Finite Group Databasebeta

← #26224440

Subgroups — #26224440

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253708C211yesmaximal, characteristic
2#20252326C211yesmaximal, characteristic, complemented
3#20252104C211yesmaximal, characteristic
4#20244367C211yesmaximal, characteristic, complemented
5#20244218C211yesmaximal, characteristic
6#20244101C211yesmaximal, characteristic, complemented
7#20244097C211yesmaximal, characteristic, complemented
8#20244093C211yesmaximal, characteristic, complemented
9#20244089C211yesmaximal, characteristic, complemented
10#20243524C211yesmaximal, characteristic, complemented
11#20243232C211yesmaximal, characteristic, complemented
12#20243227C211yesmaximal, characteristic, complemented
13#20243223C211yesmaximal, characteristic, complemented
14#20243076C211yesmaximal, characteristic, complemented
15C47×#1765C211yesmaximal, characteristic, complemented
16C2×#13094030C2211yescharacteristic
17#15689676C2211yescharacteristic
18#15689669C2211yescharacteristic
19#15689479C412yescomplemented
20#15689476C412yescomplemented
21#15689472C412yescomplemented
22#15689468C412yescomplemented
23#15689253—22no—
24#15689251—22no—
25#15689194—22no—
26#15689192—22no—
27#15689151C2211yescharacteristic
28#15689150C2211yescharacteristic
29#15689146C2211yescharacteristic
30#15689146C2211yescharacteristic
31C2×#13093952C2211yescharacteristic
32C2×#13093951C2211yescharacteristic
33C2×#13093950C2211yescharacteristic
34#15689120C2211yescharacteristic
35#15689119C2211yescharacteristic
36#15689117C2211yescharacteristic
37#15689091C2211yescharacteristic
38#15689090C2211yescharacteristic
39#15689078C2211yescharacteristic
40C2×#13093934C2211yescharacteristic
41C2×#13093934C2211yescharacteristic
42C2×C4×C47⋊C8C2211yescharacteristic
43C2×C8×Dic47C2211yescharacteristic
44C2×C8×Dic47C2211yescharacteristic
45#15688969C2211yescharacteristic
46#15688888C2211yescharacteristic
47#15688887C2211yescharacteristic
48#15688887C2211yescharacteristic
49#15688884C2211yescharacteristic
50#15688883C2211yescharacteristic
51#15688882C2211yescharacteristic
52C2×#131×C47C2211yescharacteristic
53C2×#104×C47C2211yescharacteristic
54C47×#416C2211yescharacteristic
55C47×#405C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C47×#341C2211yescharacteristic
58C47×#339C2211yescharacteristic
59C2×#10625350C2311yescharacteristic
60#13094030—22no—
61#13094030—22no—
62C2×#10625329C2311yescharacteristic
63C2×#10625329C2311yescharacteristic
64C2×#10625324—21no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C2×C8×D47—21no—
68#13094023C2×C412yes—
69#13094022C2×C411yescharacteristic
70#13094020C2×C411yescharacteristic
71#13094015C2×C412yes—
72#13094013C2×C411yescharacteristic
73#13094011C2×C411yescharacteristic
74#13094001—22no—
75#13093996—22no—
76C4×Dic94—22no—
77C4×C22⋊D47—22no—
78#13093983C2311yescharacteristic
79C2×C4×Dic47C2311yescharacteristic
80C2×C4×Dic47C2311yescharacteristic
81C4⋊(C47⋊C8)—22no—
82Dic47⋊C8—22no—
83Dic47⋊C8—22no—
84Dic47⋊C8—22no—
85#13093952—22no—
86#13093951—22no—
87#13093950—22no—
88#13093949C2×C412yes—
89#13093948C2×C412yes—
90#13093947C2×C412yes—
