Finite Group Databasebeta

← #26186896

Subgroups — #26186896

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253990C211yesmaximal, characteristic
2#20252360C211yesmaximal, characteristic
3#20252317C211yesmaximal, characteristic, complemented
4#20242284C211yesmaximal, characteristic, complemented
5#20242282C211yesmaximal, characteristic, complemented
6#20242281C211yesmaximal, characteristic, complemented
7#20242279C211yesmaximal, characteristic, complemented
8#20242208C211yesmaximal, characteristic, complemented
9#20242205C211yesmaximal, characteristic, complemented
10#20242200C211yesmaximal, characteristic, complemented
11#20242154C211yesmaximal, characteristic, complemented
12#20241859C211yesmaximal, characteristic, complemented
13#20241850C211yesmaximal, characteristic
14#20241667C211yesmaximal, characteristic, complemented
15C47×SD16⋊C8C211yesmaximal, characteristic, complemented
16#15689677C2211yescharacteristic
17#15689655C2211yescharacteristic
18#15689530C411yescharacteristic
19#15689514C411yescharacteristic
20#15689490C411yescharacteristic
21#15689481C411yescharacteristic
22#15689382C411yescharacteristic
23#15689380C411yescharacteristic
24#15689368C411yescharacteristic
25#15689366C411yescharacteristic
26#15689254C2211yescharacteristic
27#15689252—21no—
28#15689250C2211yescharacteristic
29#15689250C2211yescharacteristic
30C8×Dic94—21no—
31C8×Dic94C2211yescharacteristic
32C8×Dic94C2211yescharacteristic
33C4×Dic188C2211yescharacteristic
34C4×#10625337C2211yescharacteristic
35C4×#10625335C2211yescharacteristic
36#15689201C2211yescharacteristic
37#15689197—21no—
38C8×C22⋊D47—21no—
39#15689181—21no—
40#15689179C2211yescharacteristic
41#15689174C2211yescharacteristic
42#15689148—21no—
43#15689145C2211yescharacteristic
44#15689078—21no—
45#15689077—21no—
46C4×#10625324C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688875C2211yescharacteristic
49#15688874C2211yescharacteristic
50#15688873C2211yescharacteristic
51#15688855C2211yescharacteristic
52#15688855C2211yescharacteristic
53Q8⋊(C47⋊C8)C2211yescharacteristic
54#15688853C2211yescharacteristic
55#15688853C2211yescharacteristic
56D4⋊(C47⋊C8)C2211yescharacteristic
57#15688845C2211yescharacteristic
58C8×C47⋊C8C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C8×Q8×C47C2211yescharacteristic
61C4×SD16×C47C2211yescharacteristic
62C8×D4×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×D4⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×Dic188C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625335C2×C411yescharacteristic
70C2×#10625329—21no—
71C2×#10625324—21no—
72C2×#10625324C2×C411yescharacteristic
73C2×#10625324C2×C411yescharacteristic
74C2×C8×D47D411yescharacteristic
75C2×C8×D47D411yescharacteristic
76#13094023C2×C411yescharacteristic
77#13094022C2×C411yescharacteristic
78#13094017—21no—
79#13094015C2×C411yescharacteristic
80#13094014—21no—
81#13094011C2×C411yescharacteristic
82#13094006—21no—
83#13094004C2×C411yescharacteristic
84#13094004C2×C411yescharacteristic
85#13094002—21no—
86#13094000C2×C411yescharacteristic
87#13093995C2×C411yescharacteristic
88C4×Dic94—21no—
89C4×Dic94C2311yescharacteristic
90C4×Dic94C2311yescharacteristic
91C4×C22⋊D47—21no—
92#13093983—21no—
93#13093981—21no—
94C42×D47C2311yescharacteristic
95#13093958C2×C411yescharacteristic
