Finite Group Databasebeta

← #26245095

Subgroups — #26245095

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689122C211yesmaximal, characteristic
2C2×#15689120C211yesmaximal, characteristic
3#20246012C211yesmaximal, characteristic, complemented
4#20245981C211yesmaximal, characteristic
5#20244810C212yesmaximal, complemented
6#20244805C212yesmaximal, complemented
7#20244800C212yesmaximal, complemented
8#20244795C212yesmaximal, complemented
9#20243107C211yesmaximal, characteristic, complemented
10#20243084C211yesmaximal, characteristic, complemented
11C2×C47×#340C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C2×#13093983C2211yescharacteristic
15C2×#13093983C2211yescharacteristic
16#15689122—24no—
17#15689120—24no—
18C2×#13093949C2211yescharacteristic
19C2×#13093949C2211yescharacteristic
20C2×#13093947C2212yes—
21C2×#13093946C2211yescharacteristic
22C2×#13093946C2211yescharacteristic
23C2×#13093944C2212yes—
24#15689028C2212yes—
25#15689024C2212yes—
26#15689019—21no—
27#15689019—21no—
28#15689002—22no—
29#15688996—22no—
30#15688992C2212yes—
31#15688988C2212yes—
32#15688980—21no—
33#15688980—21no—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688877C2212yes—
37#15688877C2212yes—
38C23×C8×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C47×#340C2212yes—
42C47×#340C2212yes—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350D414yes—
49C22×Dic94C2311yescharacteristic
50C22×D188C2311yescharacteristic
51C22×C22⋊D47—21no—
52C22×C4×D47—21no—
53C2×#10625329—22no—
54C2×#10625329—22no—
55C2×#10625329—22no—
56C22×C47⋊C8C2312yes—
57#13093983—22no—
58#13093983—22no—
59#13093983—22no—
60#13093983—22no—
61C2×C4⋊Dic47D411yescharacteristic
62C2×C4⋊Dic47D411yescharacteristic
63C2×C4⋊Dic47C2311yescharacteristic
64C2×C4⋊Dic47C2311yescharacteristic
65C2×Dic47⋊C4—22no—
66C2×Dic47⋊C4—22no—
67C2×C22⋊Dic47—21no—
68C2×C22⋊Dic47D412yes—
69C2×(C2×D47)⋊C4—22no—
70C2×(C2×D47)⋊C4—22no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47C2312yes—
73#13093949—22no—
74#13093949—22no—
75#13093947—24no—
76#13093946—22no—
77#13093946—22no—
78#13093944—24no—
79C47⋊#95—22no—
80C47⋊#95—22no—
81C47⋊#95—22no—
82C47⋊#95—22no—
83C23×C4×C47—21no—
84C23×C4×C47—21no—
85C23×C4×C47C2311yescharacteristic
86C22×C8×C47—22no—
87C22×C8×C47—22no—
88C22×C8×C47—22no—
89C22×C8×C47C2311yescharacteristic
90C22×C8×C47C2311yescharacteristic
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—22no—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47C2312yes—
98C2×C4⋊C4×C47C2312yes—
99C24×C47C2×D411yescharacteristic
100C23×D47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—42no—
104C22×C4×C47—42no—
105C22×C4×C47—81no—
106C22×C4×C47—81no—
107C22×C4×C47—81no—
108C22×C4×C47—81no—
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D412yes—
111C22×C4×C47#4112yes—
112C22×C4×C47#4112yes—
113C22×C4×C47#4112yes—
114C22×C4×C47#4112yes—
115C22×C4×C47#4112yes—
116C22×C4×C47C2411yescharacteristic
117C2×C22⋊D47—24no—
118C2×C22⋊D47—42no—
119C2×C22⋊D47—42no—
120C22×Dic47—21no—
121C22×Dic47—21no—
122C22×Dic47—21no—
123C22×Dic47C2×D412yes—
124C22×Dic47C2×D412yes—
125#10625350—28no—
126#10625350—28no—
127#10625350—44no—
128#10625350—44no—
129C2×D188—21no—
130C2×D188—21no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—21no—
134C2×D188C2×D412yes—
135C2×C4×D47—24no—
136C2×C4×D47—42no—
137C2×C4×D47—42no—
138C2×Dic94—21no—
139C2×Dic94—21no—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94C2×D412yes—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—44no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—42no—
156C4⋊C4×C47—42no—
157C22⋊Dic47—24no—
158C22⋊Dic47—42no—
159(C2×D47)⋊C4—82no—
160(C2×D47)⋊C4—82no—
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167Dic47⋊C4—82no—
168Dic47⋊C4—82no—
169C4×Dic47—22no—
