Finite Group Databasebeta

← #26223228

Subgroups — #26223228

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#20253108C211yesmaximal, characteristic
2#20252289C212yesmaximal, complemented
3C2×#15689246C211yesmaximal, characteristic, complemented
4#20244286C211yesmaximal, characteristic, complemented
5#20244178C211yesmaximal, characteristic, complemented
6#20244057C212yesmaximal, complemented
7#20243889C212yesmaximal, complemented
8#20243442C211yesmaximal, characteristic, complemented
9#20243272C211yesmaximal, characteristic, complemented
10#20243189C212yesmaximal, complemented
11C47×#2875C211yesmaximal, characteristic, complemented
12#15689921C2212yes—
13C2×#13094046C2211yescharacteristic
14C2×#13094030C2212yes—
15#15689398C412yescomplemented
16#15689393C412yescomplemented
17#15689386C414yescomplemented
18#15689246—21no—
19#15689246—21no—
20#15689246—21no—
21#15689246—21no—
22#15689243—24no—
23#15689238—24no—
24#15689171C2212yes—
25C2×#13093960C2211yescharacteristic
26C2×#13093960C2211yescharacteristic
27#15689161C2211yescharacteristic
28#15689140C2212yes—
29#15689139C2212yes—
30C2×#13093952C2212yes—
31#15689105C2212yes—
32#15689096C2212yes—
33#15689078C2212yes—
34#15688895C2212yes—
35#15688893C2212yes—
36#15688883C2212yes—
37C2×#131×C47C2211yescharacteristic
38C2×#131×C47C2211yescharacteristic
39C4×#41×C47C2211yescharacteristic
40C47×#405—21no—
41C47×#405—21no—
42C47×#405C2211yescharacteristic
43C47×#405C2211yescharacteristic
44C4×M16×C47—21no—
45C4×M16×C47C2211yescharacteristic
46C2×#97×C47—21no—
47C2×#97×C47C2211yescharacteristic
48C2×#10625350C2312yes—
49#13094055—22no—
50#13094054C2×C414yes—
51#13094050—22no—
52#13094049C2×C414yes—
53#13094046—21no—
54#13094046—21no—
55#13094046—21no—
56#13094046—21no—
57#13094044—24no—
58C2×Dic188C2×C411yescharacteristic, complemented
59C2×#10625325C2×C412yescomplemented
60C2×D376C2×C411yescharacteristic, complemented
61#13094030—22no—
62#13094030—22no—
63#13094030—22no—
64#13094030—22no—
65C2×#10625329C2312yes—
66C2×#10625329C2312yes—
67C2×#10625324—22no—
68C2×C8×D47—22no—
69#13093983C2312yes—
70#13093960—21no—
71#13093960—21no—
72#13093960C2×C412yescomplemented
73#13093960C2×C412yescomplemented
74#13093959—24no—
75#13093958—21no—
76#13093956—21no—
77#13093952—22no—
78#13093952—22no—
79#13093951—24no—
80#13093950—24no—
81#13093942C2×C414yescomplemented
82#13093940C2×C414yescomplemented
83#13093938—24no—
84#13093934—22no—
85C8×Dic47—22no—
86C2×#41×C47C2311yescharacteristic
87#131×C47—21no—
88#131×C47—21no—
89#131×C47—22no—
90#131×C47C2×C412yes—
91#131×C47C2×C412yes—
92#131×C47D414yes—
93C2×M16×C47—21no—
94C2×M16×C47—21no—
95C2×M16×C47C2311yescharacteristic
96C2×M16×C47C2311yescharacteristic
97C22×C8×C47—21no—
98C22×C8×C47—21no—
99C22×C8×C47C2311yescharacteristic
100C22×C8×C47C2311yescharacteristic
101C4×Q8×C47—21no—
102C4×D4×C47—21no—
103C4×D4×C47—22no—
104#117×C47—21no—
105#117×C47C2311yescharacteristic
106C2×C42×C47—21no—
