Finite Group Databasebeta

← #26186912

Subgroups — #26186912

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253983C211yesmaximal, characteristic
2#20252539C211yesmaximal, characteristic, complemented
3#20252463C211yesmaximal, characteristic, complemented
4#20242303C211yesmaximal, characteristic, complemented
5#20242292C211yesmaximal, characteristic, complemented
6#20242280C211yesmaximal, characteristic, complemented
7#20242274C211yesmaximal, characteristic, complemented
8#20242213C211yesmaximal, characteristic, complemented
9#20242202C211yesmaximal, characteristic, complemented
10C8×#10625325C211yesmaximal, characteristic, complemented
11C8×D376C211yesmaximal, characteristic, complemented
12#20242155C211yesmaximal, characteristic, complemented
13#20241849C211yesmaximal, characteristic, complemented
14#20241840C211yesmaximal, characteristic, complemented
15C8×M16×C47C211yesmaximal, characteristic, complemented
16#15689673C2211yescharacteristic
17#15689657C2211yescharacteristic
18#15689532C411yescharacteristic
19#15689512C411yescharacteristic
20#15689488C411yescharacteristic
21#15689483C411yescharacteristic
22#15689376C411yescharacteristic
23#15689376C411yescharacteristic
24#15689369C411yescharacteristic
25#15689369C411yescharacteristic
26#15689253C2211yescharacteristic
27C8×Dic94C2211yescharacteristic
28C4×#10625325C2211yescharacteristic
29C4×#10625325C2211yescharacteristic
30C4×D376C2211yescharacteristic
31C4×D376C2211yescharacteristic
32#15689202C2211yescharacteristic
33#15689202C2211yescharacteristic
34#15689202C2211yescharacteristic
35#15689198—21no—
36#15689197—21no—
37#15689193—21no—
38#15689193—21no—
39C8×D188C2211yescharacteristic
40C8×D188C2211yescharacteristic
41C8×D188C2211yescharacteristic
42#15689183—21no—
43#15689180C2211yescharacteristic
44#15689148—21no—
45#15689147C2211yescharacteristic
46#15689088—21no—
47#15689088—21no—
48#15688876C2211yescharacteristic
49#15688876C2211yescharacteristic
50#15688874C2211yescharacteristic
51#15688874C2211yescharacteristic
52#15688855C2211yescharacteristic
53#15688855C2211yescharacteristic
54#15688852C2211yescharacteristic
55#15688852C2211yescharacteristic
56#15688852C2211yescharacteristic
57#15688852C2211yescharacteristic
58#15688852C2211yescharacteristic
59#15688852C2211yescharacteristic
60C47×#431C2211yescharacteristic
61C4×M16×C47C2211yescharacteristic
62C2×C4×C8×C47C2211yescharacteristic
63C47×#322C2211yescharacteristic
64C47×#322C2211yescharacteristic
65C82×C47C2211yescharacteristic
66C82×C47C2211yescharacteristic
67C2×#10625350—21no—
68C2×#10625325C2×C411yescharacteristic
69C2×#10625325C2×C411yescharacteristic
70C2×D376C2×C411yescharacteristic
71C2×D376C2×C411yescharacteristic
72C2×#10625329—21no—
73C2×#10625324—21no—
74C2×#10625324—21no—
75C2×#10625324—21no—
76C2×C8×D47—21no—
77C2×C8×D47—21no—
78C2×C8×D47—21no—
79#13094024C2×C411yescharacteristic
80#13094021C2×C411yescharacteristic
81#13094018—21no—
82#13094016C2×C411yescharacteristic
83#13094012—21no—
84#13094012C2×C411yescharacteristic
85#13094007—21no—
86#13094003C2×C411yescharacteristic
87#13093998—21no—
88#13093998C2×C411yescharacteristic
89#13093998C2×C411yescharacteristic
90#13093998C2×C411yescharacteristic
91C4×Dic94C2311yescharacteristic
92C4×D188—21no—
93C4×D188C2311yescharacteristic
94C4×D188C2311yescharacteristic
95C4×D188C2311yescharacteristic
96C4×C22⋊D47—21no—
97#13093981—21no—
98#13093958C2×C411yescharacteristic
99#13093958C2×C411yescharacteristic
100#13093956C2×C411yescharacteristic
101#13093956C2×C411yescharacteristic
102C4⋊(C47⋊C8)—21no—
103C4⋊(C47⋊C8)C2311yescharacteristic
