Finite Group Databasebeta

← #26223221

Subgroups — #26223221

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#20252287C212yesmaximal, complemented
3C2×#15689244C211yesmaximal, characteristic, complemented
4#20251814C211yesmaximal, characteristic, complemented
5#20244284C211yesmaximal, characteristic, complemented
6#20244176C211yesmaximal, characteristic, complemented
7#20244055C212yesmaximal, complemented
8#20243887C212yesmaximal, complemented
9#20243274C211yesmaximal, characteristic, complemented
10#20243187C212yesmaximal, complemented
11C47×#1814C211yesmaximal, characteristic, complemented
12#15689921C2212yes—
13C2×#13094046C2211yescharacteristic
14C2×#13094032C2212yes—
15C2×#13094031C2211yescharacteristic
16#41×Dic47C2211yescharacteristic
17#15689398C412yescomplemented
18#15689393C412yescomplemented
19#15689386C414yescomplemented
20#15689244—21no—
21#15689244—21no—
22#15689244—21no—
23#15689244—21no—
24#15689242—24no—
25#15689237—24no—
26#15689171C2212yes—
27C2×#13093960C2211yescharacteristic
28#15689141C2212yes—
29C2×#13093950C2212yes—
30#15689105C2212yes—
31#15689096C2212yes—
32#15689081—21no—
33#15689081C2211yescharacteristic
34#15689079C2212yes—
35#15689078—21no—
36#15689078C2211yescharacteristic
37M16×Dic47—21no—
38M16×Dic47C2211yescharacteristic
39C2×#13093934—21no—
40C2×#13093934C2211yescharacteristic
41#15688894C2212yes—
42#15688882C2212yes—
43C2×#131×C47C2211yescharacteristic
44C2×#108×C47C2211yescharacteristic
45C47×#427C2211yescharacteristic
46C47×#421C2212yes—
47C47×#344C2212yes—
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#13094032—22no—
62#13094032—22no—
63#13094032—22no—
64#13094032—22no—
65#13094031—22no—
66#13094031D414yes—
67C2×#10625329—21no—
68C2×#10625329C2312yes—
69C2×#10625329C2311yescharacteristic
70C2×#10625324—22no—
71C22×C47⋊C8—21no—
72C22×C47⋊C8C2311yescharacteristic
73C2×C8×D47—22no—
74Q8×Dic47—21no—
75D4×Dic47—21no—
76D4×Dic47—22no—
77#13093983—21no—
78#13093983C2311yescharacteristic
79C2×C4×Dic47—21no—
80C2×C4×Dic47C2311yescharacteristic
81#13093960C2×C412yescomplemented
82#13093960C2×C412yescomplemented
83#13093959—24no—
84#13093952—24no—
85#13093950—22no—
86#13093950—22no—
87#13093942C2×C414yescomplemented
88#13093940C2×C414yescomplemented
89#13093938—24no—
90#13093935—22no—
91#13093934—21no—
92#13093934—21no—
93#13093934—21no—
94#13093934D411yescharacteristic
95#13093934D411yescharacteristic
96#13093934D411yescharacteristic
97#13093934D411yescharacteristic
98C4×C47⋊C8—22no—
99C8×Dic47—21no—
100C8×Dic47—21no—
101C8×Dic47—21no—
102C2×#41×C47C2311yescharacteristic
103#131×C47—21no—
104#131×C47—21no—
105#131×C47C2×C412yes—
106#131×C47C2×C412yes—
107C2×M16×C47—21no—
108C2×M16×C47C2312yes—
109C2×M16×C47C2311yescharacteristic
110C22×C8×C47—21no—
111C22×C8×C47C2311yescharacteristic
112#117×C47C2312yes—
113#108×C47—21no—
114#108×C47—21no—
115#107×C47—21no—
116#106×C47—21no—
117#104×C47—24no—
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×C47C2411yescharacteristic
127C2×C22⋊D47—22no—
128C22×Dic47—22no—
129C22×Dic47—21no—
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—22no—
139M16×C47—22no—
140M16×C47—22no—
141M16×C47—22no—
142M16×C47C22×C412yes—
143C2×C8×C47—21no—
144C2×C8×C47—22no—
145C2×C8×C47—22no—
146C2×C8×C47—22no—
147C2×C8×C47C22×C411yescharacteristic
148C2×C8×C47C22×C411yescharacteristic
149C2×C8×C47C2×D411yescharacteristic
150C2×C8×C47C2×D411yescharacteristic
151C2×C8×C47#4111yescharacteristic