107C2×C42×C47C2311yescharacteristic
108#97×C47—21no—
109#97×C47—21no—
110#97×C47—21no—
111#97×C47D411yescharacteristic
112#97×C47D411yescharacteristic
113#97×C47D411yescharacteristic
114#97×C47D411yescharacteristic
115C4×C8×C47—21no—
116C4×C8×C47—21no—
117C4×C8×C47—21no—
118#41×C47—21no—
119#41×C47—21no—
120#41×C47C22×C412yes—
121#41×C47C2×D412yes—
122C2×Q8×C47#4111yescharacteristic
123C2×D4×C47—21no—
124C2×D4×C47#4111yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—21no—
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—22no—
134C2×D188C22×C412yes—
135C2×C4×D47—22no—
136C2×Dic94C22×C412yes—
137M16×C47—21no—
138M16×C47—21no—
139M16×C47—22no—
140M16×C47—22no—
141M16×C47—24no—
142M16×C47C22×C412yes—
143M16×C47C2×D412yes—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—22no—
147C2×C8×C47—22no—
148C2×C8×C47—24no—
149C2×C8×C47C22×C411yescharacteristic
150C2×C8×C47C22×C411yescharacteristic
151C2×C8×C47C2×D411yescharacteristic
152C2×C8×C47C2×D411yescharacteristic
153C2×C8×C47C2×D412yes—
154C2×C8×C47C2×D411yescharacteristic
155C2×C8×C47C2×D411yescharacteristic
156C2×C8×C47#4111yescharacteristic
157C2×C8×C47#4111yescharacteristic
158C2×C8×C47#4112yes—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—21no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47#4112yes—
163C42×C47—22no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47C2×D411yescharacteristic
167C42×C47C2×D411yescharacteristic
168C22⋊Dic47—22no—
169#10625337—24no—
170#10625336—24no—
171#10625335—24no—
172#10625334—24no—
173C4⋊Dic47—22no—
174C4×Dic47—22no—
175#10625329—22no—
176#10625329—22no—
177#10625329—22no—
178#10625329C22×C414yes—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181Dic188—21no—
182Dic188—21no—
183D376—21no—
184D376—21no—
185#10625325—22no—
186#10625325—22no—
187#10625324—42no—
188#10625324—42no—
189C8×D47—42no—
190C8×D47—42no—
191C23×C47—21no—
192C23×C47C2×#4111yescharacteristic
193C22×D47—22no—
194Q8×C47—21no—
195Q8×C47C4×D412yes—
196D4×C47—22no—
197D4×C47—22no—
198D4×C47—21no—
199D4×C47C4×D412yes—
200C2×C4×C47—21no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—21no—
206C2×C4×C47—81no—
207C2×C4×C47—82no—
208C2×C4×C47C23×C411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C2×C4×C47C2×#4111yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C22⋊D47—42no—
215C22⋊D47—42no—
216C2×Dic47—22no—
217C2×Dic47—42no—
218D188—22no—
219D188—22no—
220C4×D47—42no—
221C4×D47—42no—
222Dic94—22no—
223Dic94—22no—
224C8×C47—22no—
225C8×C47—22no—
226C8×C47C4×D411yescharacteristic
227C8×C47C4×D411yescharacteristic
228C8×C47C4×D411yescharacteristic
229C8×C47C4×D411yescharacteristic
230C8×C47#12112yes—
231C8×C47#12112yes—
232C47⋊C8—24no—
233C47⋊C8—42no—
234#21559—471nomaximal, Sylow
235C22×C47—22no—
236C22×C47—41no—
237C22×C47—42no—
238C22×C47C4×#4111yescharacteristic
239C22×C47#52511yescharacteristic
240C22×C47#53211yescharacteristic
241C2×D47—42no—
242C2×D47—82no—
243C4×C47—22no—
244C4×C47—42no—
245C4×C47—41no—
