Finite Group Databasebeta

← #26273295

Subgroups — #26273295

Order 12032.

1176 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689489C211yesmaximal, characteristic, complemented
2C2×#15689318C211yesmaximal, characteristic
3#20248387C211yesmaximal, characteristic, complemented
4#20244531C212yesmaximal, complemented
5#20244521C212yesmaximal, complemented
6#20244346C211yesmaximal, characteristic, complemented
7#20244329C211yesmaximal, characteristic, complemented
8#20243736C212yesmaximal, complemented
9#20243736C212yesmaximal, complemented
10#20243213C211yesmaximal, characteristic, complemented
11C2×C47×#554C211yesmaximal, characteristic, complemented
12C22×#10625334C2212yes—
13C2×#13094021C2211yescharacteristic
14C2×#13094013—21no—
15#15689489—24no—
16#15689318—24no—
17C2×C4⋊(C47⋊C8)C2211yescharacteristic
18C2×#13093945C2212yes—
19C2×#13093944C2212yes—
20C2×#13093944C2212yes—
21C2×#13093944C2212yes—
22C2×#13093935C2211yescharacteristic
23#15689040—21no—
24#15689040—21no—
25#15689025C2212yes—
26#15689025C2212yes—
27#15689024—21no—
28C4×C4⋊Dic47C2211yescharacteristic
29#15688885C2212yes—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688878C2212yes—
33#15688878C2212yes—
34C47×#554—24no—
35C47×#554—24no—
36C2×C4⋊Q8×C47C2211yescharacteristic
37C2×#125×C47C2211yescharacteristic
38C2×#125×C47C2211yescharacteristic
39C2×C22⋊Q8×C47—21no—
40C2×C22⋊Q8×C47—21no—
41C2×C22⋊Q8×C47—21no—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C2×C4×D4×C47C2211yescharacteristic
45C2×C4×D4×C47C2211yescharacteristic
46C22×C4⋊C4×C47—21no—
47C22×D188—21no—
48C22×D188—21no—
49C22×D188C2312yes—
50C2×#10625334—24no—
51C2×#10625334—24no—
52C2×#10625334—24no—
53C22×C47⋊C8C2312yes—
54C22×C47⋊C8C2312yes—
55#13094021—22no—
56#13094021—22no—
57#13094013—44no—
58C2×C4⋊Dic47C2312yes—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—22no—
63C2×C4×Dic47—21no—
64C4⋊(C47⋊C8)—22no—
65#13093945—24no—
66#13093944—24no—
67#13093944—24no—
68#13093944—24no—
69#13093935—22no—
70C47⋊#95—21no—
71C22×Q8×C47—21no—
72C22×Q8×C47D411yescharacteristic
73C22×D4×C47C2311yescharacteristic
74C23×C4×C47—21no—
75C23×C4×C47—21no—
76C23×C4×C47—21no—
77C4⋊Q8×C47—22no—
78C4⋊Q8×C47—22no—
79#125×C47—22no—
80#125×C47—22no—
81#125×C47—22no—
82#125×C47D414yes—
83C22⋊Q8×C47—28no—
84C22⋊Q8×C47—28no—
85C22⋊Q8×C47—28no—
86C4×Q8×C47—22no—
87C4×Q8×C47—22no—
88C4×D4×C47—22no—
89C4×D4×C47—22no—
90C4×D4×C47—22no—
91C4×D4×C47—22no—
92C4×D4×C47—22no—
93C4×D4×C47D414yes—
94C2×C4⋊C4×C47—28no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C4⋊C4×C47D411yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C2×C42×C47C2311yescharacteristic
112C24×C47—21no—
113C23×D47—22no—
114C2×Q8×C47—24no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47—41no—
118C2×Q8×C47—82no—
119C2×D4×C47—21no—
120C2×D4×C47—22no—
121C2×D4×C47C2×D412yes—
122C2×D4×C47#4114yes—
123C22×C4×C47—24no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47—82no—
132C22×C4×C47—82no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—22no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—42no—
148C2×D188—42no—
149C2×D188—42no—
150C2×D188—42no—
151C2×D188—82no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—24no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47C2×D412yes—
164C4⋊C4×C47C2×D412yes—
165C4⋊C4×C47C2×D412yes—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—24no—
168C22⋊C4×C47—24no—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47C2×D412yes—
174C42×C47C2×D412yes—
175#10625334—48no—
176(C2×D47)⋊C4—44no—
