Finite Group Databasebeta

← #26223220

Subgroups — #26223220

Order 12032.

876 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#20253104C211yesmaximal, characteristic
2#20252287C212yesmaximal, complemented
3C2×#15689240C211yesmaximal, characteristic, complemented
4#20251810C211yesmaximal, characteristic, complemented
5#20244178C211yesmaximal, characteristic, complemented
6#20244060C212yesmaximal, complemented
7#20243886C212yesmaximal, complemented
8#20243439C211yesmaximal, characteristic, complemented
9#20243269C211yesmaximal, characteristic, complemented
10#20243190C212yesmaximal, complemented
11C47×#1976C211yesmaximal, characteristic, complemented
12#15689921C2212yes—
13C2×#13094032C2211yescharacteristic
14C2×#13094031C2212yes—
15C2×#13094030C2211yescharacteristic
16#15689591C2211yescharacteristic
17#15689395C412yescomplemented
18#15689390C412yescomplemented
19#15689383C414yescomplemented
20#15689242—24no—
21#15689240—21no—
22#15689240—21no—
23#15689240—21no—
24#15689240—21no—
25#15689237—24no—
26#15689171C2212yes—
27C2×#13093960C2211yescharacteristic
28C2×#13093960C2211yescharacteristic
29#15689141C2212yes—
30C2×#13093951C2212yes—
31#15689107C2212yes—
32#15689080—21no—
33#15689079C2212yes—
34#15689079C2211yescharacteristic
35#15689077—21no—
36#15689075—21no—
37M16×Dic47C2211yescharacteristic
38C2×C8×Dic47C2211yescharacteristic
39C4×C8×D47—21no—
40#15688893C2212yes—
41#15688893C2212yes—
42#15688882C2212yes—
43C2×#135×C47C2211yescharacteristic
44C47×#427C2211yescharacteristic
45C47×#421C2212yes—
46C47×#410C2212yes—
47C47×#405C2211yescharacteristic
48C2×#10625350C2311yescharacteristic
49#13094055C2×C414yes—
50#13094054—22no—
51#13094050C2×C414yes—
52#13094049—22no—
53#13094044—24no—
54#13094032—21no—
55#13094032—21no—
56#13094032D412yes—
57#13094032D412yes—
58#13094031—22no—
59#13094031—22no—
60#13094031—22no—
61#13094031—22no—
62#13094030—21no—
63#13094030—21no—
64#13094030C2×C412yes—
65#13094030C2×C412yes—
66C2×#10625329—22no—
67C2×#10625329C2312yes—
68C2×#10625329C2311yescharacteristic
69C2×#10625329C2312yes—
70C2×#10625324—21no—
71C2×#10625324—21no—
72C22×C47⋊C8—22no—
73C22×C47⋊C8C2311yescharacteristic
74C2×C8×D47—21no—
75C2×C8×D47—21no—
76#13093989—21no—
77#13093986—21no—
78(C22⋊D47)⋊C4—22no—
79#13093981—21no—
80C2×C4×Dic47C2311yescharacteristic
81C42×D47—21no—
82#13093960—21no—
83#13093960—21no—
84#13093960C2×C412yescomplemented
85#13093960C2×C412yescomplemented
86#13093959—24no—
87#13093952—24no—
88#13093951—22no—
89#13093951—22no—
90#13093943C2×C414yescomplemented
91#13093941C2×C414yescomplemented
92#13093935—22no—
93#13093935—21no—
94#13093934—21no—
95C4×C47⋊C8—22no—
96C4×C47⋊C8—21no—
97C8×Dic47—21no—
98C8×Dic47D411yescharacteristic
99C8×Dic47D411yescharacteristic
100C8×Dic47D411yescharacteristic
101C8×Dic47D411yescharacteristic
102C2×#41×C47C2312yes—
103#135×C47—21no—
104#135×C47—21no—
105#135×C47—21no—
106#135×C47—21no—
107C2×Q16×C47C2×C411yescharacteristic, complemented
108C2×SD16×C47C2×C412yescomplemented
109C2×D8×C47C2×C411yescharacteristic, complemented
110C2×M16×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112#117×C47C2312yes—
