Finite Group Databasebeta

← #26252654

Subgroups — #26252654

Order 12032.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689410C211yesmaximal, characteristic, complemented
2C2×C4×#10625334C211yesmaximal, characteristic
3#20248306C211yesmaximal, characteristic, complemented
4#20244723C212yesmaximal, complemented
5#20244531C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244325C211yesmaximal, characteristic, complemented
8#20243736C212yesmaximal
9#20243649C212yesmaximal, complemented
10#20243388C211yesmaximal, characteristic, complemented
11C2×C47×#554C211yesmaximal, characteristic, complemented
12C22×#10625334C2211yescharacteristic
13C2×#13094000—21no—
14C2×C4×D188C2211yescharacteristic
15#15689410—24no—
16C4×#10625334—24no—
17C2×#13093957C2211yescharacteristic
18C2×#13093955C2212yes—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093945C2212yes—
21C2×#13093945C2211yescharacteristic
22C2×#13093944C2212yes—
23C2×#13093944C2211yescharacteristic
24C2×C4×C47⋊C8C2211yescharacteristic
25#15689045—22no—
26#15689025C2212yes—
27#15689021C2212yes—
28#15688993C2211yescharacteristic
29#15688980—21no—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34#15688878C2212yes—
35C47×#554—24no—
36C47×#554—24no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#125×C47C2212yes—
39C2×C22⋊Q8×C47—21no—
40C2×C22⋊Q8×C47—22no—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4×D4×C47C2212yes—
43C2×C4×D4×C47C2211yescharacteristic
44C22×C4⋊C4×C47—21no—
45C22×D188C2311yescharacteristic
46C22×C4×D47—21no—
47C2×#10625334—22no—
48C2×#10625334—22no—
49C2×#10625334Q814yescomplemented
50C22×C47⋊C8C2311yescharacteristic
51C22×C47⋊C8C2312yes—
52C22×C47⋊C8C2311yescharacteristic
53#13094000—44no—
54C4×D188—22no—
55C4×D188—22no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2312yes—
59C2×C4⋊Dic47C2311yescharacteristic
60C2×Dic47⋊C4—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—22no—
64#13093957—22no—
65#13093955—24no—
66C4⋊(C47⋊C8)—22no—
67#13093945—22no—
68#13093945—24no—
69#13093944—24no—
70#13093944—22no—
71C4×C47⋊C8—22no—
72C47⋊#95—22no—
73C22×Q8×C47—21no—
74C22×Q8×C47D411yescharacteristic
75C22×D4×C47C2311yescharacteristic
76C23×C4×C47—21no—
77C23×C4×C47—22no—
78C4⋊Q8×C47—22no—
79C4⋊Q8×C47D414yes—
80#125×C47—24no—
81#125×C47—24no—
82C22⋊Q8×C47—28no—
83C22⋊Q8×C47—216no—
84C4×Q8×C47—22no—
85C4×Q8×C47—22no—
86C4×D4×C47—24no—
87C4×D4×C47—24no—
88C4×D4×C47—22no—
89C4×D4×C47D414yes—
90C2×C4⋊C4×C47—28no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47D411yescharacteristic
95C2×C4⋊C4×C47D412yes—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47C2312yes—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47—21no—
100C2×C42×C47C2312yes—
101C2×C42×C47C2311yescharacteristic
102C24×C47—21no—
103C23×D47—21no—
104C2×Q8×C47—22no—
105C2×Q8×C47—24no—
106C2×Q8×C47—22no—
107C2×Q8×C47—22no—
108C2×Q8×C47—81no—
109C2×Q8×C47—81no—
110C2×D4×C47—21no—
111C2×D4×C47—21no—
112C2×D4×C47—21no—
113C2×D4×C47C2×D412yes—
114C2×D4×C47C2×Q812yes—
115C2×D4×C47#4112yes—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47—28no—
119C22×C4×C47—24no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47—82no—
123C22×C4×C47—82no—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—21no—
130C22×Dic47—22no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—21no—
134C2×D188—21no—
135C2×D188—21no—
136C2×D188C2×Q812yes—
137C2×C4×D47—42no—
138C2×C4×D47—42no—
139C2×C4×D47—42no—
140C4⋊C4×C47—28no—
141C4⋊C4×C47—28no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—24no—
144C4⋊C4×C47—24no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47C2×D414yes—
148C4⋊C4×C47C2×D412yes—
149C22⋊C4×C47—28no—
150C22⋊C4×C47—24no—
151C42×C47—22no—
152C42×C47—22no—
153C42×C47C2×D412yes—
154C42×C47C2×D412yes—
155#10625334—28no—
156(C2×D47)⋊C4—44no—
157(C2×D47)⋊C4—42no—
158(C2×D47)⋊C4—82no—
159C4⋊Dic47—21no—
160C4⋊Dic47—21no—
