Finite Group Databasebeta

← #26249563

Subgroups — #26249563

Order 12032.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689673C211yesmaximal, characteristic
2C2×#15689447C212yesmaximal, complemented
3#20244721C211yesmaximal, characteristic, complemented
4#20244716C211yesmaximal, characteristic, complemented
5#20244622C211yesmaximal, characteristic, complemented
6#20244622C211yesmaximal, characteristic, complemented
7#20243710C212yesmaximal, complemented
8#20243648C212yesmaximal, complemented
9#20243634C212yesmaximal, complemented
10#20243456C211yesmaximal, characteristic, complemented
11C47×#2296C211yesmaximal, characteristic, complemented
12C22×#10625350—21no—
13#15689673—28no—
14C2×#13094024C2211yescharacteristic
15C2×#13094021C2211yescharacteristic
16C2×#13094012—21no—
17C2×#13093998—21no—
18C2×#13093998C2212yes—
19C2×C4×D188—21no—
20C2×C4×D188C2212yes—
21#15689447—28no—
22C2×#13093957C2212yes—
23C2×#13093955C2212yes—
24C2×#13093944C2212yes—
25C2×#13093944C2212yes—
26C2×#13093944C2212yes—
27C2×#13093944C2212yes—
28C2×C22⋊(C47⋊C8)C2212yes—
29#15688885C2212yes—
30#15688885C2212yes—
31#15688878C2211yescharacteristic
32#15688878C2211yescharacteristic
33#15688878C2212yes—
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C22×C42×C47C2211yescharacteristic
37C47×#389C2211yescharacteristic
38C47×#389C2211yescharacteristic
39C47×#389C2211yescharacteristic
40C47×#389C2211yescharacteristic
41C47×#385—21no—
42C47×#385—21no—
43C4×C4⋊C4×C47—21no—
44C4×C4⋊C4×C47—21no—
45C2×#10625350—28no—
46C2×#10625350—44no—
47C2×#10625350—44no—
48C22×Dic94—21no—
49C22×D188—21no—
50C22×D188—21no—
51C22×D188C2312yes—
52C22×C22⋊D47—22no—
53C22×C4×D47—22no—
54C22×C47⋊C8C2312yes—
55C22×C47⋊C8C2312yes—
56#13094024—24no—
57#13094021—24no—
58#13094012—28no—
59#13093998—28no—
60#13093998—24no—
61#13093998D418yes—
62C4×D188—28no—
63C4×D188—24no—
64C4×D188—24no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47C2312yes—
67C2×(C2×D47)⋊C4—22no—
68C2×(C2×D47)⋊C4—22no—
69#13093957—24no—
70#13093955—24no—
71#13093944—24no—
72#13093944—24no—
73#13093944—24no—
74#13093944—24no—
75C22⋊(C47⋊C8)—24no—
76C23×C4×C47D411yescharacteristic
77C23×C4×C47D411yescharacteristic
78C23×C4×C47C2311yescharacteristic
79#117×C47—22no—
80#117×C47—22no—
81#117×C47—24no—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2311yescharacteristic
88C2×C4⋊C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47C2311yescharacteristic
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—21no—
92C2×C42×C47—24no—
93C2×C42×C47—21no—
94C2×C42×C47—21no—
95C2×C42×C47D411yescharacteristic
96C2×C42×C47D411yescharacteristic
97C2×C42×C47C2311yescharacteristic
98C2×C42×C47C2311yescharacteristic
99#95×C47—21no—
100#95×C47—21no—
101#95×C47—21no—
102#95×C47—21no—
103C24×C47C2×D411yescharacteristic
104C23×D47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—41no—
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D414yes—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C2×C22⋊D47—28no—
126C2×C22⋊D47—44no—
127C2×C22⋊D47—44no—
128C22×Dic47—22no—
129#10625350—432no—
130C2×D188—22no—
131C2×D188—22no—
132C2×D188—22no—
133C2×D188—22no—
134C2×D188—24no—
135C2×D188—24no—
136C2×D188—22no—
137C2×D188—42no—
138C2×D188—42no—
139C2×D188—42no—
140C2×D188—42no—
141C2×D188C2×D414yes—
142C2×C4×D47—28no—
143C2×C4×D47—44no—
144C2×C4×D47—44no—
145C2×Dic94—24no—
146C2×Dic94—42no—
147C2×Dic94—42no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—42no—
153C4⋊C4×C47—42no—
154C4⋊C4×C47—42no—
155C4⋊C4×C47—42no—
156C4⋊C4×C47#4112yescomplemented
157C4⋊C4×C47#4112yescomplemented
158C22⋊C4×C47—24no—
159C22⋊C4×C47—42no—
160C42×C47—24no—
161C42×C47—22no—
162C42×C47—22no—