113#117×C47C2311yescharacteristic
114#107×C47—21no—
115#106×C47—21no—
116#104×C47—24no—
117#99×C47—24no—
118#97×C47—21no—
119C4×C8×C47—21no—
120#41×C47—22no—
121#41×C47—22no—
122#41×C47—22no—
123#41×C47—22no—
124C2×Q8×C47C22×C412yes—
125C2×D4×C47—22no—
126C2×D4×C47C22×C412yes—
127C22×C4×C47—22no—
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—21no—
130C22×Dic47—21no—
131#10625350—21no—
132#10625350—21no—
133#10625350C22×C412yes—
134#10625350C2×D412yes—
135C2×D188#4111yescharacteristic
136C2×C4×D47—21no—
137C2×C4×D47—22no—
138C2×Dic94#4111yescharacteristic
139Q16×C47—21no—
140Q16×C47—21no—
141SD16×C47—22no—
142SD16×C47—22no—
143D8×C47—21no—
144D8×C47—21no—
145M16×C47—21no—
146M16×C47C2×D412yes—
147C2×C8×C47—21no—
148C2×C8×C47C22×C411yescharacteristic
149C2×C8×C47C22×C411yescharacteristic
150C2×C8×C47C2×D411yescharacteristic
151C2×C8×C47C2×D411yescharacteristic
152C2×C8×C47#4111yescharacteristic
153C2×C8×C47#4111yescharacteristic
154C4⋊C4×C47—21no—
155C4⋊C4×C47—22no—
156C22⋊C4×C47—22no—
157C22⋊C4×C47#4112yes—
158C42×C47—21no—
159C42×C47—22no—
160#10625337—24no—
161#10625336—24no—
162#10625335—24no—
163#10625334—24no—
164(C2×D47)⋊C4—22no—
165Dic47⋊C4—22no—
166C4×Dic47—22no—
167C4×Dic47—21no—
168C4×Dic47C2×D411yescharacteristic
169C4×Dic47C2×D411yescharacteristic
170#10625329—21no—
171#10625329—22no—
172#10625329—22no—
173#10625329—22no—
174#10625329—42no—
175#10625329—42no—
176#10625329C22×C412yes—
177#10625329C22×C414yes—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—42no—
182C2×C47⋊C8—42no—
183C2×C47⋊C8C2×D411yescharacteristic
184C2×C47⋊C8C2×D411yescharacteristic
185C2×C47⋊C8C2×D411yescharacteristic
186C2×C47⋊C8C2×D411yescharacteristic
187C2×C47⋊C8#4111yescharacteristic
188C2×C47⋊C8#4111yescharacteristic
189#10625324—22no—
190#10625324—22no—
191#10625324—22no—
192#10625324—22no—
193C8×D47—22no—
194C8×D47—22no—
195C8×D47—22no—
196C8×D47—22no—
197C23×C47—22no—
198C23×C47C2×#4111yescharacteristic
199C22×D47—21no—
200Q8×C47—22no—
201Q8×C47—22no—
202D4×C47—22no—
203D4×C47—22no—
204D4×C47—42no—
205D4×C47—42no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47C23×C411yescharacteristic
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C2×#4111yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C22⋊D47—22no—
218C22⋊D47—22no—
219C2×Dic47—21no—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—42no—
223D188—21no—
224D188C4×D412yes—
225C4×D47—22no—
226C4×D47—22no—
227C4×D47—44no—
228Dic94—21no—
229Dic94C4×D412yes—
230C8×C47—22no—
231C8×C47C4×D411yescharacteristic
232C8×C47C4×D411yescharacteristic
233C8×C47C4×D411yescharacteristic
234C8×C47C4×D411yescharacteristic
235C47⋊C8—22no—
236C47⋊C8—24no—
237C47⋊C8—42no—
238C47⋊C8#12111yescharacteristic
239C47⋊C8#12111yescharacteristic
240C47⋊C8#12111yescharacteristic
241C47⋊C8#12111yescharacteristic
242#21558—471nomaximal, Sylow
243C22×C47—41no—
244C22×C47—42no—
245C22×C47—82no—
246C22×C47C4×#4111yescharacteristic
247C22×C47#52511yescharacteristic
248C22×C47#53211yescharacteristic
249C2×D47—22no—
250C2×D47—42no—
251C4×C47—42no—
252C4×C47—42no—
253C4×C47—82no—