91#13093946C2×C412yes—
92#13093945C2×C412yes—
93#13093944C2×C412yes—
94C22⋊(C47⋊C8)—22no—
95(C2×D47)⋊C8—22no—
96#13093934—21no—
97#13093934—22no—
98#13093934D412yes—
99#13093934D412yes—
100C4×C47⋊C8—22no—
101C8×Dic47—21no—
102C8×Dic47—22no—
103C8×Dic47C2×C412yes—
104C8×Dic47C2×C412yes—
105C47⋊#95—22no—
106C2×#41×C47C2311yescharacteristic
107#131×C47—22no—
108#131×C47—22no—
109C2×M16×C47—21no—
110C2×M16×C47C2311yescharacteristic
111C2×M16×C47C2311yescharacteristic
112C22×C8×C47—21no—
113C22×C8×C47C2311yescharacteristic
114C22×C8×C47C2311yescharacteristic
115#117×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117#104×C47—22no—
118#103×C47C2×C412yes—
119#102×C47C2×C412yes—
120#97×C47—21no—
121#97×C47—22no—
122C4×C8×C47—21no—
123#41×C47—21no—
124#41×C47—21no—
125#41×C47—21no—
126#41×C47—21no—
127C2×Q8×C47C22×C411yescharacteristic
128C2×D4×C47—21no—
129C2×D4×C47C22×C411yescharacteristic
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47C2411yescharacteristic
133C2×C22⋊D47—21no—
134C2×C22⋊D47—22no—
135C22×Dic47—21no—
136C22×Dic47#4112yes—
137#10625350—21no—
138#10625350—21no—
139#10625350—21no—
140#10625350—21no—
141C2×D188C22×C411yescharacteristic
142C2×C4×D47—21no—
143C2×Dic94—22no—
144C2×Dic94C22×C411yescharacteristic
145M16×C47—21no—
146M16×C47—21no—
147M16×C47—21no—
148M16×C47—21no—
149M16×C47—42no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—42no—
155C2×C8×C47C22×C412yes—
156C2×C8×C47C2×D411yescharacteristic
157C2×C8×C47C2×D411yescharacteristic
158C2×C8×C47#4112yes—
159C2×C8×C47#4112yes—
160C4⋊C4×C47C22×C412yes—
161C22⋊C4×C47—21no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47#4112yes—
165C22⋊Dic47—21no—
166C22⋊Dic47—22no—
167(C2×D47)⋊C4—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47C22×C412yes—
170Dic47⋊C4—22no—
171Dic47⋊C4—22no—
172Dic47⋊C4—22no—
173C4×Dic47—21no—
174C4×Dic47—21no—
175C4×Dic47C22×C411yescharacteristic
176C4×Dic47C22×C411yescharacteristic
177C4×Dic47C2×D412yes—
178C4×Dic47#4112yes—
179#10625329—21no—
180#10625329—21no—
181#10625329—21no—
182#10625329—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8C22×C412yes—
190C2×C47⋊C8C2×D412yes—
191#10625324—42no—
192C8×D47—42no—
193C23×C47—21no—
194C23×C47C2×#4111yescharacteristic
195C22×D47—21no—
196Q8×C47—21no—
197Q8×C47—21no—
198D4×C47—21no—
199D4×C47—21no—
200D4×C47—41no—
201D4×C47—41no—
202C2×C4×C47—22no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47C23×C411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C2×C4×C47C2×#4111yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C22⋊D47—41no—
215C22⋊D47—41no—
216C22⋊D47—82no—
217C2×Dic47—22no—
218C2×Dic47—21no—
219C2×Dic47—22no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47C4×D412yes—
223C2×Dic47C4×D412yes—
224D188—21no—
225D188—21no—
226C4×D47—41no—
227C4×D47—41no—
228Dic94—21no—
229Dic94—21no—
230Dic94—82no—
231C8×C47—22no—
232C8×C47—41no—
233C8×C47—41no—
234C8×C47#12412yes—
235C8×C47#12412yes—
236C47⋊C8—22no—
237C47⋊C8—22no—
238C47⋊C8—41no—
239C47⋊C8—41no—
240#21659—471nomaximal, Sylow
241C22×C47—22no—
242C22×C47—42no—
243C22×C47—41no—
244C22×C47C2×C4×D411yescharacteristic
