Finite Group Databasebeta

← #26231935

Subgroups — #26231935

Order 12032.

1582 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689794C211yesmaximal, characteristic
2C2×#15689201C211yesmaximal, characteristic, complemented
3C2×#15689122C212yesmaximal, complemented
4#20244315C211yesmaximal, characteristic, complemented
5#20244193C211yesmaximal, characteristic, complemented
6#20243824C212yesmaximal, complemented
7#20243705C212yesmaximal, complemented
8#20243390C211yesmaximal, characteristic, complemented
9#20243326C212yesmaximal, complemented
10C4×#13093947C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2212yes—
13C22×#10625329C2212yes—
14#15689794—24no—
15#15689794C418yescomplemented
16C2×#13094021C2211yescharacteristic
17C2×#13094020—21no—
18C2×#13093998C2212yes—
19C2×#13093997—22no—
20C2×C4×Dic94C2211yescharacteristic
21C2×#13093986—21no—
22#15689201—24no—
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24#15689122—28no—
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2212yes—
27C2×#13093947C2211yescharacteristic
28C2×#13093944C2212yes—
29C2×#13093944C2212yes—
30C2×#13093944C2212yes—
31C2×#13093944C2212yes—
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×C4×C47⋊C8C2211yescharacteristic
34#15688885C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2212yes—
37C2×C4×D4×C47—21no—
38C2×C4×D4×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2212yes—
40C47×#389—21no—
41C47×#385—22no—
42C47×#384C2211yescharacteristic
43C47×#380C2212yes—
44C4×C4⋊C4×C47C2211yescharacteristic
45C2×#10625350—24no—
46C2×#10625350—24no—
47C2×#10625350—24no—
48C2×#10625350—42no—
49C2×#10625350—42no—
50C2×#10625350C2×C418yescomplemented
51C22×Dic94C2311yescharacteristic
52C22×D188—21no—
53C22×D188—21no—
54C22×D188C2312yes—
55C22×C22⋊D47—22no—
56C22×C22⋊D47—22no—
57C22×C4×D47—22no—
58C2×#10625329—24no—
59C2×#10625329—24no—
60C2×#10625329—42no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2312yes—
63C22×C47⋊C8C2311yescharacteristic
64#13094021—22no—
65#13094021C2×C414yescomplemented
66#13094020—28no—
67#13093998—24no—
68#13093998C2×C418yescomplemented
69#13093997—216no—
70C4×Dic94—22no—
71C4×Dic94C2×C414yescomplemented
72#13093986—28no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×(C2×D47)⋊C4—22no—
76C2×C4×Dic47—21no—
77C4⋊(C47⋊C8)—22no—
78#13093947—24no—
79#13093947—22no—
80#13093947D414yescomplemented
81#13093944—24no—
82#13093944—24no—
83#13093944—42no—
84#13093944—42no—
85C22⋊(C47⋊C8)—24no—
86C4×C47⋊C8—22no—
87C22×D4×C47D411yescharacteristic
88C23×C4×C47—22no—
89C23×C4×C47C2312yes—
90C4×D4×C47—44no—
91C4×D4×C47C2×C414yes—
92C4×D4×C47D414yes—
93C2×C4⋊C4×C47—24no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—42no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47C2312yes—
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47D412yes—
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106#95×C47—21no—
107#95×C47—22no—
108C24×C47C2×D412yes—
109C23×D47—22no—
110C2×D4×C47—22no—
111C2×D4×C47—22no—
112C2×D4×C47C22⋊C414yes—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—44no—
121C22×C4×C47—44no—
122C22×C4×C47—82no—