96#13093957C2×C411yescharacteristic
97#13093955C2×C411yescharacteristic
98C4⋊(C47⋊C8)—21no—
99C4⋊(C47⋊C8)C2311yescharacteristic
100C4⋊(C47⋊C8)C2311yescharacteristic
101Dic47⋊C8—21no—
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
106Dic47⋊C8—21no—
107#13093949C2×C411yescharacteristic
108#13093949C2×C411yescharacteristic
109#13093948C2×C411yescharacteristic
110#13093947C2×C411yescharacteristic
111#13093947C2×C411yescharacteristic
112#13093945C2×C411yescharacteristic
113C22⋊(C47⋊C8)—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116#13093935—21no—
117#13093934—21no—
118#13093934—21no—
119#13093934—21no—
120#13093934C2×C411yescharacteristic
121#13093934C2×C411yescharacteristic
122C4×C47⋊C8C2311yescharacteristic
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C8×Dic47—21no—
126C8×Dic47—21no—
127C8×Dic47D411yescharacteristic
128C8×Dic47D411yescharacteristic
129C2×SD16×C47C2×C411yescharacteristic
130C22×C8×C47—21no—
131C4×Q8×C47C2311yescharacteristic
132C4×D4×C47C2311yescharacteristic
133#106×C47C2×C411yescharacteristic
134C4⋊C8×C47—21no—
135C4⋊C8×C47C2311yescharacteristic
136C4⋊C8×C47C2311yescharacteristic
137#103×C47C2×C411yescharacteristic
138#102×C47C2×C411yescharacteristic
139C22⋊C8×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×D4×C47C22×C411yescharacteristic
146C22×C4×C47—21no—
147C2×C22⋊D47—21no—
148C2×C4×D47C22×C411yescharacteristic
149C2×C4×D47C2×D411yescharacteristic
150C2×C4×D47#4111yescharacteristic
151C2×Dic94—21no—
152C2×Dic94C22×C411yescharacteristic
153C2×Dic94C22×C411yescharacteristic
154SD16×C47—22no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—42no—
159C2×C8×C47C22×C411yescharacteristic
160C2×C8×C47C22×C411yescharacteristic
161C2×C8×C47C2×D411yescharacteristic
162C2×C8×C47C2×D411yescharacteristic
163C4⋊C4×C47—21no—
164C4⋊C4×C47C22×C411yescharacteristic
165C4⋊C4×C47C22×C411yescharacteristic
166C22⋊C4×C47—21no—
167C42×C47—21no—
168C42×C47C2411yescharacteristic
169C22⋊Dic47—21no—
170C22⋊Dic47—21no—
171C22⋊Dic47—21no—
172#10625337—22no—
173#10625335—22no—
174(C2×D47)⋊C4—21no—
175(C2×D47)⋊C4—21no—
176(C2×D47)⋊C4—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47C22×C411yescharacteristic
181C4⋊Dic47C22×C411yescharacteristic
182Dic47⋊C4—21no—
183Dic47⋊C4—21no—
184Dic47⋊C4—21no—
185Dic47⋊C4—21no—
186Dic47⋊C4—21no—
187Dic47⋊C4—21no—
188Dic47⋊C4—21no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47C22×C411yescharacteristic
192C4×Dic47C2×D411yescharacteristic
193C4×Dic47#4111yescharacteristic
194#10625329—42no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8C22×C411yescharacteristic
199C2×C47⋊C8C22×C411yescharacteristic
200C2×C47⋊C8C2×D411yescharacteristic
201C2×C47⋊C8C2×D411yescharacteristic
202Dic188—22no—
203#10625324—22no—
204#10625324—22no—
205#10625324—42no—
206C8×D47—22no—
207C8×D47—41no—
208C23×C47—21no—
209C22×D47C4×D411yescharacteristic
210Q8×C47—21no—
211Q8×C47—21no—
212D4×C47—21no—
213D4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C22⋊D47—42no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47C4×D411yescharacteristic
228C2×Dic47C4×D411yescharacteristic
229C2×Dic47C4×D411yescharacteristic
230C4×D47—21no—
231C4×D47—21no—
232C4×D47—21no—
233C4×D47—21no—
234C4×D47—41no—
235Dic94—21no—
236Dic94—21no—
237Dic94—21no—
238Dic94—21no—
