Finite Group Databasebeta

← #26231927

Subgroups — #26231927

Order 12032.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689777C211yesmaximal, characteristic
2C2×#15689200C211yesmaximal, characteristic, complemented
3C2×#15689121C212yesmaximal, complemented
4#20246288C211yesmaximal, characteristic, complemented
5#20244310C211yesmaximal, characteristic, complemented
6#20244189C211yesmaximal, characteristic, complemented
7#20243820C212yesmaximal, complemented
8#20243694C212yesmaximal, complemented
9#20243325C212yesmaximal, complemented
10C4×#13093945C211yesmaximal, characteristic, complemented
11C47×#1802C211yesmaximal, characteristic, complemented
12C22×#10625352C2212yes—
13C22×#10625324C2212yes—
14#15689777—24no—
15#15689777C418yescomplemented
16C2×#13094024C2211yescharacteristic
17C2×#13094008—22no—
18C2×#13094004C2212yes—
19C2×#13093998—21no—
20C2×C4×D188C2211yescharacteristic
21C2×#13093986—21no—
22C22×C4⋊Dic47—21no—
23C2×C4⋊C4×D47C2212yes—
24#15689200—24no—
25#15689121—28no—
26C2×#13093949C2212yes—
27C2×#13093949C2212yes—
28C2×#13093947C2212yes—
29C2×#13093947C2212yes—
30C2×#13093945C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×(C2×D47)⋊C8C2212yes—
33C2×C4×C47⋊C8C2211yescharacteristic
34#15689019—21no—
35#15688994—22no—
36#15688980C2212yes—
37C4×C4⋊Dic47C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40C2×C4×D4×C47C2211yescharacteristic
41C2×C4⋊C8×C47C2211yescharacteristic
42C2×#102×C47C2212yes—
43C47×#384C2211yescharacteristic
44C47×#340C2212yes—
45C2×#10625352—24no—
46C2×#10625352—24no—
47C2×#10625352—24no—
48C2×#10625352—42no—
49C2×#10625352—42no—
50C2×#10625352C2×C418yescomplemented
51C22×Dic94C2312yes—
52C22×D188D411yescharacteristic
53C22×C22⋊D47—22no—
54C23×Dic47—22no—
55C22×C4×D47—22no—
56C22×C4×D47C2312yes—
57C2×#10625324—24no—
58C2×#10625324—24no—
59C2×#10625324—42no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62#13094024—22no—
63#13094024C2×C414yescomplemented
64#13094008—216no—
65#13094004—24no—
66#13094004C2×C418yescomplemented
67#13093998—28no—
68C4×D188C2×C414yes—
69C4×D188D414yes—
70#13093986—44no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—22no—
74C2×C4⋊Dic47—28no—
75C2×C4⋊Dic47D411yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—22no—
79C4⋊C4×D47—24no—
80C4⋊C4×D47—42no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4D412yes—
84C2×C4×Dic47—21no—
85#13093949—24no—
86#13093949—42no—
87#13093947—24no—
88#13093947—42no—
89#13093945—22no—
90#13093945D414yescomplemented
91(C2×D47)⋊C8—24no—
92C4×C47⋊C8—22no—
93C47⋊#95—21no—
94C47⋊#95—22no—
95C22×D4×C47C2311yescharacteristic
96C23×C4×C47—21no—
97C22×C8×C47C2312yes—
98C4×D4×C47—22no—
99C4×D4×C47C2×C414yescomplemented
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2312yes—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C22⋊C4×C47—21no—
105C2×C42×C47C2311yescharacteristic
106C4⋊C8×C47—22no—
107#102×C47—24no—
108C24×C47—21no—
109C23×D47C2×D412yes—
110C2×D4×C47—21no—
111C2×D4×C47—21no—
112C2×D4×C47—21no—
113C2×D4×C47C22×C412yes—
114C2×D4×C47C2×D412yes—
115C2×D4×C47#4112yes—
116C22×C4×C47—21no—
117C22×C4×C47—24no—
118C22×C4×C47—22no—
119C22×C4×C47—42no—
120C22×C4×C47—42no—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2411yescharacteristic
124C2×C22⋊D47—28no—
125C2×C22⋊D47—44no—
126C2×C22⋊D47—44no—
127C22×Dic47—21no—
128C22×Dic47—22no—
129C22×Dic47—28no—
130C22×Dic47—21no—
131C22×Dic47—44no—
132C22×Dic47—44no—
133C22×Dic47C2×D412yes—
134#10625352—216no—
135#10625352—48no—
136#10625352—48no—
137#10625352—48no—
138C2×D188—22no—
139C2×D188—22no—
140C2×D188C22⋊C414yes—
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—44no—
146C2×C4×D47—44no—
147C2×C4×D47—82no—
148C2×C4×D47C22×C414yes—
149C2×C4×D47C2×D414yes—
150C2×Dic94—22no—
151C2×Dic94—22no—
152C2×Dic94—22no—
153C2×Dic94—22no—
154C2×Dic94—22no—
155C2×Dic94C22×C414yes—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—42no—
163C4⋊C4×C47—81no—
164C4⋊C4×C47C22×C412yes—
165C22⋊C4×C47—24no—
166C42×C47C22×C412yes—
167C42×C47C2×D412yes—
168(C2×D47)⋊C4—28no—
169(C2×D47)⋊C4—44no—
170(C2×D47)⋊C4C22⋊C418yes—
171C4⋊Dic47—24no—
172C4⋊Dic47—24no—
173C4⋊Dic47—21no—