123C22×C4×C47C22×C414yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D414yes—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—28no—
130C2×C22⋊D47—28no—
131C2×C22⋊D47—44no—
132C2×C22⋊D47—44no—
133C2×C22⋊D47—82no—
134C2×C22⋊D47—82no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137#10625350—216no—
138#10625350—48no—
139#10625350—48no—
140#10625350—48no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—24no—
146C2×D188—22no—
147C2×D188—24no—
148C2×D188—42no—
149C2×D188—42no—
150C2×D188—81no—
151C2×D188—81no—
152C2×D188C22×C414yes—
153C2×C4×D47—28no—
154C2×C4×D47—44no—
155C2×C4×D47—44no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94C22×C412yes—
160C2×Dic94C2×D412yes—
161C2×Dic94#4112yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—44no—
168C4⋊C4×C47—42no—
169C4⋊C4×C47—81no—
170C4⋊C4×C47C22⋊C414yes—
171C4⋊C4×C47C2×D412yes—
172C22⋊C4×C47—44no—
173C22⋊C4×C47C22⋊C418yes—
174C42×C47—42no—
175C42×C47C22×C412yes—
176C42×C47C2×D412yes—
177(C2×D47)⋊C4—28no—
178C4⋊Dic47—21no—
179C4⋊Dic47C22×C412yes—
180Dic47⋊C4—24no—
181C4×Dic47—24no—
182#10625329—48no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8C2×D412yes—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47C2×C22⋊C414yes—
199C23×C47C22×D411yescharacteristic
200C22×D47—24no—
201C22×D47—42no—
202C22×D47—42no—
203C22×D47—42no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47—42no—
207D4×C47—28no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—41no—
225C2×C4×C47—88no—
226C2×C4×C47C2×C22⋊C412yes—
227C2×C4×C47C2×C22⋊C414yes—
228C2×C4×C47C2×C22⋊C412yes—
229C2×C4×C47AGL(1,ℤ/8)12yes—
230C2×C4×C47#13712yes—
231C2×C4×C47C23×C411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47H(5,𝔽2)11yescharacteristic
236C2×C4×C47H(5,𝔽2)11yescharacteristic
237C22⋊D47—416no—
238C22⋊D47—88no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245D188—24no—
246D188—24no—
247D188—42no—
248D188—42no—
249D188—48no—
250D188—84no—
251C4×D47—416no—
252Dic94—22no—
253Dic94—22no—
254Dic94—22no—
255Dic94C4×D414yescomplemented
256C47⋊C8—22no—
257C47⋊C8—42no—
258C47⋊C8—41no—
259#22432—471nomaximal, Sylow
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—24no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47C22×C22⋊C411yescharacteristic
268C22×C47D4211yescharacteristic
269C22×C47#54611yescharacteristic
270C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C47C2×#13711yescharacteristic
272C22×C47#57611yescharacteristic
273C22×C47#57711yescharacteristic
274C2×D47—44no—
275C2×D47—44no—
276C2×D47—44no—
277C2×D47—82no—
278C2×D47—82no—
279C2×D47—82no—
280C2×D47—82no—
281C4×C47—24no—
282C4×C47—22no—
283C4×C47—42no—
284C4×C47—81no—
285C4×C47—82no—
286C4×C47#41714yescomplemented
287C4×C47C2×C4×D411yescharacteristic
288C4×C47#51811yescharacteristic
289C4×C47D4211yescharacteristic
290C4×C47#54611yescharacteristic
291Dic47—24no—
292Dic47—42no—
293C2×#550—471no—
294C2×#436—471no—
295C2×#417—472no—
296D4×C4⋊C4D4711yescharacteristic, complemented
297#1982—471no—
298#1957—471no—
299#1895—472no—
300#1874—472no—
301#1803—471no—
302#1788—472no—
303C4×#103—471no—
304C2×C47—44no—
