Finite Group Databasebeta

← #26250209

Subgroups — #26250209

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689405C211yesmaximal, characteristic, complemented
2C2×C4×#10625335C211yesmaximal, characteristic, complemented
3#20248310C211yesmaximal, characteristic, complemented
4#20244574C212yesmaximal, complemented
5#20244391C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244183C211yesmaximal, characteristic, complemented
8#20243695C212yesmaximal, complemented
9#20243581C212yesmaximal, complemented
10#20243388C211yesmaximal, characteristic, complemented
11C2×C47×#547C211yesmaximal, characteristic
12C22×#10625335C2211yescharacteristic
13C2×#13094003C2212yes—
14C2×C4×Dic94C2211yescharacteristic
15#15689405—24no—
16C4×#10625335—24no—
17C2×#13093955C2212yes—
18C2×#13093955C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093947C2212yes—
21C2×#13093947C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2212yes—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×C4×C47⋊C8C2211yescharacteristic
27#15689046—22no—
28#15689026—21no—
29#15689019—21no—
30#15688993C2211yescharacteristic
31#15688983C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C22×#41×C47—21no—
36C47×#547—24no—
37C47×#547—24no—
38C2×C4⋊Q8×C47C2211yescharacteristic
39C2×#123×C47—22no—
40C2×C22⋊Q8×C47—21no—
41C2×#121×C47C2212yes—
42C2×C4×D4×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2212yes—
45C22×Dic94C2311yescharacteristic
46C2×#10625335—22no—
47C2×#10625335—22no—
48C2×#10625335—22no—
49C22×C47⋊C8C2311yescharacteristic
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2311yescharacteristic
52#13094003—24no—
53#13094003—42no—
54C4×Dic94—22no—
55C4×Dic94—22no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2311yescharacteristic
59C2×C4⋊Dic47C2312yes—
60C2×Dic47⋊C4—22no—
61C2×Dic47⋊C4—21no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C4×Dic47—21no—
65#13093955—24no—
66#13093955—22no—
67C4⋊(C47⋊C8)—22no—
68#13093947—24no—
69#13093947—22no—
70#13093945—22no—
71#13093945—42no—
72C22⋊(C47⋊C8)—24no—
73C22⋊(C47⋊C8)—42no—
74C4×C47⋊C8—22no—
75C47⋊#95—22no—
76C2×#41×C47—28no—
77C2×#41×C47—44no—
78C2×#41×C47—44no—
79C22×Q8×C47—21no—
80C22×D4×C47—22no—
81C22×D4×C47D411yescharacteristic
82C22×D4×C47C2311yescharacteristic
83C23×C4×C47—22no—
84C23×C4×C47—21no—
85C23×C4×C47C2312yes—
86C4⋊Q8×C47—22no—
87C4⋊Q8×C47D414yes—
88#123×C47—216no—
89C22⋊Q8×C47—28no—
90#121×C47—24no—
91#121×C47—24no—
92C4×D4×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47—22no—
95C4×D4×C47D414yes—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—24no—
99C2×C4⋊C4×C47—24no—
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2312yes—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—22no—
105C2×C22⋊C4×C47D412yes—
106C2×C42×C47C2311yescharacteristic
107C24×C47—21no—
108C24×C47C2×D412yes—
109#41×C47—432no—
110C2×Q8×C47—24no—
111C2×Q8×C47—42no—
112C2×Q8×C47—42no—
113C2×D4×C47—28no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—44no—
122C2×D4×C47—44no—
123C2×D4×C47SD1614yescomplemented
124C2×D4×C47C2×D412yes—
125C22×C4×C47—28no—
126C22×C4×C47—21no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—44no—
136C22×C4×C47—44no—
137C22×C4×C47—42no—
138C22×C4×C47—42no—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C2×Dic94—21no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94#4112yescomplemented
152C4⋊C4×C47—28no—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47C2×D412yes—
158C4⋊C4×C47C2×D414yes—
159C22⋊C4×C47—28no—
160C22⋊C4×C47—24no—
161C22⋊C4×C47—24no—
162C42×C47C2×D412yes—
163C42×C47C2×D412yes—
164C22⋊Dic47—44no—
165C22⋊Dic47—82no—
