Finite Group Databasebeta

← #26217638

Subgroups — #26217638

Order 12032.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094050C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic, complemented
3C2×#15689128C211yesmaximal, characteristic, complemented
4#20245878C211yesmaximal, characteristic, complemented
5#20243736C214yesmaximal, complemented
6#20243722C214yesmaximal, complemented
7#20243107C211yesmaximal, characteristic, complemented
8#20243082C211yesmaximal, characteristic, complemented
9C22×C4×D4×C47C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C2×#13094050—24no—
12C2×#13094050—24no—
13C2×#13094050C418yescomplemented
14C22×#10625335C2212yes—
15C22×#10625334C2212yes—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19#15689128—24no—
20#15689122—24no—
21C2×#13093947C2212yescomplemented
22C2×#13093945C2214yes—
23C2×#13093944C2212yescomplemented
24#15689029C2212yescomplemented
25#15689025C2212yescomplemented
26#15689002—21no—
27#15689002—21no—
28C4×(C2×D47)⋊C4—22no—
29#15688885C2214yescomplemented
30#15688878C2214yescomplemented
31C23×D4×C47C2211yescharacteristic
32C24×C4×C47—21no—
33C2×C4×D4×C47—24no—
34C2×C4×D4×C47—24no—
35C2×C4×D4×C47—22no—
36C2×C4×D4×C47C2214yes—
37C22×C4⋊C4×C47C2211yescharacteristic
38C22×C22⋊C4×C47—21no—
39C22×C42×C47C2211yescharacteristic
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350C2×C414yescomplemented
46C22×Dic94C2311yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—21no—
49C22×C4×D47—21no—
50#13094050—232no—
51C2×#10625335—24no—
52C2×#10625335—24no—
53C2×#10625335C2×C418yescomplemented
54C2×#10625334—24no—
55C2×#10625334—24no—
56C2×#10625334C2×C418yescomplemented
57C2×#10625329—22no—
58C2×#10625329—22no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329C2×C414yescomplemented
63C22×C47⋊C8C2312yes—
64C22×C47⋊C8C2312yes—
65#13093983—22no—
66#13093983—22no—
67C2×C4⋊Dic47C2312yes—
68C2×C22⋊Dic47—21no—
69C2×(C2×D47)⋊C4—22no—
70C2×(C2×D47)⋊C4—22no—
71C2×C4×Dic47—21no—
72#13093947—24no—
73#13093945—28no—
74#13093944—24no—
75C47⋊#95—22no—
76C47⋊#95—22no—
77C25×C47—21no—
78C22×D4×C47—21no—
79C22×D4×C47—22no—
80C22×D4×C47—22no—
81C22×D4×C47—22no—
82C22×D4×C47C2×C414yes—
83C22×D4×C47D414yes—
84C22×D4×C47D414yes—
85C22×D4×C47C2312yes—
86C23×C4×C47—28no—
87C23×C4×C47—21no—
88C23×C4×C47—24no—
89C23×C4×C47—48no—
90C23×C4×C47D411yescharacteristic
91C23×C4×C47D411yescharacteristic
92C23×C4×C47C2311yescharacteristic
93C4×D4×C47—216no—
94C4×D4×C47—28no—
95C4×D4×C47—28no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47D414yes—
99C2×C4⋊C4×C47C2312yes—
100C2×C22⋊C4×C47—28no—
101C2×C22⋊C4×C47—24no—
102C2×C42×C47—22no—
103C2×C42×C47—22no—
104C2×C42×C47D412yes—
105C2×C42×C47C2312yes—
106C24×C47—24no—
107C24×C47—24no—
108C24×C47—24no—
109C24×C47—22no—
110C24×C47—44no—
111C24×C47—44no—
112C24×C47C2×D411yescharacteristic
113C23×D47—21no—
114C2×D4×C47—28no—
115C2×D4×C47—24no—
116C2×D4×C47—24no—
117C2×D4×C47—28no—
118C2×D4×C47—24no—
119C2×D4×C47—28no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47C22⋊C4116yescomplemented
124C2×D4×C47C22×C414yes—
125C2×D4×C47C2×D414yes—
126C2×D4×C47C2×D414yes—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—216no—
130C22×C4×C47—22no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—42no—
