Finite Group Databasebeta

← #25137876

Subgroups — #25137876

Order 11008.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583208C211yesmaximal, characteristic, complemented
2C2×#14582953C211yesmaximal, characteristic
3#19142903C211yesmaximal, characteristic
4C22⋊Q8×Dic43C211yesmaximal, characteristic
5#19136031C211yesmaximal, characteristic
6#19136023C211yesmaximal, characteristic
7#19135795C211yesmaximal, characteristic
8#19135549C211yesmaximal, characteristic, complemented
9#19135364C211yesmaximal, characteristic, complemented
10#19135339C211yesmaximal, characteristic, complemented
11#19135332C211yesmaximal, characteristic, complemented
12#19135287C211yesmaximal, characteristic, complemented
13#19135282C211yesmaximal, characteristic, complemented
14#19134608C211yesmaximal, characteristic, complemented
15C43×#2035C211yesmaximal, characteristic
16C2×#13063316C2211yescharacteristic
17C2×Q8×Dic43C2211yescharacteristic
18C22×C4×Dic43C2211yescharacteristic
19#14583208—24no—
20#14582953—24no—
21C2×#13063259C2211yescharacteristic
22C2×#13063258C2211yescharacteristic
23C2×#13063256C2211yescharacteristic
24C2×Dic43⋊C8C2211yescharacteristic
25C2×Dic43⋊C8C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×C22⋊(C43⋊C8)C2211yescharacteristic
29C2×#13063235C2211yescharacteristic
30C2×#13063235C2211yescharacteristic
31#14582806C2211yescharacteristic
32#14582798C2211yescharacteristic
33#14582798C2211yescharacteristic
34#14582795C2211yescharacteristic
35#14582786—21no—
36#14582785—21no—
37#14582782—21no—
38#14582774—21no—
39#14582774—21no—
40#14582771—21no—
41#14582764C2211yescharacteristic
42#14582759C2211yescharacteristic
43#14582759C2211yescharacteristic
44C4⋊C4×Dic43—21no—
45C4⋊C4×Dic43C2211yescharacteristic
46C22⋊C4×Dic43—21no—
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582661C2211yescharacteristic
54C2×C22⋊Q8×C43C2211yescharacteristic
55C2×#106×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C2×C22⋊C8×C43C2211yescharacteristic
59C43×#380C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C23×Dic43D411yescharacteristic
62C23×Dic43D411yescharacteristic
63C22×C43⋊C8C2311yescharacteristic
64C22×C43⋊C8C2311yescharacteristic
65#13063316—22no—
66#13063316—22no—
67Q8×Dic43—22no—
68Q8×Dic43—22no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43—21no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—21no—
79C2×C22⋊Dic43—21no—
80C2×C22⋊Dic43—21no—
81C2×C4×Dic43—22no—
82C2×C4×Dic43—22no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88C2×C4×Dic43C2311yescharacteristic
89#13063259—22no—
90#13063258—22no—
91#13063256—22no—
92Dic43⋊C8—22no—
93Dic43⋊C8D414yes—
94#13063249—22no—
95#13063249—22no—
96C22⋊(C43⋊C8)—22no—
97#13063235—22no—
98#13063235—22no—
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⋊#95—21no—
109C22×Q8×C43C2311yescharacteristic
110C23×C4×C43C2311yescharacteristic
111C22×C8×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113C22⋊Q8×C43—22no—
114C22⋊Q8×C43—22no—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C22⋊C4×C43—21no—
120C2×C22⋊C4×C43—21no—
121#106×C43—22no—
122#103×C43—22no—
123#103×C43—22no—
124C22⋊C8×C43—22no—
125#95×C43—21no—
126C24×C43C2×D411yescharacteristic
127C2×Q8×C43—21no—
128C2×Q8×C43—21no—
129C2×Q8×C43—21no—
130C2×Q8×C43—21no—
131C2×Q8×C43—21no—
132C2×Q8×C43#4112yes—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2411yescharacteristic
145C22×Dic43—22no—
146C22×Dic43—22no—
147C22×Dic43—22no—
148C22×Dic43—21no—
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—21no—
152C22×Dic43—41no—
153C22×Dic43—41no—
154C22×Dic43—41no—
155C22×Dic43C2×D411yescharacteristic
156C22×Dic43C2×D411yescharacteristic
157C22×Dic43C2×D411yescharacteristic
158C22×Dic43C2×D411yescharacteristic
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43C2×D412yes—
165C4⋊C4×C43—24no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43#4112yes—
172C22⋊C4×C43—24no—
173C22⋊C4×C43—42no—
174C22⋊Dic43—42no—
175C22⋊Dic43—42no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43—42no—
185C4⋊Dic43—42no—