170C4×Dic47—22no—
171C4×Dic47—42no—
172#10625329—44no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47#12112yes—
181C23×C47#12312yes—
182C23×C47#12412yes—
183C23×C47C22×D411yescharacteristic
184C22×D47—22no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—41no—
190C22×D47—41no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—84no—
198C2×C4×C47—84no—
199C2×C4×C47#12112yes—
200C2×C4×C47#12314yes—
201C2×C4×C47#12314yes—
202C2×C4×C47#12312yes—
203C2×C4×C47#12412yes—
204C2×C4×C47#12614yes—
205C2×C4×C47#12614yes—
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C2×#4111yescharacteristic
208C2×C4×C47C2×#4111yescharacteristic
209C2×C4×C47C2×#4111yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C22⋊D47—48no—
213C2×Dic47—22no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—41no—
221C2×Dic47—41no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228D188—22no—
229D188—22no—
230D188—41no—
231D188—41no—
232C4×D47—48no—
233Dic94—22no—
234Dic94—22no—
235Dic94—41no—
236Dic94—41no—
237C8×C47—41no—
238C8×C47—41no—
239C47⋊C8—42no—
240#23634—471nomaximal, Sylow
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47#39714yes—
248C22×C47C2×#12111yescharacteristic
249C22×C47C2×#12311yescharacteristic
250C22×C47C2×#12411yescharacteristic
251C22×C47#57611yescharacteristic
252C22×C47#57711yescharacteristic
253C22×C47#57711yescharacteristic
254C22×C47#57811yescharacteristic
255C2×D47—42no—
256C2×D47—42no—
257C2×D47—42no—
258C2×D47—81no—
259C2×D47—81no—
260C2×D47—81no—
261C2×D47—81no—
262C4×C47—82no—
263C4×C47—82no—
264C4×C47#39714yes—
265C4×C47C2×#12311yescharacteristic
266C4×C47C2×#12311yescharacteristic
267C4×C47C2×#12611yescharacteristic
268C4×C47C2×#12611yescharacteristic
269Dic47—42no—
270Dic47—42no—
271Dic47—42no—
272Dic47—81no—
273Dic47—81no—
274C2×#417—471no—
275C2×#416—471no—
276#2317—471no—
277#2311—471no—
278#2074—472no—
279#2073—472no—
280#2072—472no—
281#2071—472no—
282#1736—471no—
283C2×#340D4711yescharacteristic, complemented
284#1729—471no—
285C2×C47—24no—
286C2×C47C2×#39711yescharacteristic
287C2×C47#304511yescharacteristic
288C2×C47#304611yescharacteristic
289C2×C47#309811yescharacteristic
290C2×C47#309911yescharacteristic
291C2×C47#313611yescharacteristic
292C2×C47#313711yescharacteristic
293D47—82no—
294C22×#41—471no—
295C22×M16—471no—
296C23×C8C2×D4711yescharacteristic
297C2×#117—471no—
298C2×#117—471no—
299C22×C4⋊C4C2×D4711yescharacteristic
300C22×C4⋊C4C2×D4711yescharacteristic
301#417—944no—
302#416—944no—
303C2×#103—471no—
304C2×#103—471no—
305C2×#103—472no—
306C2×#102—471no—
307C2×#102—471no—
308C2×#102—472no—
309#391—472no—
310#390—472no—
311#389—941no—
312#389—941no—
313#386—942no—
314#385—942no—
315#384—472no—
316#382—472no—
317#380—941no—
318#380—941no—
319#340—472no—
320#340—472no—
321#340C2×D4712yes—
322#340C2×D4712yes—
323#336—472no—
324#336—472no—
325C47#2363411yescharacteristic, complemented, Sylow
326C2×#41—474no—
327C2×#41—942no—
328C2×#41—942no—
329C2×#41—942no—
330C2×#41—942no—
331C2×#41—942no—
332C22×Q8—471no—
333C22×D4—471no—
334C22×D4—941no—
335C23×C4—21no—
336C23×C4—21no—
337C23×C4—941no—
338C23×C4C22×D4711yescharacteristic
339C2×M16—942no—
340C2×M16—942no—
341C2×M16—942no—
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—22no—
345C22×C8—472no—
346C22×C8C22×D4711yescharacteristic
347C22×C8C22×D4711yescharacteristic
348#117—942no—
349#117—942no—
350#117—942no—
351#117—942no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—471no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4C22×D4712yes—
365C2×C4⋊C4C22×D4712yes—
366C2×C22⋊C4—472no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—941no—
370C2×C42—472no—
371C2×C42—941no—
372#103—944no—
373#103—942no—
374#103—942no—
375#102—944no—
376#102—942no—
377#102—942no—
378#95—942no—
379#95—942no—