166#10625335—28no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—42no—
172C4⋊Dic47—81no—
173Dic47⋊C4—44no—
174Dic47⋊C4—42no—
175C4×Dic47—42no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8#4112yescomplemented
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47C2×SD1614yes—
199C23×C47C22×D411yescharacteristic
200Q8×C47—48no—
201D4×C47—22no—
202D4×C47—22no—
203D4×C47—28no—
204D4×C47—416no—
205D4×C47—41no—
206D4×C47—41no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—416no—
215C2×C4×C47—41no—
216C2×C4×C47—42no—
217C2×C4×C47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—48no—
224C2×C4×C47#12012yes—
225C2×C4×C47#12012yes—
226C2×C4×C47C2×SD1612yes—
227C2×C4×C47AGL(1,ℤ/8)12yes—
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47H(5,𝔽2)11yescharacteristic
233C2×C4×C472−1+411yescharacteristic
234C2×Dic47—22no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243Dic94—22no—
244Dic94—22no—
245Dic94—41no—
246Dic94—41no—
247C47⋊C8—22no—
248C47⋊C8—41no—
249C47⋊C8—42no—
250#24120—471nomaximal, Sylow
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—81no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47—81no—
265C22×C47C2×#12011yescharacteristic
266C22×C47#54611yescharacteristic
267C22×C47#54711yescharacteristic
268C22×C47C22×SD1611yescharacteristic
269C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C47#57711yescharacteristic
271C22×C47#57811yescharacteristic
272C4×C47—22no—
273C4×C47—42no—
274C4×C47—42no—
275C4×C47—44no—
276C4×C47#45014yes—
277C4×C47#52511yescharacteristic
278C4×C47#53711yescharacteristic
279C4×C47#54811yescharacteristic
280C4×C47#55011yescharacteristic
281Dic47—42no—
282Dic47—82no—
283Dic47—81no—
284C2×#547D4711yescharacteristic
285C2×#475—471no—
286C2×C4×SD16—471no—
287#2538—471no—
288#2034—472no—
289#2000—472no—
290#1990—471no—
291#1954—471no—
292#1872—472no—
293#1853—472no—
294#1802—471no—
295C2×C47—44no—
296C2×C47—82no—
297C2×C47#240711yescharacteristic
298C2×C47C2×#45011yescharacteristic
299C2×C47#301311yescharacteristic
300C2×C47#329811yescharacteristic
301C2×C47#333211yescharacteristic
302C2×C47#333511yescharacteristic
303C2×C47#335311yescharacteristic
304C22×#41—21no—
305C22×SD16—471no—
306#547—24no—
307#547—24no—
308C2×C4⋊Q8C2×D4711yescharacteristic
309C2×#123—22no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—472no—
312C2×#121C2×D4712yes—
313C2×C4×Q8—471no—
314C2×C4×D4C2×D4711yescharacteristic
315C2×C4×D4C2×D4711yescharacteristic
316C22×C4⋊C4C2×D4712yes—
317#475—944no—
318C4×SD16—944no—
319C2×#106—472no—
320C2×#106—471no—
321C2×C4⋊C8—471no—
322C2×#103—472no—
323C2×#103—471no—
324C2×#102—471no—
325C2×#102—472no—
326C2×C22⋊C8—472no—
327C2×C22⋊C8—472no—
328C2×C4×C8—471no—
329#396—942no—
330#390—941no—
331#389—941no—
332#384—471no—
333#380—472no—
334#340—472no—
335#340—472no—
336#340—472no—
337C47#2412011yescharacteristic, complemented, Sylow
338C2×#41—28no—
339C2×#41—44no—
340C2×#41—44no—
341C22×Q8—21no—
342C22×Q8—471no—
343C22×D4—22no—
344C22×D4C22⋊D4711yescharacteristic
345C22×D4C22×D4711yescharacteristic
346C23×C4—22no—
347C23×C4—21no—
348C23×C4C22×D4712yes—
349C2×SD16—942no—
350C2×SD16—942no—
351C2×SD16—942no—
352C22×C8—471no—
353C22×C8—471no—
354C22×C8—472no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D4714yes—
357#123—216no—
358C22⋊Q8—28no—
359C22⋊Q8—944no—
360C22⋊Q8—1882no—
361#121—24no—
362#121—24no—
363C4×Q8—942no—
364C4×Q8—942no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4C22⋊D4714yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4C22⋊D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4712yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—942no—
