Finite Group Databasebeta

← #26245088

Subgroups — #26245088

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×#15689117C211yesmaximal, characteristic
3#20245976C211yesmaximal, characteristic
4#20244807C212yesmaximal, complemented
5#20244802C212yesmaximal, complemented
6#20244797C212yesmaximal, complemented
7#20244792C212yesmaximal, complemented
8#20243109C211yesmaximal, characteristic, complemented
9C2×#15688885C211yesmaximal, characteristic, complemented
10#20243085C211yesmaximal, characteristic, complemented
11C47×#2317C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C23×C47⋊C8C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16#15689122—24no—
17#15689117—24no—
18C2×#13093947C2211yescharacteristic
19C2×#13093947C2212yes—
20C2×#13093947C2211yescharacteristic
21C2×#13093944C2211yescharacteristic
22C2×#13093944C2212yes—
23C2×#13093944C2211yescharacteristic
24#15689029C2212yes—
25#15689025C2212yes—
26#15689000—22no—
27#15688980—21no—
28#15688980—21no—
29#15688885C2212yes—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34#15688878C2212yes—
35C2×#117×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C22×C4⋊C4×C47C2211yescharacteristic
38C47×#389—21no—
39C47×#389—21no—
40C47×#385—22no—
41C47×#384C2212yes—
42C47×#382C2212yes—
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—
52C23×Dic47—21no—
53C22×C4×D47—21no—
54C2×#10625329—22no—
55C2×#10625329—22no—
56C2×#10625329—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8—22no—
59C22×C47⋊C8—22no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2312yes—
63C2×C4⋊Dic47—22no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47C2312yes—
67C2×(C2×D47)⋊C4—22no—
68C2×(C2×D47)⋊C4—22no—
69#13093947—22no—
70#13093947—22no—
71#13093947—24no—
72#13093944—22no—
73#13093944—22no—
74#13093944—24no—
75C47⋊#95—22no—
76C47⋊#95—22no—
77C23×C4×C47—21no—
78C23×C4×C47C2311yescharacteristic
79#117×C47—22no—
80#117×C47—22no—
81#117×C47—22no—
82#117×C47—22no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47D411yescharacteristic
89C2×C4⋊C4×C47D411yescharacteristic
90C2×C4⋊C4×C47C2312yes—
91C2×C4⋊C4×C47C2311yescharacteristic
92C2×C4⋊C4×C47C2311yescharacteristic
93C2×C22⋊C4×C47—21no—
94C2×C22⋊C4×C47D412yes—
95C2×C42×C47—21no—
96C2×C42×C47C2312yes—
97#95×C47—22no—
98#95×C47—22no—
99C24×C47C2×D411yescharacteristic
100C23×D47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—22no—
104C22×C4×C47—42no—
105C22×C4×C47—81no—
106C22×C4×C47—81no—
107C22×C4×C47C2×D412yes—
108C22×C4×C47C2×D412yes—
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—22no—
122C22×Dic47—42no—
123C22×Dic47—81no—
124C22×Dic47—81no—
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—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—42no—
153C4⋊C4×C47—82no—
154C4⋊C4×C47—82no—
155C22⋊C4×C47—24no—
156C22⋊C4×C47—42no—
157C42×C47—22no—
158C42×C47—22no—
159C42×C47—42no—
160(C2×D47)⋊C4—82no—
161(C2×D47)⋊C4—82no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47—42no—
165#10625329—44no—
166C2×C47⋊C8—21no—
167C2×C47⋊C8—21no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—21no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—44no—
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—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—84no—
207C2×C4×C47#12112yes—
208C2×C4×C47#12312yes—
209C2×C4×C47#12314yes—
210C2×C4×C47#12314yes—
211C2×C4×C47#12412yes—
212C2×C4×C47#12614yes—
213C2×C4×C47#12614yes—
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C22⋊D47—48no—
221C2×Dic47—22no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—84no—
228D188—22no—
229D188—22no—
230D188—41no—
231D188—41no—
232C4×D47—48no—
233Dic94—22no—
234Dic94—22no—
235Dic94—41no—
236Dic94—41no—
237C47⋊C8—41no—
238C47⋊C8—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—42no—
263C4×C47—42no—
264C4×C47—81no—
265C4×C47—82no—
266C4×C47—81no—
267C4×C47#39714yes—
268C4×C47C2×#12311yescharacteristic
269C4×C47C2×#12311yescharacteristic
270C4×C47C2×#12611yescharacteristic
271C4×C47C2×#12611yescharacteristic
272Dic47—42no—
273Dic47—82no—
274C2×#417—471no—
275C2×#416—471no—
276#2317D4711yescharacteristic, complemented
277#2311—471no—
278#2074—472no—
279#2073—472no—
280#2072—472no—
281#2071—472no—
282#1736—471no—
283C2×#340—471no—
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×C8—471no—
297C2×#117C2×D4711yescharacteristic
298C2×#117C2×D4711yescharacteristic
299C22×C4⋊C4—471no—
300C22×C4⋊C4C2×D4711yescharacteristic
301#417—944no—
