Finite Group Databasebeta

← #26215881

Subgroups — #26215881

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094032C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3#20245981C211yesmaximal, characteristic
4#20245976C211yesmaximal, characteristic
5#20243422C212yesmaximal, complemented
6#20243418C212yesmaximal, complemented
7#20243414C212yesmaximal, complemented
8#20243410C212yesmaximal, complemented
9#20243104C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C2×C47×#428C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13094032—24no—
14C2×#13094032—24no—
15C2×#13094032C418yescomplemented
16C22×#10625324C2212yes—
17C23×C47⋊C8C2211yescharacteristic
18C22×C8×D47C2212yes—
19C2×#13093983C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21#15689165—24no—
22C2×#13093958C2212yes—
23C2×#13093956C2212yes—
24#15689029—21no—
25#15689028—21no—
26#15689025—21no—
27#15689024—21no—
28#15689002—22no—
29#15689000—22no—
30#15688980C2212yes—
31#15688980C2212yes—
32#15688980C2212yes—
33#15688980C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40C22×M16×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#428—24no—
44C2×#107×C47C2212yes—
45C2×#106×C47C2212yes—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350C2×C414yes—
51C2×#10625350Q814yes—
52C22×Dic94D411yescharacteristic
53C22×D188D411yescharacteristic
54C22×C22⋊D47D412yes—
55C23×Dic47—21no—
56C22×C4×D47C2312yes—
57#13094032—232no—
58C2×#10625324—24no—
59C2×#10625324—24no—
60C2×#10625324C2×C418yescomplemented
61C22×C47⋊C8—22no—
62C22×C47⋊C8—22no—
63C22×C47⋊C8C2×C414yes—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C2×C8×D47—24no—
67C2×C8×D47—24no—
68C2×C8×D47C2×C418yescomplemented
69#13093983—22no—
70#13093983—22no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2312yes—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C22⋊Dic47—21no—
78C2×(C2×D47)⋊C4—22no—
79C2×(C2×D47)⋊C4—22no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×C4×Dic47—21no—
83#13093958—24no—
84#13093956—24no—
85C47⋊#95—22no—
86C47⋊#95—22no—
87C47⋊#95—22no—
88C47⋊#95—22no—
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91C2×M16×C47C2×C414yes—
92C2×M16×C47D414yes—
93C2×M16×C47D414yes—
94C22×C8×C47C2312yes—
95#117×C47—22no—
96#117×C47—22no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47C2312yes—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—21no—
105#107×C47—24no—
106#106×C47—24no—
107C24×C47C2×D411yescharacteristic
108C23×D47C2×D412yes—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—42no—
113C22×C4×C47—81no—
114C22×C4×C47—81no—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×Q812yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×C4×C47C2411yescharacteristic
123C2×C22⋊D47—24no—
124C2×C22⋊D47—24no—
125C2×C22⋊D47C4⋊C418yes—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—22no—
129C22×Dic47—42no—
130C22×Dic47—81no—
131C22×Dic47—81no—
132C22×Dic47C2×D412yes—
133#10625350—28no—
134#10625350—28no—
135#10625350—28no—
136#10625350—28no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188C4⋊C414yes—
140C2×C4×D47—22no—
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47C22×C414yes—
145C2×C4×D47C2×Q814yes—
146C2×Dic94—22no—
147C2×Dic94—22no—
148C2×Dic94C4⋊C414yes—
149M16×C47C22⋊C4116yescomplemented
150C2×C8×C47C22×C414yes—
151C2×C8×C47C2×D414yes—
152C2×C8×C47C2×D414yes—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—42no—
160C22⋊C4×C47—42no—
161C42×C47—42no—
162C22⋊Dic47—42no—
163(C2×D47)⋊C4—82no—
164(C2×D47)⋊C4—82no—
165(C2×D47)⋊C4—82no—
166(C2×D47)⋊C4—82no—
167C4⋊Dic47—21no—
168C4⋊Dic47—22no—
169C4⋊Dic47—21no—
170C4⋊Dic47—22no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—42no—
174C4×Dic47—42no—
175C2×C47⋊C8—28no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8C22×C412yes—
181C2×C47⋊C8C22×C412yes—
182#10625324—216no—
