Finite Group Databasebeta

← #26223230

Subgroups — #26223230

Order 12032.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253109C211yesmaximal, characteristic
2#20252292C212yesmaximal, complemented
3C2×#15689246C211yesmaximal, characteristic, complemented
4#20251813C211yesmaximal, characteristic, complemented
5#20244284C211yesmaximal, characteristic, complemented
6#20244179C211yesmaximal, characteristic, complemented
7#20244058C212yesmaximal, complemented
8#20243888C212yesmaximal, complemented
9#20243439C211yesmaximal, characteristic, complemented
10#20243189C212yesmaximal, complemented
11C47×#1773C211yesmaximal, characteristic, complemented
12#15689923C2212yes—
13C2×#13094046C2211yescharacteristic
14C2×#13094030C2212yes—
15C2×#13094030C2211yescharacteristic
16C2×#13094030C2211yescharacteristic
17C4×#10625350C2211yescharacteristic
18#15689399C412yescomplemented
19#15689394C412yescomplemented
20#15689387C414yescomplemented
21#15689246—21no—
22#15689246—21no—
23#15689246—21no—
24#15689246—21no—
25#15689246—24no—
26#15689241—24no—
27#15689172C2212yes—
28C2×#13093960C2211yescharacteristic
29#15689141C2212yes—
30#15689141C2212yes—
31C2×#13093952C2212yes—
32#15689105C2212yes—
33#15689096C2212yes—
34#15689080—21no—
35#15689080—21no—
36#15689078C2211yescharacteristic
37#15689078C2211yescharacteristic
38C4×#10625329C2211yescharacteristic
39#15689075—21no—
40#97×D47—21no—
41#15688895C2212yes—
42#15688893C2212yes—
43C2×#108×C47C2211yescharacteristic
44C47×#427C2211yescharacteristic
45C47×#405C2212yes—
46C2×#97×C47C2211yescharacteristic
47C47×#339C2212yes—
48C2×#10625350C2312yes—
49C2×#10625350C2311yescharacteristic
50#13094057—22no—
51#13094057C2×C414yes—
52#13094052—22no—
53#13094052C2×C414yes—
54#13094046—24no—
55#13094046—21no—
56#13094046—21no—
57#13094046—21no—
58#13094046—21no—
59C2×Dic188C2×C411yescharacteristic, complemented
60C2×#10625325C2×C412yescomplemented
61C2×D376C2×C411yescharacteristic, complemented
62#13094030—21no—
63#13094030—21no—
64#13094030—22no—
65#13094030—22no—
66#13094030—22no—
67#13094030—22no—
68#13094030—22no—
69#13094030C2×C412yes—
70#13094030C2×C412yes—
71#13094030D414yes—
72C2×#10625329C2312yes—
73C2×#10625329C2311yescharacteristic
74C2×#10625329C2311yescharacteristic
75C2×#10625324—21no—
76C2×#10625324—21no—
77C2×#10625324—22no—
78C2×C8×D47—21no—
79C2×C8×D47—21no—
80C2×C8×D47—22no—
81C4×Dic94—21no—
82C4×D188—21no—
83C4×C22⋊D47—22no—
84#13093983C2311yescharacteristic
85#13093981—21no—
86C42×D47—21no—
87#13093960—21no—
88#13093960—21no—
89#13093960—24no—
90#13093952—24no—
91#13093952—22no—
92#13093952—22no—
93#13093952—24no—
94#13093942C2×C414yescomplemented
95#13093940C2×C414yescomplemented
96#13093938—24no—
97#13093935—21no—
98#13093934—21no—
99#13093934—21no—
100C4×C47⋊C8—21no—
101C8×Dic47—21no—
102C8×Dic47—21no—
103C2×M16×C47C2312yes—
104C22×C8×C47C2311yescharacteristic
105C22×C8×C47C2311yescharacteristic
106#117×C47C2312yes—
107C2×C42×C47C2311yescharacteristic
108#108×C47C2×C412yescomplemented
109#108×C47C2×C412yescomplemented
110#107×C47—21no—
111#106×C47—21no—