104C4⋊(C47⋊C8)C2311yescharacteristic
105C4⋊(C47⋊C8)C2311yescharacteristic
106C4⋊(C47⋊C8)C2311yescharacteristic
107Dic47⋊C8—21no—
108Dic47⋊C8—21no—
109#13093949C2×C411yescharacteristic
110#13093949C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093946C2×C411yescharacteristic
113#13093946C2×C411yescharacteristic
114#13093946C2×C411yescharacteristic
115#13093946C2×C411yescharacteristic
116#13093946C2×C411yescharacteristic
117C22⋊(C47⋊C8)—21no—
118(C2×D47)⋊C8—21no—
119(C2×D47)⋊C8—21no—
120(C2×D47)⋊C8—21no—
121(C2×D47)⋊C8—21no—
122(C2×D47)⋊C8—21no—
123(C2×D47)⋊C8—21no—
124#13093935—21no—
125#13093934—21no—
126C8×Dic47—21no—
127C2×M16×C47C2×C411yescharacteristic
128C2×M16×C47C2×C411yescharacteristic
129C22×C8×C47—21no—
130C2×C42×C47C2311yescharacteristic
131C4⋊C8×C47D411yescharacteristic
132C4⋊C8×C47D411yescharacteristic
133C22⋊C8×C47D411yescharacteristic
134C22⋊C8×C47D411yescharacteristic
135#97×C47C2×C411yescharacteristic
136#97×C47C2×C411yescharacteristic
137C4×C8×C47—21no—
138C4×C8×C47C2311yescharacteristic
139C4×C8×C47C2311yescharacteristic
140C4×C8×C47C2311yescharacteristic
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C22×C4×C47C22×C411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47#4111yescharacteristic
147C2×C22⋊D47—21no—
148C2×C22⋊D47—21no—
149C2×C22⋊D47—21no—
150#10625350—81no—
151#10625350—81no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188C22×C411yescharacteristic
155C2×D188C22×C411yescharacteristic
156C2×D188C22×C411yescharacteristic
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×Dic94—21no—
161C2×Dic94C22×C411yescharacteristic
162M16×C47—22no—
163M16×C47—22no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—42no—
167C2×C8×C47C22×C411yescharacteristic
168C2×C8×C47C22×C411yescharacteristic
169C2×C8×C47C22×C411yescharacteristic
170C2×C8×C47C22×C411yescharacteristic
171C2×C8×C47C2×D411yescharacteristic
172C2×C8×C47C2×D411yescharacteristic
173C2×C8×C47C2×D411yescharacteristic
174C2×C8×C47C2×D411yescharacteristic
175C42×C47C22×C411yescharacteristic
176C42×C47C2×D411yescharacteristic
177C42×C47#4111yescharacteristic
178C42×C47C2411yescharacteristic
179C22⋊Dic47—21no—
180(C2×D47)⋊C4—21no—
181(C2×D47)⋊C4—21no—
182(C2×D47)⋊C4—21no—
183(C2×D47)⋊C4—21no—
184(C2×D47)⋊C4—21no—
185(C2×D47)⋊C4—21no—
186(C2×D47)⋊C4—21no—
187(C2×D47)⋊C4—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C22×C411yescharacteristic
190C4⋊Dic47C22×C411yescharacteristic
191C4⋊Dic47C22×C411yescharacteristic
192C4⋊Dic47C22×C411yescharacteristic
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195Dic47⋊C4—21no—
196Dic47⋊C4—21no—
197C4×Dic47—21no—
198#10625329—81no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203D376—22no—
204D376—22no—
205#10625325—22no—
206#10625325—22no—
207#10625324—81no—
208#10625324—81no—
209#10625324—81no—
210C8×D47—81no—
211C8×D47—81no—
212C8×D47—81no—
213C23×C47C4×D411yescharacteristic
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—41no—
222C2×C4×C47C4×D411yescharacteristic
223C2×C4×C47C4×D411yescharacteristic
224C2×C4×C47C4×D411yescharacteristic
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C22⋊D47—81no—
229C22⋊D47—81no—
230C22⋊D47—81no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236D188—21no—
237D188—21no—
238D188—21no—
239D188—21no—
240D188—21no—
241D188—21no—
242D188—81no—
243D188—81no—
244C4×D47—41no—
245C4×D47—41no—
246C4×D47—41no—
247C4×D47—41no—
248C4×D47—41no—
249C4×D47—41no—
250Dic94—21no—
251Dic94—21no—
252Dic94—81no—
253C8×C47—22no—
254C8×C47—21no—
255C8×C47—21no—
256C8×C47—21no—
257C8×C47—21no—