246C4×C47—41no—
247C4×C47C2×C4×D411yescharacteristic
248C4×C47C2×C4×D411yescharacteristic
249C4×C47C2×#12111yescharacteristic
250C4×C47#52911yescharacteristic
251Dic47—42no—
252Dic47—82no—
253#3067—471no—
254#2955—472no—
255C2×#444—471no—
256#2875D4711yescharacteristic, complemented
257#1976—471no—
258#1953—471no—
259#1932—472no—
260#1906—472no—
261#1814—471no—
262#1773—471no—
263#1757—472no—
264C2×C47—21no—
265C2×C47—21no—
266C2×C47—21no—
267C2×C47—42no—
268C2×C47C4×#12111yescharacteristic
269D47—82no—
270#575—472no—
271C2×#135—471no—
272C2×#131—472no—
273C2×#131C2×D4711yescharacteristic
274C2×#131C2×D4711yescharacteristic
275C4×#41C2×D4711yescharacteristic
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×#108—471no—
288#427—471no—
289#421—472no—
290#421—472no—
291C2×#104—472no—
292#413—472no—
293#410—472no—
294#405—21no—
295#405—21no—
296#405—472no—
297#405C2×D4711yescharacteristic
298#405C2×D4711yescharacteristic
299C4×M16—21no—
300C4×M16C2×D4711yescharacteristic
301C2×#97—21no—
302C2×#97C2×D4711yescharacteristic
303#344—472no—
304#344—472no—
305#339—472no—
306C47#2155911yescharacteristic, complemented, Sylow
307C2×#41—472no—
308C2×#41C22×D4711yescharacteristic
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—21no—
322#131—21no—
323#131—22no—
324#131—942no—
325#131—942no—
326#131—942no—
327#131—942no—
328#131C4×D4712yes—
329#131C4×D4712yes—
330#131D18814yes—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—472no—
334C2×M16—472no—
335C2×M16—942no—
336C2×M16C22×D4711yescharacteristic
337C2×M16C22×D4711yescharacteristic
338C22×C8—21no—
339C22×C8—21no—
340C22×C8—942no—
341C22×C8C22×D4711yescharacteristic
342C22×C8C22×D4711yescharacteristic
343C4×Q8—21no—
344C4×D4—21no—
345C4×D4—22no—
346#117—21no—
347#117—472no—
348#117C22×D4711yescharacteristic
349C2×C42—21no—
350C2×C42C22×D4711yescharacteristic
351#108—472no—
352#108—472no—
353#108—944no—
354#108—941no—
355#108—941no—
356#107—941no—
357#106—941no—
358#104—944no—
359#104—944no—
360#104—942no—
361#104—942no—
362#101—474no—
363#100—474no—
364#99—944no—
365#97—21no—
366#97—21no—
367#97—21no—
368#97—942no—
369#97C22⋊D4711yescharacteristic
370#97C22⋊D4711yescharacteristic
371#97C22⋊D4711yescharacteristic
372#97C22⋊D4711yescharacteristic
373C4×C8—21no—
374C4×C8—21no—
375C4×C8—21no—
376C4×C8—942no—
377#41—21no—
378#41—21no—
379#41—942no—
380#41—942no—
381#41—942no—
382#41—942no—
383#41C2×C4×D4712yes—
384#41C2×D18812yes—
385C2×Q8—472no—
386C2×Q8#1062535011yescharacteristic
387C2×D4—21no—
388C2×D4—472no—
389C2×D4—942no—
390C2×D4#1062535011yescharacteristic
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
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—21no—
407M16—21no—
408M16—22no—
409M16—22no—
410M16—24no—
411M16—474no—
412M16—942no—
413M16—942no—
414M16—942no—
415M16—1882no—
416M16—1882no—
417M16C2×C4×D4712yes—
418M16C2×D18812yes—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—942no—
425C2×C8—942no—
426C2×C8—1882no—