112#97×C47—22no—
113#97×C47D411yescharacteristic
114#97×C47D411yescharacteristic
115#97×C47D411yescharacteristic
116#97×C47D411yescharacteristic
117C4×C8×C47—22no—
118C22×C4×C47—21no—
119C22×C4×C47C2411yescharacteristic
120C2×C22⋊D47—21no—
121C2×C22⋊D47—22no—
122#10625350—21no—
123#10625350—21no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350C22×C412yes—
129#10625350C2×D412yes—
130C2×D188C22×C412yes—
131C2×D188#4111yescharacteristic
132C2×C4×D47—21no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×Dic94C22×C412yes—
136C2×Dic94#4111yescharacteristic
137M16×C47—22no—
138M16×C47C22×C414yes—
139C2×C8×C47—22no—
140C2×C8×C47C22×C411yescharacteristic
141C2×C8×C47C22×C411yescharacteristic
142C2×C8×C47C2×D411yescharacteristic
143C2×C8×C47C2×D411yescharacteristic
144C2×C8×C47C2×D411yescharacteristic
145C2×C8×C47C2×D411yescharacteristic
146C2×C8×C47C2×D412yes—
147C2×C8×C47#4111yescharacteristic
148C2×C8×C47#4111yescharacteristic
149C2×C8×C47#4112yes—
150C4⋊C4×C47—22no—
151C22⋊C4×C47—22no—
152C42×C47—21no—
153C42×C47—22no—
154C42×C47C2×D411yescharacteristic
155C42×C47C2×D411yescharacteristic
156C22⋊Dic47#4112yes—
157(C2×D47)⋊C4—22no—
158C4⋊Dic47—21no—
159Dic47⋊C4—22no—
160C4×Dic47—22no—
161C4×Dic47—21no—
162#10625329—21no—
163#10625329—21no—
164#10625329—22no—
165#10625329—22no—
166#10625329C22×C412yes—
167#10625329C2×D412yes—
168C2×C47⋊C8—21no—
169C2×C47⋊C8—21no—
170C2×C47⋊C8—22no—
171Dic188—24no—
172Dic188—21no—
173Dic188—21no—
174D376—24no—
175D376—21no—
176D376—21no—
177#10625325—24no—
178#10625325—24no—
179#10625325—22no—
180#10625325—22no—
181#10625324—22no—
182#10625324—22no—
183#10625324—24no—
184#10625324—42no—
185#10625324—42no—
186C8×D47—22no—
187C8×D47—22no—
188C8×D47—24no—
189C8×D47—42no—
190C8×D47—42no—
191C23×C47C2×#4111yescharacteristic
192C22×D47—21no—
193C22×D47—22no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—42no—
197C2×C4×C47—81no—
198C2×C4×C47C23×C411yescharacteristic
199C2×C4×C47C22×D411yescharacteristic
200C2×C4×C47C22×D411yescharacteristic
201C2×C4×C47C2×#4111yescharacteristic
202C2×C4×C47C2×#4111yescharacteristic
203C2×C4×C47C2×#4111yescharacteristic
204C22⋊D47—22no—
205C22⋊D47—22no—
206C22⋊D47—42no—
207C22⋊D47—42no—
208C2×Dic47—21no—
209C2×Dic47—22no—
210C2×Dic47—22no—
211D188—22no—
212D188—22no—
213D188—21no—
214D188C4×D412yes—
215C4×D47—22no—
216C4×D47—22no—
217C4×D47—42no—
218C4×D47—42no—
219C4×D47—82no—
220Dic94—21no—
221Dic94—22no—
222Dic94—22no—
223Dic94C4×D412yes—
224C8×C47—24no—
225C8×C47C4×D411yescharacteristic
226C8×C47C4×D411yescharacteristic
227C8×C47C4×D411yescharacteristic
228C8×C47C4×D411yescharacteristic
229C8×C47#12112yes—
230C8×C47#12112yes—
231C47⋊C8—22no—
232C47⋊C8—22no—
233C47⋊C8—42no—
234#21559—471nomaximal, Sylow
235C22×C47—41no—
236C22×C47C4×#4111yescharacteristic
237C22×C47#52511yescharacteristic
238C22×C47#53211yescharacteristic
239C2×D47—22no—
240C2×D47—42no—
241C2×D47—42no—
242C2×D47—82no—
243C4×C47—41no—
244C4×C47—41no—
245C4×C47—82no—
246C4×C47C2×C4×D411yescharacteristic
247C4×C47C2×C4×D411yescharacteristic
248C4×C47C2×#12111yescharacteristic