258C8×C47—21no—
259C8×C47—21no—
260C8×C47—41no—
261C8×C47#13112yes—
262C8×C47#13112yes—
263C47⋊C8—41no—
264C47⋊C8—41no—
265C47⋊C8—41no—
266C47⋊C8—41no—
267#17968—471nomaximal, Sylow
268C22×C47—21no—
269C22×C47—21no—
270C22×C47—21no—
271C22×C47C2×C4×D411yescharacteristic
272C2×D47—41no—
273C2×D47—41no—
274C2×D47—41no—
275C2×D47—41no—
276C2×D47—41no—
277C2×D47—41no—
278C2×D47—41no—
279C2×D47—41no—
280C2×D47—41no—
281C4×C47—21no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—21no—
285C4×C47C2×#13111yescharacteristic
286C4×C47C2×#13111yescharacteristic
287C4×C47#57611yescharacteristic
288C4×C47#57711yescharacteristic
289Dic47—41no—
290Dic47—41no—
291Dic47—41no—
292Dic47—41no—
293Dic47—41no—
294#3171—471no—
295#2993—471no—
296#2981—471no—
297#1594—471no—
298#1592—471no—
299#1590—471no—
300#1589—471no—
301#1580—471no—
302SD16⋊C8—471no—
303C8×SD16—471no—
304C8×D8—471no—
305#1569—471no—
306#1494—471no—
307#1492—471no—
308C8×M16D4711yescharacteristic, complemented
309C2×C47—41no—
310C2×C47#293111yescharacteristic
311C2×C47#294611yescharacteristic
312C2×C47#295611yescharacteristic
313D47—81no—
314D47—81no—
315D47—81no—
316#532—471no—
317#529—471no—
318#501—471no—
319#496—471no—
320#490—471no—
321#489—471no—
322#469—471no—
323#469—471no—
324#468—471no—
325#468—471no—
326#446—471no—
327C8×Q8—471no—
328C4×SD16—471no—
329C4×SD16—471no—
330C4×D8—471no—
331C4×D8—471no—
332#436—471no—
333#436—471no—
334#436—471no—
335#436—941no—
336#435—941no—
337#435—941no—
338#435—941no—
339C8×D4—471no—
340C8×D4—471no—
341C8×D4—471no—
342#433—941no—
343#432—471no—
344#431C2×D4711yescharacteristic
345#424—941no—
346#423—471no—
347#408—941no—
348#408—941no—
349C4×M16C2×D4711yescharacteristic
350C2×C4×C8C2×D4711yescharacteristic
351#335—471no—
352#335—471no—
353#334—471no—
354#334—471no—
355Q8⋊C8—471no—
356Q8⋊C8—471no—
357D4⋊C8—471no—
358D4⋊C8—471no—
359D4⋊C8—471no—
360D4⋊C8—471no—
361D4⋊C8—471no—
362D4⋊C8—471no—
363#322C2×D4711yescharacteristic
364#322C2×D4711yescharacteristic
365C82C2×D4711yescharacteristic
366C82C2×D4711yescharacteristic
367C47#1796811yescharacteristic, complemented, Sylow
368C2×#41—941no—
369C2×SD16—471no—
370C2×SD16—471no—
371C2×D8—471no—
372C2×D8—471no—
373C2×M16—941no—
374C2×M16—941no—
375C2×M16—941no—
376C2×M16—941no—
377C2×M16C4×D4711yescharacteristic
378C2×M16C4×D4711yescharacteristic
379C22×C8—21no—
380C22×C8—941no—
381C22×C8—941no—
382C22×C8—941no—
383C4⋊Q8—471no—
384#127—471no—
385#126—941no—
386#125—471no—
387#124—471no—
388#124—941no—
389#123—941no—
390C22⋊Q8—471no—
391#121—471no—
392#121—471no—
393#121—471no—
394#121—941no—
395C4×Q8—471no—
396C4×D4—471no—
397C4×D4—471no—
398C4×D4—471no—
399C4×D4—941no—
400C4×D4—941no—
401#117—941no—
402C2×C42C22×D4711yescharacteristic
403#107—471no—
404#107—471no—
405#106—471no—
406#106—471no—
407C4⋊C8—471no—
408C4⋊C8—471no—
409C4⋊C8—471no—
410C4⋊C8—471no—
411C4⋊C8—941no—
412C4⋊C8—941no—
413C4⋊C8—941no—
414C4⋊C8D18811yescharacteristic
415C4⋊C8D18811yescharacteristic
416#103—471no—
417#103—471no—
418#102—471no—
419#102—471no—
420#102—471no—
421#102—471no—
422#102—471no—
423#102—471no—
424C22⋊C8—941no—
425C22⋊C8—941no—
426C22⋊C8—941no—
427C22⋊C8—941no—
428C22⋊C8—941no—
429C22⋊C8—941no—
430C22⋊C8—941no—
431C22⋊C8D18811yescharacteristic
432C22⋊C8D18811yescharacteristic
433#97—941no—
434#97—941no—
435#97C4×D4711yescharacteristic
436#97C4×D4711yescharacteristic
437C4×C8—21no—
438C4×C8—941no—
439C4×C8C22×D4711yescharacteristic
440C4×C8C22×D4711yescharacteristic
441C4×C8C22×D4711yescharacteristic
442C4×C8C22×D4711yescharacteristic
443C4×C8C22×D4711yescharacteristic