134C22×C4×C47—432no—
135C22×C4×C47—48no—
136C22×C4×C47—44no—
137C22×C4×C47—48no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D412yes—
145C22×C4×C47#4112yes—
146C22×C4×C47#4112yes—
147C22×C4×C47C2411yescharacteristic
148C2×C22⋊D47—24no—
149C2×C22⋊D47—42no—
150C2×C22⋊D47—42no—
151C22×Dic47—21no—
152C22×Dic47—22no—
153#10625350—28no—
154#10625350—28no—
155#10625350—44no—
156#10625350—44no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188C22×C412yescomplemented
163C2×C4×D47—24no—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94C22×C412yescomplemented
172C4⋊C4×C47—24no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47C2×D414yes—
175C22⋊C4×C47—216no—
176C42×C47—24no—
177C42×C47—22no—
178C42×C47—22no—
179C22⋊Dic47—42no—
180#10625335—216no—
181#10625334—216no—
182(C2×D47)⋊C4—44no—
183(C2×D47)⋊C4—44no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4×Dic47—42no—
187#10625329—28no—
188#10625329—44no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8C22×C414yescomplemented
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—216no—
201C23×C47—48no—
202C23×C47—416no—
203C23×C47—48no—
204C23×C47—416no—
205C23×C47—44no—
206C23×C47—44no—
207C23×C47C4×D412yes—
208C23×C47#12012yes—
209C23×C47#12112yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—22no—
212C22×D47—41no—
213C22×D47—41no—
214C22×D47—42no—
215C22×D47—42no—
216D4×C47—24no—
217D4×C47—24no—
218D4×C47—24no—
219D4×C47—216no—
220D4×C47C2×C22⋊C418yes—
221C2×C4×C47—28no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—432no—
229C2×C4×C47—44no—
230C2×C4×C47—48no—
231C2×C4×C47—44no—
232C2×C4×C47C2×C22⋊C414yes—
233C2×C4×C47C4×D414yes—
234C2×C4×C47C4×D412yes—
235C2×C4×C47#12012yes—
236C2×C4×C47#12114yes—
237C2×C4×C47#12112yes—
238C2×C4×C47#12314yes—
239C2×C4×C47AGL(1,ℤ/8)14yescomplemented
240C2×C4×C47C23×C411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C22⋊D47—48no—
247C2×Dic47—22no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253D188—22no—
254D188—22no—
255D188—41no—
256D188—41no—
257C4×D47—48no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C47⋊C8—24no—
263C47⋊C8—42no—
264#20925—471nomaximal, Sylow
265C22×C47—24no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—24no—
269C22×C47—28no—
270C22×C47—432no—
271C22×C47—48no—
272C22×C47—44no—
273C22×C47—44no—
274C22×C47—48no—
275C22×C47#38614yes—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×#12011yescharacteristic
278C22×C47C2×#12111yescharacteristic
279C22×C47C2×AGL(1,ℤ/8)12yes—
280C22×C47#57512yes—
281C2×D47—44no—
282C2×D47—42no—
283C2×D47—81no—
284C2×D47—81no—
285C2×D47—82no—
286C4×C47—24no—
287C4×C47—44no—
288C4×C47—42no—
289C4×C47#38614yes—
290C4×C47C22×C22⋊C411yescharacteristic
291C4×C47C2×C4×D411yescharacteristic
292C4×C47C2×#12111yescharacteristic
293C4×C47C2×#12311yescharacteristic
294Dic47—42no—
295Dic47—82no—
296C22×AGL(1,ℤ/8)—471no—
297C22×C4×D4D4711yescharacteristic, complemented
298C2×#417—471no—
299C2×#418—471no—
300#2297—471no—
301#1879—474no—
302#1877—474no—
303#1736—471no—
304#1728—471no—
305C2×C47—24no—
306C2×C47—48no—
307C2×C47C2×#38611yescharacteristic
308C2×C47C4×AGL(1,ℤ/8)12yes—
309C2×C47#300311yescharacteristic
310C2×C47#300511yescharacteristic
311C2×C47#303612yes—
312D47—82no—
313C22×#41—471no—