249C4×C47#52911yescharacteristic
250Dic47—22no—
251Dic47—42no—
252Dic47—42no—
253#3067—471no—
254#2955—472no—
255C2×#444—471no—
256#2875—471no—
257#1976—471no—
258#1953—471no—
259#1932—472no—
260#1906—472no—
261#1814—471no—
262#1773D4711yescharacteristic, complemented
263#1757—472no—
264C2×C47—21no—
265C2×C47—21no—
266C2×C47—21no—
267C2×C47C4×#12111yescharacteristic
268D47—42no—
269D47—82no—
270#575—472no—
271C2×#135—471no—
272C2×#131—471no—
273C2×#131—471no—
274C2×#131—472no—
275C4×#41—471no—
276#473—472no—
277#472—472no—
278#471—474no—
279#444—944no—
280#444—941no—
281#444—941no—
282#444—941no—
283#444—941no—
284#443—944no—
285#430—472no—
286C2×#108—471no—
287C2×#108C2×D4711yescharacteristic
288#427C2×D4711yescharacteristic
289#421—472no—
290#421—472no—
291C2×#104—472no—
292#413—472no—
293#410—472no—
294#405—471no—
295#405—471no—
296#405—941no—
297#405—941no—
298#405C2×D4712yes—
299C4×M16—471no—
300C4×M16—941no—
301C2×#97—941no—
302C2×#97C2×D4711yescharacteristic
303#344—472no—
304#344—472no—
305#339C2×D4712yes—
306C47#2155911yescharacteristic, complemented, Sylow
307C2×#41—471no—
308C2×#41—472no—
309#137—474no—
310#137—942no—
311AGL(1,ℤ/8)—474no—
312AGL(1,ℤ/8)—942no—
313#135—944no—
314#135—941no—
315#135—941no—
316#135—941no—
317#135—941no—
318C2×Q16—471no—
319C2×SD16—472no—
320C2×D8—471no—
321#131—472no—
322#131—472no—
323#131—474no—
324#131—941no—
325#131—941no—
326#131—942no—
327#131—942no—
328#131—942no—
329#131—942no—
330#131—942no—
331C2×M16—471no—
332C2×M16—471no—
333C2×M16—472no—
334C2×M16—941no—
335C2×M16—941no—
336C2×M16—942no—
337C2×M16C22×D4712yes—
338C22×C8—941no—
339C22×C8—941no—
340C22×C8—942no—
341C22×C8C22×D4711yescharacteristic
342C22×C8C22×D4711yescharacteristic
343C4×Q8—941no—
344C4×D4—941no—
345C4×D4—942no—
346#117—471no—
347#117—941no—
348#117C22×D4712yes—
349C2×C42—941no—
350C2×C42C22×D4711yescharacteristic
351#108—944no—
352#108—941no—
353#108—941no—
354#108C4×D4712yescomplemented
355#108C4×D4712yescomplemented
356#107—21no—
357#106—21no—
358#104—944no—
359#104—944no—
360#104—942no—
361#104—942no—
362#101—474no—
363#100—474no—
364#99—944no—
365#97—22no—
366#97—941no—
367#97—941no—
368#97—941no—
369#97D18811yescharacteristic
370#97D18811yescharacteristic
371#97D18811yescharacteristic
372#97D18811yescharacteristic
373C4×C8—22no—
374C4×C8—941no—
375C4×C8—941no—
376C4×C8—941no—
377#41—472no—
378#41—472no—
379#41—941no—
380#41—941no—
381#41—942no—
382#41—942no—
383#41—942no—
384#41—942no—
385C2×Q8—472no—
386C2×Q8—471no—
387C2×D4—472no—
388C2×D4—471no—
389C2×D4—941no—
390C2×D4—942no—
391C22×C4—21no—
392C22×C4—942no—
393C22×C4—941no—
394C22×C4—942no—
395C22×C4C23×D4711yescharacteristic
396Q16—944no—
397Q16—941no—
398Q16—941no—
399SD16—944no—
400SD16—944no—
401SD16—942no—
402SD16—942no—
403D8—944no—
404D8—941no—
405D8—941no—
406M16—22no—
407M16—472no—
408M16—472no—
409M16—941no—
410M16—941no—
411M16—942no—
412M16—942no—
413M16—944no—
414M16—942no—
415M16—942no—
416M16—1882no—
417M16—1882no—
418M16C2×C4×D4714yes—
419C2×C8—22no—
420C2×C8—941no—
421C2×C8—941no—