444C4×C8C22×D4711yescharacteristic
445#41—3761no—
446#41—3761no—
447C2×Q8—471no—
448C2×Q8—941no—
449C2×D4—471no—
450C2×D4—471no—
451C2×D4—471no—
452C2×D4—941no—
453C2×D4—941no—
454C2×D4—941no—
455C2×D4—941no—
456C2×D4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4C2×C4×D4711yescharacteristic
461C22×C4C2×D18811yescharacteristic
462C22×C4#1062535011yescharacteristic
463SD16—942no—
464SD16—942no—
465D8—942no—
466D8—942no—
467M16—22no—
468M16—22no—
469M16—3761no—
470M16—3761no—
471M16—3761no—
472M16—3761no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—42no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—3761no—
481C2×C8—3761no—
482C2×C8—3761no—
483C2×C8C2×C4×D4711yescharacteristic
484C2×C8C2×C4×D4711yescharacteristic
485C2×C8C2×C4×D4711yescharacteristic
486C2×C8C2×C4×D4711yescharacteristic
487C2×C8C2×D18811yescharacteristic
488C2×C8C2×D18811yescharacteristic
489C2×C8C2×D18811yescharacteristic
490C2×C8C2×D18811yescharacteristic
491C4⋊C4—471no—
492C4⋊C4—471no—
493C4⋊C4—471no—
494C4⋊C4—471no—
495C4⋊C4—941no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C22⋊C4—941no—
501C22⋊C4—941no—
502C22⋊C4—941no—
503C22⋊C4—941no—
504C22⋊C4—941no—
505C22⋊C4—941no—
506C22⋊C4—941no—
507C22⋊C4—941no—
508C22⋊C4—941no—
509C42—941no—
510C42C2×C4×D4711yescharacteristic
511C42C2×D18811yescharacteristic
512C42#1062535011yescharacteristic
513C42C23×D4711yescharacteristic
514C23—941no—
515C23—941no—
516C23—941no—
517C23C4×D18811yescharacteristic
518Q8—941no—
519Q8—941no—
520Q8—3761no—
521D4—941no—
522D4—941no—
523D4—941no—
524D4—941no—
525D4—941no—
526D4—941no—
527D4—3761no—
528D4—3761no—
529D4—3761no—
530D4—3761no—
531D4—3761no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4C4×D18811yescharacteristic
549C2×C4C4×D18811yescharacteristic
550C2×C4C4×D18811yescharacteristic
551C2×C4C22×C4×D4711yescharacteristic
552C2×C4C22×D18811yescharacteristic
553C2×C4C2×#1062535011yescharacteristic
554C8—22no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—1881no—
563C8—1881no—
564C8—1881no—
565C8—1881no—
566C8#1309403012yes—
567C8#1309403012yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—1881no—
572C22—1881no—
573C22—1881no—
574C22—1881no—
575C22—1881no—
576C22—1881no—
577C22—1881no—
578C22—1881no—
579C22—1881no—
580C22C2×C4×D18811yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—1881no—
586C4—1881no—
587C4—1881no—
588C4—1881no—
589C4—1881no—
590C4C2×#1309403011yescharacteristic
591C4C2×#1309403011yescharacteristic
592C4#1568992811yescharacteristic
593C4#1568993511yescharacteristic
594C2—41no—
595C2—3761no—
596C2—3761no—
597C2—3761no—
598C2#2025210111yescharacteristic
599C2#2025222211yescharacteristic
600C2#2025229911yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC8×M16×C47C2
socleC22×C47C2×C4×D4
prime radical#20253983C2
cosocleC42×C47C24
hypercenterC8×M16D47
hypocenterC47#17968
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #20252539C2semidirect
3. #20252463C2semidirect
4. #20242303C2semidirect
5. #20242292C2semidirect
6. #20242280C2semidirect
7. #20242274C2semidirect
8. #20242213C2semidirect
9. #20242202C2semidirect
10. C8×#10625325C2semidirect
11. C8×D376C2semidirect
12. #20242155C2semidirect
13. #20241849C2semidirect
14. #20241840C2semidirect
15. C8×M16×C47C2semidirect
308. C8×M16D47semidirect
367. C47#17968semidirect

Maximal subgroups

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

subgroupindex
#202539832
#202525392
#202524632
#202423032
#202422922
#202422802
#202422742
#202422132
#202422022
C8×#106253252
C8×D3762
#202421552
#202418492
#202418402
C8×M16×C472
#1796847