174C4⋊Dic47—28no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47C22⋊C414yes—
178C4⋊Dic47C2×D412yes—
179Dic47⋊C4—44no—
180C4×Dic47—42no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8C2×D412yes—
187#10625324—48no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47C22×D411yescharacteristic
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47C2×C22⋊C414yes—
203D4×C47—22no—
204D4×C47—22no—
205D4×C47—22no—
206D4×C47C4×D414yescomplemented
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—22no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—48no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47C2×C22⋊C412yes—
217C2×C4×C47C2×C22⋊C412yes—
218C2×C4×C47AGL(1,ℤ/8)12yes—
219C2×C4×C47#13712yes—
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47H(5,𝔽2)11yescharacteristic
225C2×C4×C47H(5,𝔽2)11yescharacteristic
226C22⋊D47—416no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—22no—
231C2×Dic47—416no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47C2×C22⋊C414yes—
240D188—28no—
241C4×D47—24no—
242C4×D47—24no—
243C4×D47—24no—
244C4×D47—24no—
245C4×D47—88no—
246Dic94—24no—
247Dic94—24no—
248Dic94—42no—
249Dic94—42no—
250C8×C47—42no—
251C47⋊C8—22no—
252C47⋊C8—41no—
253#22431—471nomaximal, Sylow
254C22×C47—24no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47C22×C22⋊C411yescharacteristic
262C22×C47D4211yescharacteristic
263C22×C47#54611yescharacteristic
264C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C47C2×#13711yescharacteristic
266C22×C47#57711yescharacteristic
267C22×C47#57811yescharacteristic
268C2×D47—24no—
269C2×D47—24no—
270C2×D47—24no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—42no—
275C4×C47—22no—
276C4×C47—42no—
277C4×C47—82no—
278C4×C47#41714yescomplemented
279C4×C47C2×C4×D411yescharacteristic
280C4×C47#51811yescharacteristic
281C4×C47D4211yescharacteristic
282C4×C47#54611yescharacteristic
283Dic47—24no—
284Dic47—44no—
285Dic47—42no—
286Dic47—81no—
287C2×#547—471no—
288C2×#436—471no—
289C2×#417—472no—
290D4×C4⋊C4—471no—
291#1981—471no—
292#1956—471no—
293#1894—472no—
294#1872—472no—
295#1802D4711yescharacteristic, complemented
296#1788—472no—
297C4×#102—471no—
298C2×C47—44no—
299C2×C47D4×C22⋊C411yescharacteristic
300C2×C47C2×#41711yescharacteristic
301C2×C47#289811yescharacteristic
302C2×C47#327211yescharacteristic
303C2×C47#327611yescharacteristic
304C2×C47#333011yescharacteristic
305C2×C47#333111yescharacteristic
306D47—44no—
307C22×#41—472no—
308C22×M16—472no—
309#547—478no—
310#547—944no—
311C2×C4⋊Q8—471no—
312C2×#123—942no—
313C2×C22⋊Q8—472no—
314C2×#121—941no—
315C2×C4×D4—471no—
316C2×C4×D4—941no—
317C2×C4×D4C2×D4711yescharacteristic
318C22×C4⋊C4—472no—
319C22×C4⋊C4—941no—
320#436—944no—
321C2×C4⋊C8C2×D4711yescharacteristic
322#417—948no—
323C2×#103—472no—
324C2×#103—472no—
325C2×#103—472no—
326C2×#103—472no—
327C2×#102—471no—
328C2×#102—471no—
329C2×#102C2×D4712yes—
330C2×C22⋊C8—472no—
331C2×C4×C8—471no—
332#389—941no—
333#385—942no—
334#384C2×D4711yescharacteristic
335#380—472no—
336C4×C4⋊C4—471no—
337#340—471no—
338#340—471no—
339#340C2×D4712yes—
340C47#2243111yescharacteristic, complemented, Sylow
341C2×#41—478no—
342C2×#41—944no—
343C2×#41—944no—
344C2×#41—944no—
345C2×#41—1882no—
346C2×#41—1882no—
347C22×Q8—472no—
348C22×D4—471no—
349C22×D4—942no—
350C22×D4C22×D4711yescharacteristic
351C23×C4—21no—
352C23×C4—472no—
353C23×C4—942no—
354C23×C4—942no—
355C2×M16—944no—
356C2×M16—944no—
357C2×M16—1882no—
358C22×C8—471no—
359C22×C8—471no—
360C22×C8C22×D4712yes—
361C4⋊Q8—474no—
362C4⋊Q8—942no—
363#123—9416no—
364C22⋊Q8—478no—
365C22⋊Q8—944no—
366#121—948no—
367C4×D4—22no—
368C4×D4—474no—
369C4×D4—474no—
370C4×D4—1884no—
371C4×D4C4×D4714yescomplemented
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—944no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—1882no—
384C2×C4⋊C4C22×D4712yes—
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—472no—
388C2×C22⋊C4—942no—
389C2×C22⋊C4—942no—
390C2×C42—941no—
391C2×C42C22×D4711yescharacteristic
392C4⋊C8—22no—
393#103—944no—
394#103—944no—
395#103—1882no—
396#103—1882no—
397#102—24no—
398#102—474no—
399#102—942no—
400C22⋊C8—944no—
