Finite Group Databasebeta

← #26244963

Subgroups — #26244963

Order 12032.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689163C211yesmaximal, characteristic, complemented
2C2×#15689149C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic
4C2×#15689120C211yesmaximal, characteristic
5#20246012C211yesmaximal, characteristic, complemented
6#20245981C211yesmaximal, characteristic
7#20244784C212yesmaximal, complemented
8#20244777C212yesmaximal, complemented
9#20244770C212yesmaximal, complemented
10#20244763C212yesmaximal, complemented
11C2×C47×#340C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C2×#13093983C2211yescharacteristic
15C2×#13093983C2211yescharacteristic
16#15689163—24no—
17C2×#13093957C2212yes—
18C2×#13093955C2212yes—
19#15689149—24no—
20C2×Dic47⋊C8C2212yes—
21C2×Dic47⋊C8C2212yes—
22#15689122—24no—
23#15689120—24no—
24C2×#13093949C2211yescharacteristic
25C2×#13093949C2211yescharacteristic
26C2×#13093947C2212yes—
27C2×#13093946C2211yescharacteristic
28C2×#13093946C2211yescharacteristic
29C2×#13093944C2212yes—
30#15689028C2212yes—
31#15689024C2212yes—
32#15689019C2212yes—
33#15689019C2212yes—
34#15689002—22no—
35#15688996—22no—
36#15688992—21no—
37#15688988—21no—
38#15688980—21no—
39#15688980—21no—
40C23×C8×C47C2211yescharacteristic
41C22×C4⋊C4×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#340C2212yes—
44C47×#340C2212yes—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350Q814yes—
51C22×Dic94C2311yescharacteristic
52C22×D188C2311yescharacteristic
53C22×C22⋊D47—21no—
54C22×C4×D47—21no—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329D414yes—
58C22×C47⋊C8C2312yes—
59#13093983—22no—
60#13093983—22no—
61#13093983—22no—
62#13093983D414yes—
63C2×C4⋊Dic47D411yescharacteristic
64C2×C4⋊Dic47D411yescharacteristic
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4—22no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47D412yes—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—22no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47C2312yes—
75#13093957—24no—
76#13093955—24no—
77Dic47⋊C8—24no—
78Dic47⋊C8—24no—
79#13093949—22no—
80#13093949—22no—
81#13093947—24no—
82#13093946—22no—
83#13093946—22no—
84#13093944—24no—
85C47⋊#95—22no—
86C47⋊#95—22no—
87C47⋊#95—22no—
88C47⋊#95—22no—
89C23×C4×C47—21no—
90C23×C4×C47—21no—
91C23×C4×C47C2311yescharacteristic
92C22×C8×C47—22no—
93C22×C8×C47—22no—
94C22×C8×C47—22no—
95C22×C8×C47C2311yescharacteristic
96C22×C8×C47C2311yescharacteristic
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47—22no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47C2312yes—
104C2×C4⋊C4×C47C2312yes—
105C24×C47C2×D411yescharacteristic
106C23×D47—21no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—81no—
112C22×C4×C47—81no—
113C22×C4×C47—81no—
114C22×C4×C47—81no—
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×Q812yes—
119C22×C4×C47#4112yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×C4×C47C2411yescharacteristic
123C2×C22⋊D47—24no—
124C2×C22⋊D47—42no—
125C2×C22⋊D47—42no—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47C2×D412yes—
130C22×Dic47C2×D412yes—
131#10625350—28no—
132#10625350—28no—
133#10625350—44no—
134#10625350—44no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188C2×Q812yes—
141C2×C4×D47—24no—
142C2×C4×D47—42no—
143C2×C4×D47—42no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—21no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94C2×Q812yes—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—44no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—42no—
162C4⋊C4×C47—42no—
163C22⋊Dic47—24no—
164C22⋊Dic47—24no—
165(C2×D47)⋊C4—82no—
166(C2×D47)⋊C4—82no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47C2×D412yes—
172C4⋊Dic47C2×D412yes—