163C42×C47—22no—
164C42×C47—22no—
165C42×C47—42no—
166(C2×D47)⋊C4—28no—
167(C2×D47)⋊C4—28no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47C2×D414yes—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—21no—
183C23×C47#12012yes—
184C23×C47C22×D411yescharacteristic
185C22×D47—24no—
186C22×D47—42no—
187C22×D47—42no—
188C22×D47—42no—
189C22×D47—42no—
190C22×D47—42no—
191C22×D47—42no—
192C2×C4×C47—28no—
193C2×C4×C47—21no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—44no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47#12012yes—
220C2×C4×C47#13512yes—
221C2×C4×C47AGL(1,ℤ/8)12yes—
222C2×C4×C47AGL(1,ℤ/8)12yes—
223C2×C4×C47C22×D412yes—
224C2×C4×C47C2×#4111yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C472−1+412yes—
227C22⋊D47—416no—
228C2×Dic47—24no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231D188—24no—
232D188—24no—
233D188—48no—
234D188—48no—
235D188—42no—
236D188—42no—
237C4×D47—416no—
238Dic94—48no—
239C47⋊C8—42no—
240C47⋊C8—42no—
241#24062—471nomaximal, Sylow
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—41no—
246C22×C47—41no—
247C22×C47—41no—
248C22×C47—41no—
249C22×C47C2×#12011yescharacteristic
250C22×C47#52511yescharacteristic
251C22×C47#53611yescharacteristic
252C22×C47C2×#13511yescharacteristic
253C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C47#57511yescharacteristic
256C2×D47—44no—
257C2×D47—44no—
258C2×D47—44no—
259C2×D47—82no—
260C2×D47—82no—
261C2×D47—82no—
262C2×D47—82no—
263C4×C47—22no—
264C4×C47—22no—
265C4×C47—22no—
266C4×C47—22no—
267C4×C47—42no—
268C4×C47—42no—
269C4×C47—41no—
270C4×C47—81no—
271C4×C47—81no—
272C4×C47#54712yes—
273C4×C47#54712yes—
274Dic47—44no—
275C2×#532—471no—
276C2×#482—472no—
277#2296D4711yescharacteristic, complemented
278#2058—471no—
279#2057—471no—
280#2041—471no—
281#2041—471no—
282#1875—472no—
283#1864—472no—
284#1862—472no—
285#1817—471no—
286C2×C47—42no—
287C2×C47#240611yescharacteristic
288C2×C47#300211yescharacteristic
289C2×C47#300311yescharacteristic
290C2×C47#310611yescharacteristic
291C2×C47#311511yescharacteristic
292C2×C47#322811yescharacteristic
293C2×C47#323511yescharacteristic
294D47—84no—
295C22×#41—941no—
296#532—948no—
297C2×C4⋊Q8—471no—
298C2×#127—471no—
299C2×#124—941no—
300C2×#121—472no—
301C2×#121—941no—
302C2×C4×D4—472no—
303C2×C4×D4—941no—
304C2×#117C2×D4711yescharacteristic
305C2×#117C2×D4711yescharacteristic
306C22×C42C2×D4711yescharacteristic
307#482—948no—
308C2×#107—472no—
309C2×#106—472no—
310C2×#102—472no—
311C2×#102—472no—
312C2×#102—472no—
313C2×#102—472no—
314C2×C22⋊C8—472no—
315#389C2×D4711yescharacteristic
316#389C2×D4711yescharacteristic
317#389C2×D4711yescharacteristic
318#389C2×D4711yescharacteristic
319#385—21no—
320#385—21no—
321C4×C4⋊C4—21no—
322C4×C4⋊C4—21no—
323#340—472no—
324#340—472no—
325#336—471no—
326#336—471no—
327#336—472no—
328C47#2406211yescharacteristic, complemented, Sylow
329C2×#41—948no—
330C2×#41—1884no—
331C2×#41—1884no—
332C22×Q8—941no—
333C22×D4—472no—
334C22×D4—942no—
335C22×D4—941no—
336C22×D4—941no—
337C23×C4—942no—
338C23×C4C22⋊D4711yescharacteristic
339C23×C4C22⋊D4711yescharacteristic
340C23×C4C22×D4711yescharacteristic
341C22×C8—472no—
342C22×C8—472no—
343C4⋊Q8—944no—
344#127—944no—
345#124—948no—
346#121—478no—
347#121—948no—
348#121—944no—
349C4×D4—944no—
350C4×D4—944no—
351C4×D4—948no—
352#117—22no—
353#117—22no—
354#117—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—472no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4C22×D4711yescharacteristic
362C2×C4⋊C4C22×D4711yescharacteristic
363C2×C4⋊C4C22×D4711yescharacteristic
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C42—24no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42C22⋊D4711yescharacteristic
