Finite Group Databasebeta

← #26243853

Subgroups — #26243853

Order 12032.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689701C211yesmaximal, characteristic
2C22×#13093956C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689089C211yesmaximal, characteristic, complemented
5#20245976C211yesmaximal, characteristic, complemented
6#20244583C214yesmaximal, complemented
7#20244576C214yesmaximal, complemented
8#20243109C211yesmaximal, characteristic, complemented
9C2×C47×#416C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689701—24no—
13#15689701—24no—
14C2×#13094003—21no—
15C2×#13094000C2212yes—
16C2×#13093998—21no—
17C2×#13093996—22no—
18C2×#13093995C2212yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21C2×#13093956—24no—
22C2×#13093956C2214yes—
23#15689124—24no—
24C2×#13093948C2212yes—
25C2×#13093945C2212yes—
26#15689089—24no—
27C2×(C2×D47)⋊C8C2214yes—
28#15689029—21no—
29#15689025—21no—
30#15689000—22no—
31#15688980C2214yes—
32#15688885C2214yes—
33C22×#41×C47C2211yescharacteristic
34C23×C8×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#416—24no—
37C2×#103×C47C2212yes—
38C2×#102×C47C2212yes—
39C2×#10625350—22no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C22×Dic94D411yescharacteristic
46C22×D188D411yescharacteristic
47C22×C22⋊D47—22no—
48C22×C22⋊D47D412yes—
49C23×Dic47—21no—
50C23×Dic47—21no—
51C22×C4×D47C2312yes—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C2×#10625329—22no—
55C22×C47⋊C8C2312yes—
56#13094003—28no—
57#13094000—24no—
58#13094000—24no—
59#13093998—28no—
60#13093996—216no—
61#13093995—24no—
62#13093995—24no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47D414yes—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—22no—
72C2×C22⋊Dic47—22no—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4—24no—
75#13093956—28no—
76#13093948—24no—
77#13093945—24no—
78(C2×D47)⋊C8—28no—
79C47⋊#95—24no—
80C2×#41×C47—22no—
81C2×#41×C47—22no—
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47D414yes—
86C22×Q8×C47C2311yescharacteristic
87C22×D4×C47—21no—
88C22×D4×C47C2311yescharacteristic
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91C22×C8×C47—22no—
92C22×C8×C47—22no—
93C22×C8×C47D414yescomplemented
94C22×C8×C47C2312yes—
95#117×C47—22no—
96#117×C47—22no—
97C2×C4⋊C4×C47C2312yes—
98C2×C22⋊C4×C47—21no—
99C2×C42×C47—21no—
100#103×C47—24no—
101#102×C47—24no—
102C24×C47—21no—
103C24×C47C2×D411yescharacteristic
104C23×D47C2×D412yes—
105#41×C47—28no—
106#41×C47—28no—
107#41×C47—44no—
108#41×C47—44no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47C2×D412yes—
115C2×D4×C47—24no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—42no—
122C2×D4×C47—42no—
123C2×D4×C47C2×D412yes—
124C22×C4×C47—21no—
125C22×C4×C47—24no—
126C22×C4×C47—22no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47#4112yes—
134C22×C4×C47#4112yes—
135C22×C4×C47#4112yes—
136C22×C4×C47C2411yescharacteristic
137C2×C22⋊D47—24no—
138C2×C22⋊D47—24no—
139C2×C22⋊D47—24no—
140C2×C22⋊D47—28no—
141C2×C22⋊D47—84no—
142C22×Dic47—22no—
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—42no—
146C22×Dic47—82no—
147C22×Dic47—82no—
148C22×Dic47C2×D412yes—
149#10625350—44no—
150#10625350—44no—
151#10625350—44no—
152#10625350—44no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×Dic94—22no—
163C2×Dic94—22no—
164C2×Dic94—22no—
165C2×C8×C47—28no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47C2×D414yes—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C22⋊C4×C47—42no—
172C42×C47—42no—
173C22⋊Dic47—28no—
174(C2×D47)⋊C4—28no—
175(C2×D47)⋊C4—84no—
176C4⋊Dic47—24no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47—42no—
181C4⋊Dic47C2×D414yes—