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47—42no—
164C4⋊Dic47—81no—
165Dic47⋊C4—44no—
166C2×C47⋊C8—21no—
167C2×C47⋊C8—21no—
168C2×C47⋊C8—21no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8C2×Q812yes—
175C23×C47—22no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—41no—
179C23×C47—41no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47C22×D411yescharacteristic
183C22×D47—22no—
184C22×D47—41no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—41no—
190Q8×C47—44no—
191Q8×C47—84no—
192D4×C47—22no—
193D4×C47—22no—
194D4×C47—22no—
195D4×C47C22⋊Q814yescomplemented
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—48no—
211C2×C4×C47—88no—
212C2×C4×C47#12012yes—
213C2×C4×C47#12012yes—
214C2×C4×C47#13512yes—
215C2×C4×C47AGL(1,ℤ/8)12yes—
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×Q811yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C472−1+411yescharacteristic
221C2×C4×C472−1+411yescharacteristic
222C2×Dic47—41no—
223C2×Dic47—42no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228D188—22no—
229D188—22no—
230D188—41no—
231D188—41no—
232C4×D47—84no—
233C47⋊C8—22no—
234C47⋊C8—41no—
235C47⋊C8—42no—
236#24327—471nomaximal, Sylow
237C22×C47—24no—
238C22×C47—42no—
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—42no—
243C22×C47—42no—
244C22×C47C2×#12011yescharacteristic
245C22×C47#54811yescharacteristic
246C22×C47D4×Q811yescharacteristic
247C22×C47C2×#13511yescharacteristic
248C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C47#57611yescharacteristic
250C22×C47#57711yescharacteristic
251C2×D47—42no—
252C2×D47—42no—
253C2×D47—42no—
254C2×D47—81no—
255C2×D47—81no—
256C2×D47—81no—
257C2×D47—81no—
258C4×C47—22no—
259C4×C47—42no—
260C4×C47—42no—
261C4×C47—42no—
262C4×C47—44no—
263C4×C47#44914yes—
264C4×C47C2×C22⋊Q811yescharacteristic
265C4×C47#53611yescharacteristic
266C4×C47#54811yescharacteristic
267C4×C47D4×Q811yescharacteristic
268Dic47—82no—
269Dic47—81no—
270C2×#554D4711yescharacteristic, complemented
271C2×#476—471no—
272C2×C4×D8—471no—
273#2538—471no—
274#2059—472no—
275#2025—472no—
276#1990—471no—
277#1984—471no—
278#1879—472no—
279#1864—472no—
280#1802—471no—
281C2×C47—44no—
282C2×C47#240911yescharacteristic
283C2×C47C2×#44911yescharacteristic
284C2×C47#301211yescharacteristic
285C2×C47#336011yescharacteristic
286C2×C47#336611yescharacteristic
287C2×C47#338011yescharacteristic
288C2×C47#338411yescharacteristic
289D47—82no—
290C22×D8—471no—
291#554—24no—
292#554—24no—
293C2×C4⋊Q8C2×D4711yescharacteristic
294C2×#125C2×D4712yes—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8—22no—
297C2×C22⋊Q8—941no—
298C2×C4×Q8C2×D4711yescharacteristic
299C2×C4×D4—471no—
300C2×C4×D4C2×D4712yes—
301C2×C4×D4C2×D4711yescharacteristic
302C22×C4⋊C4—21no—
303#476—944no—
304C4×D8—944no—
305C2×#107—471no—
306C2×#106—472no—
307C2×C4⋊C8—471no—
308C2×#102—472no—
309C2×#102—471no—
310C2×#102—472no—
311C2×#102—471no—
312C2×C4×C8—471no—
313#396—942no—
314#390—472no—
315#389—472no—
316#384—471no—
317#380—941no—
318#340—472no—
319#340—472no—
320#340—472no—
321#336—472no—
322#336—472no—
323C47#2432711yescharacteristic, complemented, Sylow
324C22×Q8—21no—
325C22×Q8C22⋊D4711yescharacteristic
326C22×D4—471no—
327C22×D4C22×D4711yescharacteristic
328C23×C4—21no—
329C23×C4—22no—
330C23×C4—941no—
331C2×D8—474no—
332C2×D8—942no—
333C2×D8—942no—
334C22×C8—471no—
335C22×C8—471no—
336C22×C8—472no—
337C4⋊Q8—22no—
338C4⋊Q8C22⋊D4714yes—
339#125—24no—
340#125—24no—
341C22⋊Q8—28no—
342C22⋊Q8—216no—
343C22⋊Q8—1884no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—24no—
347C4×D4—24no—
348C4×D4—22no—
349C4×D4—942no—
350C4×D4—942no—
351C4×D4C22⋊D4714yes—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—471no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4C22⋊D4711yescharacteristic
362C2×C4⋊C4C22⋊D4712yes—
363C2×C4⋊C4C22×D4711yescharacteristic
364C2×C4⋊C4C22×D4712yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—941no—