314C23×D4C2×D4711yescharacteristic
315C24×C4—21no—
316C2×AGL(1,ℤ/8)—478no—
317C2×AGL(1,ℤ/8)—944no—
318C2×AGL(1,ℤ/8)—944no—
319C22×SD16—472no—
320C22×D8—472no—
321C22×M16—471no—
322C22×M16—471no—
323C2×C4×D4—24no—
324C2×C4×D4—24no—
325C2×C4×D4—22no—
326C2×C4×D4C2×D4714yes—
327C2×#117—471no—
328C22×C4⋊C4C2×D4711yescharacteristic
329C22×C22⋊C4—21no—
330C22×C42C2×D4711yescharacteristic
331#418—944no—
332#417—944no—
333C2×#103—472no—
334C2×#102—474no—
335C2×#102—472no—
336#391—472no—
337#390—472no—
338#386—941no—
339#386—941no—
340C4×C22⋊C4—942no—
341#340—474no—
342#336—474no—
343C47#2092511yescharacteristic, complemented, Sylow
344C25—21no—
345C2×#41—474no—
346C2×#41—942no—
347C2×#41—942no—
348C2×#41—942no—
349C2×#41—942no—
350C2×#41—942no—
351C22×Q8—471no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—21no—
355C22×D4—22no—
356C22×D4—471no—
357C22×D4—941no—
358C22×D4C4×D4714yes—
359C22×D4D18814yes—
360C22×D4C22⋊D4714yes—
361C22×D4C22×D4712yes—
362C23×C4—28no—
363C23×C4—21no—
364C23×C4—24no—
365C23×C4—48no—
366C23×C4—941no—
367C23×C4C22⋊D4711yescharacteristic
368C23×C4C22⋊D4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370AGL(1,ℤ/8)—9432no—
371C2×SD16—478no—
372C2×SD16—944no—
373C2×SD16—944no—
374C2×D8—478no—
375C2×D8—944no—
376C2×D8—944no—
377C2×M16—474no—
378C2×M16—942no—
379C2×M16—942no—
380C2×M16—942no—
381C2×M16—942no—
382C2×M16—942no—
383C22×C8—472no—
384C22×C8—472no—
385C4×D4—216no—
386C4×D4—28no—
387C4×D4—28no—
388#117—942no—
389#117—942no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—472no—
393C2×C4⋊C4C22⋊D4714yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—28no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—941no—
403C2×C42C22⋊D4712yes—
404C2×C42C22×D4712yes—
405#103—944no—
406#102—948no—
407#102—944no—
408#95—942no—
409#95—942no—
410C24—24no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—44no—
415C24—44no—
416C24—941no—
417C24C2×C22⋊D4711yescharacteristic
418#41—948no—
419#41—948no—
420#41—1884no—
421#41—1884no—
422C2×Q8—472no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—28no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—472no—
438C2×D4—944no—
439C2×D4—941no—
440C2×D4—941no—
441C2×D4—941no—
442C2×D4—941no—
443C2×D4—941no—
444C2×D4—1882no—
445C2×D4—1882no—
446C2×D4(C2×D47)⋊C4116yescomplemented
447C2×D4C2×C4×D4714yes—
448C2×D4C2×D18814yes—
449C2×D4C2×C22⋊D4714yes—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—432no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—941no—
462C22×C4—944no—
463C22×C4—942no—
464C22×C4—1882no—
465C22×C4—1882no—
466C22×C4C2×C4×D4712yes—
467C22×C4C2×D18812yes—
468C22×C4#1062535012yes—
469C22×C4#1062535012yes—
470C22×C4C2×C22⋊D4712yes—
471C22×C4C2×C22⋊D4712yes—
472C22×C4C2×C22⋊D4712yes—
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4712yes—
475C22×C4C23×D4711yescharacteristic
476SD16—9416no—
477D8—9416no—
478M16—948no—
479M16—1884no—
480C2×C8—474no—
481C2×C8—942no—
482C2×C8—942no—
483C2×C8—942no—
484C2×C8—942no—
485C2×C8—942no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—942no—
489C4⋊C4—942no—
490C4⋊C4C2×C22⋊D4714yes—
491C22⋊C4—216no—
492C22⋊C4—1884no—
493C22⋊C4—1882no—
494C22⋊C4—1884no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—1882no—
499C23—22no—
500C23—22no—