177(C2×D47)⋊C4—44no—
178(C2×D47)⋊C4—82no—
179(C2×D47)⋊C4—82no—
180C4⋊Dic47—22no—
181C4⋊Dic47—22no—
182C4×Dic47—42no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C23×C47—22no—
190C23×C47—24no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—42no—
194C23×C47C22×D411yescharacteristic
195C22×D47—42no—
196C22×D47—42no—
197C22×D47—42no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—42no—
201C22×D47—42no—
202Q8×C47—44no—
203Q8×C47—84no—
204D4×C47—22no—
205D4×C47—24no—
206D4×C47#12414yescomplemented
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—22no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—48no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—84no—
228C2×C4×C47—84no—
229C2×C4×C47#12012yes—
230C2×C4×C47#12012yes—
231C2×C4×C47AGL(1,ℤ/8)12yes—
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4112yes—
235C2×C4×C472−1+412yes—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239D188—42no—
240D188—42no—
241D188—42no—
242D188—42no—
243D188—84no—
244D188—84no—
245C47⋊C8—42no—
246C47⋊C8—42no—
247#25946—471nomaximal, Sylow
248C22×C47—24no—
249C22×C47—44no—
250C22×C47—44no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47C2×#12011yescharacteristic
254C22×C47#54812yes—
255C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C47#57511yescharacteristic
257C22×C47#57612yes—
258C2×D47—82no—
259C2×D47—82no—
260C2×D47—82no—
261C2×D47—82no—
262C2×D47—82no—
263C2×D47—82no—
264C2×D47—82no—
265C4×C47—22no—
266C4×C47—22no—
267C4×C47—22no—
268C4×C47—41no—
269C4×C47—42no—
270C4×C47—41no—
271C4×C47—41no—
272C4×C47—42no—
273C4×C47—42no—
274C4×C47C2×#12411yescharacteristic
275C4×C47#53611yescharacteristic
276C4×C47#54812yes—
277Dic47—82no—
278C2×#554D4711yescharacteristic, complemented
279C2×#490—471no—
280C2×#459—471no—
281#2548—471no—
282#2025—472no—
283#2023—472no—
284#1989—471no—
285#1985—471no—
286#1879—472no—
287#1879—472no—
288#1762—471no—
289C2×C47—44no—
290C2×C47#243611yescharacteristic
291C2×C47#300611yescharacteristic
292C2×C47#300811yescharacteristic
293C2×C47#336612yes—
294C2×C47#337012yes—
295D47—162no—
296C22×D8—472no—
297#554—24no—
298#554—24no—
299C2×C4⋊Q8C2×D4711yescharacteristic
300C2×#127—471no—
301C2×#125C2×D4711yescharacteristic
302C2×#125C2×D4711yescharacteristic
303C2×#124—941no—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—21no—
307C2×C4×Q8C2×D4711yescharacteristic
308C2×C4×D4C2×D4711yescharacteristic
309C2×C4×D4C2×D4711yescharacteristic
310C2×C4×D4C2×D4711yescharacteristic
311C22×C4⋊C4—21no—
312#490—944no—
313#459—944no—
314C2×C4⋊C8—471no—
315C2×#102—472no—
316C2×#102—472no—
317C2×#102—472no—
318C2×#102—472no—
319C2×#97—471no—
320#394—941no—
321#394—941no—
322#390—472no—
323#390—472no—
324#390—941no—
325C4×C4⋊C4—471no—
326#340—472no—
327#340—472no—
328#340—472no—
329#336—472no—
330#336—472no—
331C47#2594611yescharacteristic, complemented, Sylow
332C22×Q8—21no—
333C22×Q8C22⋊D4711yescharacteristic
334C22×D4—472no—
335C22×D4—941no—
336C22×D4—941no—
337C22×D4C22×D4711yescharacteristic
338C23×C4—21no—
339C23×C4—21no—
340C23×C4—21no—
341C2×D8—944no—
342C2×D8—944no—
343C2×D8—944no—
344C22×C8—472no—
345C22×C8—472no—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348#127—942no—
349#127—942no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125C22⋊D4714yes—
354#124—1884no—
355C22⋊Q8—28no—
356C22⋊Q8—28no—
357C22⋊Q8—28no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4C22⋊D4714yes—
366C2×C4⋊C4—28no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—472no—
373C2×C4⋊C4C22⋊D4711yescharacteristic