254C4×C47C2×C4×D411yescharacteristic
255C4×C47C2×C4×D411yescharacteristic
256C4×C47C2×#12111yescharacteristic
257C4×C47#52911yescharacteristic
258Dic47—22no—
259Dic47—22no—
260Dic47—41no—
261#3067—471no—
262#2955—472no—
263C2×#443—471no—
264C4×#131—471no—
265#1976D4711yescharacteristic, complemented
266#1953—471no—
267#1932—472no—
268#1906—472no—
269#1814—471no—
270#1772—471no—
271#1757—472no—
272C2×C47—21no—
273C2×C47—21no—
274C2×C47—21no—
275C2×C47—82no—
276C2×C47C4×#12111yescharacteristic
277D47—42no—
278#575—472no—
279C2×#135C2×D4711yescharacteristic
280C2×#131—472no—
281C2×#131—471no—
282C2×#131—471no—
283C4×#41—471no—
284#473—472no—
285#472—472no—
286#471—474no—
287#444—944no—
288#443—941no—
289#443—941no—
290#443—941no—
291#443—941no—
292#443—944no—
293#430—472no—
294C2×#108—471no—
295C2×#108—471no—
296#427C2×D4711yescharacteristic
297#421—472no—
298#421C2×D4712yes—
299C2×#104—472no—
300#413—472no—
301#410C2×D4712yes—
302#405—471no—
303#405—472no—
304#405—941no—
305#405—941no—
306#405C2×D4711yescharacteristic
307C4×M16—471no—
308C4×M16—941no—
309C2×C4×C8—471no—
310C2×C4×C8—941no—
311#344—472no—
312#344—472no—
313#339—472no—
314C47#2155811yescharacteristic, complemented, Sylow
315C2×#41—471no—
316C2×#41C22×D4712yes—
317#137—474no—
318#137—942no—
319AGL(1,ℤ/8)—474no—
320AGL(1,ℤ/8)—942no—
321#135—21no—
322#135—21no—
323#135—21no—
324#135—21no—
325#135—944no—
326C2×Q16C4×D4711yescharacteristic, complemented
327C2×SD16C4×D4712yescomplemented
328C2×D8C4×D4711yescharacteristic, complemented
329#131—472no—
330#131—472no—
331#131—472no—
332#131—472no—
333#131—941no—
334#131—941no—
335#131—942no—
336#131—942no—
337#131—942no—
338#131—942no—
339#131—941no—
340#131—941no—
341C2×M16—471no—
342C2×M16—472no—
343C2×M16—472no—
344C2×M16—941no—
345C2×M16—942no—
346C2×M16—941no—
347C2×M16C22×D4711yescharacteristic
348C22×C8—471no—
349C22×C8—941no—
350C22×C8—942no—
351C22×C8—941no—
352C22×C8C22×D4711yescharacteristic
353C4×Q8—941no—
354C4×D4—941no—
355C4×D4—942no—
356#117—941no—
357#117C22×D4712yes—
358#117C22×D4711yescharacteristic
359C2×C42—471no—
360C2×C42—941no—
361#108—472no—
362#108—472no—
363#108—944no—
364#108—941no—
365#108—941no—
366#107—21no—
367#106—21no—
368#104—24no—
369#104—944no—
370#104—942no—
371#104—942no—
372#101—474no—
373#100—474no—
374#99—24no—
375#97—21no—
376#97—941no—
377#97—942no—
378#97—941no—
379C4×C8—21no—
380C4×C8—471no—
381C4×C8—471no—
382C4×C8—471no—
383C4×C8—471no—
384C4×C8—941no—
385C4×C8—942no—
386C4×C8—941no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—472no—
392#41—472no—
393#41—941no—
394#41—941no—
395C2×Q8—471no—
396C2×Q8C2×C4×D4712yes—
397C2×D4—22no—
398C2×D4—471no—
399C2×D4—941no—
400C2×D4C2×C4×D4712yes—
401C22×C4—22no—
402C22×C4—942no—
403C22×C4—941no—
404C22×C4—941no—
405C22×C4C23×D4711yescharacteristic
406Q16—21no—
407Q16—21no—
408Q16—944no—
409SD16—22no—
410SD16—22no—
411SD16—944no—
412SD16—944no—
413D8—21no—
414D8—21no—
415D8—944no—
416M16—21no—
417M16—472no—
418M16—474no—
419M16—942no—
420M16—942no—
421M16—942no—
422M16—941no—
423M16—942no—
424M16—942no—
425M16—942no—
426M16—942no—
427M16—1882no—
428M16—1882no—