245C22×C47#52911yescharacteristic
246C22×C47#52911yescharacteristic
247C2×D47—42no—
248C2×D47—41no—
249C4×C47—42no—
250C4×C47—42no—
251C4×C47—41no—
252C4×C47C4×#4111yescharacteristic
253C4×C47C4×#4111yescharacteristic
254C4×C47C2×#12411yescharacteristic
255C4×C47#53211yescharacteristic
256Dic47—22no—
257Dic47—22no—
258Dic47—42no—
259Dic47—42no—
260Dic47—41no—
261#3131—471no—
262#2961—471no—
263#2932—471no—
264#1995—471no—
265#1963—471no—
266#1938—471no—
267#1938—471no—
268#1937—471no—
269#1937—471no—
270#1837—471no—
271#1765—471no—
272#1765D4711yescharacteristic, complemented
273#1764—471no—
274#1764—471no—
275#1726—471no—
276C2×C47—21no—
277C2×C47—21no—
278C2×C47—81no—
279C2×C47C4×#12411yescharacteristic
280C2×C47#295712yes—
281D47—81no—
282C2×#131—471no—
283C2×#131C2×D4711yescharacteristic
284#533—471no—
285#532—471no—
286#488—472no—
287#488—472no—
288#487—472no—
289#486—472no—
290#446—942no—
291#446—942no—
292#435—942no—
293#435—942no—
294#424—471no—
295#424—471no—
296#423—471no—
297#423—471no—
298C2×#104—471no—
299C2×#104—471no—
300C2×#104—471no—
301C2×#104C2×D4711yescharacteristic
302#416—471no—
303#416—471no—
304#416—471no—
305#416C2×D4711yescharacteristic
306#408—471no—
307#408—471no—
308#405—471no—
309#405C2×D4711yescharacteristic
310C2×#97—471no—
311C2×#97—471no—
312C2×#97C2×D4711yescharacteristic
313C2×C4×C8—471no—
314C2×C4×C8—471no—
315C2×C4×C8—471no—
316#376—471no—
317#341—471no—
318#341—471no—
319#341—471no—
320#341C2×D4711yescharacteristic
321#339—471no—
322#339—471no—
323#339—471no—
324#339C2×D4711yescharacteristic
325C47#2165911yescharacteristic, complemented, Sylow
326C2×#41—471no—
327C2×#41C22×D4711yescharacteristic
328#131—22no—
329#131—22no—
330#131—942no—
331#131—942no—
332C2×M16—21no—
333C2×M16—471no—
334C2×M16—471no—
335C2×M16—941no—
336C2×M16C22×D4711yescharacteristic
337C2×M16C22×D4711yescharacteristic
338C22×C8—21no—
339C22×C8—471no—
340C22×C8—471no—
341C22×C8—941no—
342C22×C8C22×D4711yescharacteristic
343C22×C8C22×D4711yescharacteristic
344C4⋊Q8—472no—
345C4⋊Q8—471no—
346#127—471no—
347#125—472no—
348#124—471no—
349#124—471no—
350C22⋊Q8—942no—
351#121—942no—
352C4×Q8—942no—
353C4×D4—942no—
354#117—471no—
355#117C22×D4711yescharacteristic
356C2×C42—471no—
357C2×C42—471no—
358C2×C42C22×D4711yescharacteristic
359C4⋊C8—942no—
360C4⋊C8—942no—
361C4⋊C8—942no—
362C4⋊C8—942no—
363#104—22no—
364#104—942no—
365#104—942no—
366#104—942no—
367#103—472no—
368#103—472no—
369#103—472no—
370#103C4×D4712yes—
371#102—472no—
372#102—472no—
373#102—472no—
374#102C4×D4712yes—
375C22⋊C8—942no—
376C22⋊C8—942no—
377#97—21no—
378#97—22no—
379#97—472no—
380#97—472no—
381#97—942no—
382#97—941no—
383C4×C8—21no—
384C4×C8—472no—
385C4×C8—472no—
386C4×C8—942no—
387C4×C8—941no—
388C4×C8—942no—
389#95—942no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—941no—
395#41—941no—
396#41—941no—
397#41—941no—
398C2×Q8—471no—
399C2×Q8—942no—
400C2×Q8C2×C4×D4711yescharacteristic
401C2×D4—21no—
402C2×D4—471no—
403C2×D4—942no—
404C2×D4—941no—
405C2×D4C2×C4×D4711yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—472no—
409C22×C4—941no—