239Dic94—42no—
240C8×C47—21no—
241C8×C47—21no—
242C8×C47—21no—
243C8×C47—41no—
244C8×C47—41no—
245C8×C47—41no—
246C8×C47#13112yes—
247C47⋊C8—21no—
248C47⋊C8—21no—
249C47⋊C8—41no—
250C47⋊C8—41no—
251C47⋊C8—41no—
252C47⋊C8#13112yes—
253C47⋊C8#13712yes—
254#17967—471nomaximal, Sylow
255C22×C47—41no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47C2×C4×D411yescharacteristic
259C2×D47—21no—
260C2×D47—21no—
261C2×D47—21no—
262C4×C47—21no—
263C4×C47—41no—
264C4×C47—41no—
265C4×C47—41no—
266C4×C47C2×#13111yescharacteristic
267C4×C47C2×#13111yescharacteristic
268C4×C47C2×#13711yescharacteristic
269C4×C47#57511yescharacteristic
270Dic47—21no—
271Dic47—21no—
272Dic47—21no—
273Dic47—41no—
274Dic47—41no—
275Dic47—41no—
276Dic47—41no—
277#3172—471no—
278#2966—471no—
279#2960—471no—
280#1591—471no—
281#1591—471no—
282#1590—471no—
283#1590—471no—
284#1579—471no—
285#1579—471no—
286SD16⋊C8—471no—
287SD16⋊C8D4711yescharacteristic, complemented
288#1569—471no—
289#1497—471no—
290#1495—471no—
291C8×M16—471no—
292C2×C47—81no—
293C2×C47#293111yescharacteristic
294C2×C47C4×#13711yescharacteristic
295C2×C47#295511yescharacteristic
296D47—41no—
297#533—471no—
298#529—471no—
299#501—471no—
300#497—471no—
301#491—471no—
302#489—471no—
303#470—471no—
304#470—471no—
305#468—471no—
306#468—471no—
307#446—471no—
308#446—471no—
309#446—471no—
310#446—941no—
311C8×Q8—471no—
312C8×Q8—471no—
313C8×Q8—941no—
314C8×Q8C2×D4711yescharacteristic
315C4×Q16—471no—
316C4×Q16—471no—
317C4×SD16—471no—
318C4×SD16C2×D4711yescharacteristic
319#436—471no—
320#435—941no—
321C8×D4—941no—
322C8×D4C2×D4711yescharacteristic
323#433—941no—
324#432—471no—
325#431—471no—
326#424—941no—
327#423—471no—
328#405—941no—
329#405—941no—
330C4×M16—471no—
331C2×C4×C8—471no—
332#335—471no—
333#335C2×D4711yescharacteristic
334#334—471no—
335#334—471no—
336Q8⋊C8—471no—
337Q8⋊C8—471no—
338Q8⋊C8—471no—
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341Q8⋊C8C2×D4711yescharacteristic
342D4⋊C8—471no—
343D4⋊C8C2×D4711yescharacteristic
344#322—471no—
345#322C2×D4711yescharacteristic
346C82—471no—
347C82—471no—
348C47#1796711yescharacteristic, complemented, Sylow
349C2×Q16—471no—
350C2×Q16—471no—
351C2×SD16—471no—
352C2×SD16C4×D4711yescharacteristic
353C2×M16—471no—
354C2×M16—471no—
355C2×M16—941no—
356C2×M16—941no—
357C22×C8—21no—
358C22×C8—471no—
359C22×C8—471no—
360C4⋊Q8—471no—
361C4⋊Q8—471no—
362#126—941no—
363#125—471no—
364#125—941no—
365#124—471no—
366#123—941no—
367C22⋊Q8—471no—
368C22⋊Q8—471no—
369C22⋊Q8—471no—
370C22⋊Q8—941no—
371#121—471no—
372C4×Q8—471no—
373C4×Q8—471no—
374C4×Q8—941no—
375C4×Q8C22×D4711yescharacteristic
376C4×D4—941no—
377C4×D4C22×D4711yescharacteristic
378#117—941no—
379#117—941no—
380C2×C42—471no—
381#107—471no—
382#107—471no—
383#106—471no—
384#106C4×D4711yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—471no—
387C4⋊C8—471no—
388C4⋊C8—941no—
389C4⋊C8—941no—
390C4⋊C8—941no—
391C4⋊C8—941no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8C22×D4711yescharacteristic
396C4⋊C8C22×D4711yescharacteristic
397#103—471no—
398#103—471no—
399#103—471no—
400#103—471no—
401#103—471no—
402#103C4×D4711yescharacteristic
403#102—471no—
404#102C4×D4711yescharacteristic
405C22⋊C8—21no—
406C22⋊C8—941no—
407C22⋊C8—941no—
408C22⋊C8—941no—
409#97—471no—
410#97—471no—