422C2×C8—942no—
423C2×C8—942no—
424C2×C8—944no—
425C2×C8—942no—
426C2×C8—1882no—
427C2×C8—1882no—
428C2×C8C2×C4×D4711yescharacteristic
429C2×C8C2×C4×D4711yescharacteristic
430C2×C8C2×D18811yescharacteristic
431C2×C8C2×D18811yescharacteristic
432C2×C8C2×D18811yescharacteristic
433C2×C8C2×D18811yescharacteristic
434C2×C8#1062535011yescharacteristic
435C2×C8#1062535011yescharacteristic
436C2×C8D4×D4712yes—
437C2×C8#1062535412yes—
438C4⋊C4—22no—
439C4⋊C4—942no—
440C4⋊C4—941no—
441C22⋊C4—22no—
442C22⋊C4—472no—
443C22⋊C4—942no—
444C42—21no—
445C42—22no—
446C42—942no—
447C42—941no—
448C42C2×D18811yescharacteristic
449C42C2×D18811yescharacteristic
450C23—941no—
451C23—942no—
452C23#41×D4711yescharacteristic
453Q8—472no—
454Q8—941no—
455Q8—942no—
456Q8—942no—
457D4—472no—
458D4—942no—
459D4—942no—
460D4—942no—
461D4—942no—
462D4—941no—
463D4—1882no—
464D4—1882no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—81no—
469C2×C4—941no—
470C2×C4—942no—
471C2×C4—942no—
472C2×C4—942no—
473C2×C4—942no—
474C2×C4—3762no—
475C2×C4—1882no—
476C2×C4—1882no—
477C2×C4C22×C4×D4711yescharacteristic
478C2×C4C2×D4×D4711yescharacteristic
479C2×C4C22×D18811yescharacteristic
480C2×C4#41×D4711yescharacteristic
481C2×C4C2×#1062535011yescharacteristic
482C2×C4C2×#1062535411yescharacteristic
483C8—24no—
484C8—942no—
485C8—942no—
486C8—1882no—
487C8C4×D18811yescharacteristic
488C8C4×D18811yescharacteristic
489C8C4×D18811yescharacteristic
490C8C4×D18811yescharacteristic
491C8#1309399912yes—
492C8#1309399912yes—
493C22—41no—
494C22—942no—
495C22—3762no—
496C22—1882no—
497C22—1882no—
498C22C4×#1062535411yescharacteristic
499C22#1568963511yescharacteristic
500C22#1568967011yescharacteristic
501C4—41no—
502C4—41no—
503C4—82no—
504C4—942no—
505C4—1882no—
506C4—1882no—
507C4C4×D4×D4711yescharacteristic
508C4C2×C4×D18811yescharacteristic
509C4C2×#1309399911yescharacteristic
510C4#1568966011yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—3762no—
515C2—1882no—
516C2C4×#1309399911yescharacteristic
subgroupG/H
centerC4C2×#13093999
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1773C2
socleC2×C47C4×#121
prime radical#20253109C2
cosocleC22×C4×C47C24
hypercenter#1773D47
hypocenterC47#21559
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #20252292C2semidirect
3. C2×#15689246C2semidirect
4. #20251813C2semidirect
5. #20244284C2semidirect
6. #20244179C2semidirect
7. #20244058C2semidirect
8. #20243888C2semidirect
9. #20243439C2semidirect
10. #20243189C2semidirect
11. C47×#1773C2semidirect
18. #15689399C4semidirect
19. #15689394C4semidirect
20. #15689387C4semidirect
59. C2×Dic188C2×C4semidirect
60. C2×#10625325C2×C4semidirect
61. C2×D376C2×C4semidirect
94. #13093942C2×C4semidirect
95. #13093940C2×C4semidirect
108. #108×C47C2×C4semidirect
109. #108×C47C2×C4semidirect
262. #1773D47semidirect
306. C47#21559semidirect
354. #108C4×D47semidirect
355. #108C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202531092
#202522922
C2×#156892462
#202518132
#202442842
#202441792
#202440582
#202438882
#202434392
#202431892
C47×#17732
#2155947