173Dic47⋊C4—82no—
174Dic47⋊C4—82no—
175C4×Dic47—24no—
176C4×Dic47—22no—
177C4×Dic47—22no—
178#10625329—28no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8C2×D414yes—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47#12112yes—
187C23×C47#12112yes—
188C23×C47C22⋊Q812yes—
189C23×C47C22×D411yescharacteristic
190C22×D47—22no—
191C22×D47—41no—
192C22×D47—41no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—84no—
204C2×C4×C47—84no—
205C2×C4×C47#12114yes—
206C2×C4×C47#12112yes—
207C2×C4×C47#12112yes—
208C2×C4×C47C22⋊Q814yes—
209C2×C4×C47C22⋊Q814yes—
210C2×C4×C47C22⋊Q812yes—
211C2×C4×C47#12614yes—
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C22×Q811yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C22⋊D47—48no—
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234D188—22no—
235D188—22no—
236D188—41no—
237D188—41no—
238C4×D47—48no—
239Dic94—22no—
240Dic94—22no—
241Dic94—41no—
242Dic94—41no—
243C8×C47—41no—
244C8×C47—41no—
245C47⋊C8—24no—
246#23616—471nomaximal, Sylow
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47#39614yes—
254C22×C47C2×#12111yescharacteristic
255C22×C47C2×#12111yescharacteristic
256C22×C47C2×C22⋊Q811yescharacteristic
257C22×C47#57611yescharacteristic
258C22×C47#57711yescharacteristic
259C22×C47#57711yescharacteristic
260C22×C47#57811yescharacteristic
261C2×D47—42no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—81no—
265C2×D47—81no—
266C2×D47—81no—
267C2×D47—81no—
268C4×C47—82no—
269C4×C47—82no—
270C4×C47#39614yes—
271C4×C47C2×#12111yescharacteristic
272C4×C47C2×C22⋊Q811yescharacteristic
273C4×C47C2×C22⋊Q811yescharacteristic
274C4×C47C2×#12611yescharacteristic
275Dic47—42no—
276Dic47—42no—
277Dic47—42no—
278Dic47—42no—
279Dic47—42no—
280C2×#428—471no—
281C2×#424—471no—
282C2×#417—471no—
283C2×#416—471no—
284#2317—471no—
285#2311—471no—
286#2068—472no—
287#2067—472no—
288#2066—472no—
289#2065—472no—
290C2×#340D4711yescharacteristic, complemented
291C2×C47—24no—
292C2×C47C2×#39611yescharacteristic
293C2×C47#304511yescharacteristic
294C2×C47#304611yescharacteristic
295C2×C47#306211yescharacteristic
296C2×C47#306311yescharacteristic
297C2×C47#308311yescharacteristic
298C2×C47#308411yescharacteristic
299D47—82no—
300C22×#41—471no—
301C22×M16—471no—
302C23×C8C2×D4711yescharacteristic
303C2×#117—471no—
304C2×#117—471no—
305C22×C4⋊C4C2×D4711yescharacteristic
306C22×C4⋊C4C2×D4711yescharacteristic
307#428—944no—
308C2×#107—472no—
309C2×#106—472no—
310#424—944no—
311C2×C4⋊C8—472no—
312C2×C4⋊C8—472no—
313#417—944no—
314#416—944no—
315C2×#103—471no—
316C2×#103—471no—
317C2×#103—472no—
318C2×#102—471no—
319C2×#102—471no—
320C2×#102—472no—
321#391—472no—
322#390—472no—
323#389—472no—
324#389—472no—
325#386—942no—
326#385—942no—
327#384—941no—
328#382—941no—
329#380—941no—
330#380—941no—
331#340C2×D4712yes—
332#340C2×D4712yes—
333C47#2361611yescharacteristic, complemented, Sylow
334C2×#41—474no—
335C2×#41—942no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C22×Q8—471no—
341C22×D4—471no—
342C22×D4—941no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—941no—
346C23×C4C22×D4711yescharacteristic
347C2×M16—474no—
348C2×M16—942no—
349C2×M16—942no—
350C22×C8—22no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—472no—
354C22×C8C22×D4711yescharacteristic
355C22×C8C22×D4711yescharacteristic
356#117—474no—
357#117—942no—
358#117—942no—
359#117—942no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4C22×D4712yes—
373C2×C4⋊C4C22×D4712yes—
374C2×C22⋊C4—472no—
375C2×C22⋊C4—941no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4—942no—
378C2×C42—472no—
379C2×C42—941no—
380#107—944no—
381#106—944no—
382C4⋊C8—944no—
383C4⋊C8—944no—
384#103—944no—
385#103—942no—
386#103—942no—
387#102—944no—