152C2×C8×C47#4111yescharacteristic
153C4⋊C4×C47—22no—
154C22⋊C4×C47—22no—
155C42×C47—22no—
156C22⋊Dic47—22no—
157C22⋊Dic47#4112yes—
158#10625337—24no—
159#10625336—24no—
160#10625335—24no—
161#10625334—24no—
162C4⋊Dic47—22no—
163C4⋊Dic47—21no—
164C4×Dic47—22no—
165C4×Dic47—21no—
166C4×Dic47—21no—
167C4×Dic47C2×D411yescharacteristic
168C4×Dic47C2×D411yescharacteristic
169#10625329—21no—
170#10625329—22no—
171#10625329—24no—
172#10625329C22×C414yes—
173#10625329C2×D412yes—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—24no—
177C2×C47⋊C8C2×D411yescharacteristic
178C2×C47⋊C8C2×D412yes—
179C2×C47⋊C8C2×D411yescharacteristic
180C2×C47⋊C8#4112yes—
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—42no—
204C2×C4×C47C23×C411yescharacteristic
205C2×C4×C47C22×D411yescharacteristic
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C2×#4111yescharacteristic
208C2×C4×C47C2×#4111yescharacteristic
209C2×C4×C47C2×#4111yescharacteristic
210C22⋊D47—42no—
211C22⋊D47—42no—
212C2×Dic47—22no—
213C2×Dic47—21no—
214C2×Dic47—22no—
215C2×Dic47—21no—
216C2×Dic47—81no—
217C2×Dic47—82no—
218D188—22no—
219D188—22no—
220C4×D47—42no—
221C4×D47—42no—
222Dic94—22no—
223Dic94—22no—
224C8×C47—22no—
225C8×C47—42no—
226C8×C47C4×D411yescharacteristic
227C8×C47C4×D411yescharacteristic
228C8×C47C4×D411yescharacteristic
229C8×C47C4×D411yescharacteristic
230C47⋊C8—24no—
231C47⋊C8—22no—
232C47⋊C8#12112yes—
233C47⋊C8#12112yes—
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—82no—
245C4×C47C2×C4×D411yescharacteristic
246C4×C47C2×C4×D411yescharacteristic
247C4×C47C2×#12111yescharacteristic
248C4×C47#52911yescharacteristic
249Dic47—42no—
250Dic47—41no—
251Dic47—41no—
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#1814D4711yescharacteristic, complemented
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—471no—
273C2×#131—472no—
274C2×#131C2×D4711yescharacteristic
275C4×#41—471no—
276#473—472no—
277#472—472no—
278#471—474no—
279#444—941no—
280#444—941no—
281#444—941no—
282#444—941no—
283#444—944no—
284#443—944no—
285#430—472no—
286C2×#108—471no—
287C2×#108C2×D4711yescharacteristic
288#427C2×D4711yescharacteristic
289#421—472no—
290#421C2×D4712yes—
291C2×#104—472no—
292#413—472no—
293#410—472no—
294#405—471no—
295#405—471no—
296#405—472no—
297#405—941no—
298#405—941no—
299C4×M16—471no—
300C4×M16—941no—
301C2×#97—471no—
302C2×#97—941no—
303#344—472no—
304#344C2×D4712yes—
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—474no—
324#131—942no—
325#131—942no—
326#131—942no—
327#131—942no—
328#131—942no—
329#131C4×D4712yes—
330#131C4×D4712yes—
331C2×M16—21no—
332C2×M16—471no—
333C2×M16—472no—
334C2×M16—942no—
335C2×M16—941no—
336C2×M16C22×D4712yes—
337C2×M16C22×D4711yescharacteristic
338C22×C8—21no—
339C22×C8—471no—
340C22×C8—942no—
341C22×C8—941no—
342C22×C8C22×D4711yescharacteristic
343C4×Q8—941no—
344C4×D4—942no—
345C4×D4—941no—
346#117—471no—
347#117—941no—
348#117C22×D4712yes—
349C2×C42—471no—
350C2×C42—941no—
351#108—21no—
352#108—21no—
353#108—472no—
354#108—472no—
355#108—944no—
356#107—21no—
357#106—21no—
358#104—24no—
359#104—944no—
360#104—942no—
361#104—942no—
362#101—474no—
363#100—474no—
364#99—944no—
365#97—471no—
366#97—471no—
367#97—471no—
368#97—471no—
369#97—942no—
370#97—941no—
371#97—941no—
372#97—941no—
373C4×C8—942no—
374C4×C8—941no—
375C4×C8—941no—
376C4×C8—941no—
377#41—21no—
378#41—21no—
379#41—942no—
380#41—942no—
381#41—942no—