186C4⋊Dic43—42no—
187C4⋊Dic43—42no—
188Dic43⋊C4—42no—
189Dic43⋊C4—42no—
190C4×Dic43—21no—
191C4×Dic43—21no—
192C4×Dic43—22no—
193C4×Dic43—22no—
194C4×Dic43—42no—
195C4×Dic43—42no—
196C4×Dic43C2×D412yes—
197C4×Dic43#4112yes—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8C2×D412yes—
204C23×C43—21no—
205C23×C43—21no—
206C23×C43—21no—
207C23×C43—21no—
208C23×C43—21no—
209C23×C43—21no—
210C23×C43#13712yes—
211C23×C43C22×D411yescharacteristic
212Q8×C43—22no—
213Q8×C43—22no—
214Q8×C43—41no—
215Q8×C43—41no—
216C2×C4×C43—21no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
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×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43C22×D411yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C2×C4×C43C2×#4111yescharacteristic
238C2×C4×C43C2×#4111yescharacteristic
239C2×C4×C432−1+411yescharacteristic
240C2×C4×C432−1+411yescharacteristic
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—21no—
247C2×Dic43—21no—
248C2×Dic43—21no—
249C2×Dic43—21no—
250C2×Dic43—21no—
251C2×Dic43—44no—
252C2×Dic43—44no—
253C2×Dic43—41no—
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—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43#12112yes—
266C2×Dic43#12112yes—
267C8×C43—22no—
268C8×C43—41no—
269C43⋊C8—22no—
270C43⋊C8—41no—
271#24707—431nomaximal, Sylow
272C22×C43—22no—
273C22×C43—22no—
274C22×C43—22no—
275C22×C43—41no—
276C22×C43—41no—
277C22×C43—41no—
278C22×C43—41no—
279C22×C43C2×#12111yescharacteristic
280C22×C43#52511yescharacteristic
281C22×C43#53611yescharacteristic
282C22×C43C2×#13711yescharacteristic
283C22×C43#57511yescharacteristic
284C22×C43#57511yescharacteristic
285C22×C43#57511yescharacteristic
286C4×C43—42no—
287C4×C43—41no—
288C4×C43—42no—
289C4×C43—81no—
290C4×C43—81no—
291C4×C43#54711yescharacteristic
292C4×C43#54811yescharacteristic
293C4×C43#55211yescharacteristic
294C4×C43#55611yescharacteristic
295Dic43—22no—
296Dic43—22no—
297Dic43—41no—
298Dic43—41no—
299Dic43—81no—
300Dic43—81no—
301Dic43—81no—
302Dic43—81no—
303C2×#479—431no—
304C2×#432—431no—
305#2851—431no—
306C4×C22⋊Q8—431no—
307#2077—431no—
308#2076—431no—
309#2036—431no—
310#2035D4311yescharacteristic
311#1989—431no—
312#1949—431no—
313#1944—431no—
314#1943—431no—
315#1935—431no—
316#1935—431no—
317#1805—431no—
318C2×C43—42no—
319C2×C43#241511yescharacteristic
320C2×C43#305411yescharacteristic
321C2×C43#305611yescharacteristic
322C2×C43#311111yescharacteristic
323C2×C43#311711yescharacteristic
324C2×C43#323611yescharacteristic
325C2×C43#324011yescharacteristic
326C2×#125—431no—
327C2×C22⋊Q8C2×D4311yescharacteristic
328C2×C4×Q8—431no—
329C22×C42—431no—
330#479—864no—
331#432—864no—
332C2×#107—431no—
333C2×#107—431no—
334C2×#106—431no—
335C2×#106C2×D4311yescharacteristic
336C2×C4⋊C8—431no—
337C2×C4⋊C8—431no—
338C2×#103—431no—
339C2×#103—431no—
340C2×#103C2×D4311yescharacteristic
341C2×#103C2×D4311yescharacteristic
342C2×C22⋊C8—431no—
343C2×C22⋊C8C2×D4311yescharacteristic
344C2×#97—431no—
345C2×#97—431no—
346#391—431no—
347#389—431no—
348#389—431no—
349#389—431no—
350#387—861no—
351#387—861no—
352#387—861no—
353#385—861no—
354#385—861no—
355#385—861no—
356#382—431no—
357#380—431no—
358#380—431no—
359#380C2×D4311yescharacteristic
360C4×C4⋊C4—431no—
361C4×C4⋊C4—861no—
362C4×C22⋊C4—861no—
363#340—431no—
364#340—431no—
365#340—431no—
366#340—431no—
367#340—431no—
368#340—431no—
369#340—431no—
370#340C2×D4311yescharacteristic
371C43#2470711yescharacteristic, complemented, Sylow
372C22×Q8C22×D4311yescharacteristic
373C23×C4—431no—
374C23×C4—431no—
375C23×C4C22×D4311yescharacteristic
376C22×C8—431no—
377C22×C8—431no—
378C22×C8C22×D4311yescharacteristic
379C22×C8C22×D4311yescharacteristic
380#125—862no—
381#125—862no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C4×Q8—862no—
385C4×Q8—862no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4—431no—
390C2×C4⋊C4—431no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—861no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4—861no—
397C2×C4⋊C4C22×D4311yescharacteristic