429M16D4×D4712yes—
430C2×C8—21no—
431C2×C8—471no—
432C2×C8—471no—
433C2×C8—471no—
434C2×C8—471no—
435C2×C8—471no—
436C2×C8—471no—
437C2×C8—942no—
438C2×C8—942no—
439C2×C8—941no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—942no—
444C2×C8—1882no—
445C2×C8—1882no—
446C2×C8C2×C4×D4711yescharacteristic
447C2×C8C2×C4×D4711yescharacteristic
448C2×C8D4×D4711yescharacteristic
449C2×C8D4×D4711yescharacteristic
450C2×C8#1062535211yescharacteristic
451C2×C8#1062535211yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—942no—
455C22⋊C4—22no—
456C22⋊C4—942no—
457C22⋊C4#1062535212yes—
458C42—21no—
459C42—22no—
460C42—471no—
461C42—471no—
462C42—942no—
463C42—941no—
464C23—22no—
465C23—941no—
466C23C2×#1062535211yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—472no—
470Q8—941no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—42no—
475D4—472no—
476D4—942no—
477D4—942no—
478D4—941no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—941no—
485C2×C4—941no—
486C2×C4—942no—
487C2×C4—942no—
488C2×C4—941no—
489C2×C4—1884no—
490C2×C4—1882no—
491C2×C4C22×C4×D4711yescharacteristic
492C2×C4C2×D4×D4711yescharacteristic
493C2×C4C2×D4×D4711yescharacteristic
494C2×C4#41×D4711yescharacteristic
495C2×C4#41×D4711yescharacteristic
496C2×C4C2×#1062535211yescharacteristic
497C8—22no—
498C8—471no—
499C8—471no—
500C8—471no—
501C8—471no—
502C8—944no—
503C8—942no—
504C8—1882no—
505C8(C22⋊D47)⋊C411yescharacteristic
506C8(C22⋊D47)⋊C411yescharacteristic
507C8(C22⋊D47)⋊C411yescharacteristic
508C8(C22⋊D47)⋊C411yescharacteristic
509C22—41no—
510C22—42no—
511C22—82no—
512C22—942no—
513C22—1882no—
514C22C4×#1062535211yescharacteristic
515C22#1568963311yescharacteristic
516C22#1568967111yescharacteristic
517C4—42no—
518C4—42no—
519C4—82no—
520C4—942no—
521C4—942no—
522C4—1881no—
523C4C4×D4×D4711yescharacteristic
524C4C2×(C22⋊D47)⋊C411yescharacteristic
525C4#121×D4711yescharacteristic
526C4#1568965211yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—82no—
531C2—1882no—
532C2#2024590411yescharacteristic
subgroupG/H
centerC4#121×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1976C2
socleC2×C47C4×#121
prime radical#20253104C2
cosocleC22×C4×C47C24
hypercenter#1976D47
hypocenterC47#21558
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #20252287C2semidirect
3. C2×#15689240C2semidirect
4. #20251810C2semidirect
5. #20244178C2semidirect
6. #20244060C2semidirect
7. #20243886C2semidirect
8. #20243439C2semidirect
9. #20243269C2semidirect
10. #20243190C2semidirect
11. C47×#1976C2semidirect
17. #15689395C4semidirect
18. #15689390C4semidirect
19. #15689383C4semidirect
84. #13093960C2×C4semidirect
85. #13093960C2×C4semidirect
90. #13093943C2×C4semidirect
91. #13093941C2×C4semidirect
107. C2×Q16×C47C2×C4semidirect
108. C2×SD16×C47C2×C4semidirect
109. C2×D8×C47C2×C4semidirect
265. #1976D47semidirect
314. C47#21558semidirect
326. C2×Q16C4×D47semidirect
327. C2×SD16C4×D47semidirect
328. C2×D8C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202531042
#202522872
C2×#156892402
#202518102
#202441782
#202440602
#202438862
#202434392
#202432692
#202431902
C47×#19762
#2155847