302#416—944no—
303C2×#103—471no—
304C2×#103—472no—
305C2×#103—471no—
306C2×#102—471no—
307C2×#102—472no—
308C2×#102—471no—
309#391—472no—
310#390—472no—
311#389—21no—
312#389—21no—
313#386—942no—
314#385—22no—
315#384C2×D4712yes—
316#382C2×D4712yes—
317#380—941no—
318#380—941no—
319#340—472no—
320#340—472no—
321#340—472no—
322#340—472no—
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—941no—
337C23×C4—941no—
338C23×C4C22×D4711yescharacteristic
339C2×M16—942no—
340C2×M16—942no—
341C2×M16—942no—
342C22×C8—471no—
343C22×C8—471no—
344C22×C8—472no—
345C22×C8—942no—
346C22×C8—942no—
347C22×C8—942no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4C22⋊D4711yescharacteristic
362C2×C4⋊C4C22⋊D4711yescharacteristic
363C2×C4⋊C4C22×D4712yes—
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C4⋊C4C22×D4711yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4C22⋊D4712yes—
370C2×C42—21no—
371C2×C42C22×D4712yes—
372#103—942no—
373#103—942no—
374#103—944no—
375#102—942no—
376#102—942no—
377#102—944no—
378#95—22no—
379#95—22no—
380#95—942no—
381#95—942no—
382C24—941no—
383C24C2×C22⋊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—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—941no—
410C22×C4—944no—
411C22×C4—942no—
412C22×C4—3761no—
413C22×C4—3761no—
414C22×C4—1882no—
415C22×C4—1882no—
416C22×C4—1882no—
417C22×C4#1062535012yes—
418C22×C4#1062535012yes—
419C22×C4D4×D4712yes—
420C22×C4#1062535212yes—
421C22×C4#1062535412yes—
422C22×C4#1062535412yes—
423C22×C4C2×C22⋊D4712yes—
424C22×C4C2×C22⋊D4712yes—
425C22×C4C2×C22⋊D4711yescharacteristic
426C22×C4C23×D4711yescharacteristic
427M16—1884no—
428C2×C8—941no—
429C2×C8—941no—
430C2×C8—942no—
431C2×C8—941no—
432C2×C8—941no—
433C2×C8—942no—
434C2×C8—941no—
435C2×C8—941no—
436C2×C8—942no—
437C2×C8—1884no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—942no—
450C4⋊C4—942no—
451C4⋊C4—1882no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—3762no—
455C22⋊C4—3762no—
456C42—22no—
457C42—22no—
458C42—42no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—942no—
464C23—1881no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23—1881no—
469C23—1881no—
470C23#1309399512yes—
471C23#1309400712yes—
472C23#1309401312yes—
473C23C22×C22⋊D4711yescharacteristic
474Q8—942no—
475Q8—942no—
476Q8—1881no—
477Q8—1881no—
478D4—942no—
479D4—942no—
480D4—1888no—
481D4—1881no—
482D4—1881no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—84no—
499C2×C4—942no—
500C2×C4—3764no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4—1881no—
505C2×C4—1881no—
506C2×C4—1888no—
507C2×C4#1309399512yes—
508C2×C4#1309400712yes—
509C2×C4#1309400914yes—
510C2×C4#1309400914yes—
511C2×C4#1309401312yes—
512C2×C4#1309401914yes—
513C2×C4#1309401914yes—
514C2×C4C2×D4×D4711yescharacteristic
515C2×C4C2×#1062535211yescharacteristic
516C2×C4C2×#1062535011yescharacteristic
517C2×C4C2×#1062535011yescharacteristic
518C2×C4C2×#1062535411yescharacteristic
519C2×C4C2×#1062535411yescharacteristic
520C8—1881no—
521C8—1881no—
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#1568904914yes—
537C22C2×#1309399511yescharacteristic
538C22C2×#1309400711yescharacteristic
539C22C2×#1309401311yescharacteristic
540C22#1568992611yescharacteristic
541C22#1568993111yescharacteristic
542C22#1568993411yescharacteristic
543C22#1568993811yescharacteristic
544C4—42no—
545C4—42no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4—3762no—
550C4—1882no—
551C4#1568904914yes—
552C4C2×#1309400911yescharacteristic
553C4C2×#1309400911yescharacteristic
554C4C2×#1309401911yescharacteristic
555C4C2×#1309401911yescharacteristic
556C2—24no—
557C2—3762no—
558C2C2×#1568904911yescharacteristic
559C2#2025292311yescharacteristic
560C2#2025293411yescharacteristic
561C2#2025336611yescharacteristic
562C2#2025337711yescharacteristic
563C2#2025374711yescharacteristic
564C2#2025375811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2317C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenter#2317D47
hypocenterC47#23634
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #20244807C2semidirect
5. #20244802C2semidirect
6. #20244797C2semidirect
7. #20244792C2semidirect
8. #20243109C2semidirect
9. C2×#15688885C2semidirect
10. #20243085C2semidirect
11. C47×#2317C2semidirect
276. #2317D47semidirect
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×#156891172
#202459762
#202448072
#202448022
#202447972
#202447922
#202431092
C2×#156888852
#202430852
C47×#23172
#2363447