183C8×D47—216no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47C2×C4⋊C412yes—
189C23×C47#12112yes—
190C23×C47#12112yes—
191C23×C47C22×D411yescharacteristic
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47C2×C4⋊C414yes—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—84no—
209C2×C4×C47C2×C22⋊C414yes—
210C2×C4×C47C2×C4⋊C412yes—
211C2×C4×C47#11714yes—
212C2×C4×C47#12112yes—
213C2×C4×C47#12112yes—
214C2×C4×C47C22⋊Q814yes—
215C2×C4×C47C22⋊Q814yes—
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×Q811yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C22⋊D47—216no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—42no—
228C2×Dic47—41no—
229C2×Dic47—42no—
230C2×Dic47—41no—
231C2×Dic47—42no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—84no—
235C2×Dic47C2×C4⋊C414yes—
236D188—28no—
237C4×D47—24no—
238C4×D47—24no—
239C4×D47—24no—
240C4×D47—24no—
241Dic94—28no—
242C8×C47C2×C22⋊C418yes—
243C47⋊C8—22no—
244C47⋊C8—22no—
245#20664—471nomaximal, Sylow
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47#38014yes—
253C22×C47C22×C4⋊C411yescharacteristic
254C22×C47C2×#12111yescharacteristic
255C22×C47C2×#12111yescharacteristic
256C22×C47#57611yescharacteristic
257C22×C47#57711yescharacteristic
258C22×C47#57711yescharacteristic
259C22×C47#57811yescharacteristic
260C2×D47—24no—
261C2×D47—24no—
262C2×D47—24no—
263C2×D47—42no—
264C2×D47—42no—
265C2×D47—42no—
266C2×D47—42no—
267C4×C47—81no—
268C4×C47—82no—
269C4×C47—81no—
270C4×C47#38014yes—
271C4×C47C22×C22⋊C411yescharacteristic
272C4×C47C2×#11711yescharacteristic
273C4×C47C2×C22⋊Q811yescharacteristic
274C4×C47C2×C22⋊Q811yescharacteristic
275Dic47—24no—
276Dic47—81no—
277Dic47—82no—
278Dic47—81no—
279C22×#131—471no—
280C2×#428—471no—
281C2×#428D4711yescharacteristic, complemented
282#2311—471no—
283#2311—471no—
284#1809—472no—
285#1809—472no—
286#1808—472no—
287#1807—472no—
288#1735—471no—
289C2×#340—471no—
290C2×C47—24no—
291C2×C47C2×#38011yescharacteristic
292C2×C47#291211yescharacteristic
293C2×C47#291311yescharacteristic
294C2×C47#306211yescharacteristic
295C2×C47#306211yescharacteristic
296C2×C47#306311yescharacteristic
297C2×C47#306311yescharacteristic
298D47—44no—
299C22×#41—471no—
300C2×#131—478no—
301C2×#131—944no—
302C2×#131—944no—
303C22×M16—472no—
304C22×M16C2×D4711yescharacteristic
305C23×C8—471no—
306C23×C8—472no—
307C2×#117—471no—
308C2×#117C2×D4711yescharacteristic
309C22×C4⋊C4—471no—
310C22×C4⋊C4C2×D4711yescharacteristic
311#428—24no—
312#428—944no—
313C2×#107—472no—
314C2×#107C2×D4712yes—
315C2×#106—472no—
316C2×#106C2×D4712yes—
317#391—941no—
318#391—941no—
319#390—941no—
320#390—941no—
321#386—942no—
322#386—942no—
323#380—472no—
324#380—472no—
325#380—472no—
326#380—472no—
327#340—471no—
328#340—472no—
329#340—471no—
330#340—472no—
331#340—471no—
332#340—471no—
333C47#2066411yescharacteristic, complemented, Sylow
334C2×#41—474no—
335C2×#41—474no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C22×Q8—471no—
341C22×D4—472no—
342C22×D4—471no—
343C23×C4—21no—
344C23×C4—472no—
345C23×C4—941no—
346C23×C4C22×D4711yescharacteristic
347#131—9432no—
348C2×M16—478no—
349C2×M16—944no—
350C2×M16—944no—
351C2×M16C4×D4714yes—
352C2×M16D18814yes—
353C2×M16C22⋊D4714yes—
354C22×C8—474no—
355C22×C8—471no—
356C22×C8—471no—
357C22×C8—478no—
358C22×C8—942no—
359C22×C8—942no—
360C22×C8—944no—
361C22×C8—944no—
362C22×C8C22×D4712yes—
363#117—22no—
364#117—22no—
365#117—942no—
366#117—942no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—472no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22×D4712yes—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—942no—
384C2×C22⋊C4—941no—
385C2×C42—21no—
386C2×C42—941no—
387#107—24no—
388#107—944no—
389#106—24no—
390#106—944no—
391#95—942no—
392#95—942no—
393#95—942no—
394#95—942no—
395C24—472no—
