#340—431no—
366#340—431no—
367#340—431no—
368#340—431no—
369C43#2127411yescharacteristic, complemented, Sylow
370C22×Q8C22×D4311yescharacteristic
371C22×D4C22×D4311yescharacteristic
372C23×C4—861no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C4×D4314yes—
376C22×C8—431no—
377C22×C8—431no—
378C22×C8C22×D4311yescharacteristic
379C22×C8C22×D4311yescharacteristic
380C4×Q8—434no—
381C4×Q8—862no—
382C4×D4—434no—
383C4×D4—862no—
384#117—1724no—
385C2×C4⋊C4—431no—
386C2×C4⋊C4—431no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4—861no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4C22×D4311yescharacteristic
395C2×C4⋊C4C22×D4311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—861no—
399C2×C22⋊C4—861no—
400C2×C22⋊C4—861no—
401C2×C42—431no—
402C2×C42—431no—
403C2×C42—431no—
404C2×C42—431no—
405C2×C42—431no—
406C2×C42—431no—
407C2×C42—861no—
408C2×C42—861no—
409C2×C42—861no—
410C2×C42C22×D4311yescharacteristic
411#106—22no—
412#106—434no—
413#106—862no—
414#103—22no—
415#103—434no—
416#103—862no—
417#102—22no—
418#102—434no—
419#102—862no—
420C4×C8—22no—
421C4×C8—434no—
422C4×C8—862no—
423C4×C8—862no—
424C4×C8—862no—
425C4×C8—862no—
426#95—21no—
427#95—21no—
428#95—861no—
429#95—861no—
430#95—861no—
431#95—861no—
432#95—861no—
433#95—861no—
434#95—861no—
435#95—861no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D4312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D4312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—431no—
453C22×C4—431no—
454C22×C4—431no—
455C22×C4—431no—
456C22×C4—864no—
457C22×C4—861no—
458C22×C4—861no—
459C22×C4—861no—
460C22×C4—861no—
461C22×C4—1722no—
462C22×C4—1722no—
463C22×C4D4×D4311yescharacteristic
464C22×C4D4×D4311yescharacteristic
465C22×C4C23×D4311yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—432no—
470C2×C8—432no—
471C2×C8—861no—
472C2×C8—861no—
473C2×C8—861no—
474C2×C8—861no—
475C2×C8C2×C4×D4312yes—
476C2×C8#1062190012yes—
477C2×C8#1062190012yes—
478C2×C8#1062190012yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—432no—
484C4⋊C4—432no—
485C4⋊C4—864no—
486C4⋊C4—861no—
487C4⋊C4—861no—
488C4⋊C4—861no—
489C4⋊C4—861no—
490C4⋊C4—861no—
491C4⋊C4—861no—
492C4⋊C4—1722no—
493C4⋊C4—1722no—
494C4⋊C4—1722no—
495C4⋊C4—1722no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—864no—
499C22⋊C4—1722no—
500C22⋊C4—1722no—
501C42—21no—
502C42—21no—
503C42—434no—
504C42—432no—
505C42—432no—
506C42—862no—
507C42—862no—
508C42—864no—
509C42—862no—
510C42—861no—
511C42—861no—
512C42—1722no—
513C42—1722no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D4311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—432no—
544C2×C4—432no—
545C2×C4—432no—
546C2×C4—432no—
547C2×C4—432no—
548C2×C4—432no—
549C2×C4—861no—
550C2×C4—861no—
551C2×C4—861no—
552C2×C4—861no—
553C2×C4—861no—
554C2×C4—861no—
555C2×C4—862no—
556C2×C4—862no—
557C2×C4—1728no—
558C2×C4—1721no—
559C2×C4—1721no—
560C2×C4—1721no—
561C2×C4—1721no—
562C2×C4—1721no—
563C2×C4—1721no—
564C2×C4—1721no—
565C2×C4—1721no—
566C2×C4—1721no—
567C2×C4—1721no—
568C2×C4SD16×D4312yes—
569C2×C4#1306334612yes—
570C2×C4C22×C4×D4311yescharacteristic
571C2×C4#41×D4311yescharacteristic
572C2×C4#41×D4311yescharacteristic
573C2×C4C2×#1062190011yescharacteristic
574C2×C4C2×#1062190011yescharacteristic
575C2×C4C2×#1062190011yescharacteristic
576C8—22no—
577C8—862no—
578C8—862no—
579C8#1306328114yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D4311yescharacteristic
588C22#1458344411yescharacteristic
589C22#1458344711yescharacteristic
590C22C2×SD16×D4311yescharacteristic
591C22#135×D4311yescharacteristic
592C22#135×D4311yescharacteristic
593C22C2×#1306334611yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—434no—
600C4—862no—
601C4—862no—
602C4—862no—
603C4—3441no—
604C4—3441no—
605C4—1722no—
606C4—1722no—
607C4C2×#1306328111yescharacteristic
608C4C4×#1062190011yescharacteristic
609C4#1458342811yescharacteristic
610C4#1458343011yescharacteristic
611C2—82no—
612C2#1913705711yescharacteristic
613C2C4×SD16×D4311yescharacteristic
614C2C4×#1306334611yescharacteristic
615C2#1914514911yescharacteristic
616C2#1914515211yescharacteristic
617C2#1914515811yescharacteristic
618C2#1914516211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1968C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
cosocleC22×C4×C43C24
hypercenter#1968D43
hypocenterC43#21274
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #19137070C2semidirect
5. #19135323C2semidirect
6. #19135321C2semidirect
9. #19135274C2semidirect
10. #19135268C2semidirect
11. #106×Dic43C2semidirect
12. #103×Dic43C2semidirect
14. C4×C8×Dic43C2semidirect
19. SD16×Dic43C4semidirect
195. C4×Dic43SD16semidirect
267. C8×C43#117semidirect
369. C43#21274semidirect
579. C8#13063281semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic432
#191370702
#191370572
#191354512
#191353232
#191353212
#191352812
#191352752
#191352742
#191352682
#106×Dic432
#103×Dic432
#102×Dic432
C4×C8×Dic432
C43×#19682
#2127443