305C2×C47D4×C22⋊C411yescharacteristic
306C2×C47C2×#41711yescharacteristic
307C2×C47#289711yescharacteristic
308C2×C47#327411yescharacteristic
309C2×C47#327811yescharacteristic
310C2×C47#333211yescharacteristic
311C2×C47#333311yescharacteristic
312D47—84no—
313C22×#41—472no—
314C22×M16—472no—
315#550—478no—
316#550—944no—
317C2×#127—471no—
318C2×#127—941no—
319C2×#121—472no—
320C2×#121—942no—
321C2×C4×Q8—471no—
322C2×C4×D4—21no—
323C2×C4×D4—941no—
324C2×C4×D4C2×D4711yescharacteristic
325C22×C4⋊C4C2×D4712yes—
326#436—944no—
327C2×C4⋊C8—471no—
328#417—948no—
329C2×#103—471no—
330C2×#103—472no—
331C2×#103—471no—
332C2×#102—472no—
333C2×#102—472no—
334C2×#102—472no—
335C2×#102—472no—
336C2×C22⋊C8—472no—
337C2×C4×C8—471no—
338#389—21no—
339#385—22no—
340#384C2×D4711yescharacteristic
341#380C2×D4712yes—
342C4×C4⋊C4C2×D4711yescharacteristic
343#340—472no—
344#340—471no—
345#340—471no—
346C47#2243211yescharacteristic, complemented, Sylow
347C2×#41—478no—
348C2×#41—944no—
349C2×#41—944no—
350C2×#41—944no—
351C2×#41—1882no—
352C2×#41—1882no—
353C22×Q8—471no—
354C22×D4—472no—
355C22×D4—942no—
356C22×D4—942no—
357C22×D4—941no—
358C22×D4—941no—
359C22×D4C22⋊D4711yescharacteristic
360C23×C4—22no—
361C23×C4—942no—
362C23×C4C22×D4712yes—
363C2×M16—944no—
364C2×M16—944no—
365C2×M16—1882no—
366C22×C8—471no—
367C22×C8—472no—
368C22×C8—471no—
369#127—474no—
370#127—948no—
371#127—942no—
372#121—478no—
373#121—944no—
374#121—9416no—
375C4×Q8—474no—
376C4×Q8—942no—
377C4×D4—44no—
378C4×D4—948no—
379C4×D4C4×D4714yes—
380C4×D4D18814yes—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4C22⋊D4711yescharacteristic
389C2×C4⋊C4C22×D4712yes—
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C22⋊C4—22no—
393C2×C22⋊C4—942no—
394C2×C22⋊C4C22⋊D4712yes—
395C2×C42—21no—
396C2×C42—941no—
397C2×C42C22×D4711yescharacteristic
398C4⋊C8—942no—
399#103—474no—
400#103—942no—
401#103—944no—
402#102—944no—
403#102—944no—
404#102—1882no—
405#102—1882no—
406C22⋊C8—944no—
407C4×C8—942no—
408#95—21no—
409#95—22no—
410C24—942no—
411C24C2×C22⋊D4712yes—
412#41—9416no—
413#41—1888no—
414#41—1888no—
415#41—1888no—
416C2×Q8—472no—
417C2×Q8—472no—
418C2×Q8—472no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—474no—
425C2×D4—948no—
426C2×D4—948no—
427C2×D4—944no—
428C2×D4—944no—
429C2×D4—942no—
430C2×D4—942no—
431C2×D4—942no—
432C2×D4—942no—
433C2×D4—942no—
434C2×D4—3761no—
435C2×D4—3761no—
436C2×D4—3762no—
437C2×D4—3762no—
438C2×D4—1884no—
439C2×D4—1884no—
440C2×D4—1882no—
441C2×D4—1882no—
442C2×D4(C2×D47)⋊C414yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—82no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—948no—
456C22×C4—1884no—
457C22×C4—1884no—
458C22×C4C2×C4×D4714yes—
459C22×C4C2×D18814yes—
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C2×C22⋊D4712yes—
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C23×D4711yescharacteristic
464M16—1888no—
465C2×C8—472no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—942no—
469C2×C8—941no—
470C2×C8—942no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—942no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—472no—