373C2×C42C22⋊D4711yescharacteristic
374C2×C42C22×D4711yescharacteristic
375C2×C42C22×D4711yescharacteristic
376#107—944no—
377#106—944no—
378#102—944no—
379#102—944no—
380#102—944no—
381#102—944no—
382C22⋊C8—944no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387C24—942no—
388C24C2×C22⋊D4711yescharacteristic
389#41—18832no—
390C2×Q8—944no—
391C2×Q8—1882no—
392C2×Q8—1882no—
393C2×D4—474no—
394C2×D4—944no—
395C2×D4—944no—
396C2×D4—948no—
397C2×D4—942no—
398C2×D4—942no—
399C2×D4—942no—
400C2×D4—942no—
401C2×D4—942no—
402C2×D4—1882no—
403C2×D4—1882no—
404C2×D4—1884no—
405C2×D4—1884no—
406C2×D4—1882no—
407C2×D4—1882no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—948no—
422C22×C4—942no—
423C22×C4—1884no—
424C22×C4—1884no—
425C22×C4D4×D4714yes—
426C22×C4C2×C22⋊D4711yescharacteristic
427C22×C4C2×C22⋊D4711yescharacteristic
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C23×D4711yescharacteristic
432C2×C8—942no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8—942no—
436C2×C8—942no—
437C2×C8—942no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—474no—
447C4⋊C4—944no—
448C4⋊C4—942no—
449C4⋊C4#1062535012yescomplemented
450C4⋊C4#1062535012yescomplemented
451C22⋊C4—24no—
452C22⋊C4—42no—
453C22⋊C4—948no—
454C22⋊C4—948no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—42no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—944no—
468C23—1882no—
469C23—1882no—
470C23—1882no—
471C23—1882no—
472C23—1882no—
473C23—1882no—
474C23#1309399212yes—
475C23C22×C22⋊D4711yescharacteristic
476Q8—1888no—
477D4—944no—
478D4—944no—
479D4—1888no—
480D4—1888no—
481D4—18816no—
482D4—1882no—
483D4—1882no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—944no—
512C2×C4—18816no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4#1309399212yes—
516C2×C4#1309404412yes—
517C2×C4#1309404912yes—
518C2×C4#1309404912yes—
519C2×C4C2×D4×D4712yes—
520C2×C4C2×#1062535011yescharacteristic
521C2×C4C2×#1062535011yescharacteristic
522C2×C4#1309407212yes—
523C8—1882no—
524C8—1882no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—3762no—
533C22—3762no—
534C22—3762no—
535C22—3762no—
536C22—1884no—
537C22—1884no—
538C22—1884no—
539C22C2×#1309399211yescharacteristic
540C22#1568963411yescharacteristic
541C22#1568969711yescharacteristic
542C22C2×#1309404411yescharacteristic
543C22C2×#1309404911yescharacteristic
544C22C2×#1309404911yescharacteristic
545C22#1568992111yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—1884no—
556C4#1568977412yes—
557C4#1568977412yes—
558C2—42no—
559C2—3764no—
560C2#2024672311yescharacteristic
561C2#2025260411yescharacteristic
562C2#2025261411yescharacteristic
563C2#2025344811yescharacteristic
564C2#2025353911yescharacteristic
565C2#2025452411yescharacteristic
566C2#2025458611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2296C2
socleC23×C47C22×D4
prime radicalC2×#15689673C2
hypercenter#2296D47
hypocenterC47#24062
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. C2×#15689447C2semidirect
3. #20244721C2semidirect
4. #20244716C2semidirect
5. #20244622C2semidirect
6. #20244622C2semidirect
7. #20243710C2semidirect
8. #20243648C2semidirect
9. #20243634C2semidirect
10. #20243456C2semidirect
11. C47×#2296C2semidirect
156. C4⋊C4×C47#41semidirect
157. C4⋊C4×C47#41semidirect
277. #2296D47semidirect
328. C47#24062semidirect
449. C4⋊C4#10625350semidirect
450. C4⋊C4#10625350semidirect

Maximal subgroups

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

subgroupindex
C2×#156896732
C2×#156894472
#202447212
#202447162
#202446222
#202437102
#202436482
#202436342
#202434562
C47×#22962
#2406247