411#97—941no—
412#97—941no—
413#97—941no—
414#97—941no—
415C4×C8—21no—
416C4×C8—471no—
417C4×C8—471no—
418C4×C8—471no—
419C4×C8—471no—
420C4×C8—471no—
421C4×C8—941no—
422C4×C8—941no—
423C4×C8C22×D4711yescharacteristic
424C4×C8C22×D4711yescharacteristic
425C4×C8C22×D4711yescharacteristic
426C2×Q8—471no—
427C2×Q8—471no—
428C2×Q8—941no—
429C2×Q8C2×C4×D4711yescharacteristic
430C2×D4—941no—
431C2×D4C2×C4×D4711yescharacteristic
432C22×C4—21no—
433C22×C4—471no—
434C22×C4—471no—
435C22×C4—471no—
436Q16—942no—
437Q16—942no—
438SD16—22no—
439SD16—942no—
440M16—942no—
441M16—942no—
442M16—1882no—
443M16—1882no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—471no—
449C2×C8—471no—
450C2×C8—471no—
451C2×C8—471no—
452C2×C8—942no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—1881no—
457C2×C8C2×C4×D4711yescharacteristic
458C2×C8C2×C4×D4711yescharacteristic
459C2×C8D4×D4711yescharacteristic
460C2×C8D4×D4711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—471no—
463C4⋊C4—471no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4C2×C4×D4711yescharacteristic
475C4⋊C4C2×C4×D4711yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—941no—
478C22⋊C4—941no—
479C22⋊C4—941no—
480C22⋊C4—941no—
481C22⋊C4—941no—
482C22⋊C4—941no—
483C42—21no—
484C42—471no—
485C42—471no—
486C42—471no—
487C42—941no—
488C42—941no—
489C42C23×D4711yescharacteristic
490C23—21no—
491C23—471no—
492Q8—21no—
493Q8—21no—
494Q8—941no—
495Q8—941no—
496Q8—941no—
497Q8—941no—
498Q8—1882no—
499D4—21no—
500D4—21no—
501D4—1882no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—471no—
508C2×C4—471no—
509C2×C4—471no—
510C2×C4—941no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—1881no—
519C2×C4C22×C4×D4711yescharacteristic
520C2×C4C2×D4×D4711yescharacteristic
521C2×C4#41×D4711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—472no—
529C8—472no—
530C8—941no—
531C8—941no—
532C8—1881no—
533C8—1881no—
534C8—1881no—
535C8#1309403012yes—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—941no—
540C22—941no—
541C22—941no—
542C22C4×D4×D4711yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—941no—
548C4—941no—
549C4—941no—
550C4—1881no—
551C4—1881no—
552C4—1881no—
553C4—1881no—
554C4#131×D4711yescharacteristic
555C4C2×#1309403011yescharacteristic
556C4#137×D4711yescharacteristic
557C4#1568991911yescharacteristic
558C2—81no—
559C2—1881no—
560C2#2025209511yescharacteristic
561C2#2025221011yescharacteristic
562C2#2025228811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×SD16⋊C8C2
socleC22×C47C2×C4×D4
prime radical#13093995C2×C4
cosocleC42×C47C24
hypercenterSD16⋊C8D47
hypocenterC47#17967
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
3. #20252317C2semidirect
4. #20242284C2semidirect
5. #20242282C2semidirect
6. #20242281C2semidirect
7. #20242279C2semidirect
8. #20242208C2semidirect
9. #20242205C2semidirect
10. #20242200C2semidirect
11. #20242154C2semidirect
12. #20241859C2semidirect
14. #20241667C2semidirect
15. C47×SD16⋊C8C2semidirect
287. SD16⋊C8D47semidirect
348. C47#17967semidirect

Maximal subgroups

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

subgroupindex
#202539902
#202523602
#202523172
#202422842
#202422822
#202422812
#202422792
#202422082
#202422052
#202422002
#202421542
#202418592
#202418502
#202416672
C47×SD16⋊C82
#1796747