382#41—942no—
383#41C2×C4×D4712yes—
384#41D4×D4712yes—
385C2×Q8—472no—
386C2×Q8#1062535411yescharacteristic
387C2×D4—21no—
388C2×D4—472no—
389C2×D4—942no—
390C2×D4#1062535411yescharacteristic
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—21no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—22no—
411M16—474no—
412M16—472no—
413M16—944no—
414M16—941no—
415M16—942no—
416M16—1882no—
417M16—1882no—
418M16C2×C4×D4712yes—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—471no—
424C2×C8—472no—
425C2×C8—471no—
426C2×C8—472no—
427C2×C8—944no—
428C2×C8—941no—
429C2×C8—942no—
430C2×C8—1882no—
431C2×C8—1882no—
432C2×C8C2×C4×D4711yescharacteristic
433C2×C8C2×C4×D4711yescharacteristic
434C2×C8D4×D4711yescharacteristic
435C2×C8D4×D4711yescharacteristic
436C2×C8#1062535211yescharacteristic
437C2×C8#1062535211yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—942no—
440C4⋊C4—941no—
441C22⋊C4—22no—
442C22⋊C4—472no—
443C22⋊C4—942no—
444C42—22no—
445C42—471no—
446C42—471no—
447C42—942no—
448C42—941no—
449C42—941no—
450C23—21no—
451C23—942no—
452C23#41×D4711yescharacteristic
453Q8—21no—
454Q8—942no—
455Q8—942no—
456Q8#1309398612yes—
457D4—22no—
458D4—22no—
459D4—21no—
460D4—942no—
461D4—942no—
462D4—1882no—
463D4—1882no—
464D4#1309398612yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—942no—
470C2×C4—941no—
471C2×C4—941no—
472C2×C4—942no—
473C2×C4—3761no—
474C2×C4—3762no—
475C2×C4—1882no—
476C2×C4—1882no—
477C2×C4C22×C4×D4711yescharacteristic
478C2×C4C2×D4×D4711yescharacteristic
479C2×C4C2×D4×D4711yescharacteristic
480C2×C4#41×D4711yescharacteristic
481C2×C4C2×#1062535211yescharacteristic
482C2×C4C2×#1062535411yescharacteristic
483C8—22no—
484C8—42no—
485C8—472no—
486C8—472no—
487C8—944no—
488C8—942no—
489C8(C22⋊D47)⋊C411yescharacteristic
490C8(C22⋊D47)⋊C411yescharacteristic
491C8(C22⋊D47)⋊C411yescharacteristic
492C8(C22⋊D47)⋊C411yescharacteristic
493C22—22no—
494C22—41no—
495C22—42no—
496C22—3762no—
497C22—1882no—
498C22#1568959111yescharacteristic
499C22#1568963611yescharacteristic
500C22#1568966811yescharacteristic
501C4—22no—
502C4—82no—
503C4—1881no—
504C4—1881no—
505C4—1882no—
506C4—1882no—
507C4C2×(C22⋊D47)⋊C411yescharacteristic
508C4C2×#1309398611yescharacteristic
509C4#121×D4711yescharacteristic
510C4#1568965911yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—3762no—
516C2#2024589411yescharacteristic
subgroupG/H
centerC4#121×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1814C2
socleC2×C47C4×#121
prime radical#20253108C2
cosocleC22×C4×C47C24
hypercenter#1814D47
hypocenterC47#21559
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
2. #20252287C2semidirect
3. C2×#15689244C2semidirect
4. #20251814C2semidirect
5. #20244284C2semidirect
6. #20244176C2semidirect
7. #20244055C2semidirect
8. #20243887C2semidirect
9. #20243274C2semidirect
10. #20243187C2semidirect
11. C47×#1814C2semidirect
17. #15689398C4semidirect
18. #15689393C4semidirect
19. #15689386C4semidirect
58. C2×Dic188C2×C4semidirect
59. C2×#10625325C2×C4semidirect
60. C2×D376C2×C4semidirect
81. #13093960C2×C4semidirect
82. #13093960C2×C4semidirect
87. #13093942C2×C4semidirect
88. #13093940C2×C4semidirect
261. #1814D47semidirect
306. C47#21559semidirect

Maximal subgroups

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

subgroupindex
#202531082
#202522872
C2×#156892442
#202518142
#202442842
#202441762
#202440552
#202438872
#202432742
#202431872
C47×#18142
#2155947