182Dic47⋊C4—28no—
183#10625329—44no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47#12012yes—
195C23×C47#12112yes—
196C23×C47#12312yes—
197C23×C47C22×D411yescharacteristic
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—24no—
202C22×D47—24no—
203Q8×C47—22no—
204Q8×C47—22no—
205Q8×C47—41no—
206Q8×C47—41no—
207D4×C47—22no—
208D4×C47—22no—
209D4×C47—48no—
210D4×C47—41no—
211D4×C47—41no—
212C2×C4×C47—22no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—48no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47#12012yes—
220C2×C4×C47#12112yes—
221C2×C4×C47#12114yes—
222C2×C4×C47#12114yes—
223C2×C4×C47#12312yes—
224C2×C4×C47#12314yes—
225C2×C4×C47#12414yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C22⋊D47—48no—
233C22⋊D47—88no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—44no—
239C2×Dic47—42no—
240C2×Dic47—44no—
241C2×Dic47—84no—
242C2×Dic47—84no—
243D188—44no—
244C4×D47—42no—
245C4×D47—42no—
246C4×D47—42no—
247C4×D47—42no—
248Dic94—44no—
249C8×C47—24no—
250C47⋊C8—42no—
251#23489—471nomaximal, Sylow
252C22×C47—24no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—24no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—82no—
260C22×C47—82no—
261C22×C47#39414yes—
262C22×C47C2×#12011yescharacteristic
263C22×C47C2×#12111yescharacteristic
264C22×C47C2×#12311yescharacteristic
265C22×C47#57712yes—
266C22×C47#57712yes—
267C2×D47—44no—
268C2×D47—42no—
269C2×D47—42no—
270C2×D47—42no—
271C2×D47—44no—
272C4×C47—42no—
273C4×C47—82no—
274C4×C47#39414yes—
275C4×C47C2×#12111yescharacteristic
276C4×C47C2×#12111yescharacteristic
277C4×C47C2×#12311yescharacteristic
278C4×C47C2×#12411yescharacteristic
279Dic47—42no—
280Dic47—44no—
281Dic47—82no—
282C2×#537—471no—
283C22×#106—471no—
284C2×#417—471no—
285C2×#416D4711yescharacteristic, complemented
286C2×#408—471no—
287#2311—471no—
288#2035—474no—
289#2034—474no—
290#1736—471no—
291C2×C47—24no—
292C2×C47—82no—
293C2×C47C2×#39411yescharacteristic
294C2×C47#300911yescharacteristic
295C2×C47#301011yescharacteristic
296C2×C47#306012yes—
297C2×C47#309912yes—
298D47—82no—
299C22×#41—471no—
300C22×#41C2×D4711yescharacteristic
301C22×M16—471no—
302C23×C8C2×D4711yescharacteristic
303#537—944no—
304#537—944no—
305C2×C22⋊Q8—472no—
306C2×C22⋊Q8—941no—
307C2×#121—472no—
308C2×#121—941no—
309C2×#121—942no—
310C2×#117C2×D4711yescharacteristic
311C22×C4⋊C4—471no—
312C22×C4⋊C4—471no—
313C2×#106—474no—
314C2×#106—944no—
315#417—944no—
316#416—24no—
317C2×#103—472no—
318C2×#103C2×D4712yes—
319C2×#102—472no—
320C2×#102C2×D4712yes—
321#408—944no—
322C2×C22⋊C8—474no—
323#391—941no—
324#390—941no—
325#386—942no—
326#380—474no—
327#340—474no—
328C47#2348911yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—942no—
335C2×#41—942no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C2×#41D18814yes—
341C22×Q8—471no—
342C22×Q8C22×D4711yescharacteristic
343C22×D4—21no—
344C22×D4—472no—
345C22×D4—471no—
346C22×D4—942no—
347C22×D4C22×D4711yescharacteristic
348C23×C4—21no—
349C23×C4—472no—
350C23×C4—941no—
351C23×C4—941no—
352C23×C4C22×D4711yescharacteristic
353C2×M16—942no—
354C2×M16—942no—
355C2×M16—942no—
356C22×C8—22no—
357C22×C8—22no—
358C22×C8—472no—
359C22×C8C22⋊D4714yescomplemented
360C22×C8C22×D4712yes—
361C22⋊Q8—944no—
362C22⋊Q8—948no—
363C22⋊Q8—944no—
364#121—944no—
365#121—948no—
366#121—9416no—
367#121—944no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—472no—
372C2×C4⋊C4—472no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4C22×D4712yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—944no—
384C2×C42—21no—
385#106—948no—
386#103—24no—
387#103—944no—
388#102—24no—
389#102—944no—
390C22⋊C8—948no—
391#95—944no—