374C2×C4⋊C4C22⋊D4711yescharacteristic
375C2×C4⋊C4C22⋊D4711yescharacteristic
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—942no—
384C2×C22⋊C4—942no—
385C2×C22⋊C4—942no—
386C2×C42—941no—
387C2×C42C22×D4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389C2×C42C22×D4711yescharacteristic
390C4⋊C8—942no—
391#102—944no—
392#102—944no—
393#102—944no—
394#102—944no—
395#97—942no—
396#95—941no—
397C24—21no—
398C24—942no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—41no—
403C2×Q8—82no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—942no—
407C2×D4—942no—
408C2×D4—942no—
409C2×D4—942no—
410C2×D4—942no—
411C2×D4—942no—
412C2×D4—3762no—
413C2×D4—1882no—
414C2×D4—1882no—
415C2×D4—1882no—
416C2×D4—1882no—
417C2×D4#1062535414yes—
418C2×D4C2×C22⋊D4712yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—942no—
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C2×C22⋊D4711yescharacteristic
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C23×D4711yescharacteristic
437D8—1888no—
438C2×C8—942no—
439C2×C8—942no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—942no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—942no—
456C4⋊C4—942no—
457C4⋊C4C2×C22⋊D4712yes—
458C4⋊C4C2×C22⋊D4712yes—
459C4⋊C4C2×C22⋊D4712yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—3762no—
464C22⋊C4—3762no—
465C22⋊C4—1884no—
466C22⋊C4—1884no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—1882no—
472C42C2×C22⋊D4712yes—
473C42C2×C22⋊D4712yes—
474C23—24no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—1882no—
480C23—1882no—
481C23—1882no—
482C23—1882no—
483C23—1882no—
484C23—1882no—
485C23—1882no—
486C23C22×C22⋊D4711yescharacteristic
487Q8—44no—
488Q8—84no—
489D4—24no—
490D4—22no—
491D4—3764no—
492D4—3764no—
493D4—1882no—
494D4—1882no—
495D4—1882no—
496D4—1882no—
497D4#1309401314yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4#1309399312yes—
524C2×C4#1309399312yes—
525C2×C4#1309404912yes—
526C2×C4C22×C22⋊D4711yescharacteristic
527C2×C4C22×C22⋊D4711yescharacteristic
528C2×C4C2×#1062535412yes—
529C2×C4#1309407212yes—
530C8—1882no—
531C8—1882no—
532C22—24no—
533C22—44no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—3762no—
538C22—3762no—
539C22—3762no—
540C22—3762no—
541C22—3762no—
542C22—3762no—
543C22—3762no—
544C22C2×#1309399311yescharacteristic
545C22#1568978412yes—
546C22C2×#1309404911yescharacteristic
547C22#1568992111yescharacteristic
548C22#1568992612yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—3762no—
559C4C2×#1309401311yescharacteristic
560C4#1568969711yescharacteristic
561C4#1568978412yes—
562C2—44no—
563C2—7522no—
564C2#2024704211yescharacteristic
565C2#2025264011yescharacteristic
566C2#2025265611yescharacteristic
567C2#2025609512yes—
568C2#2025614112yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#554C2
socleC23×C47C22×D4
prime radicalC2×#15689318C2
hypercenterC2×#554D47
hypocenterC47#25946
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×#15689489C2semidirect
3. #20248387C2semidirect
4. #20244531C2semidirect
5. #20244521C2semidirect
6. #20244346C2semidirect
7. #20244329C2semidirect
8. #20243736C2semidirect
9. #20243736C2semidirect
10. #20243213C2semidirect
11. C2×C47×#554C2semidirect
206. D4×C47#124semidirect
278. C2×#554D47semidirect
331. C47#25946semidirect
497. D4#13094013semidirect

Maximal subgroups

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

subgroupindex
C2×#156894892
C2×#156893182
#202483872
#202445312
#202445212
#202443462
#202443292
#202437362
#202432132
C2×C47×#5542
#2594647