483C4⋊C4—944no—
484C4⋊C4—941no—
485C4⋊C4(C2×D47)⋊C414yes—
486C4⋊C4D4×D4712yes—
487C22⋊C4—44no—
488C22⋊C4—948no—
489C22⋊C4(C2×D47)⋊C418yes—
490C42—42no—
491C42—944no—
492C42C2×C4×D4712yes—
493C42C2×D18812yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—944no—
501C23—1882no—
502C23—1882no—
503C23—1882no—
504C23—1882no—
505C23—1882no—
506C23—1882no—
507C23C2×(C2×D47)⋊C414yes—
508C23C22×C22⋊D4711yescharacteristic
509Q8—474no—
510Q8—942no—
511Q8—942no—
512Q8—942no—
513D4—28no—
514D4—944no—
515D4—944no—
516D4—3764no—
517D4—3768no—
518D4—18816no—
519D4—1882no—
520D4—1882no—
521D4—1888no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—88no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—18816no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4C2×(C2×D47)⋊C412yes—
548C2×C4C2×(C2×D47)⋊C414yes—
549C2×C4C2×(C2×D47)⋊C412yes—
550C2×C4#1309405012yes—
551C2×C4#1309405512yes—
552C2×C4C22×C4×D4711yescharacteristic
553C2×C4C2×D4×D4711yescharacteristic
554C2×C4C22×D18811yescharacteristic
555C2×C4#41×D4711yescharacteristic
556C2×C4#1309406911yescharacteristic
557C2×C4#1309407011yescharacteristic
558C8—942no—
559C8—1882no—
560C8—1881no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—3762no—
569C22—3762no—
570C22—3762no—
571C22—3762no—
572C22—1884no—
573C22—1884no—
574C22—1884no—
575C22C22×(C2×D47)⋊C411yescharacteristic
576C22D4×C22⋊D4711yescharacteristic
577C22#1568976111yescharacteristic
578C22C2×#1309405011yescharacteristic
579C22C2×#1309405511yescharacteristic
580C22#1568992611yescharacteristic
581C22#1568993111yescharacteristic
582C4—22no—
583C4—24no—
584C4—42no—
585C4—82no—
586C4—81no—
587C4—944no—
588C4—1882no—
589C4#1568912214yescomplemented
590C4C4×D4×D4711yescharacteristic
591C4#1568959511yescharacteristic
592C4D4×D18811yescharacteristic
593C4#1568976811yescharacteristic
594C2—44no—
595C2—3764no—
596C2D4×(C2×D47)⋊C411yescharacteristic
597C2C2×#1568912211yescharacteristic
598C2#2025191111yescharacteristic
599C2#2025500411yescharacteristic
600C2#2025505011yescharacteristic
601C2#2025570911yescharacteristic
602C2#2025572011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingD4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689794C2
cosocleC22×C4×C47C24
hypercenterD4×C4⋊C4D47
hypocenterC47#22432
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15689201C2semidirect
3. C2×#15689122C2semidirect
4. #20244315C2semidirect
5. #20244193C2semidirect
6. #20243824C2semidirect
7. #20243705C2semidirect
8. #20243390C2semidirect
9. #20243326C2semidirect
10. C4×#13093947C2semidirect
11. D4×C4⋊C4×C47C2semidirect
15. #15689794C4semidirect
50. C2×#10625350C2×C4semidirect
65. #13094021C2×C4semidirect
68. #13093998C2×C4semidirect
71. C4×Dic94C2×C4semidirect
80. #13093947D4semidirect
255. Dic94C4×D4semidirect
286. C4×C47#417semidirect
296. D4×C4⋊C4D47semidirect
346. C47#22432semidirect
589. C4#15689122semidirect

Maximal subgroups

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

subgroupindex
C2×#156897942
C2×#156892012
C2×#156891222
#202443152
#202441932
#202438242
#202437052
#202433902
#202433262
C4×#130939472
D4×C4⋊C4×C472
#2243247