369C2×C22⋊C4—942no—
370C2×C42C22×D4712yes—
371C2×C42C22×D4711yescharacteristic
372#107—942no—
373#106—944no—
374C4⋊C8—942no—
375#102—944no—
376#102—942no—
377#102—942no—
378#102—944no—
379C4×C8—942no—
380#95—942no—
381C24—21no—
382C24—941no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—472no—
393C2×D4—941no—
394C2×D4—941no—
395C2×D4—941no—
396C2×D4—941no—
397C2×D4—941no—
398C2×D4Q8×D4712yes—
399C2×D4#1062535412yes—
400C2×D4C2×C22⋊D4712yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—941no—
410C22×C4—942no—
411C22×C4—1882no—
412C22×C4—1882no—
413C22×C4—1882no—
414C22×C4C2×C22⋊D4712yes—
415C22×C4C2×C22⋊D4712yes—
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C2×C22⋊D4711yescharacteristic
418C22×C4C23×D4711yescharacteristic
419D8—948no—
420C2×C8—472no—
421C2×C8—941no—
422C2×C8—941no—
423C2×C8—941no—
424C2×C8—942no—
425C2×C8—942no—
426C2×C8—942no—
427C2×C8—941no—
428C2×C8—941no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—941no—
437C4⋊C4—941no—
438C4⋊C4—942no—
439C4⋊C4—942no—
440C4⋊C4—3761no—
441C4⋊C4—1884no—
442C4⋊C4—1882no—
443C4⋊C4C2×C22⋊D4714yes—
444C4⋊C4C2×C22⋊D4712yes—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—3762no—
448C22⋊C4—1884no—
449C22⋊C4—1882no—
450C42—22no—
451C42—22no—
452C42C2×C22⋊D4712yes—
453C42C2×C22⋊D4712yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—942no—
462C23—1881no—
463C23—1881no—
464C23—1881no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23C22×C22⋊D4711yescharacteristic
469Q8—44no—
470Q8—84no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—942no—
475D4—942no—
476D4—1881no—
477D4—1881no—
478D4#1309400014yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—88no—
495C2×C4—3764no—
496C2×C4—1882no—
497C2×C4—1881no—
498C2×C4—1881no—
499C2×C4—1882no—
500C2×C4—1881no—
501C2×C4—1882no—
502C2×C4#1309399312yes—
503C2×C4#1309399312yes—
504C2×C4#1309404412yes—
505C2×C4#1309404912yes—
506C2×C4C22×C22⋊D4711yescharacteristic
507C2×C4C22×C22⋊D4711yescharacteristic
508C2×C4C2×Q8×D4711yescharacteristic
509C2×C4C2×#1062535411yescharacteristic
510C2×C4#1309407211yescharacteristic
511C2×C4#1309407211yescharacteristic
512C8—942no—
513C8—1882no—
514C8—1881no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—3761no—
523C22—3761no—
524C22—3761no—
525C22—3761no—
526C22—1882no—
527C22—1882no—
528C22—1882no—
529C22C2×#1309399311yescharacteristic
530C22#1568978411yescharacteristic
531C22Q8×C22⋊D4711yescharacteristic
532C22C2×#1309404411yescharacteristic
533C22C2×#1309404911yescharacteristic
534C22#1568992611yescharacteristic
535C22#1568993411yescharacteristic
536C4—22no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—3762no—
542C4—3761no—
543C4#1568926814yes—
544C4C2×#1309400011yescharacteristic
545C4#1568969711yescharacteristic
546C4#1568978411yescharacteristic
547C4Q8×C22⋊D4711yescharacteristic
548C2—44no—
549C2—3762no—
550C2#2024674711yescharacteristic
551C2C2×#1568926811yescharacteristic
552C2#2025269211yescharacteristic
553C2#2025603711yescharacteristic
554C2#2025609511yescharacteristic
555C2#2025625111yescharacteristic
556C2#2025628711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#554C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
hypercenterC2×#554D47
hypocenterC47#24327
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#15689410C2semidirect
3. #20248306C2semidirect
4. #20244723C2semidirect
5. #20244531C2semidirect
6. #20244350C2semidirect
7. #20244325C2semidirect
9. #20243649C2semidirect
10. #20243388C2semidirect
11. C2×C47×#554C2semidirect
49. C2×#10625334Q8semidirect
195. D4×C47C22⋊Q8semidirect
270. C2×#554D47semidirect
323. C47#24327semidirect
478. D4#13094000semidirect

Maximal subgroups

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

subgroupindex
C2×#156894102
C2×C4×#106253342
#202483062
#202447232
#202445312
#202443502
#202443252
#202437362
#202436492
#202433882
C2×C47×#5542
#2432747