380#95—942no—
381#95—942no—
382C24—941no—
383C24D4×D4711yescharacteristic
384#41—948no—
385#41—948no—
386#41—1884no—
387#41—1884no—
388C2×Q8—472no—
389C2×Q8—941no—
390C2×Q8—941no—
391C2×Q8—941no—
392C2×Q8—941no—
393C2×Q8—941no—
394C2×D4—472no—
395C2×D4—944no—
396C2×D4—941no—
397C2×D4—941no—
398C2×D4—941no—
399C2×D4—941no—
400C2×D4—941no—
401C2×D4—1882no—
402C2×D4—1882no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—941no—
414C22×C4—944no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—1882no—
418C22×C4—1882no—
419C22×C4#1062535012yes—
420C22×C4#1062535012yes—
421C22×C4#1062535012yes—
422C22×C4D4×D4711yescharacteristic
423C22×C4D4×D4712yes—
424C22×C4#1062535212yes—
425C22×C4#1062535412yes—
426C22×C4C23×D4711yescharacteristic
427M16—1884no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—44no—
435C2×C8—942no—
436C2×C8—942no—
437C2×C8—942no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—942no—
445C4⋊C4—942no—
446C4⋊C4—941no—
447C4⋊C4—941no—
448C4⋊C4—941no—
449C4⋊C4—941no—
450C4⋊C4—3762no—
451C4⋊C4—3762no—
452C22⋊C4—944no—
453C22⋊C4—3762no—
454C22⋊C4—3762no—
455C22⋊C4—1882no—
456C42—942no—
457C42—942no—
458C42—1882no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—942no—
464C23—1881no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23—1881no—
469C23—1881no—
470C23#1309399712yes—
471C23#1309400912yes—
472C23#1309401012yes—
473C23C2×D4×D4711yescharacteristic
474Q8—942no—
475Q8—942no—
476Q8—1881no—
477Q8—1881no—
478D4—942no—
479D4—942no—
480D4—1888no—
481D4—1881no—
482D4—1881no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—942no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—1881no—
499C2×C4—1881no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4—1881no—
506C2×C4—1888no—
507C2×C4#1309399712yes—
508C2×C4#1309400614yes—
509C2×C4#1309400912yes—
510C2×C4#1309400914yes—
511C2×C4#1309401012yes—
512C2×C4#1309401814yes—
513C2×C4#1309401914yes—
514C2×C4C2×D4×D4711yescharacteristic
515C2×C4C2×#1062535211yescharacteristic
516C2×C4C2×#1062535011yescharacteristic
517C2×C4C2×#1062535011yescharacteristic
518C2×C4C2×#1062535011yescharacteristic
519C2×C4C2×#1062535411yescharacteristic
520C8—41no—
521C8—41no—
522C8—1882no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—3761no—
530C22—3761no—
531C22—3761no—
532C22—3761no—
533C22—1882no—
534C22—1882no—
535C22—1882no—
536C22#1568905214yes—
537C22C2×#1309399711yescharacteristic
538C22C2×#1309400911yescharacteristic
539C22C2×#1309401011yescharacteristic
540C22#1568992411yescharacteristic
541C22#1568992911yescharacteristic
542C22#1568992911yescharacteristic
543C22#1568993611yescharacteristic
544C4—82no—
545C4—82no—
546C4—3761no—
547C4—3761no—
548C4—1882no—
549C4—1882no—
550C4—1882no—
551C4#1568905214yes—
552C4C2×#1309400611yescharacteristic
553C4C2×#1309400911yescharacteristic
554C4C2×#1309401811yescharacteristic
555C4C2×#1309401911yescharacteristic
556C2—24no—
557C2—3762no—
558C2C2×#1568905211yescharacteristic
559C2#2025293211yescharacteristic
560C2#2025294311yescharacteristic
561C2#2025337111yescharacteristic
562C2#2025338211yescharacteristic
563C2#2025374011yescharacteristic
564C2#2025375111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenterC2×#340D47
hypocenterC47#23634
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
3. #20246012C2semidirect
5. #20244810C2semidirect
6. #20244805C2semidirect
7. #20244800C2semidirect
8. #20244795C2semidirect
9. #20243107C2semidirect
10. #20243084C2semidirect
11. C2×C47×#340C2semidirect
283. C2×#340D47semidirect
325. C47#23634semidirect

Maximal subgroups

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

subgroupindex
C2×#156891222
C2×#156891202
#202460122
#202459812
#202448102
#202448052
#202448002
#202447952
#202431072
#202430842
C2×C47×#3402
#2363447