385C2×C22⋊C4—942no—
386C2×C22⋊C4C22⋊D4712yes—
387C2×C42—941no—
388C2×C42C22×D4711yescharacteristic
389#106—942no—
390#106—944no—
391C4⋊C8—942no—
392#103—942no—
393#103—944no—
394#102—942no—
395#102—1882no—
396C22⋊C8—944no—
397C22⋊C8—1882no—
398C4×C8—942no—
399#95—942no—
400C24—21no—
401C24C2×C22⋊D4712yes—
402#41—432no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—472no—
407C2×Q8—941no—
408C2×Q8—941no—
409C2×Q8—941no—
410C2×Q8—941no—
411C2×Q8—941no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4#1062533514yescomplemented
423C2×D4C2×C22⋊D4712yes—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—942no—
441C22×C4C2×C22⋊D4712yes—
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C2×C22⋊D4711yescharacteristic
444C22×C4C23×D4711yescharacteristic
445SD16—948no—
446C2×C8—472no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—942no—
451C2×C8—942no—
452C2×C8—941no—
453C2×C8—942no—
454C2×C8—941no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—942no—
463C4⋊C4—942no—
464C4⋊C4—3761no—
465C4⋊C4—1884no—
466C4⋊C4—1882no—
467C4⋊C4—1882no—
468C4⋊C4C2×C22⋊D4712yes—
469C4⋊C4C2×C22⋊D4714yes—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—3762no—
474C22⋊C4—1884no—
475C42—1882no—
476C42C2×C22⋊D4712yes—
477C42C2×C22⋊D4712yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×#1062533514yes—
492C23C22×C22⋊D4711yescharacteristic
493Q8—48no—
494Q8—942no—
495Q8—942no—
496Q8—1881no—
497Q8—1881no—
498D4—28no—
499D4—22no—
500D4—22no—
501D4—416no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—942no—
522C2×C4—1882no—
523C2×C4—1881no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4#1309399312yes—
531C2×C4#1309399312yes—
532C2×C4C2×#1062533512yes—
533C2×C4#1309405012yes—
534C2×C4C22×C22⋊D4711yescharacteristic
535C2×C4C22×C22⋊D4711yescharacteristic
536C2×C4#41×D4711yescharacteristic
537C2×C4#41×D4711yescharacteristic
538C2×C4#1309407011yescharacteristic
539C2×C4#1309407111yescharacteristic
540C8—942no—
541C8—1882no—
542C8—1881no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1309399311yescharacteristic
558C22#1568976111yescharacteristic
559C22#1568977211yescharacteristic
560C22C22×#1062533511yescharacteristic
561C22C2×#1309405011yescharacteristic
562C22#1568993111yescharacteristic
563C22#1568993811yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—3761no—
569C4—3762no—
570C4—1882no—
571C4#1568927414yes—
572C4#1568963011yescharacteristic
573C4#1568970111yescharacteristic
574C4#1568978011yescharacteristic
575C4#1568979211yescharacteristic
576C2—44no—
577C2—82no—
578C2#2024672711yescharacteristic
579C2C2×#1568927411yescharacteristic
580C2#2025270311yescharacteristic
581C2#2025524711yescharacteristic
582C2#2025570911yescharacteristic
583C2#2025574511yescharacteristic
584C2#2025597211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#547C2
socleC23×C47C22×D4
hypercenterC2×#547D47
hypocenterC47#24120
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#15689405C2semidirect
2. C2×C4×#10625335C2semidirect
3. #20248310C2semidirect
4. #20244574C2semidirect
5. #20244391C2semidirect
6. #20244350C2semidirect
7. #20244183C2semidirect
8. #20243695C2semidirect
9. #20243581C2semidirect
10. #20243388C2semidirect
123. C2×D4×C47SD16semidirect
151. C2×Dic94#41semidirect
184. C2×C47⋊C8#41semidirect
337. C47#24120semidirect
422. C2×D4#10625335semidirect

Maximal subgroups

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

subgroupindex
C2×#156894052
C2×C4×#106253352
#202483102
#202445742
#202443912
#202443502
#202441832
#202436952
#202435812
#202433882
C2×C47×#5472
#2412047