398C2×C4⋊C4C22×D4311yescharacteristic
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—861no—
403C2×C22⋊C4—861no—
404C2×C42—431no—
405C2×C42—431no—
406C2×C42—431no—
407C2×C42—431no—
408C2×C42—862no—
409C2×C42—862no—
410C2×C42—861no—
411C2×C42—861no—
412#107—862no—
413#107—862no—
414#106—22no—
415#106—862no—
416C4⋊C8—434no—
417C4⋊C8—862no—
418#103—22no—
419#103—22no—
420#103—862no—
421#103—862no—
422C22⋊C8—22no—
423C22⋊C8—862no—
424#97—862no—
425#97—862no—
426#95—21no—
427#95—861no—
428#95—861no—
429#95—861no—
430#95—861no—
431#95—861no—
432#95—861no—
433#95—861no—
434#95—861no—
435#95—861no—
436#95—861no—
437C24D4×D4311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1062190012yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—431no—
455C22×C4—431no—
456C22×C4—431no—
457C22×C4—431no—
458C22×C4—862no—
459C22×C4—862no—
460C22×C4—862no—
461C22×C4—861no—
462C22×C4—861no—
463C22×C4—861no—
464C22×C4—861no—
465C22×C4—1721no—
466C22×C4—1721no—
467C22×C4—1721no—
468C22×C4D4×D4311yescharacteristic
469C22×C4C23×D4311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—432no—
476C2×C8—861no—
477C2×C8—861no—
478C2×C8—861no—
479C2×C8—861no—
480C2×C8—861no—
481C2×C8D4×D4312yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—861no—
489C4⋊C4—861no—
490C4⋊C4—861no—
491C4⋊C4—861no—
492C4⋊C4—861no—
493C4⋊C4—861no—
494C4⋊C4—861no—
495C4⋊C4—861no—
496C4⋊C4—1722no—
497C4⋊C4—1722no—
498C4⋊C4—1722no—
499C4⋊C4—1722no—
500C4⋊C4—1722no—
501C4⋊C4—1722no—
502C4⋊C4#1062190012yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—1722no—
506C22⋊C4—1722no—
507C42—432no—
508C42—432no—
509C42—862no—
510C42—862no—
511C42—861no—
512C42—861no—
513C42—1722no—
514C42—1722no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1306335412yes—
522C23C2×D4×D4311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
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—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—432no—
547C2×C4—432no—
548C2×C4—861no—
549C2×C4—861no—
550C2×C4—861no—
551C2×C4—861no—
552C2×C4—861no—
553C2×C4—861no—
554C2×C4—861no—
555C2×C4—861no—
556C2×C4—861no—
557C2×C4—861no—
558C2×C4—1724no—
559C2×C4—1724no—
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×C4—1721no—
569C2×C4—1721no—
570C2×C4—1721no—
571C2×C4—1721no—
572C2×C4C2×D4×D4311yescharacteristic
573C2×C4#41×D4311yescharacteristic
574C2×C4C2×#1062190011yescharacteristic
575C2×C4C2×#1062190011yescharacteristic
576C2×C4#1306337211yescharacteristic
577C2×C4#1306337211yescharacteristic
578C8—22no—
579C8—41no—
580C8—862no—
581C8—1721no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22#121×D4311yescharacteristic
590C22#1458340511yescharacteristic
591C22#1458347111yescharacteristic
592C22C2×#1306335411yescharacteristic
593C22#1458369611yescharacteristic
594C22#1458369811yescharacteristic
595C22#1458369811yescharacteristic
596C4—42no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—862no—
602C4—862no—
603C4—3441no—
604C4—3441no—
605C4—3441no—
606C4—3441no—
607C4—1721no—
608C4—1721no—
609C4#1458354911yescharacteristic
610C4#1458355511yescharacteristic
611C4#1458358611yescharacteristic
612C4#1458361811yescharacteristic
613C2—42no—
614C2#1913801711yescharacteristic
615C2#1914420611yescharacteristic
616C2#1914421811yescharacteristic
617C2#1914470511yescharacteristic
618C2#1914476111yescharacteristic
619C2#1914579811yescharacteristic
620C2#1914583911yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2035C2
socleC23×C43C22×D4
prime radicalC24×C43C2×D4
hypercenter#2035D43
hypocenterC43#24707
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
1. C2×#14583208C2semidirect
8. #19135549C2semidirect
9. #19135364C2semidirect
10. #19135339C2semidirect
11. #19135332C2semidirect
12. #19135287C2semidirect
13. #19135282C2semidirect
14. #19134608C2semidirect
371. C43#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#145832082
C2×#145829532
#191429032
C22⋊Q8×Dic432
#191360312
#191360232
#191357952
#191355492
#191353642
#191353392
#191353322
#191352872
#191352822
#191346082
C43×#20352
#2470743