501C23—216no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—48no—
506C23—416no—
507C23—416no—
508C23—48no—
509C23—44no—
510C23—44no—
511C23—942no—
512C23—1881no—
513C23—1881no—
514C23—1882no—
515C23—1882no—
516C23C4×C22⋊D4712yes—
517C23#1309399312yes—
518C23#1309399812yes—
519C23C22×C22⋊D4711yescharacteristic
520Q8—942no—
521Q8—942no—
522Q8—1881no—
523Q8—1881no—
524D4—216no—
525D4—24no—
526D4—24no—
527D4—24no—
528D4—942no—
529D4—942no—
530D4—1888no—
531D4—1881no—
532D4—1881no—
533D4C2×(C2×D47)⋊C418yes—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—432no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—942no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4—1882no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1888no—
552C2×C4C2×(C2×D47)⋊C414yes—
553C2×C4C4×C22⋊D4712yes—
554C2×C4C4×C22⋊D4714yes—
555C2×C4#1309399312yes—
556C2×C4#1309399812yes—
557C2×C4#1309399814yes—
558C2×C4#1309400714yes—
559C2×C4#1309405014yescomplemented
560C2×C4C22×C4×D4711yescharacteristic
561C2×C4C22×C22⋊D4711yescharacteristic
562C2×C4C22×C22⋊D4711yescharacteristic
563C2×C4C22×D18811yescharacteristic
564C2×C4C2×#1062535011yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C8—944no—
567C8—1882no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—28no—
573C22—432no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22—3762no—
579C22—3761no—
580C22—3761no—
581C22—1884no—
582C22—1882no—
583C22#1568900214yes—
584C22C2×C4×C22⋊D4711yescharacteristic
585C22C2×#1309399311yescharacteristic
586C22C2×#1309399811yescharacteristic
587C22C2×#1309405012yes—
588C22#1568992112yes—
589C4—24no—
590C4—44no—
591C4—42no—
592C4—3762no—
593C4—1882no—
594C4#1568900214yes—
595C4C22×(C2×D47)⋊C411yescharacteristic
596C4C2×C4×C22⋊D4711yescharacteristic
597C4C2×#1309399811yescharacteristic
598C4C2×#1309400711yescharacteristic
599C2—24no—
600C2—48no—
601C2—3762no—
602C2C2×#1568900211yescharacteristic
603C2C4×#1309405012yes—
604C2#2025261511yescharacteristic
605C2#2025263011yescharacteristic
606C2#2025285212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C4×D4×C47C2
socleC23×C47C22×D4
prime radicalC22×#13094050C2
cosocleC22×C4×C47C24
hypercenterC22×C4×D4D47
hypocenterC47#20925
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15689122C2semidirect
3. C2×#15689128C2semidirect
4. #20245878C2semidirect
5. #20243736C2semidirect
6. #20243722C2semidirect
7. #20243107C2semidirect
8. #20243082C2semidirect
9. C22×C4×D4×C47C2semidirect
13. C2×#13094050C4semidirect
21. C2×#13093947C22semidirect
23. C2×#13093944C22semidirect
24. #15689029C22semidirect
25. #15689025C22semidirect
29. #15688885C22semidirect
30. #15688878C22semidirect
45. C2×#10625350C2×C4semidirect
53. C2×#10625335C2×C4semidirect
56. C2×#10625334C2×C4semidirect
62. C2×#10625329C2×C4semidirect
123. C2×D4×C47C22⋊C4semidirect
162. C2×D188C22×C4semidirect
171. C2×Dic94C22×C4semidirect
194. C2×C47⋊C8C22×C4semidirect
239. C2×C4×C47AGL(1,ℤ/8)semidirect
297. C22×C4×D4D47semidirect
343. C47#20925semidirect
446. C2×D4(C2×D47)⋊C4semidirect
559. C2×C4#13094050semidirect

Maximal subgroups

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

subgroupindex
C22×#130940502
C2×#156891222
C2×#156891282
#202458782
#202437362
#202437222
#202431072
#202430822
C22×C4×D4×C472
#2092547