427C2×C8—1882no—
428C2×C8C2×C4×D4711yescharacteristic
429C2×C8C2×C4×D4711yescharacteristic
430C2×C8C2×D18812yes—
431C2×C8#1062535011yescharacteristic
432C2×C8#1062535011yescharacteristic
433C2×C8#1062535012yes—
434C2×C8C2×C22⋊D4711yescharacteristic
435C2×C8C2×C22⋊D4711yescharacteristic
436C2×C8C2×C22⋊D4711yescharacteristic
437C2×C8C2×C22⋊D4711yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—942no—
441C22⋊C4—22no—
442C22⋊C4—942no—
443C22⋊C4#1062535012yes—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—942no—
448C42C2×C22⋊D4711yescharacteristic
449C42C2×C22⋊D4711yescharacteristic
450C23—21no—
451C23—942no—
452C23C2×#1062535011yescharacteristic
453Q8—21no—
454Q8—942no—
455Q8—942no—
456Q8C4×D18812yes—
457D4—22no—
458D4—22no—
459D4—21no—
460D4—942no—
461D4—942no—
462D4—1882no—
463D4—1882no—
464D4C4×D18812yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—942no—
474C2×C4—1882no—
475C2×C4—1882no—
476C2×C4—1882no—
477C2×C4C22×C4×D4711yescharacteristic
478C2×C4C22×C22⋊D4711yescharacteristic
479C2×C4C22×D18811yescharacteristic
480C2×C4C2×#1062535011yescharacteristic
481C2×C4C2×#1062535011yescharacteristic
482C2×C4C2×#1062535011yescharacteristic
483C8—22no—
484C8—22no—
485C8—944no—
486C8—1882no—
487C8C4×C22⋊D4711yescharacteristic
488C8C4×C22⋊D4711yescharacteristic
489C8C4×C22⋊D4711yescharacteristic
490C8C4×C22⋊D4711yescharacteristic
491C8#1309399812yes—
492C8#1309399812yes—
493C22—22no—
494C22—41no—
495C22—42no—
496C22—3762no—
497C22—1882no—
498C22C4×#1062535011yescharacteristic
499C22#1568963411yescharacteristic
500C22#1568967311yescharacteristic
501C4—22no—
502C4—41no—
503C4—42no—
504C4—41no—
505C4—3762no—
506C4—1882no—
507C4C2×C4×C22⋊D4711yescharacteristic
508C4C2×C4×D18811yescharacteristic
509C4C2×#1309399811yescharacteristic
510C4#1568965711yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—3762no—
516C2C4×#1309399811yescharacteristic
subgroupG/H
centerC4C2×#13093998
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2875C2
socleC2×C47C4×#121
prime radical#20253108C2
cosocleC22×C4×C47C24
hypercenter#2875D47
hypocenterC47#21559
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #20252289C2semidirect
3. C2×#15689246C2semidirect
4. #20244286C2semidirect
5. #20244178C2semidirect
6. #20244057C2semidirect
7. #20243889C2semidirect
8. #20243442C2semidirect
9. #20243272C2semidirect
10. #20243189C2semidirect
11. C47×#2875C2semidirect
15. #15689398C4semidirect
16. #15689393C4semidirect
17. #15689386C4semidirect
58. C2×Dic188C2×C4semidirect
59. C2×#10625325C2×C4semidirect
60. C2×D376C2×C4semidirect
72. #13093960C2×C4semidirect
73. #13093960C2×C4semidirect
81. #13093942C2×C4semidirect
82. #13093940C2×C4semidirect
256. #2875D47semidirect
306. C47#21559semidirect

Maximal subgroups

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

subgroupindex
#202531082
#202522892
C2×#156892462
#202442862
#202441782
#202440572
#202438892
#202434422
#202432722
#202431892
C47×#28752
#2155947