410C22×C4—941no—
411C22×C4C23×D4711yescharacteristic
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—42no—
417M16—941no—
418M16—941no—
419M16—941no—
420M16—941no—
421M16—1882no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—42no—
427C2×C8—472no—
428C2×C8—472no—
429C2×C8—941no—
430C2×C8—941no—
431C2×C8—941no—
432C2×C8—941no—
433C2×C8—941no—
434C2×C8—941no—
435C2×C8—1882no—
436C2×C8C2×C4×D4712yes—
437C2×C8#1062535012yes—
438C2×C8D4×D4711yescharacteristic
439C2×C8D4×D4711yescharacteristic
440C2×C8#1062535212yes—
441C4⋊C4—472no—
442C4⋊C4—942no—
443C4⋊C4—942no—
444C4⋊C4—942no—
445C4⋊C4—942no—
446C4⋊C4C2×C4×D4712yes—
447C22⋊C4—21no—
448C22⋊C4—942no—
449C22⋊C4—942no—
450C22⋊C4—941no—
451C42—21no—
452C42—21no—
453C42—471no—
454C42—471no—
455C42—472no—
456C42—472no—
457C42—941no—
458C42—941no—
459C42#1062535012yes—
460C23—21no—
461C23—941no—
462C23#41×D4711yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—941no—
466Q8—941no—
467Q8—3762no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—941no—
473D4—941no—
474D4—3762no—
475D4—1881no—
476D4—1881no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—472no—
484C2×C4—472no—
485C2×C4—942no—
486C2×C4—941no—
487C2×C4—942no—
488C2×C4—1882no—
489C2×C4—1882no—
490C2×C4—1881no—
491C2×C4—1881no—
492C2×C4C22×C4×D4711yescharacteristic
493C2×C4C2×D4×D4711yescharacteristic
494C2×C4#41×D4711yescharacteristic
495C2×C4C2×#1062535211yescharacteristic
496C2×C4C2×#1062535011yescharacteristic
497C2×C4C2×#1062535011yescharacteristic
498C8—22no—
499C8—41no—
500C8—41no—
501C8—942no—
502C8—942no—
503C8—1881no—
504C8—1881no—
505C8#1309401012yes—
506C8#1309401012yes—
507C22—22no—
508C22—42no—
509C22—41no—
510C22—1882no—
511C22—1881no—
512C22C4×D4×D4711yescharacteristic
513C22#1568965111yescharacteristic
514C22#1568965311yescharacteristic
515C4—42no—
516C4—42no—
517C4—41no—
518C4—942no—
519C4—942no—
520C4—1882no—
521C4—1882no—
522C4—1881no—
523C4C4×#1062535211yescharacteristic
524C4C4×#1062535011yescharacteristic
525C4C2×#1309401011yescharacteristic
526C4#1568967011yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—3761no—
531C2C4×#1309401011yescharacteristic
532C2#2025230212yes—
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1765C2
socleC22×C47C2×C4×D4
prime radical#15689669C22
cosocleC22×C4×C47C24
hypercenter#1765D47
hypocenterC47#21659
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #20252326C2semidirect
4. #20244367C2semidirect
6. #20244101C2semidirect
7. #20244097C2semidirect
8. #20244093C2semidirect
9. #20244089C2semidirect
10. #20243524C2semidirect
11. #20243232C2semidirect
12. #20243227C2semidirect
13. #20243223C2semidirect
14. #20243076C2semidirect
15. C47×#1765C2semidirect
19. #15689479C4semidirect
20. #15689476C4semidirect
21. #15689472C4semidirect
22. #15689468C4semidirect
272. #1765D47semidirect
325. C47#21659semidirect

Maximal subgroups

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

subgroupindex
#202537082
#202523262
#202521042
#202443672
#202442182
#202441012
#202440972
#202440932
#202440892
#202435242
#202432322
#202432272
#202432232
#202430762
C47×#17652
#2165947