388#102—942no—
389#102—942no—
390#95—942no—
391#95—942no—
392#95—942no—
393#95—942no—
394C24—941no—
395C24D4×D4711yescharacteristic
396#41—948no—
397#41—948no—
398#41—1884no—
399#41—1884no—
400C2×Q8—472no—
401C2×Q8—941no—
402C2×Q8—941no—
403C2×Q8—941no—
404C2×Q8—941no—
405C2×Q8—941no—
406C2×D4—472no—
407C2×D4—944no—
408C2×D4—941no—
409C2×D4—941no—
410C2×D4—941no—
411C2×D4—941no—
412C2×D4—941no—
413C2×D4—1882no—
414C2×D4—1882no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—944no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4#1062535012yes—
432C22×C4#1062535012yes—
433C22×C4#1062535012yes—
434C22×C4D4×D4711yescharacteristic
435C22×C4D4×D4712yes—
436C22×C4D4×D4712yes—
437C22×C4Q8×D4712yes—
438C22×C4C23×D4711yescharacteristic
439M16—948no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—44no—
447C2×C8—474no—
448C2×C8—942no—
449C2×C8—942no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—472no—
457C4⋊C4—472no—
458C4⋊C4—942no—
459C4⋊C4—942no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—3762no—
463C4⋊C4—3762no—
464C22⋊C4—944no—
465C22⋊C4—944no—
466C22⋊C4—3762no—
467C22⋊C4—3762no—
468C42—944no—
469C42—942no—
470C42—942no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—942no—
476C23—1881no—
477C23—1881no—
478C23—1881no—
479C23—1881no—
480C23—1881no—
481C23—1881no—
482C23#1309399712yes—
483C23#1309399712yes—
484C23#1309400212yes—
485C23C2×D4×D4711yescharacteristic
486Q8—942no—
487Q8—942no—
488Q8—1881no—
489Q8—1881no—
490D4—942no—
491D4—942no—
492D4—1888no—
493D4—1881no—
494D4—1881no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—942no—
504C2×C4—942no—
505C2×C4—942no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—942no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1888no—
519C2×C4#1309399712yes—
520C2×C4#1309399712yes—
521C2×C4#1309399714yes—
522C2×C4#1309400212yes—
523C2×C4#1309400214yes—
524C2×C4#1309400214yes—
525C2×C4#1309401814yes—
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4C2×D4×D4711yescharacteristic
528C2×C4C2×Q8×D4711yescharacteristic
529C2×C4C2×#1062535011yescharacteristic
530C2×C4C2×#1062535011yescharacteristic
531C2×C4C2×#1062535011yescharacteristic
532C8—41no—
533C8—41no—
534C8—944no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—3761no—
542C22—3761no—
543C22—3761no—
544C22—3761no—
545C22—1882no—
546C22—1882no—
547C22—1882no—
548C22#1568904714yes—
549C22C2×#1309399711yescharacteristic
550C22C2×#1309399711yescharacteristic
551C22C2×#1309400211yescharacteristic
552C22#1568992411yescharacteristic
553C22#1568992911yescharacteristic
554C22#1568992911yescharacteristic
555C22#1568993611yescharacteristic
556C4—82no—
557C4—82no—
558C4—1882no—
559C4—1882no—
560C4—1882no—
561C4—1882no—
562C4—1882no—
563C4#1568904714yes—
564C4C2×#1309399711yescharacteristic
565C4C2×#1309400211yescharacteristic
566C4C2×#1309400211yescharacteristic
567C4C2×#1309401811yescharacteristic
568C2—24no—
569C2—3762no—
570C2C2×#1568904711yescharacteristic
571C2#2025293211yescharacteristic
572C2#2025294311yescharacteristic
573C2#2025306611yescharacteristic
574C2#2025307711yescharacteristic
575C2#2025324411yescharacteristic
576C2#2025325511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenterC2×#340D47
hypocenterC47#23616
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
1. C2×#15689163C2semidirect
2. C2×#15689149C2semidirect
5. #20246012C2semidirect
7. #20244784C2semidirect
8. #20244777C2semidirect
9. #20244770C2semidirect
10. #20244763C2semidirect
11. C2×C47×#340C2semidirect
290. C2×#340D47semidirect
333. C47#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#156891632
C2×#156891492
C2×#156891222
C2×#156891202
#202460122
#202459812
#202447842
#202447772
#202447702
#202447632
C2×C47×#3402
#2361647