392C24—21no—
393C24—472no—
394C24D4×D4711yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399#41—1884no—
400#41—1884no—
401#41—1884no—
402#41—1884no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—942no—
409C2×Q8—942no—
410C2×Q8—942no—
411C2×Q8C2×D18812yes—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—948no—
421C2×D4—944no—
422C2×D4—944no—
423C2×D4—944no—
424C2×D4—942no—
425C2×D4—942no—
426C2×D4—942no—
427C2×D4—3764no—
428C2×D4C2×D18812yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—472no—
435C22×C4—944no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—3762no—
445C22×C4—3762no—
446C22×C4—1882no—
447C22×C4C2×D18812yes—
448C22×C4#1062535012yes—
449C22×C4D4×D4711yescharacteristic
450C22×C4D4×D4712yes—
451C22×C4#1062535212yes—
452C22×C4#1062535212yes—
453C22×C4C2×C22⋊D4712yes—
454C22×C4C23×D4711yescharacteristic
455M16—1884no—
456C2×C8—28no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—942no—
460C2×C8—942no—
461C2×C8—942no—
462C2×C8C2×C22⋊D4714yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—474no—
466C4⋊C4—948no—
467C4⋊C4—944no—
468C4⋊C4—942no—
469C4⋊C4—942no—
470C4⋊C4—942no—
471C4⋊C4—1882no—
472C22⋊C4—42no—
473C22⋊C4—948no—
474C22⋊C4—948no—
475C22⋊C4—3764no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—942no—
485C23—942no—
486C23—942no—
487C23—944no—
488C23—944no—
489C23#1309399412yes—
490C23#1309399512yes—
491C23#1309400612yes—
492C23C2×D4×D4711yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—1884no—
498D4—22no—
499D4—22no—
500D4—48no—
501D4—41no—
502D4—41no—
503D4—3768no—
504D4—1888no—
505D4—1884no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—944no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—3764no—
518C2×C4—3764no—
519C2×C4—1882no—
520C2×C4—1884no—
521C2×C4—1884no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4#1309399412yes—
527C2×C4#1309399512yes—
528C2×C4#1309399614yes—
529C2×C4#1309399814yes—
530C2×C4#1309400612yes—
531C2×C4#1309400814yes—
532C2×C4#1309401014yes—
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C22×C22⋊D4711yescharacteristic
535C2×C4C22×D18811yescharacteristic
536C2×C4C2×#1062535211yescharacteristic
537C2×C4C2×#1062535211yescharacteristic
538C2×C4C2×#1062535011yescharacteristic
539C8—24no—
540C8—1882no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—1882no—
551C22—1882no—
552C22—1884no—
553C22—1884no—
554C22—1882no—
555C22#1568903914yes—
556C22C2×#1309399411yescharacteristic
557C22C2×#1309399511yescharacteristic
558C22C2×#1309400611yescharacteristic
559C22#1568993012yes—
560C22#1568993012yes—
561C4—42no—
562C4—82no—
563C4—3762no—
564C4—1882no—
565C4—1884no—
566C4#1568903914yes—
567C4C2×#1309399611yescharacteristic
568C4C2×#1309399811yescharacteristic
569C4C2×#1309400811yescharacteristic
570C4C2×#1309401011yescharacteristic
571C2—24no—
572C2—82no—
573C2—3762no—
574C2C2×#1568903911yescharacteristic
575C2#2025266811yescharacteristic
576C2#2025267711yescharacteristic
577C2#2025304612yes—
578C2#2025337512yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#416C2
socleC23×C47C22×D4
prime radicalC2×#15689701C2
hypercenterC2×#416D47
hypocenterC47#23489
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×#13093956C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689089C2semidirect
5. #20245976C2semidirect
6. #20244583C2semidirect
7. #20244576C2semidirect
8. #20243109C2semidirect
9. C2×C47×#416C2semidirect
93. C22×C8×C47D4semidirect
285. C2×#416D47semidirect
328. C47#23489semidirect
359. C22×C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897012
C22×#130939562
C2×#156891242
C2×#156890892
#202459762
#202445832
#202445762
#202431092
C2×C47×#4162
#2348947