396C24D4×D4711yescharacteristic
397#41—948no—
398#41—948no—
399#41—948no—
400#41—948no—
401C2×Q8—474no—
402C2×Q8—942no—
403C2×Q8—942no—
404C2×D4—478no—
405C2×D4—474no—
406C2×D4—944no—
407C2×D4—944no—
408C2×D4—942no—
409C2×D4—942no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—472no—
417C22×C4—474no—
418C22×C4—474no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—942no—
426C22×C4—3761no—
427C22×C4—3761no—
428C22×C4—1882no—
429C22×C4C2×C4×D4712yes—
430C22×C4C2×D18812yes—
431C22×C4D4×D4711yescharacteristic
432C22×C4#1062535212yes—
433C22×C4Q8×D4712yes—
434C22×C4#1062535412yes—
435C22×C4C2×C22⋊D4712yes—
436C22×C4C23×D4711yescharacteristic
437M16—9416no—
438M16(C2×D47)⋊C4116yescomplemented
439C2×C8—472no—
440C2×C8—472no—
441C2×C8—948no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—9416no—
447C2×C8C2×C4×D4714yes—
448C2×C8C2×D18814yes—
449C2×C8C2×C22⋊D4714yes—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—941no—
458C4⋊C4—942no—
459C4⋊C4—941no—
460C4⋊C4—942no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—1882no—
464C22⋊C4—42no—
465C22⋊C4—3762no—
466C22⋊C4—3762no—
467C22⋊C4—3762no—
468C22⋊C4—3762no—
469C22⋊C4—1882no—
470C42—42no—
471C42—1882no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—474no—
477C23—942no—
478C23—942no—
479C23—942no—
480C23—942no—
481C23—942no—
482C23—942no—
483C23C4⋊C4×D4712yes—
484C23#1309399512yes—
485C23#1309399912yes—
486C23C2×D4×D4711yescharacteristic
487Q8—948no—
488D4—9416no—
489D4—948no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4—474no—
501C2×C4—942no—
502C2×C4—942no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—944no—
507C2×C4—3764no—
508C2×C4—1881no—
509C2×C4—1881no—
510C2×C4—1882no—
511C2×C4—1881no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4C2×(C2×D47)⋊C414yes—
518C2×C4C4⋊C4×D4712yes—
519C2×C4#1309398214yes—
520C2×C4#1309399512yes—
521C2×C4#1309399912yes—
522C2×C4#1309400014yes—
523C2×C4#1309400414yes—
524C2×C4C22×C4×D4711yescharacteristic
525C2×C4C22×C22⋊D4711yescharacteristic
526C2×C4C22×D18811yescharacteristic
527C2×C4C2×Q8×D4711yescharacteristic
528C2×C4C2×#1062535211yescharacteristic
529C2×C4C2×#1062535411yescharacteristic
530C8—942no—
531C8—942no—
532C8C2×(C2×D47)⋊C418yes—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—944no—
540C22—944no—
541C22—944no—
542C22—1882no—
543C22—1882no—
544C22—1882no—
545C22—1882no—
546C22#1568898014yes—
547C22C2×C4⋊C4×D4711yescharacteristic
548C22C2×#1309399511yescharacteristic
549C22C2×#1309399911yescharacteristic
550C22#1568992711yescharacteristic
551C22#1568993211yescharacteristic
552C22#1568993311yescharacteristic
553C22#1568993911yescharacteristic
554C4—82no—
555C4—81no—
556C4—81no—
557C4—944no—
558C4—3762no—
559C4—3761no—
560C4—3761no—
561C4#1568898014yes—
562C4C22×(C2×D47)⋊C411yescharacteristic
563C4C2×#1309398211yescharacteristic
564C4C2×#1309400011yescharacteristic
565C4C2×#1309400411yescharacteristic
566C2—24no—
567C2—1884no—
568C2C2×#1568898011yescharacteristic
569C2#2025198211yescharacteristic
570C2#2025198711yescharacteristic
571C2#2025306211yescharacteristic
572C2#2025306911yescharacteristic
573C2#2025307311yescharacteristic
574C2#2025308011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47#20664
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
5. #20243422C2semidirect
6. #20243418C2semidirect
7. #20243414C2semidirect
8. #20243410C2semidirect
9. #20243104C2semidirect
10. C2×#15688885C2semidirect
11. C2×C47×#428C2semidirect
15. C2×#13094032C4semidirect
60. C2×#10625324C2×C4semidirect
68. C2×C8×D47C2×C4semidirect
149. M16×C47C22⋊C4semidirect
281. C2×#428D47semidirect
333. C47#20664semidirect
438. M16(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940322
C2×#156891652
#202459812
#202459762
#202434222
#202434182
#202434142
#202434102
#202431042
C2×#156888852
C2×C47×#4282
#2066447