401C4×C8—942no—
402#95—942no—
403#95—941no—
404C24—21no—
405C24—472no—
406#41—9416no—
407#41—1888no—
408#41—1888no—
409#41—1888no—
410C2×Q8—474no—
411C2×Q8—942no—
412C2×Q8—942no—
413C2×Q8—942no—
414C2×Q8—942no—
415C2×Q8—942no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—474no—
420C2×D4—948no—
421C2×D4—942no—
422C2×D4—942no—
423C2×D4—1884no—
424C2×D4—1884no—
425C2×D4C2×C4×D4712yes—
426C2×D4D4×D4712yes—
427C2×D4#1062535412yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—472no—
434C22×C4—474no—
435C22×C4—474no—
436C22×C4—948no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—941no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—3762no—
445C22×C4—1884no—
446C22×C4—1884no—
447C22×C4—1884no—
448C22×C4—1884no—
449C22×C4D4×D4711yescharacteristic
450C22×C4D4×D4711yescharacteristic
451C22×C4C23×D4711yescharacteristic
452M16—1888no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—472no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—941no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—474no—
468C4⋊C4—472no—
469C4⋊C4—948no—
470C4⋊C4—944no—
471C4⋊C4—944no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—1884no—
476C4⋊C4C2×C4×D4712yes—
477C22⋊C4—24no—
478C22⋊C4—478no—
479C22⋊C4—948no—
480C22⋊C4—1884no—
481C42—1882no—
482C42C2×C4×D4712yes—
483C42D4×D4712yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—474no—
492C23—942no—
493C23—942no—
494C23—942no—
495C23—942no—
496C23—942no—
497C23—942no—
498C23C2×D4×D4711yescharacteristic
499Q8—944no—
500Q8—944no—
501Q8—1882no—
502Q8—1882no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—948no—
507D4—18816no—
508D4#1309398614yescomplemented
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—474no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—942no—
526C2×C4—944no—
527C2×C4—3768no—
528C2×C4—18816no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4C22⋊C4×D4712yes—
537C2×C4C22⋊C4×D4712yes—
538C2×C4#1309404812yes—
539C2×C4#1309405312yes—
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C2×D4×D4711yescharacteristic
543C2×C4C2×#1062535411yescharacteristic
544C2×C4#1309407011yescharacteristic
545C2×C4#1309407011yescharacteristic
546C8—42no—
547C8—942no—
548C8—1881no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—944no—
557C22—944no—
558C22—944no—
559C22—1882no—
560C22—1882no—
561C22—1882no—
562C22—1882no—
563C22C2×C22⋊C4×D4711yescharacteristic
564C22#1568975711yescharacteristic
565C22#1568976411yescharacteristic
566C22C2×#1309404811yescharacteristic
567C22C2×#1309405311yescharacteristic
568C22#1568993311yescharacteristic
569C22#1568993911yescharacteristic
570C4—22no—
571C4—42no—
572C4—82no—
573C4—944no—
574C4—3761no—
575C4—1882no—
576C4—1884no—
577C4#1568912114yescomplemented
578C4C2×#1309398611yescharacteristic
579C4#1568959711yescharacteristic
580C4#1568975711yescharacteristic
581C4#1568976411yescharacteristic
582C2—44no—
583C2—1884no—
584C2#2024620811yescharacteristic
585C2C2×#1568912111yescharacteristic
586C2#2025191911yescharacteristic
587C2#2025498311yescharacteristic
588C2#2025503711yescharacteristic
589C2#2025569011yescharacteristic
590C2#2025570111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC2×#15689777C2
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#22431
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#15689200C2semidirect
3. C2×#15689121C2semidirect
4. #20246288C2semidirect
5. #20244310C2semidirect
6. #20244189C2semidirect
7. #20243820C2semidirect
8. #20243694C2semidirect
9. #20243325C2semidirect
10. C4×#13093945C2semidirect
11. C47×#1802C2semidirect
15. #15689777C4semidirect
50. C2×#10625352C2×C4semidirect
63. #13094024C2×C4semidirect
66. #13094004C2×C4semidirect
90. #13093945D4semidirect
99. C4×D4×C47C2×C4semidirect
206. D4×C47C4×D4semidirect
278. C4×C47#417semidirect
295. #1802D47semidirect
340. C47#22431semidirect
371. C4×D4C4×D47semidirect
508. D4#13093986semidirect
577. C4#15689121semidirect

Maximal subgroups

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

subgroupindex
C2×#156897772
C2×#156892002
C2×#156891212
#202462882
#202443102
#202441892
#202438202
#202436942
#202433252
C4×#130939452
C47×#18022
#2243147